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Division of Engineering Services

BRIDGE DESIGN PRACTICE 4 th Edition

2015

State of California Department of Transportation

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

PREFACE The first edition of the Bridge Design Practice Manual (BDP) was published in 1960, and the second and third editions were published in 1963 and 1971, respectively. The BDP has been published as a live document continuously since the 1990s. The primary purpose of the BDP is to provide bridge design engineers with application of the California Department of Transportation (Caltrans) design standards and practices that lead to consistency in the design of bridge and highway structure projects on the California highway systems. The fourth edition of the BDP is divided into three volumes and covers the major areas in bridge and highway structure design. The BDP conforms to the AASHTO LRFD Bridge Design Specifications (Customary US units), sixth Edition with the 2014 California Amendments, except as noted; describes the basic design concepts and assumptions; provides step-by-step design examples; introduces innovative practice; and serves as a comprehensive reference manual for Caltrans bridge design engineers. A total of 15 chapters are published in February 2015, with more to follow. Development of the fourth edition of the BDP was a team effort and product of the Caltrans Division of Engineering Services Technical Organization. Many people gave unselfishly of their time and talent; their effort is gratefully acknowledged. Recognition of those individuals and groups who have made major contributions is as follows: VOLUME I: FUNDAMENTALS AND SUPERSTRUCTURE DESIGN Chapter 1, Chapter 2,

Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 10,

Preface

“Bridge Design Specifications” was written by Lian Duan. “Bridge Architecture and Aesthetics” originally written by Javier Chavez, was revised and updated for this edition by Bob Travis and Vanessa Gehringer. “Loads and Load Combinations” originally written by Marc Friedheim, was revised and updated for this edition by Kammy Bhala. “Structural Modeling and Analysis” was written by Mina Pezeshpour, Lian Duan, and Paul Chung. “Concrete Design Theory” was written by Jinrong Wang. “Steel Design Theory” was written by Lian Duan. “Post-Tensioned Concrete Girders” was written by Bartt Gunter, Gabriel Galo, Edward Mercado, and Daryoush Balbas. “Precast Pretensioned Concrete Girders” was written by Say-Gunn Low, Eric Matsumoto, Bartt Gunter, and Jim Ma. “Steel Plate Girders” was written by Lian Duan, Yusuf Saleh, and YongPil Kim. “Concrete Decks” originally written by Newton Armstrong, was revised and updated for this edition by Lian Duan.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

VOLUME II: SUBSTRUCTURE DESIGN Chapter 12,

“Concrete Bent Caps” was written by Don Nguyen-Tan, Krishnakant Andurlekar, and Ahmed Ibrahim. Chapter 13, “Concrete Columns” was written by Ashraf Ahmed. Chapter 15, “Shallow Foundations” was written by Amir Malek and Hernan Perez with supports from Mohammed Islam and Jinxing Zha. Chapter 16, “Deep Foundations” was written by Amir Malek, Sam Ataya, Ryan Stiltz, and Mohey El-Mously with support from Mark Mahan. VOLUME III: SEISMIC DESIGN Chapter 21, “Seismic Design of Concrete Bridges” was written by Christian Unanwa, Mark Mahan, Surjit Dhillon, Tariq Masroor, and Jay Quiogue.

The fourth edition of the BDP was prepared under the direction of Roberto Lacalle, BDP Manager; and Lian Duan, BDP Editor. Division of Engineering Services Technical Committees performed technical reviews; Tom Ruckman, James Choi, and Don Reding conducted independent quality assurance reviews; and Janet Barnett performed grammar review.

Dolores M. Valls Interim State Bridge Engineer Deputy Division Chief Structure Policy and Innovation Division of Engineering Services

Preface

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CHAPTER 1 BRIDGE DESIGN SPECIFICATIONS TABLE OF CONTENTS 1.1

INTRODUCTION ........................................................................................................... 1-1

1.2

LIMIT STATES .............................................................................................................. 1-2

1.3

ALLOWABLE STRESS DESIGN (ASD) ...................................................................... 1-2

1.4

LOAD FACTOR DESIGN (LFD) .................................................................................. 1-3

1.5

LOAD AND RESISTANCE FACTOR DESIGN (LRFD) ............................................. 1-4 1.5.1

Probability-Based Design ............................................................................... 1-4

1.5.2

Probabilistic Basis of the LRFD Specifications .............................................. 1-4

1.5.3

Calibration of Load and Resistance Factors.................................................... 1-6

1.5.4

Load and Resistance Factors ........................................................................... 1-8

1.5.5

General Design Requirements......................................................................... 1-9

1.5.6

Serviceability Requirements ......................................................................... 1-10

1.5.7

Constructability Requirements ...................................................................... 1-11

NOTATION ............................................................................................................................... 1-12 REFERENCES ........................................................................................................................... 1-13

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CHAPTER 1 BRIDGE DESIGN SPECIFICATIONS 1.1

INTRODUCTION The main purpose of bridge design specifications is to ensure bridge safety such that minimum resistances or capacities, in terms of strength, stiffness, and stability of each bridge component and the whole bridge structural system exceed the potential maximum demands or force effects due to various loads during its design life. The first national standard for highway bridge design and construction in the United States, “Standard Specifications for Highway Bridges and Incidental Structures” was published by the American Association of State Highway Officials (AASHO) in 1927, the predecessor to the American Association of State Highway and Transportation Officials (AASHTO). Design theory and practice have evolved significantly due to increased understanding of structural behavior and loading phenomena gained through research. Prior to 1970, the sole design philosophy was allowable stress design (ASD). Beginning in early 1970, a new design philosophy referred to as load factor design (LFD) was introduced. The latest version entitled “Standard Specifications for Highway Bridges” (Standard Specifications) is the final 17th Edition (AASHTO, 2002) and includes both ASD and LFD philosophies. Reliability-based and probability-based load and resistance factor design (LRFD) philosophy was first adopted in “AASHTO LRFD Bridge Design Specifications” (LRFD Specifications) (AASHTO 1994) and continues in the 7th Edition (AASHTO 2014). The LRFD Specifications had not been widely used until AASHTO discontinued updating of its Standard Specifications in 2003. ASD, LFD, and LRFD are distinct design philosophies and methods. ASD does not recognize that some loads are more variable than others. LFD brings the major philosophical change of recognizing that some loads are more accurately represented than others. LRFD is a logical extension of the LFD procedure and provides a mechanism to more systematically and rationally select the load and resistance factors with uniform margins of safety. The LRFD Specifications with California Amendments has been implemented for the all new bridge designs in the State of California since 2006. The latest version of the California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition (AASHTO, 2012) was published in 2014 (Caltrans, 2014). This chapter will briefly describe the general concepts and backgrounds of ASD and LFD but primarily discuss the LRFD philosophy. A detailed discussion may be found in Kulicki (2014).

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1.2

LIMIT STATES The design specifications are written to establish an acceptable level of safety for different loading cases. “Limit states” is a terminology for treating safety issues in modern specifications. A limit state is a condition beyond which the bridge or component ceases to satisfy the provisions for which it was designed. Limit states may be expressed by functional requirements such as the limiting deformation, stress or cracks, or by safety requirements such as the maximum strength. The design provisions make certain that the probability of exceeding a limit state is acceptably small by stipulating combinations of nominal loads and load factors, as well as resistances and resistance factors that are consistent with the design assumptions. The following four limit states are specified in the LRFD Specifications (AASHTO, 2012):

1.3



Service Limit State: Deals with restrictions on stress, deformation and crack width under regular service conditions. These provisions are intended to ensure the bridge performs acceptably during its design life.



Fatigue and Fracture Limit State: The fatigue limit state deals with restrictions on stress range under specified truck loading reflecting the number of expected stress range excursions. The fracture limit state is to establish a set of material toughness requirements. These provisions are intended to limit crack growth under repetitive loads to prevent fracture during its design life.



Strength Limit State: Ensures that strength and stability, both local and global, are provided to resist the statistically-significant load combinations during the life of a bridge. The overall structural integrity is expected to be maintained.



Extreme Event Limit State: Ensures the structural survival of a bridge during a major earthquake, collision by a vessel, vehicle or ice flow, or floods. These provisions deal with circumstances considered to be unique occurrences whose return period is significantly greater than the design life of a bridge. The probability of a joint occurrence of these events is extremely low, and, therefore, they are applied separately. Under these extreme conditions, the structure is expected to undergo considerable inelastic deformation.

ALLOWABLE STRESS DESIGN (ASD) ASD, also known as working stress design (WSD) or service load design, is based on the concept that the maximum applied stress in a structural component not exceed a certain allowable stress under normal service or working conditions. The general ASD design equation can be expressed as:

Q

i



Rn FS

(1.3-1)

where Qi is a load effect; Rn is the nominal resistance and FS is a factor of safety.

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The left side of Equation 1.3-1 represents working stress or service load effects. The right side of Equation 1.3-1 means allowable stress. The load effect Qi is obtained by an elastic structural analysis for a specified load, while the allowable stress (Rn/FS) is the nominal limiting stress such as yielding, instability or fracture divided by a safety factor. The magnitude of a factor of safety is primarily based on past experience and engineering judgment. For example, the factors of safety for axial tension and axial compression in structural steel are 1.82 and 2.12, respectively in the Standard Specifications (AASHTO, 2002). The ASD treats each load in a given load combination as equal from the view point of statistical variability. It does not consider the probability of both a higher than expected load and a lower than expected strength occurring simultaneously. They are both taken care of by the factor of safety. Although there are some drawbacks to ASD, bridges designed based on ASD have served very well with safety inherent in the system.

1.4

LOAD FACTOR DESIGN (LFD) LFD, also known as ultimate or strength design, mainly recognizes that the live load such as vehicular loads and wind forces, in particular, is more variable than the dead load. This concept is achieved by using different multipliers, i.e., load factors on dead and live loads. The general LFD design equation can be expressed as:

 Q i

i

  Rn

(1.4-1)

where i is a load factor and  is the strength reduction factor. The nominal resistance is usually based on either loss of stability of a component or inelastic cross-sectional strength. In some cases, the resistance is reduced by a “strength reduction factor”, , which is based on the possibility that a component may be undersized, the material may be under strength, or the method of calculation may be less accurate. It should be pointed out, however, that the probability of a joint occurrence of higher than expected loads and less than the expected resistance is not considered. One major disadvantage of LFD is that the load factors and resistance factors were not calibrated in a manner that takes into account the statistical variability of design parameters in nature, although the calibration for a simple-span of a 40-foot steel girder was performed by Vincent (1969).

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1.5

LOAD AND RESISTANCE FACTOR DESIGN (LRFD)

1.5.1

Probability-Based Design Probability-based design is to ensure that probability of failure of a structure is less than a level acceptable to society. It directly takes into account the statistical mean resistance, the statistical mean loads, the nominal or notional value of resistance, the nominal or notional value of the loads and the dispersion of resistance and loads as measured by either the standard deviation or the coefficient of variation. Direct probability-based design that computes the probability of failure for a given set of loads, statistical data and the estimate of the nominal resistance of the component has been used in numerous engineering disciplines but had not been widely used in bridge engineering.

1.5.2

Probabilistic Basis of the LRFD Specifications The probability-based LRFD Specifications center around the load effects Q and the resistances R modeled as statistically independent random variables (Ravindra and Galambos, 1978; Ellingwood, et. al., 1982; Kulicki, et. al., 1994). Figure 1.5-1 shows the relative frequency distributions for Q and R as separate curves. The mean value of the load effects ( Q ) and the mean value of the resistance ( R ) are also shown. Qn and Rn are the nominal value of the load effects and the resistance, respectively.  and  are the load and the resistance factor, respectively.

Figure 1.5-1 Relative Frequency Distribution of Load Effect Q and Resistance R As long as the resistance R is greater than the load effects Q, a margin of safety for a particular limit state exists. However, since Q and R are random variables in reality, there is a small probability that R may be less than Q. In other words, the probability Chapter 1 – Bridge Design Specifications

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of R < Q as shown as the overlap shadow area in Figure 1.5-1 is related to the relative positioning of R and Q and their dispersions. For both the load effect and the resistance, a second value somewhat offset from the mean value, is the “nominal” value. Designers calculate these values for the load effect and the resistance. The objective of the reliability-based or probability-based design philosophy is to separate the distribution of resistance from the distribution of load effect, such that the area of overlap, i.e., the area where load effect is greater than resistance, is acceptably small. In the LRFD Specifications, load factors and resistance factors were developed together in a way that forces the relationship between the resistance and load effect to be such that the area of overlap in Figure 1.5-1 is less than or equal to the value that AASHTO accepts. Probability of “exceedance” or “achievement of a limit state” can be examined by comparing R and Q as shown in Figure 1.5-2. Potential structural failure is represented by the left side region. The distance between the “exceedance” line and the mean value of the function of R-Q is defined as , where  is the standard deviation of the function of R-Q and  is called the “reliability index” or “safety index”. The larger  is, the greater the margin of safety.

Figure 1.5-2 Reliability Index The probability of R < Q depends on the distribution shapes of each of many variables (material, loads, etc.). Usually, the mean values and the standard deviations or the coefficients of variation of many variables involved in R and Q can be estimated. By applying the simple advanced first-order second-moment method (Ravindra and Galambos, 1978; Kulicki et. al., 1994) and assuming that both the resistance and load effect are normal random variables, the following reliability index equation can be obtained:

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R Q

 

(1.5-1)

 2R   Q2

where R and Q are the coefficients of variation of the resistance R and the load effect Q, respectively. Considering variations of both the load effect and the resistance, the basic design equations can be expressed as: R  Q 



i xi

(1.5-2)

Introducing  as the ratio of the mean value divided by the nominal value called the "bias" leads to:

R 

1  



i xi

(1.5-3)

Solving for the resistance factor  from Equations 1.5-3 and 1.5-1 yields: 



 

Q 

i

 2R

i

  Q2

(1.5-4)

It is seen that there are three unknowns, i.e., the resistance factor, , the reliability index, , and the load factor, . The reliability index is very useful. It can give an indication of the consistency of safety for a bridge designed using traditional methods. It also can be used to establish new methods which will have consistent margins of safety. Most importantly, it is a comparative indicator. One group of bridges having a reliability index which is greater than a second group of bridges has more inherent safety. A group of existing bridges designed by either ASD or LFD formed the basis for determining the target, or codespecified reliability index and the load and resistance factors in the LRFD Specifications (Kulicki et. al., 1994).

1.5.3

Calibration of Load and Resistance Factors A target value of the reliability index , usually denoted T, is chosen by a codewriting body. Equation 1.5-4 still indicates that both the load and resistance factors must be found. One way to solve this problem is to select the load factors and then calculate the resistance factors. This process has been used by several code-writing authorities (AASHTO, 1994; OMTC, 1994; CSA, 1998). The basic steps of calibration (Nowak, 1993) of the load and resistance factors for the LRFD Specifications were:   

Develop a database of sample current bridges Extract load effects as percentage of total load Estimate the reliability indices implicit in current designs

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  

Quantify variability in loads and materials by deciding on coefficients of variation Assume load factors Vary resistance factors until suitable reliability indices result

Approximately 200 representative bridges (Nowak, 1993) were selected from various regions of the United States by requesting sample bridge plans from various states. The selection was based on structural-type, material, and geographic location to represent a full-range of materials and design loads and practices as they vary around the country. Statistically-projected live load and the notional values of live load effects were calculated. Resistance was calculated in terms of the moment and the shear capacity for each structure according to the prevailing requirements, in this case the AASHTO Standard Specifications (AASHTO, 1989) for load factor design. Based on the relative amounts of the loads identified for each of the combinations of span and spacing and type of construction indicated by the database, a simulated set of 175 bridges was developed. The simulated group was comprised of non-composite steel girder bridges, composite steel girder bridges, reinforced concrete T-beam bridges, and prestressed concrete I-beam bridges. The reliability indices were calculated for each simulated and each actual bridge for both the shear and the moment. The range of reliability indices which resulted from this calibration process is presented in Figure 1.5-3 (Kulicki, et. al., 1994). It can be seen that a wide-range of values were obtained using the Standard Specifications, but this was anticipated based on previous calibration work done for the Ontario Highway Bridge Design Code (Nowak, 1979). These calculated reliability indices, as well as past calibration of other specifications, served as a basis for the selection of the target reliability index, T. A target reliability index of 3.5 was selected for the Ontario Highway Bridge Design Code (OMTC, 1994) and other reliability-based specifications. A consideration of the data shown in Figure 1.5-3 indicates that a  of 3.5 is representative of past LFD practice. Hence, this value was selected as a target for the calibration of the LRFD Specifications.

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0 0

30 30

60 60

90 90

120 120

160 160

Span Length (ft)(ft) Figure 1.5-3

1.5.4

Reliability Indices Inherent in the 1989 AASHTO Standard Specifications

Load and Resistance Factors The recommended values of load factors are simplified to be practical for bridge design. One factor is specified for weight of both shop-built and field-built components:  = 1.25. For weight of asphalt and utilities,  = 1.50, a higher value is used. For live load, the calibrated value of load factor was 1.60. However, a more conservative value of  = 1.75 is utilized in the LRFD Specifications. A detailed discussion of load factors and load combinations is in Chapter 3. The acceptance criterion, in the selection of resistance factors, is how close the calculated reliability indices are to the target value of the reliability index, T. Calculations were performed using the load components for each of the 175 simulated bridges using the range of resistance factors shown in Table 1.5-1 (Nowak, 1993). Reliability indices were recalculated for each of the 175 simulated cases and each of the actual bridges from which the simulated bridges were produced. The range of values obtained using the new load and resistance factors are indicated in Figure 1.5-4 (Kulicki, et. al., 1994). It is seen from Figure 1.5-4 that the new calibrated load and resistance factors, and new load models and load distribution techniques work together to produce very narrowly-clustered reliability indices. This was the objective of developing the new factors. Correspondence to a reliability index of 3.5 can now be altered by AASHTO when either a higher level of safety or taking more risk is appropriate. If the target reliability index is to be raised or lowered, the factors need to

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be recalculated accordingly. This ability to adjust the design parameters in a coordinated manner is one of the benefits of a probability-based reliability design. Table 1.5-1 Considered Resistance Factors in LRFD Calibration MATERIAL Non-Composite Steel

Moment Shear Moment Shear Moment Shear Moment Shear

Composite Steel Reinforced Concrete Prestressed Concrete

0

RESISTANCE FACTOR  LOWER UPPER 0.95 1.00 0.95 1.00 0.95 1.00 0.95 1.00 0.85 0.90 0.90 0.90 0.95 1.00 0.90 0.95

LIMIT STATE

30

60

90

120

160

Span Length (ft) Figure 1.5-4 Reliability Indices Inherent in LRFD Specifications

1.5.5

General Design Requirements Public safety is the primary responsibility of the design engineer. All other aspects of design, including serviceability, maintainability, economics and aesthetics are secondary to the requirement for safety. The LRFD Specifications specifies that each component and connection shall satisfy the following equation for each limit state:

  Q i

i

i

  Rn  Rr

(AASHTO 1.3.2.1-1)

where i is a load modifier relating to ductility, redundancy, and operational importance and Rr is the factored resistance. Chapter 1 – Bridge Design Specifications

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For loads for which a maximum value of i is appropriate:

i   D  R  I  0.95 For loads for which a minimum value of i is appropriate: 1 i   1.0 D R I

(AASHTO 1.3.2.1-2)

(AASHTO 1.3.2.1-3)

where D , R and I are factors relating to ductility, redundancy, and operational importance, respectively. The California Amendments (Caltrans 2014) specifies that the value of 1.0 shall be used for D, R and I until their applications are further studied.

1.5.6

Serviceability Requirements The LRFD Specifications address serviceability from the view points of durability, restriction of stresses, cracking, corrosion, and deformation - all in conjunction with contract documents to achieve the desired design life. Bridge designers also need to be mindful of ease in inspection and maintainability, as addressed in the Manual for Bridge Evaluation (AASHTO, 2011). Durability is to be assured through contract documents calling for high quality materials and requiring that those materials that are subject to deterioration from moisture content and/or salt attack be protected. Good workmanship is also important for good durability. Maintainability is treated in the specifications in a similar manner to durability; a list of desirable attributes to be considered is provided. Inspectability is to be assured by providing adequate means for inspectors to view all parts of the structure which have structural or maintenance significance. Bridge inspection can be very expensive and is a recurring cost. Therefore, the cost of providing walkways and other means of access and adequate room for people and inspection equipment to be moved about on the structure is usually a good investment. Rider comfort is often rationalized as a basis for deflection control. As a compromise between the need for establishing comfort levels and the lack of compelling evidence that deflection was the cause of structural distress, the deflection criteria, other than those pertaining to relative deflections of ribs of orthotropic decks and components of some wood decks, were written as voluntary provisions to be activated by those states that so choose. Deflection limits, stated as span length divided by some number, were established for most cases, and additional provisions of absolute relative displacement between planks and panels of wooden decks and ribs of orthotropic decks were also added. Similarly, optional criteria were established for a span-to-depth ratio for guidance primarily in starting preliminary designs, but also as

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a mechanism for checking when a given design deviates significantly from past successful practice. User comfort on pedestrian bridges is addressed in the LRFD Guide Specifications for the Design of Pedestrian Bridges (AASHTO, 2009).

1.5.7

Constructability Requirements The following provisions in the LRFD Specifications are related to constructability: 

Design bridges so that they can be fabricated and built without undue difficulty and with control over locked in construction force effects,



Document one feasible method of construction in the contract documents, unless the type of construction is self-evident, and



Indicate clearly the need to provide strengthening and/or temporary bracing or support during erection, but not requiring the complete design thereof.

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NOTATION FS

=

factor of safety

Q

=

load effects

Q

=

mean value of the load effect

Qi

=

a load effect

R

=

resistance

Rn

=

nominal resistance

R

=

mean value of the resistance

Rr

=

factored resistance

i

=

a load factor



=

load factor



=

ratio of the mean value divided by the nominal value, called the “bias”



=

reliability index

T

=

target reliability index



=

resistance factor

D

=

a factor relating to ductility

R

=

a factor relating to redundancy

I

=

a factor relating to operational importance

i

=

a load modifier, a factor relating to ductility, redundancy, and operation importance



=

standard deviation

R

=

coefficient of variation of the resistance R

Q

=

coefficient of variation of the load effect Q

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REFERENCES 1. AASHO, (1927). Standard Specifications for Highway Bridges and Incidental Structures, American Association of State Highway Officials, Washington, D.C. 2. AASHTO, (1989). Standard Specifications for Highway Bridges, 14th Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 3. AASHTO, (1994). AASHTO LRFD Bridge Design Specifications, 1st Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 4. AASHTO, (2002). Standard Specifications for Highway Bridges, 17th Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 5. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 6. AASHTO, (2011). The Manual for Bridge Evaluation, 2nd Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 7. AASHTO, (2009). AASHTO LRFD Guide Specifications for the Design of Pedestrian Bridges, 2nd Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 8. AASHTO, (2014). AASHTO LRFD Bridge Design Specifications, 7th Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 9. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 10. CSA, (1998). Canadian Highway Bridge Design Code, Canadian Standards Association, Rexdale, Ontario, Canada. 11. Ellingwood, B.E., MacGregor, J.G., Galambos, T.V., and Cornell, C.A. (1982). “Probability-Based Load Criteria: Load Factors and Load Combinations,” Journal of Structural Division, ASCE, 108(5), 1337-1353. 12. Kulicki, J.M. (2014). “Chapter 5: Highway Bridge Design Specifications,” Bridge Engineering Handbook, 2nd Edition: Fundamentals, Edited by Chen, W.F. and Duan, L., CRC Press, Boca Raton, FL. 13. Kulicki, J.M., Mertz, D.R. and Wassef, W.G. (1994). “LRFD Design of Highway Bridges,” NHI Course 13061, Federal Highway Administration, Washington, D. C.

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14. Nowak, A.S. (1993). “Calibration of LRFD Bridge Design Code,” Department of Civil and Environmental Engineering Report UMCE 92-25, University of Michigan, Ann Arbor, MI. 15. Nowak, A.S. and Lind, N.C. (1979). “Practical Bridge Code Calibration,” ASCE, Journal of the Structural Division, ASCE, 105(ST12), 2497-2510. 16. OMTC, (1994). Ontario Highway Bridge Design Code, Ontario Ministry of Transportation and Communications, Toronto, Ontario, Canada. 17. Ravindra, M.K., and Galambos, T.V. (1978). “Load and Resistance Factor Design for Steel,” Journal of Structural Division, ASCE, 104(ST9), 1337-1353. 18. Vincent, G.S. (1969). “Tentative Criteria for Load Factor Design of Steel Highway Bridges,” AISI Bulletin 15, American Iron and Steel Institute, Washington, DC.

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CHAPTER 2 BRIDGE ARCHITECTURE AND AESTHETICS TABLE OF CONTENTS 2.1 

INTRODUCTION ............................................................................................................ 2-1 

2.2

BRIDGE ARCHITECTURE AND AESTHETICS ......................................................... 2-1 2.2.1   The Bridge Architecture & Aesthetics Design Branch ...................................... 2-1  2.2.2   Context Sensitive Design ................................................................................... 2-2  2.2.3   Route and Corridor Themes ............................................................................... 2-2  2.2.4   Products and Services......................................................................................... 2-3  2.2.5   Recommended Levels of Aesthetic Treatments ................................................. 2-4 

2.3 

BRIDGE ARCHITECTURE AND AESTHETICS DELIVERY MILESTONES........... 2-4  2.3.1   Advanced Planning Study .................................................................................. 2-5

2.4 

2.3.2

Structures General Plan ...................................................................................... 2-6

2.3.3

Structure Plans, Specifications, and Estimate Development .............................. 2-8 

 STRUCTURE TYPES AND COMPONENTS ............................................................... 2-9  2.4.1   Water Crossing Bridges...................................................................................... 2-9  2.4.2   Valley Crossing Bridges................................................................................... 2-10 2.4.3   Overcrossings ................................................................................................... 2-11  2.4.4   Undercrossings ................................................................................................. 2-12  2.4.5   Pedestrian Bridges  .......................................................................................... 2-13 2.4.6     Viaducts ........................................................................................................... 2-15 2.4.7     Interchanges ..................................................................................................... 2-16 2.4.8   Bridge Columns/Bents ..................................................................................... 2-17  2.4.9 

Bridge Abutments and Wingwalls .................................................................. 2-18

2.4.10 Slope Paving .................................................................................................... 2-19  2.4.11 Bridge Barriers ................................................................................................. 2-20  2.4.12   Fences and Railings  ........................................................................................ 2-23 2.4.13    Light Fixtures ................................................................................................ 2-25 2.4.14    Retaining Walls ............................................................................................... 2-26

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2.4.15

Sound Wall Pilaster Design ........................................................................... 2-29

2.4.16 Tunnel Portal Design ........................................................................................ 2-30 2.4.17

Rocksheds ...................................................................................................... 2-32

ATTACHMENT 1 BRIDGE ARCHITECTURE AND AESTHETICS DELIVERY FLOWCHART .................................................................... 2-34  ATTACHMENT 2 PRELIMNARY ARCHITECTURAL AESTHETIC RECOMMENDATION FORM ................................................................ 2-35  ATTACHMENT 3 FINAL ARCHITECTURAL AESTHETIC RECOMMENDATION FORM ............................................................... 2-36  REFERENCES ........................................................................................................................... 2-37 

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CHAPTER 2 BRIDGE ARCHITECTURE AND AESTHETICS 2.1

INTRODUCTION Well executed architectural design and aesthetic treatment solutions are vital to developing a legacy of safe, functional, and beautiful Caltrans’ structures. The Bridge Architecture and Aesthetics Design Branch works in a coordinated effort with the Division of Engineering Services Project Engineers, and District design personnel, to insure quality, safe, and beautiful structures on the state’s highway system. The coordination and integration of complimentary design disciplines (e.g., bridge architecture and bridge engineering), are the keys for successful structure design and delivery. Utilizing an orderly design effort, the Division of Engineering Services Project Engineers, Bridge Architecture & Aesthetics Design Branch, and District staff, reduce the chance of late scope changes for aesthetics during the structure’s design phase. Late changes are undesirable and may cause delays in project delivery, increase costs, and can result in projects with poor visual appearance. This chapter presents current Caltrans bridge architecture and aesthetics design guidelines and practice. For general bridge aesthetics, references are made to AASHTO (2010), Billington (1983 and 2003), Gottemoeller (2004 and 2014), and Leonhardt (1983 and 2014).

2.2

BRIDGE ARCHITECTURE AND AESTHETICS

2.2.1

The Bridge Architecture & Aesthetics Design Branch The Bridge Architecture and Aesthetics (BA&A) Design Branch is an accomplished team of architects, graphic artists, and model makers at the forefront of bridge architectural and aesthetics design. The architects’ primary goals are visual, and the primary value of the architect in structures design is to create beautiful and pleasing structures. The BA&A Design Branch typically reviews and provides aesthetics recommendations for the following types of projects: bridges over bodies of water, valleys and canyons; highway overcrossings and undercrossings; highway interchanges, pedestrian overcrossings; highway viaducts; tunnels and tunnel portals; retaining walls; and rocksheds. While the term “bridge” has a specific meaning as structures built over bodies of water and canyons, the use of the terms “bridge” and “structure” are used interchangeably within the body of this chapter. Every structure receives multiple architectural and aesthetic design reviews and recommendations during the design phase. The BA&A Design Branch also provides

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research and development of material specifications, estimates for application, and feasibility of aesthetic recommendations for structures.

2.2.2

Context Sensitive Design The BA&A Design Branch utilizes a context sensitive design approach to structure design aesthetics. Context sensitive design acknowledges a concern for local architectural identity, and investment. The aesthetic design objective is to build a visual legacy of structures that recognizes the diverse and varied character of communities along the highway system. The BA&A Design Branch strives to create structure aesthetics, distinctive in their forms, as well as designs, that relate to a continuity of existing architectural traditions and aspirations. The approach to context sensitive structure aesthetics is a an iterative process, with numerous aspects of aesthetic design reviewed many times over, as a cross-check, to build upon on a new framework for color, scale, style, direction, proportion, shape, form, balance, etc. These aspects of design are synthesized into parts of the structure, by what lies at the organic core of contextual design: harmony. Correspondingly, the key element drawing everything into a harmonious whole is the structure’s site location. The site represents the foundation of local traditions, the built backdrop, the historical past, and present community aspirations for the future. In view of these core values affecting the site, the organic aspects of design (e.g., color, scale style, direction, proportion, shape, form, balance, setting, etc.), are then analyzed for the aesthetics of various structure components in the course of the design process. Context sensitive design begins with a circle drawn around the project site that surveys and observes the established complexity of parts, and intricate patterning within the fabric of the surrounding community. The BA&A Design Branch then designs complimentary forms and shapes for structures that fit back into the local architectural fabric of the community. This process inspires a historical attitude in a structure’s aesthetic design, that emanates from the beauty of new highway structures themselves; creating a Déjà vu affect. The new expression in design becomes, at the same time, a celebration of the aspirations of the people most concerned for the project and the site; and their real concerns for identity, continuity, and a sense of occasion. This design approach reinforces and satisfies those aspirations, when applied to new bridge structures.

2.2.3

Route and Corridor Themes Route and corridor themes are established by developing a complimentary appearance between varying bridge types and components along the route. The BA&A Design Branch has the responsibility for integrating, designing, and recommending visual solutions for bridges and other structures. The Branch uses its expertise in structure component design, architectural perspective drawing, and physical model making to lead the discussion and develop recommendations regarding opportunities and limitations that may affect a structure’s design and appearance. The BA&A Design

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Branch is also instrumental in the development of aesthetic guidelines, and details for planned structures that would be acceptable to both the Department and the community. The BA&A Design Branch solicits and receives aesthetic input from the Districts, public groups, and from local officials, as a necessary part for gaining acceptance and cooperation of major highway projects affecting the highway corridor. The Branch’s proactive design approach addresses future visual impacts by providing anticipated aesthetic strategies to address and/or mitigate those impacts. The Department is then able to achieve an overall strategic plan for visual continuity between geographically related highway structures, and correspondingly, aesthetically pleasing highway corridors. Route and corridor themes are established by what is made visible to the traveling motorist. Overcrossing structures usually represent the aesthetic theme for state highway routes. Undercrossings and viaducts may vary from a particular route theme since these structures are usually not within the highway driver’s focused viewing area. Other variations to the overall route theme are applied when local needs are considered during the aesthetics design phase. Water and valley-crossings, and structures on scenic routes are usually considered “special designs,” and may also vary from the aesthetic route theme.

2.2.4

Products and Services Graphically prepared materials are used primarily as tools for exhibiting a project’s design features, and expenditures to the Caltrans Project Development Team, Project Managers, funding agencies, the public, and other interested parties and stakeholders. Typically, prepared materials include: drawings, illustrations, and physical models. Drawings:

Sketches, preliminary drawings, and finished contract drawings.

Illustrations:

3D Drawings - Three dimensional computer drawings that can be rendered into a photo-realistic representation of the proposed project. Photographic Simulations – Retouched and manipulated photos of the existing project site with an insert of the proposed structure. The purpose of this illustration type is to give a visual indication of the proposed project, within its setting.

Models:

Physical models are produced by the BA&A Design Branch as requested. There are two types of models: the detailed presentation model and the study model. The detailed presentation models show the project as proposed. Requests for these types of models are becoming rare due to modern 3D CADD drawing and illustration technology. 3D CADD drawings and illustrations satisfy the need for demonstrating various detailed aspects of proposed projects in their environment.

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Study models are the second type of models produced by the Branch. This model type is developed using 3D CADD printing technology. These models are limited in size, and therefore, are mainly produced to focus on various components of structure design (e.g., column detail studies, railing studies, superstructure studies, etc.) during the structure’s design phase. Services:

Project Presentations to Caltrans, public meetings, outside agencies, and stakeholders.

Construction support is provided as required.

2.2.5

Recommended Levels of Aesthetic Treatments The BA&A Design Branch designates a level of architectural and aesthetic treatment for all structure design projects. This information is transmitted to the structure project engineer at the beginning of the design process: Level Designation

2.3

Level 1 =

Standard Aesthetics – Standard treatments applied to one or more parts of the structure.

Level 2 =

Moderate Aesthetics – Elevated aesthetic considerations. Custom design and graphics, or a level of standardized aesthetics applied to multiple components on the structure.

Level 3 =

Complex Architectural Forms and Aesthetics - Highly elevated architectural forms and shapes: a custom-designed structure along with corresponding components and aesthetics.

Level 4 =

Complex Environmental & Community Sensitive Project – These complex projects may be located within corridors of environmentally sensitive areas, projects that are politically sensitive, or projects having substantial community involvement or external agency reviews.

BRIDGE ARCHITECTURE AND AESTHETICS DELIVERY MILESTONES The BA&A Design Branch coordinates the aesthetic and architectural recommendations with the project delivery schedule for each project. The Branch may be involved with project support form project initiation through construction completion. See Attachment 1 for the BA&A Delivery Flowchart.

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2.3.1

Advanced Planning Study

2.3.1.1

General The Advanced Planning Study (APS) phase is a planning activity that may occur several years in advance of initiating structure design. The basic objective of the APS is to develop a feasible structure type with an appropriate cost for a future structure project. This activity includes identifying potential applicable structure design/cost alternatives, and reaching consensus with internal/external stakeholders on those alternatives addressed. The BA&A Design Branch supports the APS effort through early identification of aesthetic requirements that may affect the appearance and cost of structures. Projects that may have substantial requirements for architectural treatment, thus increasing costs, are those projects located within corridors of environmentally and politically sensitive areas (e.g., large retaining wall projects, or any project having substantial community involvement). Where there is less environmental and public sensitivity, the BA&A Design Branch continues using a disciplined approach for making the best possible aesthetic recommendations. All structures on State routes receive aesthetic recommendations that reflect the latest artistic, technological, and safety innovations in structure design. The aesthetic recommendation may be in the form of a brief comment, or it may result in a thorough investigation into several alternatives with preliminary design and drawing support.

2.3.1.2

Preliminary Architectural/Aesthetic Recommendation Form The BA&A Design Branch responds to the Division of Engineering Services (DES) Project Engineer’s request for aesthetic design concepts in the form of sketches and the Preliminary Architectural/Aesthetic Recommendation Form (PAAR), see Attachment 2. The PAAR is a checklist indicating the project’s aesthetic requirements, including the level of aesthetic complexity; existing route conditions and themes; and the conceptual design recommendation for project aesthetics. A PAAR form for each aesthetic alternative is attached to the APS for consideration by the Project Engineer. The BA&A Design Branch may coordinate its review efforts with District’s aesthetics representatives during the APS phase; however, this interaction depends largely on the complexity and sensitivity of the proposed project. Usually, the time allocated for determining aesthetic criteria during the planning phase is short; therefore, design interaction is typically between the BA&A Design Branch and the DES Project Engineer.

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2.3.2

Structures General Plan

2.3.2.1

Bridge Site Data Submittal The Bridge Site Data Submittal package (BSDS) is a checklist of pertinent layouts, environmental criteria, aesthetics considerations, site information, and other constraints needed for the design of structures. District delivery of this package to DES essentially initiates the start of structure design. The BA&A Design Branch starts preliminary aesthetics design concepts and recommendations, upon receiving the BSDS from the DES Project Engineer.

2.3.2.2

Preliminary Design and Details The draft General Plan work includes the preparation of preliminary structure plans, estimates, foundation recommendations, and aesthetic recommendations. The Type Selection Meeting is also a part of this activity. The BA&A Design Branch responds to the DES Project Engineer’s request for aesthetic recommendations at this stage of design in the form of sketches and the PAAR form. The Branch reviews and updates previous PAAR forms submitted for projects during the APS phase, if an APS was done. A new PAAR form and sketches for each alternative-design/cost scenario are produced and attached to the Structure Type Selection Memo for consideration in the Type Selection Meeting. The BA&A Design Branch typically interacts with the structure’s project engineer in one of two ways in order to develop appropriate aesthetic recommendations for a project. The design engineer usually provides the first orientation to the proposed project by developing one, or several structural alternatives, and introducing them to the BA&A Design Branch for review and comment. The second form of interaction may be a request for the Branch to work jointly with an engineering team comprised of both DES and District design personnel, to develop several architectural/aesthetic recommendations. The format can be fluid, depending on the complexity and magnitude of the project.

2.3.2.3

District Review Coordination The BA&A Design Branch typically initiates communication and coordinates its design efforts with District landscape architects during the draft General Plan development phase. The District provides BA&A with reviews and comments of the BA&A’s design recommendations; which may include the District’s emphasis for aesthetics development on the project. These reviews are usually in the form of written text, pictures, references to physical examples, thumbnail sketches, verbal descriptions, etc. In some instances, the District may not have a preconceived notion about the project’s aesthetics prior to the BA&A Design Branch’s initial contact. In this case, the District typically relies on the BA&A Design Branch to develop design criteria, aesthetic emphasis, and solutions for joint DES and District consideration, and inclusion into the PAAR.

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The District may be involved in discussions of context sensitive design issues, environmental impact statements, visual impact assessments, public hearings/meetings/workshops, or in writing various project planning documents pertinent to the project corridor. The objective is to have these activities concluded prior to the Type Selection Meeting. All pertinent material that may impact the preliminary aesthetic design is shared with the BA&A Design Branch for consideration and inclusion into the structure aesthetics recommendations package for Type Selection. 2.3.2.4

Structure Type Selection The structure type selection process is a fundamental step in the design of structures. At the Type Selection Meeting, the structures project engineer presents the proposed structure and briefly discusses issues pertinent to the selection of the preferred structure type; particularly requirements for foundations, hydraulics, construction (including falsework), seismic design, retrofit strategy, aesthetics, traffic handling, safety, and other information needed to support the selected structure type. The BA&A Design Branch prepares an aesthetics recommendation package for the project engineer prior to the Type Selection Meeting. The aesthetic recommendations require coordination with the District, DES project engineers, and BA&A staff. The BA&A Design Branch prepares project sketches and details for the DES Project Engineer. The goal of the Type Selection Meeting is to approve the structure type. This approval is based on satisfactorily addressing all issues raised in prior design reviews, and in the course of the meeting itself. In many instances the aesthetics package will have a direct bearing on major components of the preferred structure type. If the aesthetics package is regarded as incomplete, it may preclude the definitive determination for components of the structure and estimated costs. As with all other issues pertinent to the design, major aesthetic issues are resolved prior to the Type Selection Meeting to the satisfaction of all stakeholders concerned. Failure to accomplish this task could jeopardize the project schedule.

2.3.2.5

Final Architectural Aesthetics Recommendation Aesthetic features may change as a result of discussions at the Type Selection Meeting. If this occurs, the BA&A Design Branch will assist the DES Project Engineer in revising the aesthetics design details prior to General Plan distribution. The BA&A Design Branch prepares final aesthetics recommendations in the form of drawings, and the Final Architectural Aesthetic Recommendation Form (FAAR, see Attachment 3). The FAAR indicates the project’s aesthetic requirements as a result of the structure’s type selection process.

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2.3.2.6

General Plan Distribution The approved General Plan, which includes the final aesthetics design recommendation, is distributed by the DES Project Engineer to all DES functional units involved in the design of the project, and also to the District for their review and comment.

2.3.3

Structure Plans, Specifications, and Estimate Development The Plans, Specifications, and Estimate (PS&E) phase includes the development of structures aesthetics plans, specifications, and estimates. Comments received during the General Plan distribution period are incorporated into the aesthetics design during this phase. The BA&A develops and prepares complete details and plan sheets for structures aesthetics in concert with the DES project design team of bridge designers, specification engineers, and structure estimators. The aesthetic details are coordinated for conformance with standard design practices for: safety, engineering, specifications, constructability, budget, and aesthetic requirements.

2.3.3.1

Plans and Quantities The BA&A Branch design plans and details for structure aesthetics are quantified for all items of aesthetics work and are included as part of the Plans and Quantities (P&Q) distribution package.

2.3.3.2

Draft Specifications and Estimate Support The BA&A Design Branch aids in developing draft specifications and estimates for structures’ aesthetics in concert with the DES design team.

2.3.3.3

Draft Plans, Specifications, and Estimate The Draft PS&E, which includes all aesthetic design recommendations, is distributed by the DES Project Engineer to all DES functional units involved in the design of the project, and also to the District for their review and comment. The BA&A Design Branch reviews the Draft PS&E package for aesthetics design intent and conformance with structures’ details, specifications, and estimates practices. The Branch coordinates this review with the District aesthetics representatives and addresses District aesthetic review comments prior to the Final Structures PS&E.

2.3.3.4

Final Structure Plans, Specifications and Estimate The Final PS&E consists of complete sets of project plans, specifications, and estimates to advertise and construct a project. As part of preparing the Final Structures

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PS&E Expedite Package, BA&A addresses all aesthetics comments on the Draft Structures PS&E for incorporation into the Final Structures PS&E.

2.4

STRUCTURE TYPES AND COMPONENTS The following photographs and photographic simulations illustrate various architectural and aesthetic treatments used on a variety of structures types and components.

2.4.1

Water Crossing Bridges  

Figure 2.4.1-1 Smith River Bridge Rendering – District 1 – Route 199  

Figure 2.4.1-2 Antler’s Bridge Rendering – District 2 – Route 5

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2.4.2

Valley Crossing Bridges

Figure 2.4.2-1 Archie Stevenot Bridge – District 10 – Route 49

Figure 2.4.2-2 Devil’s Slide Bridge – District 4 – Route 1

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2.4.3

Overcrossings

Figure 2.4.3-1 Donner Park Overcrossing – District 3 – Route 80

Figure 2.4.3-2 Linden Avenue Overcrossing Rendering – District 5 – Route 101

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2.4.3-3 Casitas Pass Overcrossing (Proposed) – District 5 – Route 101

2.4.4

Undercrossings  

Figure 2.4.4-1 Cacique Street Undercrossing Rendering – District 5 – Route 101

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Figure 2.4.4-2 Puente Avenue Undercrossing (Proposed) – District 7 – Route10

2.4.5

Pedestrian Bridges

Figure 2.4.5-1 Bedford Avenue Pedestrian Overcrossing – District 3 – Route 50

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Figure 2.4.5-2 Bess Avenue Pedestrian Overcrossing Rendering – District 7 – Route 10

 

Figure 2.4.5-3 White Rock Road Pedestrian Overcrossing – District 3 – Route 50

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2.4.6

Viaducts

 

Figure 2.4.6-1 Doyle Drive High Viaduct Rendering – District 4 – Route 101

Figure 2.4.6-2 HOV Viaduct #2 – District 7 – Route 110

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2.4.7

Interchanges

Figure 2.4.7-1 5/14 Interchange – District 7  

Figure 2.4.7-2 280/680 Interchange – District 4

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2.4.8

Bridge Columns/Bents  

(a)  

(b)  

 

(d) Figure 2.4.8-1

(c)

(e)

(f)

(a) Schuyler Heim Bridge Rendering – Dist 7 – Route 47 (b) Bess Ave POC Rendering – Dist 7 – Route 10 (c) White Rock POC – Dist 3 – Route 50 (d) Devil’s Slide – Dist 4 – Route 1 (e) Antlers Bridge Rendering – Dist 2 – Route 5 (f) Bedford POC – Dist 3 – Route 50

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2.4.9

Bridge Abutments and Wingwalls

Figure 2.4.9-1 Carmenita Avenue Overcrossing Rendering – District 7 – Route 5

Figure 2.4.9-2 Rosencrans Avenue Overcrossing Rendering – District 7 – Route 5

Figure 2.4.9-3 Route 101/41 Separation Rendering – District 5 Chapter 2 – Bridge Architecture and Aesthetics

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2.4.10

Slope Paving  

Figure 2.4.10-1 Placerville Project - District 3 – Route 50

Figure 2.4.10-2 Carmenita Avenue Overcrossing Rendering - District 7 – Route 5

Figure 2.4.10-3 San Jose Airport Slope Paving – District 4 – Route 87

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2.4.11 Bridge Barriers

Figure 2.4.11-1 Devil’s Slide – District 4 – Route 1

Figure 2.4.11-2 Ten Mile River Bridge – District 1 – Route 1  

Figure 2.4.11-3 Emerald Bay – District 3 – Route 89

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Figure 2.4.11-4 Pitkins Curve – District 5 – Route 1

 

Figure 2.4.11-5 Pitkins Curve – District 5 – Route 1

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Figure 2.4-11-6 Placerville Project – District 3 – Route 50  

Figure 2.4.11-7 Placerville Project – District 3 – Route 50

Figure 2.4.11-8 Bear River Bridge – District 3 – Route 49 Chapter 2 – Bridge Architecture and Aesthetics

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2.4.12 Fences and Railings

Figure 2.4.12-1 Placerville Project – District 3 – Route 50

Figure 2.4.12-2 Cold Spring Canyon Bridge – District 5 – Route 154

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Figure 2.4.12-3 La Conchita Rendering – District 7 – Route 101

 

Figure 2.4.12-4 10th Street Bridge – District 3 – Route 20

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2.4.13 Light Fixtures

(a)

(b)

(c)

Figure 2.4-13-1 (a) Doyle Drive Rendering - District 4 – Route 101 (b) Mace Boulevard Overcrossing – District 3 – Route 80 (c) Bedford POC – District 3 – Route 50

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2.4.14 Retaining Walls

Figure 2.4.14-1 District 4 – Route 37  

Figure 2.4.14-2 Buena Park Rendering - District 12 – Route 5  

Figure 2.4.14-3 Mission Avenue UC – District 8 – Route 61

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Figure 2.4.14-4 Route 605/405/22 Project Rendering – District 12  

Figure 2.4.14-5 San Juan Capistrano – District 12 – Route 5

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Route 10 Mountain Motif Rendering – District 8

Fractured Rib

Split Slate

Split Face

Heavy Sandblast

Combined Textures Figure 2.14-6 Concrete Retaining Wall Surface Treatment Examples Chapter 2 – Bridge Architecture and Aesthetics

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2.4.15 Sound Wall Pilaster Design

 

Figure 2.4.15-1 Mobility Project Rendering – District 7 – Route 5

Figure 2.4.15-2 Orange County Pilaster – District 12 – Route 22  

Figure 2.4.15-3 District 7 Rendering – District 7 – Route 710 Chapter 2 – Bridge Architecture and Aesthetics

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2.4.16 Tunnel Portal Design

Figure 2.4.16-1 Devil’s Slide North Tunnel Portals – District 4 – Route 1

Figure 2.4.16-2 Devil’s Slide South Tunnel Portals – District 4 – Route 1

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Figure 2.4.16-3 Devil’s Slide Rendering – District 4 – Route 1

 

Figure 2.4.16-4 Devil’s Slide Rendering – District 4 – Route 1

 

Figure 2.4.16-5 Doyle Drive Rendering – District 4 – Route 101

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2.4.17 Rocksheds  

Figure 2.4.17-1 Pitkins Curve Rockshed Rendering – District 5 – Route 1

 

Figure 2.4.17-2 Pitkins Curve Rockshed – District 5 – Route 1

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(a)  

(b)

 

 

( c)

(d)

Figure 2.4.17-3 Pitkin’s Curve (Proposed Alternatives) – District 5 – Route 1

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ATTACHMENT 1: BRIDGE ARCHITECTURE AND AESTHETICS DELIVERY FLOWCHART

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ATTACHMENT 2: PRELIMNARY ARCHITECTURAL AESTHETIC RECOMMENDATION FORM

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ATTACHMENT 3: FINAL ARCHITECTURAL AESTHETIC RECOMMENDATION FORM

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REFERENCES 1. AASHTO. (2010). Bridge Aesthetics Sourcebook. American Association of State Highway and Transportation Officials, Washington, DC. 2. Billington, D. P. 1983. The Tower and The Bridge: The New Art of Structural Engineering. Basic Books, Inc., New York, NY. 3. Billington, D. P. (2003). The Art of Structural Design, a Swiss Legacy. Princeton University Art Museum, Princeton, NJ. 4. Gottemoeller, F. (2004). Bridgescape, The Art of Designing Bridges. 2nd Ed., John Wiley & Sons, Inc., New York, NY. 5. Gottemoeller, F. (2014). “Chapter 3: Bridge Aesthetics: Achieving Structural Art in Bridge Design,” Bridge Engineering Handbook, 2nd Edition: Fundamentals, Ed. Chen, W.F. and Duan, L., CRC Press, Boca Raton, FL. 6. Leonhardt, F. (1983). Bridges—Aesthetics and Design, MIT Press, Cambridge, MA. 7. Leonhardt, F. (2014). “Chapter 2: Aesthetics: Basics,” Bridge Engineering Handbook, 2nd Edition: Fundamentals, Ed. Chen, W.F. and Duan, L., CRC Press, Boca Raton, FL.

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CHAPTER 3 LOADS AND LOAD COMBINATIONS TABLE OF CONTENTS 3.1

INTRODUCTION .................................................................................................. 3-1 3.1.1

3.2

3.3

3.4

Load Path................................................................................................ 3-1

LOAD DEFINITIONS ............................................................................................ 3-3 3.2.1

Permanent Loads ..................................................................................... 3-3

3.2.2

Transient Loads ....................................................................................... 3-3

PERMANENT LOAD APPLICATION WITH EXAMPLES .................................... 3-3 3.3.1

Dead Load of Components, DC ................................................................ 3-5

3.3.2

Dead Load of Wearing Surfaces and Utilities, DW ..................................... 3-5

3.3.3

Downdrag, DD ........................................................................................ 3-5

3.3.4

Horizontal Earth Pressure, EH .................................................................. 3-6

3.3.5

Vertical Pressure from Dead Load of Earth Fill, EV ................................... 3-7

3.3.6

Earth Surcharge, ES ................................................................................. 3-7

3.3.7

Force Effect Due to Creep, CR ................................................................. 3-7

3.3.8

Force Effect Due to Shrinkage, SH ........................................................... 3-7

3.3.9

Forces from Post-Tensioning, PS .............................................................. 3-8

3.3.10

Miscellaneous Locked-in Force Effects Resulting from the Construction Process, EL ............................................................................................. 3-9

TRANSIENT LOAD APPLICATION WITH EXAMPLES ...................................... 3-9 3.4.1

Vehicular Live Load, LL .......................................................................... 3-9

3.4.2

Vehicular Dynamic Load Allowance, IM ................................................ 3-15

3.4.3

Vehicular Braking Force, BR .................................................................. 3-15

3.4.4

Vehicular Centrifugal Force, CE ............................................................. 3-16

3.4.5

Live Load Surcharge, LS ........................................................................ 3-17

3.4.6

Pedestrian Live Load, PL ....................................................................... 3-18

3.4.7

Uniform Temperature, TU ...................................................................... 3-18

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3.5

3.6

3.4.8

Temperature Gradient, TG...................................................................... 3-20

3.4.9

Settlement, SE ....................................................................................... 3-20

3.4.10

Water Load and Stream Pressure, WA ..................................................... 3-20

3.4.11

Wind Load on Structure, WS .................................................................. 3-22

3.4.12

Wind on Live Load, WL ......................................................................... 3-24

3.4.13

Friction, FR........................................................................................... 3-25

3.4.14

Ice Load, IC .......................................................................................... 3-26

3.4.15

Vehicular Collision Force, CT ................................................................ 3-26

3.4.16

Vessel Collision Force, CV..................................................................... 3-27

3.4.17

Earthquake, EQ ..................................................................................... 3-27

LOAD DISTRIBUTION FOR BEAM-SLAB BRIDGES ........................................ 3-27 3.5.1

Permanent Loads ................................................................................... 3-27

3.5.2

Live Loads on Superstructure ................................................................. 3-28

3.5.3

Live Loads on Substructure .................................................................... 3-36

3.5.4

Skew Modification of Shear Force in Superstructures .............................. 3-40

LOAD FACTORS AND COMBINATION ............................................................ 3-43

NOTATION..................................................................................................................... 3-45 REFERENCES ................................................................................................................ 3-48

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CHAPTER 3 LOADS AND LOAD COMBINATIONS 3.1

INTRODUCTION Properly identifying bridge loading is fundamental to the design of each component. Bridge design is iterative in the sense that member sizes are a function of loads and loads are a function of member sizes. It is, therefore, necessary to begin by proportioning members based on prior experience and then adjusting for actual loads and bridge geometry. This chapter summarizes the loads to be applied to bridges specified in the AASHTO LRFD Bridge Design Specifications, 6th Edition (AASHTO, 2012) and the California Amendments to the AASHTO LRFD Bridge Design Specifications (CA) (Caltrans, 2014). It is important to realize that not every load listed will apply to every bridge. For example, a bridge located in Southern California may not need to consider ice loads. A pedestrian overcrossing structure may not have to be designed for vehicular live load.

3.1.1

Load Path The Engineer must provide a clear load path. The following illustrates the pathway of truck loading into the various elements of a box girder bridge.

Figure 3.1-1 Truck Load Path from Deck Slab to Girders

The weight of the truck is distributed to each axle of the truck. One half of the axle load then goes to each wheel or wheel tandem. This load will be carried by the deck slab which spans between girders, see Figure 3.1-1.

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Once the load has been transferred to the girders, the direction of the load path changes from transverse to longitudinal. The girders carry the load by spanning between bents and abutments (Figure 3.1-2).

Figure 3.1-2 Truck Load Path from Girders to Bents

12'

12'

12'

12'

Figure 3.1-3 Truck Load on Bent Cap

When the girder load reaches the bent caps or abutments, it once again changes direction from longitudinal to transverse. The bent cap beam transfers the load to the columns. Load distribution in the substructure is covered in Section 3.5.3. The columns are primarily axial load carrying members and carry the load to the footing and finally to the piles. The piles transfer the load to the soil where it is carried by the soil matrix. Load distribution can be described in a more refined manner, however, the basic load path from the truck to the ground is as described above. Each load in Table CA 3.4.1-1 has a unique load path. Some are concentrated loads, others are uniform line loads, while still others, such as wind load, are pressure forces on a surface.

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3.2

LOAD DEFINITIONS

3.2.1

Permanent Loads Permanent loads are defined as loads and forces that are either constant or varying over a long time interval upon completion of construction. They include dead load of structural components and nonstructural attachments (DC), dead load of wearing surfaces and utilities (DW), downdrag forces (DD), horizontal earth pressure loads (EH), vertical pressure from dead load of earth fill (EV), earth surcharge load (ES), force effects due to creep (CR), force effects due to shrinkage (SH), secondary forces from post-tensioning (PS), and miscellaneous locked-in force effects resulting from the construction process (EL).

3.2.2

Transient Loads Transient loads are defined as loads and forces that are varying over a short time interval. A transient load is any load that will not remain on the bridge indefinitely. This includes vehicular live loads (LL) and their secondary effects including dynamic load allowance (IM), braking force (BR), centrifugal force (CE), and live load surcharge (LS). Additionally, there are pedestrian live loads (PL), force effects due to uniform temperature (TU), and temperature gradient (TG), force effects due to settlement (SE), water loads and stream pressure (WA), wind loads on structure (WS), wind on live load (WL), friction forces (FR), ice loads (IC), vehicular collision forces (CT), vessel collision forces (CV), and earthquake loads (EQ).

3.3

PERMANENT LOAD APPLICATION WITH EXAMPLES The following structure, shown in Figures 3.3-1 to 3.3-3, is used as an example throughout this chapter, unless otherwise indicated, for use in determining individual loads.

Figure 3.3-1 Elevation View of Example Bridge

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Figure 3.3-2 Typical Section View of Example Bridge

Railroad Railroad

Figure 3.3-3 Plan View of Example Bridge

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3.3.1

Dead Load of Components, DC The dead load of the structure is a gravity load and is based on structural member geometry and material unit weight. It is generally calculated by modeling the structural section properties in a computer program such as CTBRIDGE. Additional loads such as intermediate diaphragms, hinge diaphragms, and barriers must be applied separately. Be aware of possibly “double counting” DC loads. For example, when the weight of the bent cap is included in the longitudinal frame analysis, this weight shall not be included again in a transverse analysis of the bent. Normal weight concrete is assigned a density of 150 pcf which includes the weight of bar reinforcing steel and lost formwork in cast-in-place (CIP) box girder superstructures. Adjustments need not be made for the presence of prestressing tendons, soffit access openings, vents and other small openings for utilities. For this example bridge, the weight of a Type 732 barrier and Type 7 chain link fence is modeled as a line load in a longitudinal frame analysis as follows: Type 732 barrier: A = 2.73 ft2 wc = 0.15 kcf wbarrier = Awc = 2.73 (0.15) = 0.41 kip/ft

(AASHTO C5.4.2.4)

Type 7 chain link fence: wchain = 16 lb/ft (this weight is essentially negligible) Total weight of two barriers w = (0.41 + 0.02)(2) = 0.86 kip/ft

3.3.2

Dead Load of Wearing Surfaces and Utilities, DW Future wearing surfaces are generally asphalt concrete. New bridges require designing for a thickness of 3 in., which results in a load of 35 psf as specified in MTD 15-17 (Caltrans, 1988). Therefore, the weight of the wearing surface to be considered is: Uniform weight: 35 psf Width of bridge with AC: 58.83-2(1.42) = 55.99 ft Line Load: w = 55.99(0.035) = 1.96 kip/ft The bridge has a utility opening in one of the interior bays. It will be assumed that the weight of this utility is 0.100 kip/ft.

3.3.3

Downdrag, DD Downdrag, or negative skin friction, can add to the permanent load on the piles. Therefore, if piles are located in an area where a significant amount of fill is to be

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placed over a compressible soil layer (such as at an abutment), this additional load on the piles needs to be considered. The geotechnical engineer is responsible for determining the additional load due to DD and incorporating that load with all other loads provided in the CA, Section 10 (Caltrans, 2014).

3.3.4

Horizontal Earth Pressure, EH Horizontal earth pressure is a load that affects the design of the abutment including the footing, piles and wing walls. Application follows standard soil mechanics principles. As an example, the horizontal earth pressure resultant force acting on Abutment 1 of the example bridge is calculated below. This calculation is necessary to determine the total moment demand at the bottom of the abutment stem wall. Assume: ka = 0.3, γs = 120 pcf and abutment height, H = 30 ft.

Note: Refer to the geotechnical report for actual soil properties for a given bridge.

p

Figure 3.3-4 Abutment 1 with EH Load Pressure, p = ka γs z

(AASHTO 3.11.5.1-1)

where: z = depth below ground surface 1 2

Resultant line= load = ka γ s z 2

1 = (0.3)(0.12)(30) 2 16.2 kip ft 2

58.83 = 62.6 ft cos 20o Total Force = 16.2 (62.6) = 1,014 kips Abutment length =

This force acts at a distance = H/3 from the top of footing.  30  Moment about base of stem wall = 1,014   = 10,140 kip-ft  3 

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3.3.5

Vertical Pressure from Dead Load of Earth Fill, EV Similar to horizontal earth pressure, vertical earth pressure can be calculated using basic principles. For the 30 ft tall abutment, the weight of earth on the heel at the Abutment 1 footing is obtained as: Assume distance from heel to back of stem wall = 10.5 ft

 58.83  = EV 10.5 = (30)(0.12) 2,366 kips  o   cos 20 

3.3.6

Earth Surcharge, ES This force effect is the result of a concentrated load or uniform load placed near the top of a retaining wall. For Abutment 1, the approach slab is considered an ES load. ∆p =ks qs ks = 0.3; qs = (0.15)(1.0) = 0.150 ksf; ∆p = 0.3×0.150 = 0.045 ksf (ES Load)

(AASHTO 3.11.6.1-1) (approach slab thickness = 1 ft)

Δp

Figure 3.3-5 Abutment 1 with ES Load

3.3.7

Force Effect Due to Creep, CR Creep is a time dependent phenomenon of concrete structures due to sustained compression load. Generally creep has little effect on the strength of structures, but it will cause prestress losses and leads to increased deflections for service loads (affecting camber calculations). Refer to Chapters 7 and 8 for more information.

3.3.8

Force Effect Due to Shrinkage, SH Shrinkage of concrete structures occurs as they cure. Shrinkage, like creep, creates a loss in prestress force as the structure shortens beyond the initial elastic shortening due to the axial compressive stress of the prestressing. Refer to Chapters 6 to 9 for more information.

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3.3.9

Forces from Post-Tensioning, PS Post tensioning introduces axial compression into the superstructure. The primary post-tensioning forces counteract dead load forces. Secondary PS forces introduce load into the members of a statically indeterminate structure as the structure shortens elastically toward the point of no movement. These forces can be calculated using the longitudinal frame analysis program, CTBRIDGE. Table 3.3-1 shows the Span 1 and Bent 2 output due to these forces. Table 3.3-1 PS Secondary Force Effects PS Secondary Effects After Long Term Losses in Span 1 (All Frames) Location (ft) 1.5 12.60 25.20 37.80 50.40 63.00 75.60 88.20 100.80 113.40 123.00

AX (kips) VY (kips) -7.6 70.7 -6.7 69.4 -5.7 67.7 -5.2 67.7 -4.8 67.0 -4.3 66.8 -4.1 66.9 -3.9 66.9 2.8 66.4 1.9 21.9 10.0 -9.5

VZ (kips) TX (kip-ft) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

MY (kip-ft) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

MZ (kip-ft) 103.1 819.4 1,519.0 2,246.6 3,007.0 3,638.7 4,461.8 5,157.2 6,364.6 6,842.7 6,895.4

PS Secondary Effects After Long Term Losses in Bent 2, Column 1 (All Frames) Location (ft) 0.00 11.00 22.00 33.00 44.00

AX (kips) VY (kips) 31.7 31.7 31.7 31.7 31.7

1.9 1.9 1.9 1.9 1.9

VZ (kips) TX (kip-ft) 0.0 0.0 0.0 0.0 0.0

-0.0 -0.0 -0.0 -0.0 -0.0

MY (kip-ft)

MZ (kip-ft)

0.0 0.0 0.0 0.0 0.0

0.0 20.4 40.8 61.2 81.7

PS Secondary Effects After Long Term Losses in Bent 2, Column 2 (All Frames) Location (ft) 0.00 11.00 22.00 33.00 44.00

AX (kips) VY (kips) 31.7 31.7 31.7 31.7 31.7

1.9 1.9 1.9 1.9 1.9

VZ (kips) TX (kip-ft) 0.0 0.0 0.0 0.0 0.0

-0.0 -0.0 -0.0 -0.0 -0.0

MY (kip-ft)

MZ (kip-ft)

0.0 0.0 0.0 0.0 0.0

0.0 20.4 40.8 61.2 81.7

Note: Location is shown from the left end of the span to the right. AX = axial force, VY = vertical shear, VZ = transverse shear, TX = torsion, MY = transverse bending, MZ = longitudinal bending

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3.3.10

Miscellaneous Locked-in Force Effects Resulting from the Construction Process, EL There are instances when a bridge design requires force to be “locked” into the structure in order to be built. These forces are considered permanent loads and must be included in the analysis. Such an example might be found in a segmental bridge where the cantilever segments are jacked apart before the final closure pour is cast at the midspan. For the example bridge shown above, EL forces do not need to be considered.

3.4

TRANSIENT LOAD APPLICATION WITH EXAMPLES For most ordinary bridges there are a few transient loads that should always be considered. Vehicular live loads (LL) and their secondary effects including braking force (BR), centrifugal force (CE), and dynamic load allowance (IM) are the most important to consider. These secondary effects shall always be combined with the gravity effects of live loads in an additive sense. Uniform Temperature (TU) can be quite significant, especially for bridges with long frames and/or short columns. Wind load on structure (WS) and wind on live load (WL) are significant on structures with tall single column bents over 30 feet. Earthquake load (EQ) is specified by Caltrans Seismic Design Criteria (SDC) and generally controls the majority of column designs in California. Refer to Volume III of this practice manual for seismic design.

3.4.1

Vehicular Live Load, LL Vehicular live load consists of two types of vehicle groups. These are: design vehicular live load – HL-93 and permit vehicles – P loads. For both types of loads, axles that do not contribute to extreme force effects are neglected.

3.4.1.1

HL-93 Design Load The AASHTO HL-93 (Highway Loading adopted in 1993) load includes variations and combinations of truck, tandem, and lane loading. The design truck is a 3-axle truck with variable rear axle spacing and a total weight of 72 kips (Figure 3.41). The design lane load is 640 plf (Figure 3.4-2). The design tandem is a two-axle vehicle, 25 kips per axle, spaced 4 ft apart (Figure 3.4-2). When loading the superstructure with HL-93 loads, only one vehicle per lane is allowed on the bridge at a time, except for Cases 3 and 4 (Figure 3.4-2). Trucks shall be placed transversely in as many lanes as practical. Multiple presence factors shall be used to account for the improbability of multiple fully loaded lanes side by side.

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Figure 3.4-1 HL-93 Design Truck

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The following 4 cases represent, in general, the requirements for HL-93 loads as shown in Figure 3.4-2. Cases 1 and 2 are for positive moments and Cases 3 and 4 are for negative moments and bent reactions only. 50 kip 4′ 640 plf

Case 1: tandem + lane 72 kip 640 plf

Case 2: design truck + lane

64.8 kip

50′

64.8 kip 576 plf

14

ft

Case 3: two design trucks + lane 50 kip 4′

26′-40′

50 kip 4′ 640 plf

Case 4: two tandem trucks + lane Figure 3.4-2 Four Load Cases for HL-93

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Tables 3.4-1 to 3.4-4 list maximum positive moments in Span 2 obtained by the CTBRIDGE program by applying HL-93 loads to the example bridge. Looking at the Span 2 maximum positive moment only, Cases 1 and 2 apply. Case 1 moment is 6,761 + 4,510 = 11,271 kip-ft while Case 2 moment is 8,696 + 4,510 = 13,206 kip-ft. Case 2 controls (truck + lane). The example bridge has 4.092 live load lanes for maximum positive moment design. Live load distribution will be discussed in detail in Section 3.5. Dynamic load allowance (IM) is included in these tables. IM will be covered in Section 3.4.2. Table 3.4-1 HL-93 Design Truck Forces in Span 2 with IM = 1.33 Location (ft) 3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

Positive Moment and Associate Shear # Lanes 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092

MZ+ (kip-ft) 1,394.54 1,675.23 4,546.13 6,836.51 8,272.60 8,696.09 8,215.33 6,730.25 4,419.12 1,570.81 1,584.83

AssocVY (kips) -41.07 187.00 135.41 77.46 14.47 -194.92 -259.64 -322.23 -379.54 -430.11 46.37

# Lanes 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

Negative Moment and Associate Shear # Lanes 4.231 4.231 4.231 4.092 4.092 4.092 4.092 4.092 4.092 4.260 4.260

MZAssoc VY (kip-ft) (kips) -5,950.39 321.25 -3,537.61 47.32 -2,944.20 47.32 -2,276.08 46.92 -1,707.10 46.92 -1,138.12 46.78 -1,523.62 -41.62 -2,028.19 -41.62 -2,535.09 -42.00 -3,189.65 -252.06 -6,238.79 -329.02

# Lanes 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

Table 3.4-2 HL-93 Tandem Forces in Span 2 with IM = 1.33 Location (ft) 3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

Positive Moment and Associate Shear # Lanes 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092

MZ+ Assoc VY (kip-ft) (kips) 995.30 -29.31 1,812.59 156.97 3,802.35 121.81 5,408.62 81.97 6,435.94 38.32 6,760.62 -184.42 6,394.19 -229.54 5,333.83 -272.87 3,715.51 -312.24 1,744.93 -346.66 1,126.76 32.97

Chapter 3 – Loads and Load Combinations

# Lanes 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

Negative Moment and Associate Shear # Lanes 4.231 4.231 4.231 4.092 4.092 4.092 4.092 4.092 4.092 4.260 4.260

MZAssoc VY (kip-ft) (kips) -4,199.54 229.49 -2,515.07 33.64 -2,093.18 33.64 -1,618.18 33.36 -1,213.66 33.36 -809.14 33.26 -1,087.36 -29.70 -1,447.47 -29.70 -1,809.23 -29.98 -2,265.75 -178.53 -4,400.94 -229.07

# Lanes 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

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Table 3.4-3 HL-93 Lane Forces in Span 2 with IM = 1.0 Positive Moment and Associate Shear Location (ft)

# Lanes

MZ+ (kip-ft)

3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092

741.54 852.85 1,720.76 3,069.57 4,159.10 4,509.60 4,123.37 2,998.13 1,709.11 942.33 894.24

Assoc VY (kips) -12.38 37.25 103.01 120.20 59.39 -5.00 -62.34 -123.10 -95.18 -29.62 17.29

Negative Moment and Associate Shear

# Lanes

# Lanes

MZ(kip-ft)

Assoc VY (kips)

# Lanes

5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

4.231 4.231 4.231 4.092 4.092 4.092 4.092 4.092 4.092 4.260 4.260

-6,369.34 -3,687.00 -1,874.86 -1,280.05 -1,226.01 -1,172.13 -1,120.20 -1,068.46 -1,591.42 -3,513.85 -6,220.63

308.97 209.28 82.58 4.59 4.44 4.28 4.28 4.08 -84.70 -211.24 -308.23

5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

Table 3.4-4 HL-93 Design Vehicle Enveloped Forces in Span 2 with IM = 1.33 Positive Moment and Associate Shear

3.4.1.2

Location (ft)

# Lanes

MZ+ (kip-ft)

3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092 4.092

2,136.08 2,665.43 6,266.89 9,906.08 12,431.70 13,205.69 12,338.69 9,728.38 6,128.23 2,687.26 2,479.07

Assoc VY (kips) -53.45 194.22 238.42 197.65 73.85 -199.92 -321.97 -445.33 -474.72 -376.27 63.66

Negative Moment and Associate Shear

# Lanes

# Lanes

MZ(kip-ft)

5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

4.231 4.231 4.231 4.092 4.092 4.092 4.092 4.092 4.092 4.260 4.260

-14,708.31 -9,177.36 -5,787.08 -3,556.13 -2,933.11 -2,310.25 -2,643.82 -3,096.65 -4,126.50 -8,884.28 -14,643.57

Assoc VY (kips) 613.18 454.12 145.45 51.52 51.37 51.06 -37.33 -37.53 -126.71 -457.04 -755.57

# Lanes 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671 5.671

Permit Load The California P-15 permit (CA 3.6.1.8) vehicle is used in conjunction with the Strength II limit state. For superstructure design, if refined methods are used, either 1 or 2 permit trucks shall be placed on the bridge at a time, whichever controls. If simplified distribution is used (AASHTO 4.6.2.2), girder distribution factors shall be the same as the design vehicle distribution factors.

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Table 3.4-5 shows the maximum positive moments in Span 2 obtained by the CTBRIDGE program.

4′-6″

Figure 3.4-3 P-15 Truck Table 3.4-5 Permit Moments in Span 2 with IM = 1.25 Location (ft) 3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

Positive Moment and Associate Shear # MZ+ Assoc VY # Lanes (kip-ft) (kips) Lanes 4.092 4301.45 -126.88 5.671 4.092 3037.35 -126.88 5.671 4.092 8953.10 595.24 5.671 4.092 18103.38 500.79 5.671 4.092 24145.10 155.37 5.671 4.092 26029.03 -34.87 5.671 4.092 23859.67 -498.73 5.671 4.092 17812.72 -498.73 5.671 4.092 8607.76 -798.23 5.671 4.092 3707.44 153.29 5.671 4.092 5233.96 153.29 5.671

Negative Moment and Associate Shear # MZAssoc VY # Lanes (kip-ft) (kips) Lanes 4.231 -24408.50 1094.85 5.671 4.231 -13982.51 920.77 5.671 4.231 -9737.33 156.42 5.671 4.092 -7528.62 155.11 5.671 4.092 -5647.79 155.11 5.671 4.092 -3766.96 154.62 5.671 4.092 -4712.93 -128.55 5.671 4.092 -6271.59 -128.55 5.671 4.092 -7837.45 -129.75 5.671 4.260 -13797.91 -947.21 5.671 4.260 -24485.67 -1462.71 5.671

Notice that the maximum P-15 moment of 26,029 kip-ft exceeds the HL-93 moment of 13,206 kip-ft. Although load factors have not yet been applied, Strength II will govern over Strength I in the majority of bridge superstructure design elements. When determining the force effects on a section due to live load, the maximum moment and its associated shear, or the maximum shear and its associated moment should be considered. Combining maximum moments with maximum shears simultaneously for a section is too conservative. 3.4.1.3

Fatigue Load There are two fatigue load limit states used to insure the structure withstands cyclic loading. A single HL-93 design truck with rear axle spacing of 30 ft shall be run across the bridge by itself for the first case. The second case is a P-9 truck by itself. Dynamic load allowance shall be 15% for these cases.

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3.4.1.4

Multiple Presence Factors (m) To account for the improbability of fully loaded trucks crossing the structure sideby-side, MPFs are applied as follows: Table 3.4-6 Multiple Presence Factors

3.4.2

Number of Loaded Lanes

Multiple Presence Factors, m

1 2 3 >3

1.2 1.0 0.85 0.65

Vehicular Dynamic Load Allowance, IM To capture the “bouncing” effect and the resonant excitations due to moving trucks, the static truck live loads or their effects shall be increased by the percentage of the vehicular dynamic load allowance, IM as specified by CA 3.6.2. For example, the maximum HL-93 static moment at the midspan of Span 2 due to the design truck is 6,538 kip-ft. The static moment due to the lane load is 4,510 kipft. The dynamic load allowance for the HL-93 load case is 33%. Therefore, LL + IM = 1.33(6,538) + 4,510 = 13,206 kip-ft. Note that IM does not apply to the lane load. The Permit static moment at the midspan of Span 2 is 20,823 kip-ft. Dynamic load allowance for Permit is 25%. Therefore, LL+IM = 1.25(20,823) = 26,029 kip-ft.

3.4.3

Vehicular Braking Force, BR This force accounts for traction (acceleration) and braking. It is a lateral force acting in the longitudinal direction and primarily affects the design of columns and bearings. For the example bridge, BR is the greater of the following (AASHTO 3.6.4): 1) 25% of the axle weight of the Design Truck or Design Tandem 2) 5% of (Design Truck + Lane Load) or 5% of (Design Tandem + Lane Load) There are 4 cases to consider. Calculating BR force for one lane of traffic results in the following: Case 1) Case 2) Case 3) Case 4)

25% of Design Truck: 25% of Design Tandem: 5% of truck + lane: 5% of tandem + lane:

Chapter 3 – Loads and Load Combinations

0.25(72) = 18.0 kips 0.25(50) = 12.5 kips 0.05(72 + (412)(0.64)) = 16.8 kips 0.05(50 + (412)(0.64)) = 15.7 kips

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

It is seen that Case 1 controls at 18.0 kips. For column design, this one lane result must be multiplied by as many lanes as practical considering the multiple presence factor, m. The maximum number of lanes that can fit on this structure is determined by using 12.0 ft traffic lanes: 58.83 − 2(1.42) Number of lanes: = 4.66 lanes 12 Dropping the fractional portion, 4 lanes will fit. The controlling BR force is therefore the maximum of: 1) 2) 3) 4)

One lane only: Two lanes: Three lanes: Four lanes:

(18.0)(1.2)(1) (18.0)(1.0)(2) (18.0)(0.85)(3) (18.0)(0.65)(4)

= 21.6 kips = 36.0 kips = 45.9 kips = 46.8 kips

Four lanes control at 46.8 kips. This force is a horizontal force to be applied at deck level in the longitudinal direction resulting in shear and bending moments in the columns. In order to determine these column forces, a longitudinal frame model can be used, as in CTBRIDGE. Apply a user load and input the load factors to a superstructure member in the longitudinal direction. When a percentage of the truck weight is used to determine BR, only that portion of the truck that fits on the bridge shall be utilized. For example, if the bridge total length is 25 ft, then only the two 32 kip axles that fit shall be used for BR calculations.

3.4.4

Vehicular Centrifugal Force, CE Horizontally curved bridges are subject to CE forces. These forces primarily affect substructure design. The sharper the curve, the higher these forces will be. These forces act in a direction that is perpendicular to the alignment and toward the outside of the curve. Centrifugal forces apply to both HL-93 live load (truck and tandem only) and Permit live load. Dynamic load allowance does not apply to these calculations.

R = 400 ft

Figure 3.4-4 Centrifugal Force Example

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

 v2    gR 

C= f 

(AASHTO 3.6.3-1)

Example Assume:

v f

= 70 mph (Highway Design Speed) = 4/3 (Strength I Load combination)

Reaction of one lane of HL-93 truck at Bent 2 = 71.6 kips Reaction of one lane of HL-93 tandem at Bent 2 = 50.0 kips R

= 400 ft

Convert v to feet per second:  miles  1 hr   5280ft.  v =  70  = 102.7 ft/sec   hr  3600sec   1 mile  

4  102.7 2   = 1.092 3  (32.2)(400) 

C= 

Total shear for 4 lanes over Bent 2 simultaneously: Shear = 1.092(71.6)(4)(0.65) = 203.3 kips

3.4.5

Live Load Surcharge, LS This load shall be applied when trucks can come within one half of the wall height at the top of the wall on the side of the wall where earth is being retained.

< H/2 H

Figure 3.4-5 Applicability of Live Load Surcharge

Chapter 3 – Loads and Load Combinations

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When the condition of Figure 3.4-5 is met, then the following constant horizontal earth pressure shall be applied to the wall: ∆p =kγs heq

(AASHTO 3.11.6.4-1)

An equivalent height of soil is used to approximate the effect of live load acting on the fill. Refer to AASHTO Table 3.11.6.4-1. For the example bridge, the live load surcharge for Abutment 1 is calculated as follows: Abutment Height = 30 ft

heq = 2.0 ft ∆p =0.3(0.12)(2.0)=0.072 ksf Loading is similar to ES as shown in Figure 3.3-5.

3.4.6

Pedestrian Live Load, PL Pedestrian live loads (PL) are assumed to be a uniform load accounting for the presence of large crowds, parades, and regular use of the bridge by pedestrians. Pedestrian live load can act alone or in combination with vehicular loads if the bridge is designed for mixed use. This load is investigated when pedestrians have access to the bridge. Either the bridge will be designed as a pedestrian overcrossing or will have a sidewalk where both vehicles and pedestrians utilize the same structure. The PL load is 75 psf vertical pressure on sidewalks wider than 2 ft. For pedestrian overcrossings (POCs) the vertical pressure is 90 psf. The example bridge does not have a sidewalk and would therefore not need to be designed for pedestrian live load.

3.4.7

Uniform Temperature, TU Superstructures will either expand or contract due to changes in temperature. This movement will introduce additional forces in statically indeterminate structures and results in displacements at the bridge joints and bearings that need to be taken into account. These effects can be rather large in some instances. The design thermal range for which a structure must be designed is shown in AASHTO Table 3.12.2.1-1.

Chapter 3 – Loads and Load Combinations

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AASHTO Table 3.12.2.1-1 Procedure A Temperature Ranges Climate

Steel or Aluminum

Concrete

Wood

Moderate Cold

0° to 120°F -30° to 120°F

10° to 80°F 0° to 80°F

10° to 75°F 0° to 75°F

For the example bridge, column movements due to a uniform temperature change are calculated below. This can be accomplished using a frame analysis program such as CSiBridge or CTBRIDGE. A hand method is shown below. To start, calculate the point of no movement. The following relative stiffness method can be used to accomplish this. Table 3.4-7 Center of Stiffness Calculation P@1″ (kip/in.) D (ft) PD/100

Abut 1

Bent 2

Bent 3

Abut 4

SUM

0 0 0

206 126 260

169 294 497

0 412 0

375 757

Force to deflect the top of column by 1 in. (P@1 in.) can be determined from: 3EI col ∆ (for pinned columns) P= L3 where ∆ = 1 in.; E = 3,834 ksi; Icol =

πr 4 ; L = 44 ft at Bent 2, 47 ft at Bent 3; 4

r = 3.0 ft PD 757 100 = (100) = (100) 201.8ft ΣP 375

Σ

The point of no movement =

The factor of 100 is used to keep the numbers small and can be factored out if preferred. This point of no movement is the location from Abutment 1 where no movement is expected due to uniform temperature change. Next determine the rise or fall in temperature change. From AASHTO Table 3.12.2.1-1, assuming a moderate climate, the temperature range is 10 to 80°F. Design thermal movement is determined by the following formula: ∆T =αL(TMaxDesign−TMinDesign)/2 Using a temperature change of +/-40°F, we can now determine a movement factor using concrete properties. Movement Factor = α × ∆T α = coefficient of thermal expansion for a given material

Chapter 3 – Loads and Load Combinations

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in. ) = 0.29 in / 100 ft 100 ft The movement at each bent is then calculated (movement at abutments is determined in a similar fashion): Movement Factor = (0.000006/°F)(40°F) (1200

(201.8 − 126) = Bent 2 (0.29) = 0.220 in. 100 (294 − 201.8) = Bent 3 (0.29) = 0.267 in. 100 The factored load is calculated using γTU = 0.5. For joint displacements the larger factor γTU = 1.2 is used. Refer to Chapter 14 for expansion joint calculations.

3.4.8

Temperature Gradient, TG Bridge decks are exposed to the sunlight thereby causing them to heat up much faster than the bottom of the structure. This thermal gradient can induce additional stresses in the statically indeterminate structure. For simply-supported or wellbalanced framed bridge types with span lengths less than 200 ft this effect can be safely ignored. If, however, your superstructure is built using very thick concrete members, or for structures where mass concrete is used, thermal gradients should be investigated especially in an environment where air temperature fluctuations are extreme.

3.4.9

Settlement, SE Differential settlement of supports causes force effects in statically indeterminate structures. A predefined maximum settlement of 1 in. or 2 in. at Service-I Limit State is generally assumed for foundation design. At this level of settlement, ordinary bridges will not be significantly affected if the actual differential settlement is not expected to exceed ½ inch. If, however, this criterion makes the foundation cost unacceptable, larger settlements may be allowed. In that case, settlement analysis will be required. For example, if an actual settlement of one inch for the example bridge is assumed, one would have to consider loads generated by SE and check the superstructure under Strength load combinations. To perform this analysis, assume Bent 2 doesn’t settle. Then allow Bent 3 to settle one inch. Force effects that result from this scenario become SE loads.

3.4.10

Water Load and Stream Pressure, WA The example bridge can be modified by assuming Bent 2 is a pier in a stream as shown in Figure 3.4-6. See the figure below for the pier configuration.

Chapter 3 – Loads and Load Combinations

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Figure 3.4-6 Stream Flow Example Assume the angle between stream flow and the pier is 10 degrees and the stream flow velocity is 6.0 fps. The pressure on the pier in the direction of the longitudinal axis of the pier is calculated by: C DV 2 1000 0.7 × 6 2 p= = 0.0252 ksf 1000 p=

(AASHTO 3.7.3.1-1)

10o

Figure 3.4-7 Longitudinal to Pier Forces due to Stream Flow

Figure 3.4-8 Transverse to Pier Forces due to Stream Flow

Chapter 3 – Loads and Load Combinations

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This pressure is applied to the pier’s projected area, assuming the distance from the river bottom to the high water elevation is 12 ft. Total pier force = 0.0252 (56) sin (10°)(12) = 2.94 kips Then, pressure on the pier in the direction perpendicular to the axis of the pier is calculated using the following: C LV 2 1000 0.7 × 6 2 p= = 0.0252 ksf 1000 p=

(AASHTO 3.7.3.2-1)

Total pressure on the pier in the lateral direction is therefore: Total pier force = 0.0252(56)(12) = 16.93 kips

3.4.11

Wind Load on Structure, WS Wind load is based on a base wind velocity that is increased for bridges taller than 30 ft from ground to top of barrier. Wind load primarily affects the substructure design. Using the example bridge, calculate wind load on the structure as shown below. First calculate the design wind velocity:

Z  V  VDZ = 2.5V0  30  l n   (AASHTO 3.8.1.1-1)  VB   Z0  Assume the bridge is in ‘open country’ with an average height from ground to top of barrier equal to 50.25 ft.  100   50.25  2.5 × 8.2  110.4 mph VDZ =  l n  0.23  = 100    

Next, a design wind pressure, PD is calculated. V  PD = PB  DZ   VB 

2

(AASHTO 3.8.1.2.1-1)

For the superstructure with wind acting normal to the structure (skew = 0 degree), 2

 110.4  = PD 0.05 =   0.061 ksf  100  For the columns 2

 110.4  = PD 0.04 =   0.049 ksf  100 

Chapter 3 – Loads and Load Combinations

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0.049 ksf

0.049 ksf

0.061 ksf

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 3.4-9 WS Application Table 3.4-8 Wind Load at Various Angles of Attack Superstructure Skew 0 15 30 45 60

PD lat 0.061 0.054 0.050 0.040 0.021

PD long 0.000 0.007 0.015 0.020 0.023

In order to use these pressures, it is convenient to turn these into line loads for application to a frame analysis model. Load on the spans = (6.75 + 2.67) × 0.061 = 0.575 klf > 0.30 klf (min) Load on columns = 6.0 × 0.049 = 0.294 klf WS load application within a statically indeterminate frame model is shown in Figure 3.4-10. For the superstructure use table 3.8.1.2.2-1 to calculate the pressure from various angles skewed from the perpendicular to the longitudinal axis. Results are shown above in Table 3.4-8. The “Trusses, Columns, and Arches” heading in the AASHTO table refers to superstructure elements. The table refers to spandrel columns in a superstructure not pier/substructure columns. Transverse and longitudinal pressures should be applied simultaneously. For application to the substructure, the transverse and longitudinal superstructure wind forces are resolved into components aligned relative to the pier axes.

Chapter 3 – Loads and Load Combinations

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Load perpendicular to the plane of the pier: FL = FL,super cos(20°) + FT,super sin(20°) At 0 degrees: FL = (0)cos(20°) + 0.061(6.75 + 2.67) sin(20°) = 0.196 klf At 60 degrees: FL = 0.023(6.75 + 2.67) cos (20°) + 0.021(6.75 +2 .67) sin(20°) = 0.204 klf + 0.068 klf = 0.272 klf And, load in the plane of the pier (parallel to the columns): FT = FL,super sin(20°) + FT,super cos(20°) At 0 degrees: FT = (0) sin(20°) + 0.061(9.42)cos(20°) = 0.540 klf At 60 degrees: FT = 0.023(9.42) sin(20°) + 0.021(9.42) cos(20°) = 0.074 klf + 0.186 klf = 0.260 klf The wind pressure applied directly to the substructure is resolved into components perpendicular to the end and front elevations of the substructure. The pressure perpendicular to the end elevation of the pier is applied simultaneously with the wind load from the superstructure.

3.4.12

Wind on Live Load, WL This load is applied directly to vehicles traveling on the bridge during periods of a moderately high wind of 55 mph. This load is to be 0.1 klf applied transverse to the bridge deck. WL load application is shown in Figure 3.4-11.

Chapter 3 – Loads and Load Combinations

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Figure 3.4-10 Wind on Structure

Figure 3.4-11 Wind on Live Load

3.4.13

Friction, FR Friction loading can be any loading that is transmitted to an element through a frictional interface. There are no FR forces for the example bridge.

Chapter 3 – Loads and Load Combinations

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3.4.14

Ice Load, IC The presence of ice floes in rivers and streams can result in extreme event forces on the pier. These forces are a function of the ice crushing strength, thickness of ice floe, and width of pier. For equations and commentary on ice load, see AASHTO 3.9. Snow load/accumulation on a bridge need not be considered in general.

3.4.15

Vehicular Collision Force, CT Vehicle collision refers to collisions that occur with the barrier rail or at unprotected columns (AASHTO 3.6.5). Referring to AASHTO Section 13, the design loads for CT forces on barrier rails are as shown in AASHTO Table A13.2-1. Test Level Four (TL-4) will apply most of the time. These forces are applied to our Type 732 barrier rail from our example bridge as follows: F FT 10 ft 10 ft

Figure 3.4-12 CT Force on Barrier FT = 54 kips FL = 18 kips Load from this collision force spreads out over a width calculated based on detailing of the barrier bar reinforcement and yield line theory. Caltrans policy is to assume this distance to be 10 ft at the base of the barrier for Standard Plan barriers that are solid. Given that the barrier height is 2′-8″, we can calculate the moment per foot as follows:

54 × 2.67 = M CT = 14.4 kip-ft/ft 10 Applying a 20% factor of safety (CA A13.4.2) results in: 1.2 × 14.4 = 17.28 kip-ft/ft

Chapter 3 – Loads and Load Combinations

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Standard plan barriershave already been designed for these CT forces. However, these forces must be carried into the overhang and deck. Caltrans deck design charts in MTD 10-20 (Caltrans, 2008) were developed to include these CT forces in the overhang. For a bridge with a long overhang or an unusual typical section configuration, for which the deck design charts do not apply, calculations for CT force should be performed. Post-type (see-through) barriers require special analysis for various failure modes and are not covered here.

3.4.16

Vessel Collision Force, CV Generally, California bridges over navigable waterways are protected by a fender system. In these instances, the fender system is then subject to the requirements of AASHTO 3.14 and/or the AASHTO Guide Specifications and Commentary for Vessel Collision Design of Highway Bridges (AASHTO, 2010). Due to the infrequent occurrence of these bridges, an example of CV force calculations will not be made here.

3.4.17

Earthquake, EQ In California, a high percentage of bridges are close enough to a major fault to be controlled by EQ forces. EQ loads are a function of structural mass, structural period, and the Acceleration Response Spectrum (ARS). The ARS curve is determined from a Caltrans online mapping tool or supplied by the Office of Geotechnical Services. These requirements will be covered in detail in Volume III of this practice manual. It is recommended that EQ forces be considered early in the design process in order to properly size members.

3.5

LOAD DISTRIBUTION FOR BEAM-SLAB BRIDGES

3.5.1

Permanent Loads Load distribution for permanent loads follows standard structure mechanics methods. There are, however, a few occasions where assumptions are made to simplify the design process, rather than follow an exact load distribution pathway.

3.5.1.1

Barriers Barrier loads are generally distributed equally to all girders in the superstructure section (Figure 3.5-1). The weight of the barrier is light enough that a more detailed method of distribution is not warranted. For the example bridge, DC load for barriers is 0.86 klf for two barriers. The barrier load to each girder is simply 0.86/5 = 0.172 klf (Figure 3.5-1).

Chapter 3 – Loads and Load Combinations

3-27

0.172 kip/ft

0.172 kip/ft

0.172 kip/ft

0.172 kip/ft

0.172 kip/ft

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 3.5-1 Barrier Distribution 3.5.1.2

Soundwalls Since a soundwall has a much higher load per lineal length than a barrier, a more refined analysis should be performed to obtain more accurate distribution. The following procedure can be found in MTD 22-2 (Caltrans, 2004) for non-seismic design. Soundwall distribution is simplified by applying 100% of the soundwall shear demand on the exterior girder. Secondly, apply 1/n to the first interior girder; where n = number of girders. For moment, apply 60% to the exterior girder and 1/n to the first interior girder. It is assumed that other girders in the bridge are unaffected by the presence of the soundwall. For the example bridge, assume a soundwall 10 ft tall using 8-inch blocks on the north side of the bridge. The approximate weight per foot assuming solid grouting is 88 psf × 10 ft = 880 plf. Applying this load in a 2-D frame program such as CTBRIDGE, the results are shown in Table 3.5-1.

3.5.2

Live Loads on Superstructure

3.5.2.1

Cantilever Overhang Loads Live load distribution on the overhang is determined using an equivalent strip width method. The overhang is designed for Strength I and Extreme Event II only (AASHTO A13.4) Consider the case of maximum overhang moment due to the HL-93 design truck (Strength I). Since the overhang is designed on a lineal length basis it is, therefore, necessary to determine how much of the overhang is effective at resisting this load. Wheel loads can be placed up to 1 ft from the face of the barrier. The 32-kip axle weight of the HL-93 truck is divided by two to get a 16-kip point load, 1 ft from the barrier. See Figure 3.5-2.

Chapter 3 – Loads and Load Combinations

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Table 3.5-1 Soundwall Forces Whole Bridge Location

VY (kips)

MZ (kip-ft)

Apply to Exterior Girder VY (kips)

MZ (kip-ft)

Apply to First Interior Girder VY (kips) MZ (kip-ft)

Span 1 1.50 12.60 25.20 37.80 50.40 63.00 75.60 88.20 100.80 113.40 123.00

38.3 28.5 17.5 6.4 -4.7 -15.8 -26.9 -38.0 -49.0 -60.1 -68.6

58.5 429.8 719.8 870.0 880.4 751.1 482.1 73.3 -475.1 -1,162.9 -1,780.8

38.3 28.5 17.5 6.4 -4.7 -15.8 -26.9 -38 -49 -60.1 -68.6

35.1 258.0 432.0 522.0 528.0 451.0 289.0 44.0 -285.0 -698.0 -1,068.0

7.66 5.7 3.5 1.28 -0.94 -3.16 -5.38 -7.6 -9.8 -12.0 -13.7

11.7 86.0 144.0 174.0 176.0 150.0 96.4 14.7 -95.0 -233.0 -356.2

-1,041.0 -504.0 14.0 383.0 603.0 674.0 597.0 370.0 -5.7 -530.0 -1,072.0

14.2 11.7 8.8 5.84 2.88 -0.06 -3.02 -5.98 -8.92 -11.9 -14.3

-347.0 -168.0 4.7 128.0 201.0 225.0 199.0 123.0 -1.9 -177.0 -357.0

-931.0 -613.0 -251.0 37.9 253.0 395.0 463.0 457.0 378.0 226.0 32.8

12.8 11.3 9.20 7.12 5.04 2.96 0.88 -1.20 -3.26 -5.34 -7.16

-310.0 -204.0 -83.6 12.6 84.4 132.0 154.0 152.0 126.0 75.3 10.9

Span 2 3.00 16.80 33.60 50.40 67.20 84.00 100.80 117.60 134.40 151.20 165.00

70.9 58.7 44.0 29.2 14.4 -0.3 -15.1 -29.9 -44.6 -59.4 -71.5

-1,734.7 -839.8 23.4 638.4 1,005.3 1,123.9 994.3 616.4 -9.5 -883.3 -1,786.9

70.9 58.7 44.0 29.2 14.4 -0.3 -15.1 -29.9 -44.6 -59.4 -71.5 Span 3

3.00 11.80 23.60 35.40 47.20 59.00 70.80 82.60 94.40 106.20 116.50

64.1 56.3 46.0 35.6 25.2 14.8 4.4 -6.0 -16.3 -26.7 -35.8

-1,551.0 -1,021.0 -418.0 63.2 422.0 658.0 771.0 762.0 631.0 377.0 54.7

Chapter 3 – Loads and Load Combinations

64.1 56.3 46.0 35.6 25.2 14.8 4.4 -6.0 -16.3 -26.7 -35.8

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X 5'-0''

Figure 3.5-2 Overhang Wheel Load The moment arm for this load is: X = 5.0 −1.42 −1.0 = 2.58 ft The strip width is therefore: Strips = 45.0 + 10X = 45 + 10(2.58) = 70.8 in.

(AASHTO Table 4.6.2.1.3-1)

Overhang moment for design is therefore: = M LL

(16)(2.58)

= 7.0 kip-ft/ft 70.8 / 12

Include dynamic load allowance: MLL = 7.0(1.33) = 9.31 kip – ft/ft Include the Strength I load factor of 1.75: MLL = 9.31 (1.75) = 16.3 kip – ft/ft 3.5.2.2

CIP Box Girder Live load distribution to each girder in a box girder bridge is accomplished using empirical formulas to determine how many live load lanes each girder must be designed to carry. Empirical formulas are used because a bridge is generally modeled in 2D. Refined methods can be used in lieu of empirical methods whereby a 3D model is used to develop individual girder live load distribution. These expressions were developed by exponential curve-fitting of force effects from a large bridge database and comparing to results from more refined analyses. Because flexural behavior differs from shear behavior, and force effects in exterior girders differ from those in interior girders, different formulae are provided for each.

Chapter 3 – Loads and Load Combinations

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Due to the torsional rigidity and load sharing capability of a box girder, the box is often considered as a single girder. The formula for interior girders then applies to all girders. 1.

Live Load Distribution for Interior Girder Moment Span 1 S ≈ 12 ft, L = 126 ft, Nc = 4 (falls within the range of applicability of AASHTO Table 4.6.2.2.2b-1) One lane loaded case:

 1.75 + S   1  gM =    3.6   L  

0.35

 1  N   c

0.45

 1.75 + 12   1  =    3.6   126  

0.35

1   4

0.45

0.501 =

Fatigue limit state: = gM

0.501 = 0.418 1.2

Two or more lanes loaded case:

 13   S   1   13   12   1  = g M = 0.880    =       4   5.8   126   N c   5.8   L  0.3

0.25

0.3

0.25

The distribution factors for all spans are listed in Table 3.5-2. Table 3.5-2 Girder Live Load Distribution for Moment Span

Fatigue Limit State*

All other Limit States

1 0.418 2 0.378 3 0.428 *m of 1.2 has been divided out for the Fatigue Limit State

0.880 0.818 0.894

For a whole bridge design method (such as is used in CTBRIDGE), multiply by the number of girders. For span 1, (gM)total = 4.400. 2. Live Load Distribution for Interior Girder Shear Span 1 Depth of member, d = 81 in. (falls within the range of applicability of AASHTO Table 4.6.2.2.3a-1) One lane loaded case:

 S   d   12.0=   81  = g S = 0.859         9.5   12.0 L   9.5   12 × 126.0  0.6

0.1

Chapter 3 – Loads and Load Combinations

0.6

0.1

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Fatigue limit state: 0.859 gS = = 0.716 1.2 Two or more lanes loaded case: S   d  =  12.0=   81  1.167          7.3   12.0 L   7.3   12 × 126.0  0.9

= gS

0.1

0.9

0.1

The distribution factors for all spans are listed in Table 3.5-3. Table 3.5-3 Girder Live Load Distribution for Shear Span 1 2 3

Fatigue Limit State*

All other Limit States

0.716 0.695 0.720

1.167 1.134 1.175

*m of 1.2 has been divided out for the Fatigue Limit State

The total for the whole bridge for span 1 would be: (gS)total = 5.835 3.5.2.3

Precast I, Bulb-Tee, or Steel Plate Girder In general, the live load distribution at the exterior girder is not the same as that for the interior girder. However, in no instance should the exterior girder be designed for fewer live load lanes than the interior girder, in case of future widening. A precast I-girder bridge is shown in Figure 3.5-3. Calculations for live load distribution factors for interior and exterior girders follow. Given: S = 9.67 ft; L = 110 ft; ts = 8 in.; Kg = longitudinal stiffness parameter (in.4); Nb = 6 Calculation of the longitudinal stiffness parameter, Kg: Kg = n (I + Ae2g) (AASHTO 4.6.2.2.1-1) EB 4696 = n = = 1.225 ED 3834 A = 1,085 in.2 beam only I = 733,320 in.4; eg = vertical distance from c.g. beam to c.g. deck = 39.62 in. Kg = 1.225(733,320 + 1,085 × 39.622) = 2,984,704 in.4

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

110'-0''

110'-0''

H-Piles

Integral Abutment

22'-0''

55'-4 ½ '' Total Width 52'-0 '' 1'-8 1/4 '' 1'-10'

5 spaces at 9' -8'' 8 '' Reinforced Concrete Deck

9 ''

Figure 3.5-3 Precast Bulb-Tee Bridge to be Used for Distribution Calculations

1. Live Load Distribution for Interior Girder Moment One lane loaded case:

 S   S   Kg  = g M 0.06 +      3   14   L   12.0 Lts  0.4

0.3

0.1

0.1

0.4 0.3  9.67   9.67   2,984, 704  = = 0.06 +  0.542     3   14   110   (12 )(110 )( 8 ) 



 3   12.0 Lts 

Note: The term 

Kg

0.1

could have been taken as 1.09 for preliminary

design (AASHTO 4.6.2.2.1-2), but was not used here. 0.542 Fatigue limit state:= gM = 0.452 1.2

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Two or more lanes loaded case:  S  = g M 0.075 +    9.5 

0.6

S L  

0.2

0.6

 Kg  3  12.0 Lts

  

0.1

0.2

 9.67   9.67   2,984,704  = 0.075 +      3   9.5   110   (12)(110)(8) 

0.1

= 0.796

2. Live Load Distribution for Exterior Girder Moment One lane loaded case: Use the lever rule. The lever rule assumes the deck is a simply supported member between girders. Live loads shall be placed to maximize the reaction of one lane of live load (Figure 3.5-4). 2'

6'

3.5'

LL

A

9.67'

B

Exterior Girder Interior Girder

Figure 3.5-4 Lever Rule Example for Exterior Girder Distribution Factor

0 ΣM B = LL (3.5 + 9.5) =RA × 9.67 2 RA = 0.672 lanes Therefore, for exterior girder moment, gM = 0.672 lanes. Use for the Fatigue Limit State. For other limit states, gM = 1.2 (0.672) = 0.806 lanes. Two or more lanes loaded case: g M = e ( g M )interior = e 0.77 +

de 9.1

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

d e = 1.83ft e = 0.77 +

1.83

= 0.971 9.1 = g M 0.971(0.796) = 0.773 lanes

It is seen that the one lane loaded case controls for all limit states. 3. Live Load Distribution for Interior Girder Shear One lane loaded case: g S =0.36 +

9.67 S =0.36 + =0.747 25.0 25.0

Fatigue limit state:= gM

0.747 = 0.623 1.2

Two or more lanes loaded case: 9.67  9.67  S g S =0.2 + −   =0.2 + −  =0.929 12.0  35  12.0  35  S

2

2

4. Live Load Distribution for Exterior Girder Shear One lane loaded case: This case requires the lever rule once again. The result is exactly the same for moment as for shear. Therefore (gS)exterior = 0.672 for the Fatigue Limit State and (gS)exterior = 0.806 for all other limit states. Two or more lanes loaded case: ( g s )exterior = e( g s )interior de 1.83 = 0.6 + = 0.783 10 10 g S 0.783 = = ( 0.929 ) 0.727 e = 0.6 +

However, because the exterior girder cannot be designed for fewer live load lanes than the interior girders, use (gS)exterior = 0.929 for all other limit states. The complete list of distribution factors for this bridge is shown in Tables 3.5-4 and 3.5-5.

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 3.5-4 Girder Live Load Distribution for Moment Girder

Fatigue Limit State

All other Limit States

Interior Exterior

0.452 0.672

0.796 0.806

Table 3.5-5 Girder Live Load Distribution for Shear

3.5.3

Girder

Fatigue Limit State

All other Limit States

Interior Exterior

0.623 0.672

0.929 0.929

Live Loads on Substructure Substructure elements include the bent cap beam, columns, footings, and piles. To calculate the force effects on these elements a “transverse” analysis shall be performed. In order to properly load the bent with live load, results from the longitudinal frame analysis are used. In this section, live load forces affecting column design are discussed. For column design there are 3 cases to consider: 1) (MT)max + (ML)assoc + Passoc 2) (ML)max + (MT)assoc + Passoc 3) Pmax + (ML)assoc + (MT)assoc Each of these three cases applies to both the Design Vehicle live load and the Permit load. In the Permit load case, up to two permit trucks are placed in order to produce maximum force effects. These loads are then used in a column design program such as Caltrans’ WINYIELD (2007).

3.5.3.1

Example Consider the following bridge with a single column bent as shown in Figure 3.55 and 3.5-6 to calculate the force effects at the bottom of the column: 285' 150'

135'

Figure 3.5-5 Example Bridge Elevation for Substructure Calculations

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

51' -10'' 1' -5'' 5' -0''

12' -0''

12' -0''

12' -0''

8' -0''

Figure 3.5-6 Example Bridge Typical Section for Substructure Calculations Live load effects from a longitudinal frame analysis are tabulated: Table 3.5-6 CTBRIDGE Live Load Effects Bottom of Column Live Load Forces (one lane + IM) Vehicle class Design Truck+IM Design Lane Permit Truck+IM

Case Pmax (ML)max Pmax (ML)max Pmax (ML)max

P (kips) 154 100 103 61 455 333

ML(kip-ft) 66 465 39 228 240 1,319

1. Design Vehicle Maximum Transverse Moment (MT)max Case To obtain the moments in the transverse direction, the axial forces due to one lane of live load listed above are placed on the bent to produce maximum effects. By inspection, placing two design vehicle lanes on one side of the bent will produce maximum transverse moments in the column (Figure 3.5-7). When not obvious, cases with one, two, three, and four vehicles should be evaluated. Note that wheel lines must be placed 2 ft from the face of the barrier. The edge of deck to edge of deck case should also be checked. Longitudinally, the vehicles are located over the bent thus maximizing MT.

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

12'

12'

LL

LL

6'

6'

12'

12'

2' 22' -6''

Figure 3.5-7 Vehicle Position for (MT)max LL = 154 + 103 = 257 kips Multiple presence factor, m = 1.0 for two lanes. (M T= ) max

257 2

= ( 22.5 + 16.5 + 10.5 + 4.5 )

6,939 kip-ft

(ML)associated =(66 + 39) × 2 = 210 kip-ft Passociated = 257× 2 = 514 kips Maximum Axial Force Pmax Case To maximize axial forces on the column, place as many lanes as can fit on the bridge. In this case four lanes are required: 12'

12'

12'

12'

LL

LL

LL

LL

6'

6'

6'

6'

Figure 3.5-8 Vehicle Position for (P)max

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Multiple presence, m = 0.65 for four-lanes loaded.  257  = ( M T ) associated (0.65)   (22.5 + 16.5 + 10.5 + 4.5 − 1.5 − 7.5 − 13.5 − 19.5)  2  = 1, 002 kip-ft

(ML)associated = (0.65)(66+39)(4) = 273 kip-ft Pmax = (0.65)(257)(4) = 668 kips Maximum Longitudinal Moment (ML)max Case Load the bridge with as many lanes as possible but this time, the vehicles are located longitudinally somewhere within the span: ( M T ) associated (0.65) =

(100 + 61) (22.5 + 16.5 + 10.5 + 4.5 − 1.5 − 7.5 − 13.5 − 19.5)

2 = 628 kip-ft

(ML)max = (0.65)(465 + 228)(4) = 1,802 kip-ft Passociated = (0.65)(100 + 61)(4) = 419 kips

2. Permit Vehicle Next calculate the live load forces at the bottom of the column due to the Permit vehicle. Note: Multiple presence, m = 1.0 when using either one or two lanes (Article CA 3.6.1.8.2). (MT)max Case Two lanes of Permit load are placed on one side of the bent cap as shown in Figure 3.5-7. ( M T )= max

455 = ( 22.5 + 16.5 + 10.5 + 4.5 ) 12, 285 kip-ft 2

(ML)associated = 240 (2) = 480 kip-ft Passociated = 455(2) = 910 kips

Pmax Case Again, to maximize the axial force, the trucks are located right over the bent and a maximum of 2 lanes of Permit vehicles are placed on the bridge. This results in the same configuration as in the (MT)max case. Therefore, the results are the same.

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

(ML)max Case ( M T ) associated =

333 2

= ( 22.5 + 16.5 + 10.5 + 4.5 )

8, 991 kip-ft

(ML)max = 1,319(2) = 2,638 kip-ft Passociated = 333(2) = 666 kips

Summary of the live load forces at the bottom of column for all live load cases are shown in Tables 3.5-7 and 3.5-8. Table 3.5-7 Summary of Design Vehicle Forces for Column Design Load

(MT)max Case (kip-ft)

(ML)max Case (kip-ft)

Pmax Case (kips)

MT ML P

6,939 210 514

628 1,802 419

1,002 273 668

Table 3.5-8 Summary of Permit Vehicle Forces for Column Design

3.5.4

Load

(MT)max Case (kip-ft)

(ML)max Case (kip-ft)

Pmax Case (kips)

MT ML P

12,285 480 910

8,991 2,638 666

12,285 480 910

Skew Modification of Shear Force in Superstructures To illustrate the effect of skew modification, the example bridge shown in Figure 3.5-9 is used. Because load takes the shortest pathway to a support, the girders at the obtuse corners of the bridge will carry more load. A 2-D model cannot capture the effects of skewed supports. Therefore, shear forces must be amplified according to Table 3.5-9. Table 3.5-9 Skew correction of shear forces Type of Superstructure

Applicable Cross-Section from Table 4.6.2.2.1-1

Cast-in-place Concrete Multicell Box

d

Correction Factor θ

1.0 +

0 < θ < 60o

for

50

exterior girder 1.0 +

θ

6.0 < S < 13.0 20 < L < 240

for first

300

interior girder

Chapter 3 – Loads and Load Combinations

Range of Applicability

35 < d < 110 Nc > 3

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The example bridge has a 20 degree skew. Correction Factors are as follows: Exterior Girder: 1.0 +

20 1.4 = 50

First Interior Girder: 1.0 +

20 = 1.067 300

To illustrate the application of these correction factors, apply them to dead load (DC) shear forces only on the northern most exterior girder. Correction would also be made to DW and LL in general (as well as the other exterior girders). Figure 3.5-9 shows the girder layout and Table 3.5-10 lists DC correction factors for the example bridge.

Figure 3.5-9 Girder Layout

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 3.5-10 Example Bridge DC Skew Correction (Northern Most Girder) Span

1

2

3

Tenth Point

VDC (kips)

(VDC)per girder (kips)

Correction

(VDC)corrected (kips)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

721 540 334 128 -78 -284 -490 -696 -906 -1,121 -1,285 1,357 1,124 840 562 288 13.2 -261 -536 -815 -1,098 -1,331 1,219 1,068 866 669 476 284 90.6 -102 -295 -488 -656

144 108 66.8 25.6 -15.6 -56.8 -98.0 -139 -181 -224 -257 271 225 168 112 57.5 2.6 -52.3 -107 -163 -220 -266 244 214 173 134 95.3 56.7 18.1 -20.4 -59.0 -97.6 -131

1.39 1.32 1.24 1.16 1.08 1.00 1.00 1.00 1.00 1.00 1.00 1.386 1.32 1.24 1.16 1.08 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.32 1.24 1.16 1.08 1.00 1.00 1.00 1.00 1.00 1.00

201 143 82.8 29.7 -16.8 -56.8 -98.0 -139 -181 -224 -257 376 297 208 130 62.1 2.6 -52.3 -107 -163 -220 -266 336 282 215 155 103 56.7 18.1 -20.4 -59.0 -97.6 -131

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

3.6

LOAD FACTORS AND COMBINATION The Limit States of AASHTO (2012) and CA (Caltrans, 2014) Section 3 require combining the individual loads with specific load factors to achieve design objectives. The example bridge shown in Figure 3.3-1 is used to determine the maximum positive moments in the superstructure by factoring all relevant load effects in the appropriate limit states. Tables 3.6-1, 3.6-2 and 3.6-3 summarize load factors used for the example bridge Span 2. For Span 2, unfactored midspan positive moments are as follows: MDC = 20,936 kip-ft MDW = 2,496 kip-ft MHL-93 = 13,206 kip-ft MPERMIT = 26,029 kip-ft MPS = 7,023 kip-ft Factored positive moments are calculated as follows: Strength I: M = 1.25(20,936) + 1.5(2,496) + 1.0(7,023) + 1.75(13,206) = 60,047 kip-ft Strength II: M = 1.25(20,936) + 1.5(2,496) + 1.0(7,023) + 1.35(26,029) = 72,076 kip-ft Therefore the Strength II Limit State controls for positive moment at this location. Table 3.6-1 Load Combinations for Span 2 +M Load Combination

Limit State

STRENGTH I STRENGTH II

DC LLHL-93 LLPermit DD IM IM DW CE CE EH BR EV PL ES LS EL PS CR SH

γp

1.75

γp

-

1.35

Chapter 3 – Loads and Load Combinations

WA

WS

WL

FR

TU

TG

SE

EQ BL IC CT CV (use only one)

1.0

-

-

1.0

0.50/1.20

γTG

γSE

-

1.0

-

-

1.0

0.50/1.20

γTG

γSE

-

3-43

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 3.6-2 Load Factors for Permanent Loads, γp Load Factor

Type of Load, Foundation Type, and Method Used to Calculate Downdrag

Maximum Minimum

DC: Component and Attachments DC: Strength IV, only

1.25 1.50

0.90 0.90

1.4 1.05 1.25

0.25 0.30 0.35

DW: Wearing Surfaces and Utilities

1.50

0.65

EH: Horizontal Earth Pressure • Active • At-Rest • AEP for Anchored Walls

1.50 1.35 1.35

0.90 0.90 N/A

EL: Locked-in Construction Stresses

1.00

1.00

1.00 1.35 1.30 1.35

N/A 1.00 0.90 0.90

1.5 1.3 1.95

0.9 0.9 0.9

1.50

0.75

DD: Downdrag

Piles, α Tomlison Method Piles, λ Method Drilled Shafts, O’Neill and Reese (1999) Method

EV: Vertical Earth Pressure • Overall Stability • Retaining Walls and Abutments • Rigid Buried Structure • Rigid Frames • Flexible Buried Structures o Metal Box Culverts and Structural Plate Culverts with Deep Corrugations o Thermoplastic Culverts o All Others ES: Earth Surcharge

Table 3.6-3 Load Factors for Permanent Loads Due to Superimposed Deformations, γp Bridge Component

PS

CR,SH

Superstructures–Segmental Concrete Substructures supporting Segmental Superstructures (see 3.12.4, 3.12.5)

1.0

See γp for DC, Table 3.6-2

Concrete Superstructures–non-segmental

1.0

1.0

Substructures supporting non-segmental Superstructures • using Ig • using Ieffective

0.5 1.0

0.5 1.0

Steel Substructures

1.0

1.0

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

NOTATION Load Designations BR

=

vehicular braking force

CE

=

vehicular centrifugal force

CR

=

force effects due to creep

CT

=

vehicular collision force

CV

=

vessel collision force

DC

=

dead load of components

DD

=

downdrag

DW

=

dead load of wearing surfaces and utilities

EH

=

horizontal earth pressure load

EQ

=

earthquake

ES

=

earth surcharge load

EV

=

vertical pressure from dead load of earth fill

FR

=

friction

IC

=

ice load

IM

=

vehicular dynamic load allowance

LL

=

vehicular live load

LS

=

live load surcharge

PL

=

pedestrian live load

PS

=

secondary forces from post-tensioning

SE

=

force effects due to settlement

SH

=

force effects due to shrinkage

TG

=

force effects due to temperature gradient

TU

=

force effects due to uniform temperature

WA

=

water load and stream pressure

WL

=

wind on live load

WS

=

wind load on structure

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

General Symbols A

=

area of section (ft2)

C

=

centrifugal force factor

CD

=

drag coefficient

CL

=

lateral drag coefficient

d

=

structure depth (in.)

de

=

distance from cl exterior girder and face of barrier (ft)

e

=

girder LL distribution factor multiplier for exterior girders

eg

=

vertical distance from c.g. beam to c.g. deck (in.)

E

=

modulus of elasticity (ksi)

f

=

CE fatigue factor

Ft

=

transverse barrier collision force (kip)

FL

=

longitudinal barrier collision force (kip)

g

=

gravitational acceleration (32.2 ft/sec)

gM

=

girder LL distribution factor for moment

gS

=

girder LL distribution factor for shear

heq

=

equivalent height of soil for vehicular load (ft)

H

=

height of element (ft)

I

=

moment of inertia (ft4)

k

=

coefficient of lateral earth pressure

ka

=

active earth pressure coefficient

ks

=

earth pressure coefficient due to surcharge

Kg

=

longitudinal stiffness parameter (in.4)

L

=

span length (ft)

MCT

=

vehicular collision moment on barrier (kip-ft)

MLL

=

moment due to live load (kip-ft)

MT

=

transverse moment on column (kip-ft)

ML

=

longitudinal moment on column (kip-ft)

MDC

=

moment due to dead load (kip-ft)

MDW

=

moment due to dead load wearing surface (kip-ft)

MHL-93

=

moment due to design vehicle (kip-ft)

MPERMIT =

moment due to permit vehicle (kip-ft)

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

MPS

=

moment due to secondary pre-stress forces (kip-ft)

n

=

modular ratio

Nb

=

number of beams

Nc

=

number of cells in the box girder section

p

=

stream force pressure (ksf)

p

=

pressure against wall

P

=

axial load on column (k)

PB

=

base wind pressure (ksf)

PD

=

wind pressure (ksf)

qs

=

uniform surcharge applied to upper surface of the active earth wedge (ksf)

R

=

radius of curvature of traffic lane (ft)

S

=

center to center girder spacing (ft)

ts

=

top slab thickness (in.)

v

=

highway design speed (ft/sec)

V

=

design velocity of water (ft/sec)

VDC

=

shear due to dead load (kip)

VDZ

=

design wind velocity at elevation z (mph)

Vo

=

friction velocity (mph)

V30

=

wind velocity at 30 ft above ground (mph)

VB

=

base wind velocity of 100 mph at 30 ft height (mph)

w

=

uniform load (kip/ft)

X

=

moment arm for overhang load (ft)

z

=

depth to point below ground surface (ft)

Z

=

height of structure at which wind loads are being calculated (ft)

Zo

=

friction length of upstream fetch (ft)

α

=

coefficient of thermal expansion

∆p

=

earth surcharge load

γs

=

density of soil (pcf)

θ

=

skew angle (degrees)

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 2. AASHTO, (2010). Guide Specifications and Commentary for Vessel Collision Design of Highway Bridges, 2nd Edition, with 2010 Interim Revisions, American Association of State Highway and Transportation Officials, Washington, D.C. 3. CSI, (2015). CSiBridge 2015, v. 17.0.0, Computers and Structures, Inc., Walnut Creek, CA. 4. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 5. Caltrans, (2014). CTBRIDGE, Caltrans Bridge Analysis and Design v. 1.6.1, California Department of Transportation, Sacramento, CA. 6. Caltrans, (2007). WINYIELD, Column Design Program v. 3.0.10, California Department of Transportation, Sacramento, CA. 7. Caltrans, (2008). Memo to Designers 10-20: Deck and Soffit Slab, California Department of Transportation, Sacramento, CA. 8. Caltrans, (2004). Memo to Designers 22-2: Soundwall Load Distribution, California Department of Transportation, Sacramento, CA. 9. Caltrans, (1988). Memo to Designers 15-17: Future Wearing Surface, California Department of Transportation, Sacramento, CA.

Chapter 3 – Loads and Load Combinations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

CHAPTER 4 STRUCTURAL MODELING AND ANALYSIS TABLE OF CONTENTS 4.1

INTRODUCTION ........................................................................................................... 4-1

4.2

STRUCTURE MODELING ........................................................................................... 4-1

4.3

4.4

4.2.1

General ............................................................................................................... 4-1

4.2.2

Structural Modeling Guidelines ......................................................................... 4-5

4.2.3

Material Modeling Guidelines............................................................................ 4-7

4.2.4

Types of Bridge Models ..................................................................................... 4-7

4.2.5

Slab-Beam Bridges ............................................................................................. 4-9

4.2.6

Abutments ........................................................................................................ 4-15

4.2.7

Foundation........................................................................................................ 4-16

4.2.8

Examples .......................................................................................................... 4-18

STRUCTURAL ANALYSIS ........................................................................................ 4-27 4.3.1

General ............................................................................................................. 4-27

4.3.2

Analysis Methods ............................................................................................. 4-28

BRIDGE EXAMPLES – 3-D VEHICLE LIVE LOAD ANALYSIS ........................... 4-36 4.4.1

Background ...................................................................................................... 4-36

4.4.2

Moving Load Cases .......................................................................................... 4-37

4.4.3

Live Load Distribution For One And Two-Cell Box Girders Example ........... 4-40

NOTATION ............................................................................................................................... 4-53 REFERENCES .......................................................................................................................... 4-54

Chapter 4 – Structural Modeling and Analysis

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4-ii

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CHAPTER 4 STRUCTURAL MODELING AND ANALYSIS 4.1

INTRODUCTION Structural analysis is a process to analyze a structural system to predict its responses and behaviors by using physical laws and mathematical equations. The main objective of structural analysis is to determine internal forces, stresses and deformations of structures under various load effects. Structural modeling is a tool to establish three mathematical models, including (1) a structural model consisting of three basic components: structural members or components, joints (nodes, connecting edges or surfaces), and boundary conditions (supports and foundations); (2) a material model; and (3) a load model. This chapter summarizes the guidelines and principles for structural analysis and modeling used for bridge structures.

4.2

STRUCTURE MODELING

4.2.1

General For designing a new structure, connection details and support conditions shall be made as close to the computational models as possible. For an existing structure evaluation, structures shall be modeled as close to the actual as-built structural conditions as possible. The correct choice of modeling and analysis tools/methods depends on: a) Importance of the structure b) Purpose of structural analysis c) Required level of response accuracy This section will present modeling guidelines and techniques for bridge structures.

4.2.1.1

Types of Elements Different types of elements may be used in bridge models to obtain characteristic responses of a structure system. Elements can be categorized based on their principal structural actions. a) Truss Element A truss element is a two-force member that is subjected to axial loads either tension or compression. The only degree of freedom for a truss (bar) element is axial displacement at each node. The cross sectional dimensions and material properties of each element are usually assumed constant along its length. The element may interconnect in a two-dimensional (2-D) or three-

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dimensional (3-D) configuration. Truss elements are typically used in analysis of truss structures. b) Beam Element A beam element is a slender member subject to lateral loads and moments. In general, it has six degrees of freedom at each node including translations and rotations. A beam element under pure bending has only four degrees of freedom. c) Frame Element A frame element is a slender member subject to lateral loads, axial loads and moments. It is seen to possess the properties of both truss and beam elements and also called a beam-column element. A three-dimensional frame formulation includes the effects of biaxial bending, torsion, axial deformation, and biaxial shear deformations. A frame element is modeled as a straight line connecting two joints. Each element has its own local coordinate system for defining section properties and loads. d) Plate Element A plate element is a two dimensional solid element that acts like a flat plate. There are two out-of-plane rotations and the normal displacement as Degree of Freedom (DOF). These elements model plate-bending behavior in two dimensions. The element can model the two normal moments and the cross moment in the plane of the element. The plate element is a special case of a shell element without membrane loadings. e) Shell Element A shell element (Figure 4.2-1) is a three-dimensional solid element (one dimension is very small compared with another two dimensions) that carries plate bending, shear and membrane loadings. A shell element may have either a quadrilateral shape or a triangular shape. Shell element internal forces are reported at the element mid-surface in force per unit length and are reported both at the top and bottom of the element in force per unit area. It is primarily used to determine local stress levels in cellular superstructure or in cellular pier and caissons. It is generally recommended to use the full behavior unless the entire structure is planar and is adequately restrained.

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Figure 4.2-1 Shell and Solid Elements

f)

Plane Element The plane element is a two-dimensional solid, with translational degrees of freedom, capable of supporting forces but not moments. One can use either plane stress elements or plane strain elements. Plane stress element is used to model thin plate that is free to move in the direction normal to the plane of the plate. Plane strain element is used to model a thin cut section of a very long solid structure, such as walls. Plain strain element is not allowed to move in the normal direction of the element’s plane.

g) Solid Element A solid element is an eight-node element as shown in Figure 4.2-1 for modeling three-dimensional structures and solids. It is based upon an isoparametric formulation that includes nine optional incompatible bending modes. Solid elements are used in evaluation of principal stress states in joint regions or complex geometries (CSI, 2014). h) The NlLink Element A NlLink element (CSI, 2014) is an element with structural nonlinearities. A NlLink element may be either a one-joint grounded spring or a two-joint link and is assumed to be composed of six separate springs, one for each degree of deformational degrees of freedom including axial, shear, torsion, and pure bending. Non-linear behavior is exhibited during nonlinear time-history analyses or nonlinear static analyses. 4.2.1.2

Types of Boundary Elements Selecting the proper boundary condition has an important role in structural analysis. Effective modeling of support conditions at bearings and expansion joints requires a careful consideration of continuity of each translational and rotational component of displacement. For a static analysis, it is common to use a simpler assumption of supports (i.e. fixed, pinned, roller) without considering the soil/ foundation system stiffness. However for dynamic analysis, representing the soil/foundation stiffness is essential. In most cases choosing a [6×6] stiffness matrix is adequate.

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For specific projects, the nonlinear modeling of the system can be achieved by using nonlinear spring/damper. Some Finite Element programs such as ADINA (ADINA, 2014) have more capability for modeling the boundary conditions than others. 4.2.1.3

Types of Materials Different types of materials are used for bridge structure members such as concrete, steel, prestressing tendons, etc. For concrete structures, see Article C5.4.1 and for steel structures see Article 6.4 of the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012). The material properties that are usually used for an elastic analysis are: modulus of elasticity, shear modulus, Poisson’s ratio, the coefficient of thermal expansion, the mass density and the weight density. One should pay attention to the units used for material properties.

4.2.1.4

Types of Loads There are two types of loads in a bridge design: Permanent Loads: Loads and forces that are assumed to be either constant upon completion of construction or varying only over a long time interval (AASHTO 3.2). Such loads include the self weight of structure elements, wearing surface, curbs, parapets and railings, utilities, locked-in force, secondary forces from posttensioning, force effect due to shrinkage and due to creep, and pressure from earth retainments (CA 3.3.2). Transient Loads: Loads and forces that can vary over a short time interval relative to the lifetime of the structure (AASHTO 3.2). Such loads include gravity loads due to vehicular, railway and pedestrian traffic, lateral loads due to wind and water, ice flows, force effect due to temperature gradient and uniform temperature, and force effect due to settlement and earthquakes (CA 3.3.2). Loads are discussed in Chapter 3 in detail.

4.2.1.5

Modeling Discretization Formulation of a mathematical model using discrete mathematical elements and their connections and interactions to capture the prototype behavior is called Discretization. For this purpose: a) Joints/Nodes are used to discretize elements and primary locations in structure at which displacements are of interest. b) Elements are connected to each other at joints. c) Masses, inertia, and loads are applied to elements and then transferred to joints. Figure 4.2-2 shows a typical model discretization for a bridge bent.

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Figure 4.2-2 Model Discretization for Monolithic Connection

4.2.2

Structural Modeling Guidelines

4.2.2.1

Lumped-Parameter Models (LPMs) •

•

4.2.2.2

Mass, stiffness, and damping of structure components are usually combined and lumped at discrete locations. It requires significant experience to formulate equivalent force-deformation with only a few elements to represent structure response. For a cast-in-place prestressed (CIP/PS) concrete box girder superstructure, a beam element located at the center of gravity of the box girder can be used. For non-box girder structures, a detailed model will be needed to evaluate the responses of each separate girder.

Structural Component Models (SCMs) - Common Caltrans Practice • • •

Based on idealized structural subsystems/elements to resemble geometry of the structure. Structure response is given as an element force-deformations relationship. Gross moment of inertia is typically used for non-seismic analysis of concrete column modeling. Effective moment of inertia can be used when analyzing large deformation under loads, such as prestressing and thermal effects. Effective moment of inertia is the range between gross and cracked moment of inertia. To calculate effective moment of inertia, see AASHTO LRFD 5.7.3.6.2 (AASHTO, 2012).

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•

4.2.2.3

Cracked moment of inertia is obtained using section moment - curvature analysis (e.g. xSection or CSiBridge Section Designer), which is the moment of inertia corresponding to the first yield curvature. For seismic analysis, refer to Seismic Design Criteria (SDC) 5.6 “Effective Section Properties” (Caltrans, 2013).

Finite Element Models (FEMs) •

A bridge structure is discretized with finite-size elements. Element characteristics are derived from the constituent structural materials (AASHTO 4.2).

Figure 4.2-3 shows the levels of modeling for seismic analysis of bridge structures.

Figure 4.2-3 Levels of Modeling for Seismic Analysis of Bridge (Priestley, et al 1996) The importance of the structure, experience of the designer and the level of needed accuracy affects type of model, location of joints and elements within the selected model, and number of elements/joints to describe geometry of the structure. For example, a horizontally curved structure should be defined better by shell elements in comparison with straight elements. The other factors to be considered are: a) Structural boundaries - e.g., corners b) Changes in material properties c) Changes in element sectional properties d) Support locations e) Points of application of concentrated loads - Frame elements can have inspan loads

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4.2.3

Material Modeling Guidelines Material models should be selected based on a material’s deformation under external loads. A material is called elastic, when it returns to its original shape upon release of applied loads. Otherwise it is called an inelastic material. For an elastic body, the current state of stress depends only on the current state of deformation while, in an inelastic body, residual deformation and stresses remain in the body even when all external tractions are removed. The elastic material may show linear or nonlinear behavior. For linear elastic materials, stresses are linearly proportional to strains (σ = Eє) as described by Hooke’s Law. The Hooke’s Law is applicable for both homogeneous and isotropic materials. • •

Homogeneous means that the material properties are independent of the coordinates. Isotropic means that the material properties are independent of the rotation of the axes at any point in the body or structure. Only two elastic constants (modulus of elasticity E and Poisson’s ratio ν) are needed for linear elastic materials.

For a simple linear spring, the constitutive law is given as: Fs = kξ where ζ is the relative extension or compression of the spring, while Fs and k represent the force in the spring and the spring stiffness, respectively. Stiffness is the property of an element which is defined as force per unit displacement. For a nonlinear analysis, nonlinear stress-strain relationships of structural materials should be incorporated. • • •

For unconfined concrete a general stress-strain relationship proposed by Hognestad is widely used. For confined concrete, generally Mander’s model is used (Akkari and Duan, 2014). For structural steel and reinforcing steel, the stress-strain curve usually includes three segments: elastic, perfectly plastic, and a strain-hardening region. For prestressing steel, an idealized nonlinear stress-strain model may be used.

4.2.4

Types of Bridge Models

4.2.4.1

Global Bridge Models A global bridge model includes the entire bridge with all frames and connecting structures. It can capture effects due to irregular geometry such as curves in plane and elevation, effects of highly-skewed supports, contribution of ramp structures, frames interaction, expansion joints, etc. It is primarily used in seismic design to verify design parameters for the individual frame. The global model may be in question because of spatially varying ground motions for large, multi-span, and multi-frame

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bridges under seismic loading. In this case a detailed discretization and modeling force-deformation of individual element is needed. 4.2.4.2

Tension and Compression Models The tension and compression models are used to capture nonlinear responses for bridges with expansion joints (MTD 20-4, Caltrans, 2007) to model the non-linearity of the hinges with cable restrainers. Maximum response quantities from the two models are used for seismic design. a) Tension Model Tension model is used to capture out-of-phase frame movement. The tension model allows relative longitudinal movement between adjacent frames by releasing the longitudinal force in the rigid hinge elements and abutment joints and activating the cable restrainer elements. The cable restrainer unit is modeled as an individual truss element with equivalent spring stiffness for longitudinal movement connecting across expansion joints. b) Compression Model Compression model is used to capture in-phase frame movement. The compression model locks the longitudinal force and allows only moment about the vertical and horizontal centerline at an expansion joint to be released. All expansion joints are rigidly connected in longitudinal direction to capture effects of joint closing-abutment mobilized.

4.2.4.3

Frame Models A frame model is a portion of structure between the expansion joints. It is a powerful tool to assess the true dynamic response of the bridge since dynamic response of stand-alone bridge frames can be assessed with reasonable accuracy as an upper bound response to the whole structure system. Seismic characteristics of individual frame responses are controlled by mass of superstructure and stiffness of individual frames. Transverse stand-alone frame models shall assume lumped mass at the columns. Hinge spans shall be modeled as rigid elements with half of their mass lumped at the adjacent column (SDC Figure 5.4.1-1, Caltrans, 2013). Effects from the adjacent frames can be obtained by including boundary frames in the model.

4.2.4.4

Bent Models A transverse model of bent cap and columns is needed to obtain maximum moments and shears along bent cap. Dimension of bent cap should be considered along the skew. Individual bent model should include foundation flexibility effects and can be combined in frame model simply by geometric constraints. Different ground motion can be input for individual bents. The high in-plane stiffness of bridge superstructures allows rigid body movement assumption which simplifies the combination of individual bent models.

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4.2.5

Slab-Beam Bridges

4.2.5.1

Superstructures For modeling slab-beam bridges, either Spine Model or a Grillage Model should be used.

Figure 4.2-4 Superstructure Models (Priestley, et al 1996) a) Spine Model Spine Models with beam elements are usually used for ordinary bridges. The beam element considers six DOF at both ends of the element and is modeled at their neutral axis.  The effective stiffness of the element may vary depending on the structure type.  Use SDC V1.7 to define effective flexural stiffness EIeff for reinforced concrete box girders and pre-stressed box girders as follows: − For reinforced concrete (RC) box girder, (0.5~0.75) EIg − For prestressed concrete (PS) box girder, 1.0 EIg and for tension it considers Ig,  where Ig is the gross section moment of inertia.  The torsional stiffness for superstructures can be taken as: GJ for uncracked section and 0.5 GJ for cracked section.  Spine model can’t capture the superstructure carrying wide roadway, high-skewed bridges. In these cases use grillage model.

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b) Grillage Models/3D Finite Element Model  Grillage Models are used for modeling steel composite deck superstructures and complicated structures where superstructures can’t be considered rigid such as very long and narrow bridges, interchange connectors. 4.2.5.2

Bents If the bridge superstructure can be assumed to move as a rigid body under seismic load, the analysis can be simplified to modeling bents only. Frame elements, effective bending stiffness, cap with large torsional and transverse bending stiffness to capture superstructure, and effective stiffness for outriggers should be considered. Figure 4.2-5 shows single column bent models.

Figure 4.2-5 Single-Column Bent Models (Priestley et al, 1996) 4.2.5.3

Superstructure - Bents Connection In modeling the superstructure bent connections, two different connections as shown in Figures 4.2-2 and 4.2-6 may be considered: a) Monolithic connections for cast-in-place box girders and integral bent cap for precast girders. b) Bearing supported connections for precast concrete girders or steel superstructures on drop cap. Different types of bearings are: PTFE, stainless steel sliders, rocker bearings and elastomeric bearings. With the bearingsupported connections, one may use the isolated bearing by using special seismic bearings and energy-dissipating devices to reduce resonant buildup of displacement. In monolithic connections all the degrees of freedom are restrained (three degrees of translations and three degrees for rotation); however, in bearing supported

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connections, only three degrees of translations are restrained but the rotational degrees of freedom are free. In the bearing supported structures, the superstructure is not subjected to seismic moment transferred through the column. However the design is more sensitive to seismic displacement than with the monolithic connection. The energy dissipation devices in the isolated bearing reduce the seismic displacement significantly in comparison with bearing-supported structures. The designer should pay attention to the possibility of increased acceleration when using the bearing-supported connections with or without energy-dissipation devices in soft soils.

Figure 4.2-6 Superstructure-Bent Connection

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4.2.5.4

Hinges Hinges separate frames in long structures to allow for movements due to thermal, initial pre-stress shortening and creep without large stresses and strains in members. A typical hinge should be modeled as 6 degrees of freedom, i.e., free to rotate in the longitudinal direction and pin in the transverse direction to represent shear (Figure 4.2-7). It is Caltrans practice to use Linear Elastic Modal Analysis with two different structural models, Tension and Compression, to take care of this analysis issue.

Figure 4.2-7 Span Hinge Force Definitions (Priestley et al, 1996) 4.2.5.5

Substructures Figures 4.2-8 and 4.2-9 show a multi-column bent model and a foundations spring model at a bent, respectively. Figure 4.2-10 shows a multi bridge frame model. a) Column-Pier Sections  Prismatic - same properties or Non-Prismatic  Shapes Circular Column, Rectangular, Hollow-Section Column

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Figure 4.2-8 Multi-Column Bent Model (Priestley et al, 1996)

b) Bent-Foundation Connection  Pin base: Generally used for multi-column bents.  Fixed Base: For single column base.

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Figure 4.2-9 Foundation Spring Definition at a Bent

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Figure 4.2-10 Multi Bridge Frame (Priestley et al, 1996)

4.2.6

Abutments When modeling bridge structure, abutment can be modeled as pin, roller or fixed boundary condition. For modeling the soil-structure interaction, springs can be used.

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Figure 4.2-11 shows end restraint with springs to model soil-structure interaction for seat and rigid abutments. Abutment stiffness, capacities, and damping affect seismic response. Seismic Design Criteria V1.7, Section 7.8 discusses the longitudinal and transverse abutment responses in an earthquake. For modeling gap, back wall and piles effective stiffness is used with non-linear behavior. Iterative procedure should be used to find a convergence between stiffness and displacement.

Figure 4.2-11 Foundation Spring Definition

4.2.7

Foundation

4.2.7.1

Group Piles Supports can be modeled using: • • •

Springs - 6 × 6 stiffness matrix - defined in global/joint local coordinate system. Restraints - known displacement, rotation - defined in global DOF. Complete pile system with soil springs along with the bridge.

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4.2.7.2

Pile shaft When modeling the pile shaft for non-seismic loading, an equivalent fixity model can be used (Figure 4.2-12c). For seismic loading, a soil-spring model (Figure 4.212b) should be considered to capture the soil-structure interaction. Programs such as Wframe, L-Pile, CSiBridge or ADINA can be used.

a) Prototype

b) Soil-Spring Model

c) Equivalent Fixity Model

Figure 4.2-12 CIDH Pile Shaft Models (Priestley et al, 1996) 4.2.7.3

Spread Footing Spread footings are usually built on stiff and competent soils, fixed boundary conditions are assumed for the translational springs, and rotation is considered only when uplift and rocking of the entire footing are expected.

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4.2.8

Examples

4.2.8.1

CTBridge CTBridge (Caltrans, 2014b) is a Finite Element Analysis and Design software using a 3D spine model for the bridge structure. This allows description of skewed supports, horizontal and vertical curves, and multi-column bents. CTBridge allows user manipulation of various settings such as: • Number of Elements • Live Load Step Sizes • Prestress Discretization • P-Jack Design Limits For non-skewed bridges, the abutment can be considered pinned or roller. For skewed bridges, springs should be used at the abutments. The stiffness of the springs shall be based on the stiffness of the bearing pads. If bearing stiffness is not available, slider can be used instead of pin or roller. For bridges with curved alignments and skewed supports or straight bridges with skews in excess of 60 degrees, a full 3-D analysis model, such as a grillage or shell model may be required to more accurately capture the true distribution of the load. Note that in order to get the result at each 0.1 span, you should define the offset from begin and end span, i.e. from CL abutment to face of abutment. The following structure shown in Figures 4.2-13a to 4.2-13c is used as an example for CTBridge.

Figure 4.2-13a Elevation View of Example Bridge

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Figure 4.2-13b Typical Section View of Example Bridge

Figure 4.2-13c Plan View of Example Bridge

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Figure 4.2-14 shows CTBridge model for example bridge.

Figure 4.2-14 Example Bridge - CTBridge Model

Figure 4.2-15 shows sign convention for CTBridge.

Figure 4.2-15 Sign Convention at CTBridge

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Figure 4.2-16 shows two spine models.

Figure 4.2-16 3D Frame in CTBridge

4.2.8.2

CSiBridge CSiBridge is the latest and one of the most powerful versions of the well-known Finite Element Program SAP series of Structural Analysis Programs, which offers the following features: • • • • • • • • • •

Static and Dynamic Analysis Linear and Nonlinear Analysis Dynamic Seismic Analysis and Static Pushover Analysis Vehicle Live-Load Analysis for Bridges, Moving Loads with 3D Influence Surface, Moving Loads with Multi-Step Analysis, Lane Width Effects P-Delta Analysis Cable Analysis Eigen and Ritz Analyses Fast Nonlinear Analysis for Dampers Energy Method for Drift Control Segmental Construction Analysis

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The following are the general steps to be defined for analyzing a structure using CSiBridge: • • • • • •

Geometry (input nodes coordinates, define members and connections) Boundary Conditions/Joint Restraints (fixed, free, roller, pin or partially restrained with a specified spring constant) Material Property (Elastic Modulus, Poisson’s Ratio, Shear Modulus, damping data, thermal properties and time-dependent properties such as creep and shrinkage) Loads and Load cases Stress-strain relationship Perform analysis of the model based on analysis cases

Bridge Designers can use CSiBridge templates for generating Bridge Models, Automated Bridge Live Load Analysis and Design, Bridge Base Isolation, Bridge Construction Sequence Analysis, Large Deformation Cable Supported Bridge Analysis, and Pushover Analysis. The user can either model the structure as a Spine Model (Frame) or a 3D Finite Element Model. Concrete Box Girder Bridge: In this section, we create a CSiBridge model for the Example Bridge using the Bridge Wizard (BrIM-Bridge Information Modeler). The Bridge Modeler has 13 modeling step processes which the major steps are described below: a) Layout line The first step in creating a bridge object is to define highway layout lines using horizontal and vertical curves. Layout lines are used as reference lines for defining the layout of bridge objects and lanes in terms of stations, bearings and grades considering super elevations and skews. b) Deck Section Various parametric bridge sections (Box Girders & Steel Composites) are available for use in defining a bridge. See Figure 4.2-17. User can specify different Cross Sections along Bridge length.

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Figure 4.2-17 Various Bridge Sections c) Abutment Definition Abutment definitions specify the support conditions at the ends of the bridge. The user defined support condition allows each six DOF at the abutment to be specified as fixed, free or partially restrained with a specified spring constant. Those six Degrees of Freedom are: U1- Translation Parallel to Abutment U2- Translation Normal to Abutment U3- Translation Vertical R1- Rotation about Abutment R2- Rotation about Line Normal to Abutment R3- Rotation about Vertical For Academy Bridge consider U2, R1 and R3 DOF directions to have a “Free” release type and other DOF fixed. d) Bent Definition This part specifies the geometry and section properties of bent cap beam and bent cap columns (single or multiple columns) and base support condition of the bent columns.

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The base support condition for a bent column can be fixed, pinned or user defined as a specified link/support property which allows six degrees of freedom. For Example Bridge enter the column base supports as pinned. All units should be kept consistent (kip-ft for this example). The locations of columns are defined as distance from left end of the cap beam to the centerline of the column and the column height is the distance from the mid-cap beam to the bottom of the column. For defining columns use Bent definition under bridge wizard, then go to define/show bents and go to Modify/show column data. The base column supports at top and bottom will be defined here. e) Diaphragm Definition Diaphragm definitions specify properties of vertical diaphragms that span transverse across the bridge. Diaphragms are only applied to area objects and solid object models and not to spine models. Steel diaphragm properties are only applicable to steel bridge sections. f)

Hinge Definition Hinge definitions specify properties of hinges (expansion joints) and restrainers. After a hinge is defined, it can be assigned to one or more spans in the bridge object. A hinge property can be a specified link/support property or it can be user-defined spring. The restrainer property can be also a link/support or user defined restrainer. The user-restrainer is specified by a length, area and modulus of elasticity.

g) Parametric Variation Definition Any parameter used in the parametric definition of the deck section can be specified to vary such as bridge depth, thickness of the girders and slabs along the length of the bridge. The variation may be linear, parabolic or circular. h) Bridge Object Definition The main part of the Bridge Modeler is the Bridge Object Definition which includes defining bridge span, deck section properties assigned to each span, abutment properties and skews, bent properties and skews, hinge locations are assigned, super elevations are assigned and pre-stress tendons are defined.

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The user has two tendon modeling options for pre-stress data:  

Model as loads Model as elements

Since we calculate the pre-stress jacking force from CTBridge, use option (a) (layout line) to input the Tendon Load force. The user can input the Tendon loss parameters which have two parts: 1) Friction and Anchorage losses (Curvature coefficient, Wobble coefficient and anchorage setup). 2) Other loss parameters (Elastic shortening stress, Creep stress, Shrinkage stress and Steel relaxation stress). When you input values for Friction and Anchorage losses, make sure the values match your CTBridge which should be based on “CALIFORNIA Amendments Table 5.9.5.2.2b-1 (Caltrans, 2014) and there is no need to input other loss parameters. If the user decides to model tendon as elements, the values for other loss parameters shall be input; otherwise, leave the default values. Note: •

•

i)

If you model the bridge as a Spine Model, only define one single tendon with total Pjack load. If you model the bridge with shell element, then you need to specify tendon in each girder and input the Pjack force for each girder which should be calculated as Total Pjack divided by number of the girders. Anytime a bridge object definition is modified, the link model must be updated for the changes to appear in /CSiBridge model.

Update Linked Model The update linked model command creates the CSiBridge object-based model from the bridge object definition. Figures 4.2-18 and 4.2-19 show an area object model and a solid object model, respectively. Note that an existing object will be deleted after updating the linked model. There are three options in the Update Linked Model including:   

Update a Spine Model using Frame Objects Update as Area Object Model Update as Solid Object Model

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Figure 4.2-18 Area Object Model

Figure 4.2-19 Solid Object Model Other analysis steps include: 

Parametric Bridge Modeling − Layered Shell Element − Lane Definition Using Highway Layout or Frame Objects − Automatic Application of Lane Loads to Bridge − Predefined Vehicle and Train Loads



Bridge Results & Output − Influence Lines and Surfaces − Forces and Stresses Along and Across Bridge − Displacement Plots − Graphical and Tabulated Outputs

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CSiBridge also has an Advanced Analysis Option that is not discussed in this section including:           

Segmental Construction Effects of Creep, Shrinkage Relaxation Pushover Analysis using Fiber Models Bridge Base Isolation and Dampers Explicitly Model Contact Across Gaps Nonlinear Large Displacement Cable Analysis Line and Surface Multi-Linear Springs (P-y curves) High Frequency Blast Dynamics using Wilson FNA Nonlinear Dynamic Analysis & Buckling Analysis Multi-Support Seismic Excitation Animated Views of Moving Loads

The program has the feature of automated line constraints that enforce the displacement compatibility along the common edges of meshes as needed.

4.3

STRUCTURAL ANALYSIS Structural Analysis provides the numerical mathematical process to extract structure responses under service and seismic loads in terms of structural demands such as member forces and deformations.

4.3.1

General For any type of structural analysis, the following principles should be considered.

4.3.1.1

Equilibrium a) Static Equilibrium In a supported structure system when the external forces are in balance with the internal forces, or stresses, which exactly counteract the loads (Newton’s Second Law), the structure is said to be in equilibrium. Since there is no translatory motion, the vector sum of the external forces must   be zero ( Σ F = 0 ). Since there is no rotation, the sum of the moments of the   external forces about any point must be zero ( Σ M = 0 ). b) Dynamic Equilibrium When dynamic effects need to be included, whether for calculating the dynamic response to a time-varying load or for analyzing the propagation of waves in a structure, the proper inertia terms shall be considered for analyzing the dynamic equilibrium:

ΣF = mu

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4.3.1.2

Constitutive Laws The constitutive laws define the relationship between the stress and strain in the material of which a structure member is made.

4.3.1.3

Compatibility Compatibility conditions are referred to continuity or consistency conditions on the strains and the deflections. As a structure deforms under a load, we want to ensure that: a) Two originally separate points do not merge into a single point. b) Perimeter of a void does not overlap as it deforms. c) Elements connected together remain connected as the structure deforms.

4.3.2

Analysis Methods Different types of analysis are discussed in this section.

4.3.2.1

Small Deflection Theory If the deformation of the structure doesn’t result in a significant change in force effects due to an increase in the eccentricity of compressive or tensile forces, such secondary force effects may be ignored. Small deflection theory is usually adequate for the analyses of beam-type bridges. Suspension bridges, very flexible cable-stayed bridges and some arches rather than tied arches and frames in which flexural moments are increased by deflection are generally sensitive to deflections. In many cases the degree of sensitivity can be evaluated by a single-step approximate method, such as moment magnification factor method (AASHTO 4.5.3.2.2).

4.3.2.2

Large Deflection Theory If the deformation of the structure results in a significant change in force effects, the effects of deformation shall be considered in the equations of equilibrium. The effect of deformation and out-of-straightness of components shall be included in stability analysis and large deflection analyses. For slender concrete compressive components, time-dependent and stress-dependent material characteristics that cause significant changes in structural geometry shall be considered in the analysis. Because large deflection analysis is inherently nonlinear, the displacements are not proportional to applied load, and superposition cannot be used. Therefore, the order of load application are very important and should be applied in the order experienced by the structure, i.e. dead load stages followed by live load stages, etc. If the structure undergoes nonlinear deformation, the loads should be applied incrementally with consideration for the changes in stiffness after each increment.

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4.3.2.3

Linear Analysis In the linear relation of stress-strain of a material, Hooke’s law is valid for small stress-strain range. For linear elastic analysis, sets of loads acting simultaneously can be evaluated by superimposing (adding) the forces or displacements at the particular point.

4.3.2.4

Non-linear Analysis The objective of non-linear analysis is to estimate the maximum load that a structure can support prior to structural instability or collapse. The maximum load which a structure can carry safely may be calculated by simply performing an incremental analysis using non-linear formulation. In a collapse analysis, the equation of equilibrium is for each load or time step. Design based on assumption of linear stress-strain relation will not always be conservative due to material or physical non-linearity. Very flexible bridges, e.g. suspension and cable-stayed bridges, should be analyzed using nonlinear elastic methods (LRFD C4.5.1, AASHTO, 2012). P-Delta effect is an example of physical (geometrical) non-linearity, where principle of superposition doesn’t apply since the beam-column element undergoes large changes in geometry when loaded.

4.3.2.5

Elastic Analysis Service and fatigue limit states should be analyzed as fully elastic, as should strength limit states, except in the case of certain continuous girders where inelastic analysis is permitted, inelastic redistribution of negative bending moment and stability investigation (LRFD C4.5.1, AASHTO, 2012). When modeling the elastic behavior of materials, the stiffness properties of concrete and composite members shall be based upon cracked and/or uncracked sections consistent with the anticipated behavior (LRFD 4.5.2.2, AASHTO, 2012). A limited number of analytical studies have been performed by Caltrans to determine effects of using gross and cracked moment of inertia. The specific studies yielded the following findings on prestressed concrete girders on concrete columns: 1) Using Igs or Icr in the concrete columns do not significantly reduce or increase the superstructure moment and shear demands for external vertical loads, but will significantly affect the superstructure moment and shear demands from thermal and other lateral loads (CA C4.5.2.2, Caltrans, 2014a). Using Icr in the columns can increase the superstructure deflection and camber calculations (CA 4.5.2.2, Caltrans, 2014a). Usually an elastic analysis is sufficient for strength-based analysis.

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4.3.2.6

Inelastic Analysis Inelastic analysis should be used for displacement-based analysis (Akkari and Duan, 2014). The extreme event limit states may require collapse investigation based entirely on inelastic modeling. Where inelastic analysis is used, a preferred design failure mechanism and its attendant hinge locations shall be determined (LRFD 4.5.2.3, AASHTO, 2012).

4.3.2.7

Static Analysis Static analysis mainly used for bridges under dead load, vehicular load, wind load and thermal effects. The influence of plan geometry has an important role in static analysis (AASHTO 4.6.1). One should pay attention to plan aspect ratio and structures curved in plan for static analysis. •

Plan Aspect Ratio If the span length of a superstructure with torsionally stiff closed crossed section exceeds 2.5 times its width, the superstructure may be idealized as a single-spine beam. Simultaneous torsion, moment, shear and reaction forces and the attendant stresses are to be superimposed as appropriate. In all equivalent beam idealizations, the eccentricity of loads should be taken with respect to the centerline of the equivalent beam.

•

Structure curved in plan Horizontally cast-in-place box girders may be designed as single spine beam with straight segments, for central angles up to 34° within one span, unless concerns about other force effects dictate otherwise. For I-girders, since equilibrium is developed by the transfer of load between the girders, the analysis must recognize the integrated behavior of all structure components.

Small deflection theory is adequate for the analysis of most curved-girder bridges. However curved I-girders are prone to deflect laterally if not sufficiently braced during erection. This behavior may not be well recognized by small deflection theory.

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4.3.2.8

Equivalent Static Analysis (ESA) It is used to estimate seismic demands for ordinary bridge structures as specified in Caltrans SDC (Caltrans, 2013). A bridge is usually modeled as Single-Degree-ofFreedom (SDOF) and seismic load applied as equivalent static horizontal force. It is suitable for individual frames with well balanced spans and stiffness. Caltrans SDC (Caltrans, 2013) recommends stand-alone “Local” Analysis in Transverse & Longitudinal direction for demands assessment. Figure 4.3-1 shows a stand-alone model with lumped masses at columns, rigid body rotation, and half span mass at adjacent columns.

Transverse Stand-Alone Model

Longitudinal Stand-Alone Model Figure 4.3-1 Stand Alone Model Types of Equivalent Static Analysis such as Lollipop Method, Uniform Load Method and Generalized Coordinate Method can be used. 4.3.2.9

Nonlinear Static Analysis (Pushover Analysis) Nonlinear Incremental Static Procedure is used to determine displacement capacity of a bridge structure. Horizontal loads are incrementally increased until a structure reaches collapse condition or collapse mechanism. Change in structure stiffness is modeled as member stiffness due to cracking, plastic hinges, yielding of soil spring at each step (event).

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Analysis Programs are available such as: WFRAME, CSiBridge, STRUDL, SCPush 3D, ADINA. Figures 4.3-2 and 4.3-3 shows typical force-displacement and moment-curvature for a concrete column.

Figure 4.3-2 Pushover curve a) Pushover Analysis - Requirements  Linear Elastic Structural Model  Initial or Gravity loads  Characterization of all Nonlinear actions - multi-linear forcedeformation relationships (e.g. plastic hinge moment-curvature relationship)  Limits on strain based on design performance level to compute moment curvature relationship of nonlinear hinge elements.  Section Analysis─> Strain─> Curvature  Double Integration of curvature─> Displacements  Track design performance level strain limits in structural response

Figure 4.3-3 A Typical Moment-Curvature Curve for a Concrete Column

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4.3.2.10

General Dynamic Equilibrium Equation The dynamic equation of motion for a typical SDOF is:

FInput = FI + FD + FS where: F1 FD FS

= mass × acceleration = mü = Damping const × Velocity = m u = Stiffness × Deformation = ku

m

= mass = ρsV =

ρs

= Material mass density

V

= Element volume = A × L

K

= stiffness

c ccr z EDC U

Weight g

= damping constant = z × ccr = critical damping = 2 × m × w = damping-ratio = 0.5 × p × EDC

ku 2 = Energy dissipated per cycle = displacement

In addition to earthquakes, wind and moving vehicles can cause dynamic loads on bridge structures. Wind load may induce instability and excessive vibration in long-span bridges. The interaction between the bridge vibration and wind results in two kind of forces including motion-dependent and motion-independent. The motion dependent force causes aerodynamic instability with emphasis on vibration of rigid bodies. For short span bridges the motion dependent part is insignificant and there is no concern about aerodynamic instability. The bridge aerodynamic behavior is controlled by two types of parameters: structural and aerodynamics. The structure parameters are the bridge layout, boundary condition, member stiffness, natural modes and frequencies. The aerodynamic parameters are wind climate, bridge section shape. The aerodynamic equation of motion is expressed as:

mü + cu + ku= FU md + Fmi where: FUmd Fmi

= motion-dependent aerodynamic force vector = motion-independent wind force vector

For a detailed analytical solution for effect of wind on long span bridges and cable vibration, see (Cai, etl al., 2014).

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4.3.2.11

Free Vibration Analysis Vehicles such as trucks and trains passing bridges at certain speed will cause dynamic effects. The dynamic loads for moving vehicles on bridges are counted for by a dynamic load allowance, IM. See (Duan, et al., 1999). Major characteristics of the bridge dynamic response under moving load can be summarized as follows: Impact factor increases as vehicle speed increases, impact factor decreases as bridge span increases. Under the condition of “Very good” road surface roughness (amplitude of highway profile curve is less than 0.4 in.) the impact factor is well below the design specifications. But the impact factor increases tremendously with increasing road surface roughness from “good” to “poor” (the amplitude of the roadway profile is more than 1.6 in.) beyond the impact factor specified in AASHTO LRFD Specifications. Field tests indicate that in the majority of highway bridges, the dynamic component of the response does not exceed 25% of the static response to vehicles with the exception of deck joints. For deck joints, 75% of the impact factor is considered for all limit states due to hammer effect, and 15% for fatigue and fracture limit states for members vulnerable to cyclic loading such as shear connectors, see CA - C3.6.2.1 (Caltrans, 2014a) to AASHTO LRFD (AASHTO, 2012). Dynamic effects due to moving vehicles may be attributed to two sources:  

Hammering effect is the dynamic response of the wheel assembly to riding surface discontinuities, such as deck joints, cracks, potholes and delaminations. Dynamic response of the bridge as a whole to passing vehicles, which may be due to long undulations in the roadway pavement, such as those caused by settlement of fill, or to resonant excitation as a result of similar frequencies of vibration between bridge and vehicle. (AASHTO LRFD C3.6.2.1)

The magnitude of dynamic response depends on the bridge span, stiffness and surface roughness, and vehicle dynamic characteristics such as moving speed and isolation systems. There have been two types of analysis methods to investigate the dynamic response of bridges due to moving load:  

Numerical analysis (Sprung mass model). Analytical analysis (Moving load model).

The analytical analysis greatly simplifies vehicle interaction with bridge and models a bridge as a plate or beam with a good accuracy if the ratio of live load to self weight of the superstructure is less than 0.3. Free vibration analysis assuming a sinusoidal mode shape can be used for the analysis of the superstructure and calculating the fundamental frequencies of slabbeam bridges (Zhang, et al., 2014).

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For long span bridges or low speed moving load, there is little amplification which does not result in much dynamic responses. Maximum dynamic response happens when load frequency is near the bridge fundamental frequency. The aspect ratios of the bridge deck play an important role. When they are less than 4.0 the first mode shape is dominant, when more than 8.0, other mode shapes are excited. Free-Vibration Properties are shown in Figure 4.3-4.

¼ cycles m

π /2 K

R=1. π

0 2*π

T = 2 sec f = 1/T = ½ = 0.5 cycle/sec ω= 2*π/T = π rad/sec T/2

T = 1sec f = 1/T = 1/1= 1.0 cycle/sec T T

T

T = 0.5 sec f = 1/T = 1/0.5= 2.0 cycle/sec ω= 2*π/T = 4* π rad/sec

Figure 4.3-4 Natural Period a) Cycle:

When a body vibrates from its initial position to its extreme positive position in one direction, back to extreme negative position, and back to initial position (i.e., one revolution of angular displacement of 2 π ) (radians)

b) Frequency (ω): If a system is disturbed and allowed to vibrate on its own, without external forces and damping (free Vibration).

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A system having n degrees of freedom will have, in general, n distinct natural frequencies of vibration.

ω= K/m 2π

= ω

= distance/time T ω = 2π f

c) Period (T): Is the time taken to complete one cycle of motion. It is equal to the time required for a vector to rotate 2 π (one round) d) Frequency (f): The number of cycles per unit time, f = 1/T (H.Z)

4.4

BRIDGE EXAMPLES – 3-D VEHICLE LIVE LOAD ANALYSIS

4.4.1

Background The United States has a long history of girder bridges being designed “girder-bygirder”. That is, the girder is designed for some fraction of live loads, depending on girder spacing and structure type. The method is sometimes referred to as “girder line” or “beam line” analysis and the fraction of live load lanes used for design is sometimes referred to as a grid or Load Distribution Factor (LDF). The approximate methods of live-load distribution in the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) use “girder load distribution factors” (LDFs) to facilitate beam analysis of multiple vehicular live loads on a three-dimensional bridge structural system. The formal definition of LDF: “a factor used to multiply the total longitudinal response of the bridge due to a single longitudinal lane load in order to determine the maximum response of a single girder” (Barker and Puckett 2013). A more practical definition: the ratio Mrefined /Mbeam or Vrefined /Vbeam, where the numerator is the enveloped force effect at one location, and the denominator is force effect at the same location in a single girder due to the same load. •

Although each location within a girder can have a different LDF, the expressions in the tables of AASHTO LRFD BRIDGE SPECIFICATIONS, Articles 4.6.2.2.2 and 4.6.2.2.3 are based on the critical locations for bending and shear, respectively. Critical locations refer to maximum absolute positive moments, negative moments, and maximum absolute shear. For cast-in-place (CIP) concrete multicell box girders, the AASHTO tables only apply to typical bridges, which refer to: − − −

Girder spacing, S: 7′ < S < 13′ Span length, L: 60′ < L < 240′ (AASHTO 4.6.2.2.2b-1) The CA Amendments (CA Table 4.6.2.2.2b-2) provides the LDF for one cell, and two cell boxes based on:

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− −

•

Two or three-girder beam-slab structures; Spans greater than 240 ft in length; Structures with extra-wide overhangs (greater than one-half of the girder spacing or 3 ft).

Three-dimensional (3D) finite element analysis (FEA) must be used to determine the girder LDFs of these less-typical bridges. The following cases may also require such refined analysis: − − −

•

60' < L < 240' 35" < d < 110"

The use of approximate methods on less-typical structures is prohibited. The less-typical structures refer to either one of the following cases: − − −

•

Span Length, L: Structure Depth, d,

Skews greater than 45°; Structures with masonry sound walls; Beam-slab structures with beams of different bending stiffness.

A moving load analysis on a 3D finite element (FE) model provides accurate load distribution. However, for routine design of commonly used bridge superstructure system, 3D FEA requires the familiarity with sophisticated, usually also expensive, finite element methods. FEM software. It may not be economical due to the additional time required to build and run the 3D model, and analyze the results, comparing to simple FEM program, e.g. Caltrans.

•

4.4.2

CTBridge. In addition, in terms of the reliability of an FE model, 3D FEM model may not be as reliable as a simple 2D FE model due to the much greater number of details in a 3D FE model. Based on Caltrans experience, a combination of LDF formula with the in-house 2D FEM design program, CTBridge, provides sufficient, reliable and efficient design procedure and output. The latest version of CTBridge includes the LDF values for a one- or two-cell box-girder bridge.

Moving Load Cases In many situations, the one- or two-cell box girders are for widening of existing bridges. If they are new bridges, it is also possible that they will be widened in the future. Both cases imply that the traffic loads may be applied anywhere across the bridge width, i.e., edge to edge, and this shall be taken into account in design. This also means that one wheel line of the truck can be on the new/widened bridge, while the other one on the existing bridge. As one can imagine, for certain bridge width, the maximum force effect may be due to, say, 1.5 or 2.5 lanes. For particularly narrow bridge, e.g., 6 or 8 ft. bridge, probably only one wheel line load can be applied.

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CSiBridge (CSI, 2015) has the capability to permute all the possible vehicular loading patterns once a set of lanes is defined. First, the entire bridge response due to a single lane loaded, without the application of the Multiple Presence Factor (MPF), can be easily obtained by arbitrarily defining a lane of any width within the bridge. Then, lane configurations that would generate the maximum shear and moment effects would be defined and the MPF would be defined. The cases where one lane is loaded are important for fatigue design; in addition, the cases where one lane is loaded may control over the cases where two lanes are loaded. Therefore, the cases where one lane is loaded are separated from the permutation and are defined based on a single lane of the whole bridge width. AASHTO standard design vehicular live loads, HL-93, are used as the traffic load for the CSiBridge analyses of the live load distribution factor. Figure 4.4-1 shows the elevation view of the four types of design vehicles per lane, including the details of the axle load and axle spacing. Transverse spacing of the wheels for design truck and design tandem is 6 ft. The transverse width of the design lane load is 10 ft. The extreme force effect, moment and shear in girders for this study, at any location of any girder, are the largest from the 4 design vehicles: • • • •

HL-93K: HL-93M: HL-93S: HL-93LB:

design tandem and design lane load; design truck and design lane load; 90% of two design trucks and 90% of the design lane load; pair of one design tandem and one design lane load.

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(a) HL-93K

32k (b) HL-93M

32k

8k

14’ 14’-30’

(c) HL-93S minimum

14’

14’

(d) HL93-LB

Figure 4.4-1 Elevation View of AASHTO Standard HL-93 Vehicular Live Loads (Caltrans) Cases (c) and (d) in Figure 4.4-1 are for maximum negative moment over bent caps. A dynamic load allowance of 33% is applied and only applied to the design truck and design tandem in all cases. Multiple Presence Factor as shown in Table 4.4-1 is applied in accordance to AASHTO LRFD Bridge Design Specifications.

Table 4.4-1 Multiple Presence Factor (MPF) Number of Loaded Lanes

Multiple Presence Factor

≤1

1.20

>1 and ≤ 2

1

>2 and ≤ 3

0.85

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4.4.3

Live Load Distribution For One And Two-Cell Box Girders Example Model Bridge in CSiBridge as given data below: In this example, the method of calculating LLDF is shown for a two-cell box girder by using a 3D FEM-CSiBridge model for different lane loading (Figures 4.4-3 to 4.4-6). The bridge data is given as shown below: • • •

Girder spacing, S: 6′ < S =13′< 13′ Span length, L: 60′ < L=180′ < 220′ Structure depth, D: 35″ < d =96″< 110″

Single span, simply supported, 180 foot long, 8-foot depth two-cell Box Girder Bridge with the following cross section as shown in Figure 4.4-2.

Figure 4.4-2 Live Load Distribution For Two-Cell Box Girders Snap Shot

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1) Load groups Load Group 1

Figure 4.4-3 Live Load Distribution For Two-Cell Box Girders Snap Shot In Group 1 Load Group 2

Figure 4.4-4 Live Load Distribution For Two-Cell Box Girders Snap Shot In Group 2

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Load Group 3

Figure 4.4-5 Live Load Distribution For Two-Cell Box Girders Snap Shot In Group 3

Load Group 4

Figure 4.4-6 Live Load Distribution For Two-Cell Box Girders Snap Shot In Group 4 In order to calculate the LDF, both spine model and area object model were run for different lane loading using BrIM.

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2) Bridge Modeler (Figure 4.4-7) •

Update Bridge Structural Model as Area Object Model

BrIM

Update Link Model

Update as Area Object

Figure 4.4-7 Bridge Modeler Snap Shot 3) Define Lane (Figure 4.4-8) Define

Bridge loads

Lanes

Figure 4.4-8 Define Lane Snap Shot Chapter 4 – Structural Modeling and Analysis

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•

Maximum Lane Load Discretization Lengths: Along Lane 10 ft Across Lane 2 ft

4) Define Vehicle (Figure 4.4-9) Define

Bridge loads

Vehicles

Figure 4.4-9 Define Vehicle Snap Shot 5) Define Vehicle Classes (Figure 4.4-10) Define

Bridge loads

Vehicle classes:

Figure 4.4-10 Define Vehicle Classes Snap Shot

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6) Analysis Cases (Figure 4.4-11) •

Group1: 1 Lane loaded

Define

Load cases

Define Load cases

Figure 4.4-11 Analysis Cases Snap Shot in One-Lane Loaded •

Group 2: 2 Lane loaded (Lanes 1, 2 & 3) (Figure 4.4-12)

Figure 4.4-12 Analysis Cases Snap Shot in Two-Lane Loaded

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•

Group 3: 2 Lane loaded (Lanes 4 & 5)(Figure 4.4-13)

Figure 4.4-13 Analysis Cases Snap Shot in Two-Lane Loaded •

Group 4: 3 Lane loaded (Lanes 1, 2 & 3) (Figure 4.4-14)

Figure 4.4-14 Analysis Cases Snap Shot in Three-Lane Loaded

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7) Analysis Single Lane Loaded (MPF = 1) with running updated Bridge Structural Model as Spine Model (Figure 4.4-15) BrIM Lane

Update Link Model Define Load cases

Update as Spine Model

Define

Figure 4.4-15 Analysis Single Lane Loaded Snap Shot Results: A) Spine Model The maximum moments and shears at entire bridge width for one lane loaded are shown in Figure 4.4-16 and 4.4-17.

The Maximum moment value = 6,527 Kips-ft at x = 90 ft Figure 4.4-16 Maximum Moment Snap Shot

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The Maximum shear value =148.35 Kips Figure 4.4-17 Maximum Shear Snap Shot B) Area Model The maximum moments at exterior and interior girders for one lane loaded are shown in Figure 4.4-18 to 4.4-20.

Left Exterior Girder, M3 = 2566 Kips-ft at x = 90 ft Figure 4.4-18 Maximum Moment for One-Lane Loaded at Left Exterior Girder

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Interior Girder, M3 = 3410 Kips-ft at x = 90 ft Figure 4.4-19 Maximum Moment for One-Lane Loaded at Interior Girder

Right Exterior Girder, M3 = 2566 Kips-ft at x = 90 ft Figure 4.4-20 Maximum Moment for One-Lane Loaded at Right Exterior Girder The maximum shears at exterior and interior girders for one lane loaded are shown in Figure 4.4-21 to 4.4-23.

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Shear at Left Exterior girder = 153.91 Kips at x = 0 Figure 4.4-21 Maximum Shear for One-Lane Loaded at Left Exterior Girder

Shear at Interior Girder = 109.27 Kips at x = 0 Figure 4.4-22 Maximum Shear for One-Lane Loaded at Interior Girder

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Shear at Right Exterior girder = 153.91 Kips at x = 0 Figure 4.4-23 Maximum Shear for One-Lane Loaded at Right Exterior Girder C) Actual, Modified LLDF: Shear (Table 4.4-2)  

Actual LLDF = (VL.Ext. + VInt. +VR.Ext.)/ VSingle lane Modified LLDF = (Max (VL.Ext., VInt. ,VR.Ext.)) × 3/ VSingle lane

Table 4.4-2 Live Load Distribution Factor for Shear Case # 1 2 3

Cell Type X2C8 X2C8 X2C8

L (ft) 180 180 180

# VSingle lane Lanes (Kips) 1 148.35 2 148.35 3 148.35

VL.Ext.

VInt.

VR.Ext.

(Kips)

(Kips)

(Kips)

156.43 172.18 153.69

110.45 165.82 157.25

156.43 172.18 153.69

LLDFActual

LLDFModified

2.85 3.44 3.13

3.16 3.48 3.18

Moment (Table 4.4-3)  

Actual LLDF = (ML.Ext. + MInt. +MR.Ext.)/ MSingle lane Modified LLDF = (Max (ML.Ext. , MInt. , MR.Ext.))× 3/ MSingle lane

Table 4.4-3 Live Load Distribution Factor for Moment Case #

Cell Type

L (ft)

# Lanes

MSingle lane

ML.Ext.

MInt.

MR.Ext.

(Kips-ft)

(Kips-ft)

(Kips-ft)

(Kips-ft)

1 2 3

X2C8 X2C8 X2C8

180 180 180

1 2 3

6527 6527 6527

2566.39 4045.82 4920.23

3410.52 5656.70 7074.28

2566.39 4045.82 4920.23

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LLDFActual LLDFModified 1.31 2.11 2.59

1.57 2.60 3.25

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Although the CSiBridge analysis provides a more exact distribution of force effects in the girders, it doesn’t calculate the amounts of prestressing, longitudinal, or shear reinforcement required on the contract plans. Different two-dimensional tools such as CTBridge are used for design. The girders are considered individually, or, lumped together into a single-spine model. Caltrans prefers the latter in the case of post-tensioned box girders because posttensioning in one girder has an effect on the adjacent girder. If the individual demands were simply lumped together and used in twodimensional software for design and the girders design equally, at least one girder would be under-designed. Hence, the value from the girder with the highest demand is used for all girders–as shown above, so it is recommended to consider LDF Modified, as the Live Load Lanes input for CTBridge.

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NOTATION A

=

area of section (ft2)

d

=

structure depth (in.)

E

=

Young’s modulus (ksi)

g

=

gravitational acceleration (32.2 ft/sec)

gM

=

girder LL distribution factor for moment

gS

=

girder LL distribution factor for shear

H

=

height of element (ft)

I

=

moment of inertia (ft4)

Kg

=

longitudinal stiffness parameter (in.4)

L

=

span length (ft)

MLL

=

moment due to live load (kip-ft)

MT

=

transverse moment on column (kip-ft)

ML

=

longitudinal moment on column (kip-ft)

MDC

=

moment due to dead load (kip-ft)

MDW

=

moment due to dead load wearing surface (kip-ft)

MHL-93

=

moment due to design vehicle (kip-ft)

MPERMIT =

moment due to permit vehicle (kip-ft)

MPS

=

moment due to Secondary prestress forces (kip-ft)

n

=

modular ratio

Nb

=

number of beams

Nc

=

number of cells in the box girder section

S

=

center-to-center girder spacing (ft)

ts

=

top slab thickness (in.)

tdeck

=

deck thickness (in.)

tsoffit

=

soffit thickness (in.)

tgirder

=

girder stem thickness (in.)

w

=

uniform load (kip/ft)

X

=

moment arm for overhang load (ft)

α

=

coefficient of thermal expansion

θ

=

skew angle (degrees)

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, 4th Edition, Washington, D.C. 2. ADINA, (2014). ADINA System, ADINA R & D, Inc., Watertown, MA. 3. Akkari, M. and Duan, L., “Chapter 5: Nonlinear Analysis,” Bridge Engineering Handbook, 2nd Edition: Seismic Design, Chen, W.F. and Duan, L., Ed., CRC Press. Boca Raton, FL. 4. Barker, R. M. and Puckett, J. A., (2013). Design of Highway Bridges: A LRFD Approach, 3rd Edition, John Wiley & Sons, Inc., New York, NY. 5. Cai, C.S., Zhang, W. and Montens, S., (2014) “Chapter 22: Wind Effects on Long Span Bridges,” Bridge Engineering Handbook, 2nd Edition: Fundamentals, Chen, W.F. and Duan, L., Ed., CRC Press. Boca Raton, FL. 6. Caltrans, (2014a). California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 7. Caltrans, (2014b). CTBRIDGE, Caltrans Bridge Analysis and Design v. 1.6.1, California Department of Transportation, Sacramento, CA. 8. Caltrans, (2007). Memo to Designers 20-4 – Earthquake Retrofit for Bridges, California Department of Transportation, Sacramento, CA. 9. Caltrans, (2013). Seismic Design Criteria, v. 1.7, California Department of Transportation, Sacramento, CA. 10. CSI, (2015). CSiBridge 2015, v. 17.0.0, Computers and Structures, Inc., Walnut Creek, CA. 11. Duan, M., Perdikaris, P.C. and Chen, W. F., (1999). “Chapter 56: Impact Effect of Moving Vehicle,” Bridge Engineering Handbook, Chen, W.F. and Duan, L., Ed., CRC Press. Boca Raton, FL. 12. Priestley, Seible and Calvi (1996). Seismic Design and Retrofit of Bridges, John Wiley & Sons, Inc., New York, NY. 13. Zhang, W., Vinyagamoorth, M. and and Duan, L., “Chapter 3: Dynamic,” Bridge Engineering Handbook, 2nd Edition: Seismic Design, Chen, W.F. and Duan, L., Ed., CRC Press. Boca Raton, FL.

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CHAPTER 5 CONCRETE DESIGN THEORY TABLE OF CONTENTS 5.1 

INTRODUCTION ........................................................................................................... 5-1 

5.2 

STRUCTURAL MATERIALS ....................................................................................... 5-2  5.2.1 

Concrete .......................................................................................................... 5-2 

5.2.2 

Reinforcing Steel............................................................................................. 5-2 

5.2.3 

Prestressing Steel ............................................................................................ 5-2 

5.3 

DESIGN LIMIT STATES............................................................................................... 5-2 

5.4 

FLEXURE DESIGN ....................................................................................................... 5-3 

5.5 

5.6 

5.4.1 

Strength Limit States ....................................................................................... 5-3 

5.4.2 

Service Limit States ........................................................................................ 5-8 

5.4.3 

Fatigue Limit States ...................................................................................... 5-10 

SHEAR DESIGN .......................................................................................................... 5-10  5.5.1 

Basic Concept of Modified Compression Field Theory................................ 5-10 

5.5.2 

Shear Strength ............................................................................................... 5-11 

5.5.3  

Flexure – Shear Interaction ........................................................................... 5-14 

5.5.4 

Transverse Reinforcement Limits ................................................................. 5-16 

COMPRESSION DESIGN ........................................................................................... 5-16  5.6.1 

Factored Axial Compression Resistance – Pure Compression ..................... 5-17 

5.6.2 

Combined Flexure and Compression ............................................................ 5-17 

5.6.3 

Reinforcement Limits ................................................................................... 5-20 

NOTATION ............................................................................................................................... 5-21  REFERENCES ........................................................................................................................... 5-24 

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CHAPTER 5 CONCRETE DESIGN THEORY 5.1

INTRODUCTION Concrete is the most commonly used material in California highway structures, especially after the wide acceptance of prestressing technology in the 1950s. Nowadays, concrete bridges, prestressed or non-prestressed, account for about 90% of all bridges in the California highway system. Such dominancy is attributable to the many advantages that concrete offers:       

Ability to be cast in almost any shape Low cost Durability Fire resistance Energy efficiency On-site fabrication Aesthetic properties

Concrete design has evolved from Allowable Stress Design (ASD), also Working Stress Design (WSD), to Ultimate Strength Design (USD) or Load Factor Design (LFD), to today’s Limit State Design (LSD) or Load and Resistance Factor Design (LRFD). Concrete design takes on a whole new look and feel in the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012). New concepts that had been ruminating amongst concrete experts for decades reached a level of maturity appropriate for implementation. While not perfect, the new methods are more rational than those in the AASHTO Standard Specifications for Highway Bridges (AASHTO, 2002) and entail an amount of effort appropriate given today’s technology compared to that available when the LFD was developed. Changes include:      

Unified design provisions for reinforced and prestressed concrete Modified compression field theory for shear and torsion Alternative Strut and Tie modeling techniques for shear and flexure End zone analysis for tendon anchorages New provisions for segmental construction Revised techniques for estimating prestress losses

Chapter 5 will summarize the general aspects of concrete component design using the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) with the California Amendments (Caltrans, 2014a), while Chapter 7 will give a detailed description of the design procedure for post-tensioned box girder bridges, and Chapter 8 will cover the design of precast prestressed girder bridges. Concrete decks are covered in Chapter 10. Chapter 5 – Concrete Design Theory

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5.2

STRUCTURAL MATERIALS

5.2.1

Concrete The most important property of concrete is the compressive strength. Concrete with 28-day compressive strength f c = 3.6 ksi is commonly used in conventionally reinforced concrete structures while concrete with higher strength is used in prestressed concrete structures. The California Amendments (Caltrans, 2014a) specify minimum design strength of 3.6 ksi for concrete, and AASHTO Article 5.4.2.1 (AASHTO, 2012) requires minimum design strength of 4.0 ksi for prestressed concrete. When a higher strength is specified for a project, designers should consider various factors including cost and local availability.

5.2.2

Reinforcing Steel Steel reinforcing bars are manufactured as plain or deformed bars. In California, the main reinforcing bars are always deformed. Plain bars are usually used for spirals or hoops. Reinforcing bars must be low-alloy steel deformed bars conforming to requirements in ASTM A 706/A 706M with a 60 ksi yield strength, except that deformed or plain billet-steel bars conforming to the requirements on ASTM A 615/A 615M, Grade 40 or 60, may be used as reinforcement in some minor structures as specified in Caltrans Standard Specifications (Caltrans, 2015).

5.2.3

Prestressing Steel Two types of high-tensile strength steel used for prestressing steel are: 1. Strands: ASTM A 416 Grade 270, low relaxation. 2. Bars: ASTM A 722 Type II All Caltrans designs are based on low relaxation strands using either 0.5 in. or 0.6 in. diameter strands.

5.3

DESIGN LIMIT STATES Concrete bridge components are designed to satisfy the requirements of service, strength, and extreme-event limit states for load combinations specified in AASHTO Table 3.4.1-1 (AASHTO, 2012) with Caltrans revisions (Caltrans, 2014). The following are the four limit states into which the load combinations are grouped:

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I.

Service Limit States Concrete stresses, deformations, cracking, distribution of reinforcement, deflection, and camber are investigated at service limit states. Service I: Crack control and limiting compression in prestressed concrete Service III: Crack control/tension in prestressed concrete Service IV: Post-tensioned precast column sections

II. Strength Limit States Axial, flexural, shear strength, and stability of concrete components are investigated at strength limit states. Resistance factors are based on AASHTO Article 5.5.4.2 (AASHTO, 2012). Strength I: Basic load (HL-93) Strength II: Owner specified load (Permit) Strength III: Wind on structure Strength IV: Structure with high DL/LL (>7) Strength V: Wind on structure and live load III. Extreme Event Limit States Concrete bridge components and connections must resist extreme event loads due to earthquake and appropriate collision forces, but not simultaneously. IV. Fatigue Limit States Fatigue of the reinforcement need not be checked for fully prestressed concrete members satisfying requirements of service limit state. Fatigue need not be investigated for concrete deck slab on multi-girder bridges. For fatigue requirements, refer to AASHTO Article 5.5.3 (AASHTO, 2012).

5.4

FLEXURE DESIGN

5.4.1

Strength Limit States

5.4.1.1

Design Requirement In flexure design, the basic strength design requirement can be expressed as follows: M u  M n  M r  where Mu is the factored moment at the section (kip-in.); Mn is the nominal flexural resistance (kip-in.); and Mr is the factored flexural resistance of a section in bending (kip-in.).

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In assessing the nominal resistance for flexure, the AASHTO LRFD provisions (AASHTO, 2012) unify the strength design of conventionally reinforced and prestressed concrete sections based on their behavior at ultimate limit state. In the old LFD Specifications, a flexure member was designed so that the section would fail in a tension-controlled mode. Thus, there was a maximum reinforcement ratio. Whereas, in the new LRFD specifications, there is no explicit upper bound for reinforcement. There is a distinction of compression and tension-controlled section based on the strain in the extreme tension steel. To penalize for the undesirable behavior of compression-controlled sections, a lower value of resistance reduction factor  is assigned to “compression-controlled” sections compared to “tensioncontrolled” sections. The new procedure defines a transition behavior region in which the resistance factor , to be used for strength computation, varies linearly with the strain in the extreme steel fibers. The design of sections falling in this behavior region may involve an iterative procedure. Here are a few terms used to describe the flexural behavior of the reinforced section: Balanced strain condition: Strain in extreme tension steel reaches its yielding strain as the concrete in compression reaches its assumed ultimate strain of 0.003. Compression-controlled strain limit: Net Tensile Strain (NTS) (excluding effect of prestressing, creep, etc.) in the extreme tension steel at the balanced condition. It may be assumed equal to 0.002 for Grade 60 reinforcement and all prestressed reinforcement. Compression-controlled section: NTS ≤ compression-controlled strain limit just as the concrete in compression reaches its assumed strain limit of 0.003. When a section falls into this situation, it behaves more like a column than a beam. Thus, the component shall be properly reinforced with ties and spirals as required by AASHTO Article 5.7.2.1 with appropriate resistance factor (Caltrans, 2014a).  Tension-controlled section: NTS  0.005 just as the concrete in compression reaches its assumed strain limit of 0.003. Resistance factors are as follow:  for precast prestressed members  for cast-in-place prestressed members  for non-prestressed members Transition region: Compression controlled strain limit < NTS < 0.005. For the transition region, the resistance factor is calculated by using linear interpolation. Caltrans Amendments require that reinforced concrete sections in flexure be designed

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so that NTS  0.004. This requirement is to ensure that the section will not fail in compression-controlled modes. Figure 5.4-1 Illustrates those three regions and equations for resistance factors of flexural resistance (Caltrans, 2014a).

Figure 5.4-1 5.4.1.2

Resistance Factor Variation for Grade 60 Reinforcement and Prestressing Steel

Nominal Flexural Resistance The provisions for conventionally reinforced and prestressed concrete are now one-and-the-same. The basic assumptions used for flexural resistance (AASHTO Article 5.7.2.2) are as follows:     

Plane section remains plane after bending, i.e., strain is linearly proportional to the distance from the neutral axis, except the deep members. For unconfined concrete, maximum usable strain at the extreme concrete compression fiber is not greater than 0.003. For confined concrete, the maximum usable strain exceeding 0.003 may be used if verified. Stress in the reinforcement is based on its stress-strain curve. Tensile strength of concrete is neglected. Concrete compressive stress-strain distribution is assumed to be rectangular, parabolic, or any shape that results in predicted strength in substantial agreement with the test results. An equivalent rectangular compression stress block of 0.85 f c' over a zone bounded by the edges of the cross-section and a straight line located parallel to the neutral axis at the distance a = β1c from the extreme compression fiber may be used in lieu of a more exact concrete stress distribution, where c is the distance measured perpendicular to the neutral axis and

0.65  1  1.05  0.05 f c'  0.85

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'

where fc is in ksi For a T-beam section, there are two cases (Figure 5.4-2) depending on where the neutral axis falls into:  

Case 1: flanged section when the neutral axis falls into the web Case 2: rectangular section when the neutral axis falls into the flange

 Figure 5.4-2

Stress and Strain Distribution of T-Beam Section in Flexure (shown with mild reinforcement only)

For flanged sections, the Mn can be calculated by the following equation assuming the compression flange depth is less than a = β1c :



a



a

'

'



'

a

'

a

hf 

M n  Aps f ps  d p    As f s  d s    As f s  d s    0.85 fc (b  bw )h f       2 2 2 2 2

(AASHTO 5.7.3.2.2-1) where a is the depth of equivalent rectangular stress block (in.); c is the distance from the extreme compression fiber to the neutral axis (in.); b is the width of the compression face of the member (in.); bw is the web width (in.); hf is the thickness of flange (in.); d is the distance from compression face to centroid of tension reinforcement (in.); ds is the distance from compression face to centroid of mild tensile reinforcement (in.) and dp is the distance to the centroid of prestressing steel (in.); As is the area of mild tensile reinforcement (in.2) and Aps is the area of prestressing steel (in.2); A′s is the area of mild compressive reinforcement (in.2); f s is the stress in mild tensile steel (ksi); f s is the stress in the mild steel compression reinforcement (ksi) and fps is the stress in prestressing steel (ksi). For rectangular sections, let bw = b. The last term of the above equation will be dropped. Chapter 5 – Concrete Design Theory

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For circular and other nonstandard cross-sections, strain-compatibility must be used. WinConc (Caltrans, 2014b) is a suitable tool, and has been modified for LRFD. To evaluate the prestressing stresses, the following equations can be used: For bonded reinforcement and tendons:

 c f ps  f pu  1  k  dp  

(AASHTO 5.7.3.1.1-1)

in which:

 f py  k  2  1.04  f pu  

(AASHTO 5.7.3.1.1-2)

For flanged sections:

A ps f pu  As f s  As f s  0.85 f c( b  bw )h f f pu 0.85 f c 1bw  kA ps dp

c 

(AASHTO 5.7.3.1.1-3)

For rectangular sections: c

A ps f pu  As f s  As f s f pu 0.85 f c 1b  kA ps dp

For unbonded tendons:  dp  c  f py f ps  f pe  900   le  in which:

le 

2li

(AASHTO 5.7.3.1.1-4)

(AASHTO 5.7.3.1.2-1)

(AASHTO 5.7.3.1.2-2)

2  Ns

For flanged sections:

c

A ps f ps  As f s  As f s  0.85 f c( b  bw )h f 0.85 f c  1 bw

(AASHTO 5.7.3.1.2-3)

For rectangular sections:

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c 

A ps f ps  As f s  As f s 0.85 f c  1 b

(AASHTO 5.7.3.1.2-4)

where fpy and fpu are the yield and ultimate tensile strength of prestressing steel respectively; fpe is the effective stress in prestressing steel after loss (ksi); le is the effective tendon length (in.); li is the length between anchorage (in.); and Ns is the number of support hinges crossed by the tendon between anchorages. 5.4.1.3

Reinforcement Limits

As mentioned before, there is no explicit limit on maximum reinforcement. Sections are allowed to be over reinforced but shall be compensated for reduced ductility in the form of a reduced resistance reduction factor. The minimum reinforcement shall be provided so that, Mr, is at least equal to the lesser of Mcr and 1.3 Mu .

5.4.2

Service Limit States Service limit states are used to satisfy stress limits, deflection, and cracking requirements. To calculate the stress and deflection, the designer can assume concrete behaves elastically. The modulus of elasticity can be evaluated according to the code specified formula such as AASHTO Article 5.4.2.4. The reinforcement and prestressing steel are usually transformed into concrete. For normal weight concrete with wc = 0.145 kcf, the modulus of elasticity, may be taken as: E c  1,820

f c

(AASHTO C5.4.2.4-1)

For prestressed concrete members, prestressing force and concrete strength are determined by meeting stress limits in the service limit states, and then checked in the strength limit states for ultimate capacity. All other members are designed in accordance with the requirements of strength limit states first, the cracking requirement is satisfied by proper reinforcement distribution. To design the prestressed members, the following stress limits listed in Table 5.4-1 should be satisfied.

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Table 5.4-1 Stress Limits for Concrete Condition

Stress

Location

Temporary Stress before loss

Tensile

In area other than Precompressed Tensile Zone and without bonded tendons or reinforcement

Compression Final Stress after loss at service load

Tensile

Compression

Permanent loads only

Tensile

Chapter 5 – Concrete Design Theory

In area with bonded tendons or reinforcement sufficient to resist the tensile force in the concrete computed assuming an uncracked section, where reinforcement is proportioned using a stress of 0.5fy, not to exceed 30 ksi All locations In the Precompressed Tensile Zone, assuming uncracked section: • Components with bonded tendons or reinforcement, and/or are located in Caltrans’ Environment Areas I and II • Components with bonded tendons or reinforcement, and/or are located in Caltrans’ Environment Area III • Components with unbonded tendons All locations due to: • Permanent loads and effective prestress loads • All load combinations Precompressed Tensile Zone with bonded prestressing tendons or reinforcement

Allowable Stress f ci ≤0.2 (ksi)

0.0948

f ci (ksi)

0.24

0.6 f c

0.19

f c (ksi) f c (ksi)

0.0948

0 0.45 f c 0.6 f c 0

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

5.4.3

Fatigue Limit States As per AASHTO 5.5.3.1 (AASHTO, 2012), the stress range in reinforcing bars due to the fatigue load combination should be checked and should satisfy:

F TH

 24  0.33 f min

(AASHTO 5.5.3.2-1)

where: (F)TH = stress range (ksi) f min = algebraic minimum live load stress (ksi) resulting from the fatigue load combined with the more severe stress from either the unfactored permanent loads or the unfactored permanent loads, shrinkage, and creep-induced external loads; positive if tension, negative if compression For the fatigue check: 

   

The fatigue load combination is given in the California Amendment Table 3.4.1-1 (Caltrans, 2014a). A load factor of 1.0 is specified on the live load (Fatigue truck) for finite fatigue life and a load factor of 1.75 for the infinite fatigue life. A fatigue load is one design truck with a constant 30-ft spacing between the 32.0-kip axles as specified in AASHTO Article 3.6.1.4 (AASHTO, 2012). Apply the IM factor to the fatigue load. There is no permanent load considered in this check. Check both top and bottom reinforcements to ensure that the stress range in the reinforcement under the fatigue load stays within the range specified in the above equation.

5.5

SHEAR DESIGN

5.5.1

Basic Concept of Modified Compression Field Theory Perhaps the most significant change for concrete design in AASHTO LRFD Bridge Design Specifications is the shear design methodology. It provides two methods: Sectional Method, and Strut and Tie Method. Both methods are acceptable to Caltrans. The Sectional Method, which is based on the Modified Compression Field Theory (MCFT), provides a unified approach for shear design for both prestressed and reinforced concrete components. For a detailed derivation of this method, please refer to the book by Collins and Mitchell (1991). The two approaches are summarized as follows: 

Sectional Method - Plane section remains plane – Basic Beam Theory - Based on Modified Compression Field Theory (MCFT)

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

Used for most girder design, except disturbed-end regions Used for any undisturbed regions

Strut and Tie Method - Plane section does not remain plane - Used in “disturbed regions” and deep beams - Examples of usage: Design of Bent Caps (clear span to depth ratio less than 4); pile caps; anchorage zones (general or local); area around openings

In this chapter, only the Sectional Method will be outlined. The Strut and Tie Method will be discussed in other chapters. Compression Field Theory (CFT) is highlighted as follows:      

Angle for compressive strut (or crack angle) is variable Plane section remains plane (for strain compatibility) Strength of concrete in tension is ignored Element level strains incorporate the effects of axial forces, shear and flexure Equations are based on element level stresses and strains The shear capacity is related to the compression in diagonally cracked concrete through equilibrium

This theory is further modified by including the strength of concrete in tension, and it is referred to as the Modified Compression Field Theory (MCFT).

5.5.2

Shear Strength According to AASHTO LRFD 5.8.3.3, the nominal shear resistance, Vn , shall be determined as: Vn = Vc + Vs + Vp

(AASHTO 5.8.3.3-1)

But total resistance by concrete and steel: Vc + Vs should be no greater than 0.25 f c bvdv . In the end region of the beam-type element when it is not built integrally with the support, Vc + Vs should not exceed 0.18 f c bvdv . If it exceeds this value, this region should be designed using the Strut and Tie Method and special consideration should be given to detailing.

Vc  0.0316 f cbv d v Vs 

Av f y d v (cot   cot  ) sin  s

Chapter 5 – Concrete Design Theory

(AASHTO 5.8.3.3-3) (AASHTO 5.8.3.3-4)

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where Vp is the component in the direction of applied shear of the effective prestressing force (kip); bv is the effective web width (in.); and dv is the effective shear depth (in.). 5.5.2.1

Simplified Procedure for Nonprestressed Sections

For concrete footings in which the distance from point of zero shear to the face of the column, pier, or wall is less than 3dv, and for other nonprestressed concrete sections not subjected to axial tension and containing at least the minimum amount of transverse reinforcement, or having an overall depth of less than 16.0 in.,  = 45o and  = 2.0 may be used (AASHTO, 2012). The current California Amendments (Caltrans, 2014) do not allow the use of simplified procedure for prestressed and nonprestressed sections. 5.5.2.2

General Procedure

Parameter  and  may be determined either by the following general procedure or the MCFT method specified in AASHTO Appendix B5 (AASHTO, 2012).

  29  3500  s Mu

s 

dv

(AASHTO 5.8.3.4.2-3)

 0.5 N u  Vu  V p  A ps f po E s As  E p A ps

(AASHTO 5.8.3.4.2-4)

For sections with minimum transverse reinforcement:



4 .8 (1  750 s )

(AASHTO 5.8.3.4.2-1)

For sections without minimum transverse reinforcement:



 51  4.8   (1  750 s )  39  s xe 

(AASHTO 5.8.3.4.2-2)

The crack spacing parameter sxe , is determined as: 1.38 (AASHTO 5.8.3.4.2-5) 12 .0  s xe  s x  80 in. a g  0.63 where ag is maximum aggregate size (in.); and sx is the lesser of either dv , or the maximum distance between layers of longitudinal crack control reinforcement (in.). 5.5.2.2

MCFT Method

Unlike in the old LFD code where the angle of crack was assumed as a constant 45º, the MCFT method assumes it is a variable, which is a more accurate depiction of actual behavior.

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For sections containing at least the minimum transverse reinforcement,  and  values calculated from the MCFT are given as functions of x, shear stress vu , and f c in AASHTO Table B5.2.1 (AASHTO, 2012). x is taken as the calculated longitudinal strain at mid-depth of the member when the section is subjected to Mu , Nu , and Vu .  Mu     0 . 5 N  0 . 5 V  V cot  A f  u u p ps po  d  v   x  (AASHTO B5.2-1) 2( Es As  E p Aps ) For sections containing less than the minimum transverse reinforcement,  and  values calculated from the MCFT are given as functions of x, and the crack spacing sxe in AASHTO Table B5.2-2 (AASHTO, 2012). x is taken as the largest calculated longitudinal strain which occurs within the web of the member when the section is subjected to Mu , Nu , and Vu .  Mu     0 . 5 N  0 . 5 V  V cot  A f  u u p ps po  d  v   x  ( E s As  E p A ps )

(AASHTO B5.2-2)

If the value of x from AASHTO Equations (B5.2-1) or (B5.2-2) is negative, the strain is taken as:  Mu         0 . 5 N 0 . 5 V V cot A f u u p ps po  d  v   x  2( Ec Ac  E s As  E p A ps )

The crack spacing parameter sxe , is determined as: 1.38  80 in. s xe  s x a g  0.63

(AASHTO B5.2-3)

(AASHTO B5.2-4)

As one can see, x,  and  are all inter-dependent. So, design is an iterative process: 1. Calculate shear stress demand vu at a section and determine the shear ratio (vu/f ′c) 2. Calculate x at the section based on normal force (including p/s), shear and bending and an assumed value of  3. Longitudinal strain x is the average strain at mid-depth of the cross section 4. Knowing vu /f ′c & x, obtain the values of  and  from the table 5. Recalculate x based on revised value of ; repeat iteration until convergence in  is achieved.

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where vu , the shear stress on the concrete, should be determined as: vu 

Vu  V p

bv d v

(AASHTO 5.8.2.9-1)

To simplify this iterative approach, Professors Bentz, Vecchio, and Collins have proposed a simplified method (Bentz, E. C. et al, 2006).

5.5.3

Flexure - Shear Interaction In the MCFT model, the concrete is essentially modeled as a series of compression struts in resisting shear forces. The horizontal components of these diagonal forces have to be resisted by horizontal ties - longitudinal reinforcement. Therefore, after the design of flexure and shear is completed, the longitudinal reinforcement is checked for such interaction. Provide additional reinforcement if required. The following equations should be used for checking the adequacy of longitudinal reinforcement:  V Mu N A ps f ps  As f y   0.5 u   u  V p  0.5Vs  cot   d v f c  v  (AASHTO 5.8.3.5-1)

V Vs  u  V  Aps f ps  As f y   u  0.5Vs  V p  cot   v     

(AASHTO 5.8.3.5-2)

Requirement for the interaction check depends on the support / load transfer mechanism (direct supports or indirect supports) Maximum flexural steel based on moment demand need not be exceeded in / near direct supports Interaction check is required for simple spans made continuous for live load or where longitudinal steel is not continuous Equation (5.8.3.5-2) is required to be satisfied at the inside edge of the bearing area of simple support

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Direct Support / Direct Loading Figure 5.5-1 shows some of the examples of direct support and direct loading:

Figure 5.5-1 Examples of Direct Support and Direct Loading

Special Notes:  

If flexural reinforcement is not curtailed (eg: in bent cap), then there is no need to check for interaction. If flexural reinforcement is curtailed (eg: in a superstructure), check for interaction: - Check at 1/10 points and/or at curtailment locations. - If reinforcement per equation is inadequate, extend primary flexural reinforcement. - Area of tensile reinforcement need not exceed that required for maximum moment demand acting alone.

Indirect Support / Integral Girders Figure 5.5-2 shows some of the examples of indirect support and integral girders. The girders framing into the bent cap are indirectly supported while the bent cap itself is directly supported by the columns.

Figure 5.5-2

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Special Notes:  In bent caps, check interaction at 10 points, and at the girder locations of major concentrated loads  Check for interaction at face of integral supports  If interaction is not satisfied, then adopt one of the following: - Increase flexural reinforcement - Increase shear reinforcement - Combination of the above

5.5.4

Transverse Reinforcement Limits 

Minimum Transverse Reinforcement

Except for segmental post-tensioned concrete box girder bridges, the area of steel should satisfy: bs Av  0.0316 f c v (AASHTO 5.8.2.5-1) fy



Maximum Spacing of Transverse Reinforcement

The spacing of the transverse reinforcement should not exceed the maximum spacing, smax , determined as: 

If νu < 0.125 f ′c then: smax = 0.8dv ≤ 18.0 in.



If νu ≥ 0.125 f ′c then: smax = 0.4dv ≤ 12.0 in.

5.6

(CA 5.8.2.7-1)

(AASHTO 5.8.2.7-2)

COMPRESSION DESIGN As stated previously, when a member is subjected to a combined moment and compression force its resulting strain can be in a compression-controlled state. Compression design procedure applies. The following effects are considered in addition to bending: degree of end fixity; member length; variable moment of inertia; deflections; and duration of loads. This chapter will only cover the two basic cases: pure compression, and combined flexure and compression ignoring slenderness. AASHTO LRFD 5.7.4.3 provides an approximate method for evaluating slenderness effect.

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5.6.1

Factored Axial Compression Resistance – Pure Compression The factored axial resistance of concrete compressive members, symmetrical about both principal axes, is taken as: Pr   Pn

(AASHTO 5.7.4.4.-1)

In which: Pn , the nominal compression resistance, can be evaluated for the following two cases: For members with spiral reinforcement: Pn = 0.85[0.85 f ′c (Ag – Ast – Aps)+ fy Ast – Aps (fpe – Ep εcu)] (AASHTO 5.7.4.4-2) For members with tie reinforcement: Pn = 0.80[0.85 f ′c (Ag – Ast – Aps)+ fy Ast – Aps (fpe – Ep εcu)] (AASHTO 5.7.4.4-3) where Ag is the gross area of the section (in.2); Ast is the total area of longitudinal mild reinforcement (in.2); Aps is the area of prestressing steel (in.2); Ep is the modulus of elasticity of prestressing steel (ksi); and  cu is the failure strain of concrete in compression. In order to achieve the above resistance, the following minimum spiral shall be supplied:  Ag  f c  s  0.45  1 (AASHTO 5.7.4.6-1)  Ac  f yh where fyh is the specified yield strength of transverse reinforcement (ksi). To achieve more ductility for seismic resistance, Caltrans has its own set of requirements for spirals and ties. For further information, please refer to the current version of the Caltrans Seismic Design Criteria (Caltrans, 2013).

5.6.2

Combined Flexure and Compression When a member is subjected to a compression force, end moments are often induced by eccentric loads. The end moments rarely act solely along the principal axis. So at any given section for analyzing or design, the member is normally subjected to biaxial bending as well as compression. Furthermore, to analyze or design a compression member in a bridge substructure, many load cases need to be considered.

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Under special circumstances, the Specifications allow designers to use an approximate method to evaluate biaxial bending combined with axial load (AASHTO Article 5.7.4.5). Generally, designers rely on computer programs based on equilibrium and strain compatibility, such as WinYield (Caltrans, 2014b), to generate a moment-axial interaction diagram. For cases like noncircular members with biaxial flexure, an interaction surface is required to describe the behavior. Figure 5.6-1 shows a typical moment-axial load interaction surface for a concrete section (Park and Pauley 1975).

Figure 5.6-1

Moment-Axial Interaction Surface of a Noncircular Section

In day-to-day practice, such a surface has little value to designers. Rather, the design program normally gives out a series of lines, basically slices of the surface, at fixed intervals, such as 15º. Figure 5.6-2 is an example plot from WinYield (Caltrans, 2014b).

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Figure 5.6-2 Interaction Diagrams Generated by WinYield

From these lines, it can be seen that below the balanced condition the moment capacity increases with the increase of axial load. So, when designing a column, it is not enough to simply take a set of maximum axial load with maximum bending moments. The following combination needs to be evaluated: 1. 2. 3. 4.

Mux max , corresponding Muy and Pu Muy max , corresponding Mux and Pu A set of Mux and Muy that gives largest Mu combined, and corresponding Pu Pu max and corresponding Mux and Muy

Special Notes:     

Columns will be more thoroughly covered in Chapter 13. For load cases 1 through 3, the load factor γp corresponding to the minimum shall be used. Pn and Mn shall be multiplied by a single  factor depending on whether it is compression controlled or tension controlled, as illustrated previously and as shown in AASHTO Figure C5.5.4.2.1-1. Slenderness effect shall be evaluated with an appropriate nonlinear analysis program or the use of approximate methods such as AASHTO Article 5.7.4.3. In California, the column design is normally controlled by seismic requirements. That topic is not covered in this chapter.

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5.6.3

Reinforcement Limits The maximum area of prestressed and non-prestressed longitudinal reinforcement for non-composite compression members is as follows:

As A ps f pu   0.08 Ag Ag f y

(AASHTO 5.7.4.2-1)

and

A ps f pe Ag f c

 0.30

(AASHTO 5.7.4.2-2)

The minimum area of prestressed and non-prestressed longitudinal reinforcement for non-composite compression members is as follows:

As f y Ag f c



A ps f pu A g f c

 0.135

(AASHTO 5.7.4.2-3)

Due to seismic concerns, Caltrans put further limits on longitudinal steel in columns. For such limits, please refer to the latest version of the Caltrans Seismic Design Criteria (Caltrans, 2013).

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NOTATION Ac

= area of core of spirally reinforced compression member measured to the outside diameter of the spiral (in.2)

Ag

= gross area of section (in.2)

Aps

= area of prestressing steel (in.2)

As

= area of non-prestressed tension steel (in.2)

A′s

= area of compression reinforcement (in.2)

Ash

= cross-sectional area of column tie reinforcement (in.2)

Ast

= total area of longitudinal mild steel reinforcement (in.2)

Av

= area of transverse reinforcement within distances (in.2)

a

= depth of equivalent rectangular stress block (in.)

b

= width of compression face of the member (in.)

bv

= effective web width taken as the minimum web width (in.)

bw

= web width (in.)

c

= distance from the extreme compression fiber to the neutral axis (in.)

D

= external diameter of the circular members (in.)

d

= distance from compression face to centroid of tension reinforcement (in.)

db

= nominal diameter of a reinforcing bar (in.)

de

=

dp

= distance from extreme compression fiber to centroid of prestressing strand (in.)

ds

= distance from extreme compression fiber to centroid of non-prestressed tensile reinforcement (in.)

dv

= effective shear depth (in.)

Ec

= modulus of elasticity of concrete (ksi)

Ep

= modulus of elasticity of prestressing tendons (ksi)

Es

= modulus of elasticity of reinforcing bars (ksi)

f c

effective depth from extreme compression fiber to the centroid of tensile force in the tensile reinforcement (in.)

= specified compressive strength of concrete (ksi)

fcpe

= compressive stress in concrete due to effective prestress force only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi)

fpe

= effective stress in prestressing steel after losses (ksi)

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fps

= average stress in prestressing steel at the time for which the nominal resistance of members is required (ksi)

fpu

= specified tensile strength of prestressing steel (ksi)

fpy

= yield strength of prestressing steel (ksi)

fr

= modulus of rupture of concrete (ksi)

fs

= stress in mild tensile reinforcement at nominal flexural resistance (ksi)

f s

= stress in mild compression reinforcement at nominal flexural resistance (ksi)

fy

= specified minimum yield strength of reinforcing bars (ksi)

fyh

= specified yield strength of transverse reinforcement (ksi)

hf

= thickness of flange (in.)

le

= effective tendon length (in.)

li

= length of tendon between anchorages (in.)

Mb

= nominal flexural resistance at balanced condition (kip-in.)

Mcr

= cracking moment (kip-in.)

Mn

= nominal flexural resistance (kip-in.)

Mdnc

= total unfactored dead load moment acting on the monolithic or non-composite section (kip-ft.)

Mr

= factored flexural resistance of a section in bending (kip-in.)

Mu

= factored moment at the section (kip-in.)

Mux

= factored moment at the section in respect to principal x axis (kip-in.)

Muy

= factored moment at the section in respect to principal y axis (kip-in.)

Nu

= factored axial force (kip)

Ns

= number of support hinges crossed by the tendon between anchorages or discretely bonded points

Pn

= nominal axial resistance of a section (kip)

Po

= nominal axial resistance of a section at 0 eccentricity (kip)

Pr

= factored axial resistance of a section (kip)

Pu

= factored axial load of a section (kip)

s

= spacing of reinforcing bars (in.)

Vc

= nominal shear resistance provided by tensile stresses in the concrete (kip)

Vn

= nominal shear resistance of the section considered (kip)

Vp

= component in the direction of the applied shear of the effective prestressing forces; positive if resisting the applied shear (kip)

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Vr

= factored shear resistance (kip)

Vs

= shear resistance provided by the shear reinforcement (kip)

Vu

= factored shear force (kip)

vu

= average factored shear stress on the concrete (ksi)

Sc

= section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3)

Snc

= section modulus for the extreme fiber of the monolithic or non-composite section where tensile stress is caused by externally applied loads (in.3)



= angle of inclination of transverse reinforcement to longitudinal axis (º)



= factor relating effect of longitudinal strain on the shear capacity of concrete, as indicated by the ability of diagonally cracked concrete to transmit tension

1

=

γ

= load factor

cu

= failure strain of concrete in compression (in./in.)

x

= Longitudinal strain in the web reinforcement on the flexural tension side of the member (in./in.)



= angle of inclination of diagonal compressive stress (º)



= resistance factor

c

= resistance factor for compression

f

= resistance factor for moment

v

= resistance factor for shear

s

=

ratio of the depth of the equivalent uniformly stressed compression zone assumed in the strength limit state to the depth of the actual compression zone

ratio of spiral reinforcement to total volume of column core

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 2. AASHTO, (2002). Standard Bridge Design Specifications, 17th Edition, American Association of State Highway and Transportation Officials, Washington, D.C. 3. Bentz, E. C. et al, (2006). “Simplified Modified Compression Field Theory for Calculating Shear Strength of Reinforced Concrete Elements”, ACI Structural Journal/July-August 2006. 4. Caltrans, (2015). Standard Specifications, California Department of Transportation, Sacramento, CA. 5. Caltrans, (2014a). California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 6. 7. Caltrans, (2014b). WinYield, California Department of Transportation, Sacramento, CA. 8. Caltrans, (2014c). WinConc, California Department of Transportation, Sacramento, CA. 9. Caltrans, (2013). Caltrans Seismic Design Criteria, Version 1.7, California Department of Transportation, Sacramento, CA. 10. Collins, M. P. and Mitchell, D. (1991). Prestressed Concrete Structures, Prentice Hall, Englewood Cliffs, NJ. 11. Park, R. and Paulay, T. (1975). Reinforced Concrete Structures, John Willey & Sons, New York, NY.

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CHAPTER 6 STEEL DESIGN THEORY TABLE OF CONTENTS 6.1 

INTRODUCTION ............................................................................................................ 6-1 

6.2  

STRUCTURAL STEEL MATERIALS ........................................................................... 6-1 

6.3 

DESIGN LIMIT STATES ................................................................................................ 6-2 

6.4 

FLEXURE DESIGN ........................................................................................................ 6-2  6.4.1 

Design Requirements ......................................................................................... 6-2 

6.4.2   Composite Sections in Positive Flexure ............................................................. 6-4  6.4.3   Steel Sections ..................................................................................................... 6-9  6.5 

SHEAR DESIGN ........................................................................................................... 6-13  6.5.1   Design Requirements ....................................................................................... 6-13  6.5.2   Nominal Shear Resistance ................................................................................ 6-13  6.5.3   Transverse Stiffeners ........................................................................................ 6-14 6.5.4   Shear Connectors.............................................................................................. 6-14

6.6 

COMPRESSION DESIGN ............................................................................................ 6-15  6.6.1   Design Requirements ....................................................................................... 6-15  6.6.2   Axial Compressive Resistance ......................................................................... 6-16 

6.7 

TENSION DESIGN ....................................................................................................... 6-16  6.7.1   Design Requirements ....................................................................................... 6-16  6.7.2   Axial Tensile Resistance .................................................................................. 6-17 

6.8 

FATIGUE DESIGN ....................................................................................................... 6-18 

6.9 

SERVICEABILITY STATES ........................................................................................ 6-20 

6.10 

CONSTRUCTIBILITY .................................................................................................. 6-21 

NOTATION ............................................................................................................................... 6-22  REFERENCES ........................................................................................................................... 6-26 

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CHAPTER 6 STEEL DESIGN THEORY 6.1

INTRODUCTION Steel has higher strength, ductility and toughness than many other structural materials such as concrete or wood, and thus makes a vital material for bridge structures. In this chapter, basic steel design concepts and requirements for I-sections specified in the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) and the California Amendments (Caltrans, 2014) for flexure, shear, compression, tension, fatigue, and serviceability and constructibility are discussed. Design considerations, procedure and example for steel plate girders will be presented in Chapter 9.

6.2

STRUCTURAL STEEL MATERIALS AASHTO M 270 (Grade 36, 50, 50S, 50W, HPS 50W, HPS 70W and 100/100W) structural steels are commonly used for bridge structures. AASHTO material property standards differ from ASTM in notch toughness and weldability requirements. When these additional requirements are specified, ASTM A 709 steel is equivalent to AASHTO M 270 and is pre-qualified for use in welded steel bridges. The use of ASTM A 709 Grade 50 for all structural steel, including flanges, webs, bearing stiffeners, intermediate stiffeners, cross frames, diaphragms and splice plates is preferred. The use of ASTM A 709 Grade 36 for secondary members will not reduce material unit costs. The use of ASTM A 709 Grade 100 or 100W steel is strongly discouraged. The hybrid section consisting of flanges with a higher yield strength than that of the web may be used to save materials and is becoming more promoted due to the new high performance steels. Using HPS 70W top and bottom flanges in negative moment regions and bottom flanges in positive moment regions and Grade 50 top flanges in positive moment regions, and Grade 50 for all webs may provide the most efficient hybrid girder. The use of HPS (High Performance Steel) and weathering steel is encouraged if it is acceptable for the location. FHWA Technical Advisory T5140.22 (FHWA, 1989) provides guidelines on acceptable locations. In some situations, because of a particularly harsh environment, steel bridges must be painted. Although weathering steel will perform just as well as conventional steel in painted applications, it will not provide superior performance, and typically costs more than conventional steel. Therefore, specifying weathering steel in painted applications does not add value and should be avoided. HPS and weathering steel should not be used for the following conditions: 

The atmosphere contains concentrated corrosive industrial or chemical fumes.



The steel is subject to heavy salt-water spray or salt-laden fog.

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6.3



The steel is in direct contact with timber decking, because timber retains moisture and may have been treated with corrosive preservatives.



The steel is used for a low urban-area overcrossing that will create a tunnellike configuration over a road on which deicing salt is used. In these situations, road spray from traffic under the bridge causes salt to accumulate on the steel.



The location has inadequate air flow that does not allow adequate drying of the steel.



The location has very high rainfall and humidity or there is constant wetness.



There is low clearance (less than 8 to 10 ft) over stagnant or slow-moving waterways.

DESIGN LIMIT STATES Steel girder bridges shall be designed to meet the requirements for all applicable limit states specified by AASHTO (2012) and California Amendments (Caltrans, 2014). For a typical steel girder bridges, Strength I and II, Service II, Fatigue and Constructibility are usually controlling limit states.

6.4

FLEXURE DESIGN

6.4.1

Design Requirements The AASHTO 6.10 and its Appendices A6 and B6 provide a unified flexural design approach for steel I-girders. The provisions combine major-axis bending, minor-axis bending and torsion into an interaction design formula and are applicable to straight bridges, horizontally curved bridges, or bridges combining both straight and curved segments. The AASHTO flexural design interaction equations for the strength limit state are summarized in Table 6.4-1. Those equations provide an accurate linear approximation of the equivalent beam-column resistance with the flange lateral bending stress less than 0.6Fy as shown in Figure 6.4-1 (White and Grubb 2005).

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Table 6.4-1 I-Section Flexural Design Equations (Strength Limit State) Section Type Compact Composite Sections in Positive Flexure

Composite Sections in Negative Flexure and Noncomposite Sections

Noncompact

Discretely braced

Continuously braced fbu fl Fnc Fnt Fyf Mu Mn f Rh Sxt

= = = = = = = = = =

Design Equation 1 M u  f l S xt  f M n 3

Compression flange fbu   f Fnc Tension flange 1 f bu  f l   f Fnt 3 Compression flange 1 f bu  fl   f Fnc 3 Tension flange 1 f bu  f l   f Fnt 3

fbu   f Rh Fyf

(AASHTO 6.10.7.1.1-1)

(AASHTO 6.10.7.2.1-1) (AASHTO 6.10.7.2.1-2)

(AASHTO 6.10.8.1.1-1)

(AASHTO 6.10.8.1.2-1)

(AASHTO 6.10.8.1.3-1)

flange stress calculated without consideration of the flange lateral bending (ksi) flange lateral bending stress (ksi) nominal flexural resistance of the compression flange (ksi) nominal flexural resistance of the tension flange (ksi) specified minimum yield strength of a flange (ksi) bending moment about the major axis of the cross section (kip-in.) nominal flexural resistance of the section (kip-in.) resistance factor for flexure = 1.0 hybrid factor elastic section modulus about the major axis of the section to the tension flange taken as Myt/Fyt (in.3)

Figure 6.4 -1 AASHTO Unified Flexural Design Interaction Equations

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For compact sections, since the nominal moment resistance is generally greater than the yield moment capacity, it is physically meaningful to design in terms of moment. For noncompact section, since the nominal resistance is limited to the yield strength, stress format is used. For composite I-sections in negative flexure and for noncomposite I-sections with compact or noncompact webs in straight bridges, when the web slenderness is well below the noncompact limit, the provisions specified in AASHTO Appendix 6A are encouraged to be used. However, when the web slenderness approaches the noncompact limit, Appendix 6A provides only minor increases in the nominal resistance.

6.4.2

Composite Sections in Positive Flexure

6.4.2.1

Nominal Flexural Resistance At the strength limit state, the compression flange of composite sections in positive flexure is continuously supported by the concrete deck and lateral bending does not need to be considered. For compact sections, the flexural resistance is expressed in terms of moment, while for noncompact sections, the flexural resistance is expressed in terms of the elastically computed stress. The compact composite section shall meet the following requirements: 

Straight bridges



Fyf  70 ksi



D  150 tw



2Dcp tw

 3.76

(AASHTO 6.10.2.1.1-1)

E Fyc

(AASHTO 6.10.6.2.2-1)

where Dcp is the web depth in compression at the plastic moment (in.); E is modulus of elasticity of steel (ksi); Fyc is specified minimum yield strength of a compression flange (ksi). Composite sections in positive flexure not satisfying one or more of above four requirements are classified as noncompact sections. The nominal flexural resistances are listed in Table 6.4-2.

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Table 6.4-2 Nominal Flexural Resistance for Composite Sections in Positive Flexure (Strength Limit State) Section Type

Nominal Flexural Resistance

Compact

 M p    My   M n  min  M p 1  1      M p  1.3Rh M y

for D p  0.1 Dt   D p / Dt  0.1     for D p  0.1 Dt   0.32    for a continous span

(AASHTO 6.10.7.1.2-1, 3) and (CA 6.10.7.1.2-2)

Noncompact

Compression flange Fnc  Rb Rh Fyc Tension flange Fnt  Rh Fyt

Ductility Requirement

For both compact and noncompact sections D p  0.42Dt

(AASHTO 6.10.7.2.2-1) (AASHTO 6.10.7.2.2-2) (AASHTO 6.10.7.3-1)

Dp = distance from the top of the concrete deck to the neutral axis of the composite section at the plastic moment (in.) Dt = total depth of the composite section (in.) Fyt = specified minimum yield strength of a tension flange (ksi) Mp = plastic moment of the composite section (kip-in.) My = yield moment of the composite section (kip-in.) Rb = web load-shedding factor 6.4.2.2 Yield Moment The yield moment My for a composite section in positive flexure is defined as the moment which causes the first yielding in one of the steel flanges. My is the sum of the moments applied separately to the appropriate sections, i.e., the steel section alone, the short-term composite section, and the long-term composite section. It is based on elastic section properties and can be expressed as: M y  M D1  M D2  M AD

(AASHTO D6.2.2-2)

where MD1 is moment due to factored permanent loads applied to the steel section alone (kip-in.); MD2 is moment due to factored permanent loads such as wearing surface and barriers applied to the long-term composite section (kip-in.); MAD is additional live load moment to cause yielding in either steel flange applied to the short-term composite section and can be obtained from the following equation (kipin.):

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Fyf 

M D1 M D2 M   AD S NC S LT S ST

 M M D2  M AD  S ST  Fyf  D1   S NC S LT  

(AASHTO D6.2.2-1) (6.4-1)

where SNC, SST and SLT are elastic section modulus for steel section alone, short-term composite and long-term composite sections, respectively (in.3). 6.4.2.3 Plastic Moment The plastic moment Mp for a composite section is defined as the moment which causes the yielding of the entire steel section and reinforcement and a uniform stress distribution of 0.85 f c' in the compression concrete slab. f c' is minimum specified 28-day compressive strength of concrete. In positive flexure regions the contribution of reinforcement in the concrete slab is small and can be neglected. Table 6.4-3 summarizes calculations of Mp.

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Table 6.4-3 Plastic Moment Calculation Regions

Case and PNA I - In Web

Condition and

Mp

Y

Pt  Pw  Pc  Ps  Prb  Prt  D   P  Pc  Ps  Prt  Prb  Y    t 1 Pw  2  

II - In Top Flange

Positive Figure 6.4.2

Pt  Pw  Pc  Ps  Prb  Prt

t Y  c 2

   Pw  Pt  Ps  Prt  Prb 1  Pc  









2 Pw  2 Y  D  Y   2 D    Ps d s  Prt d rt  Prb d rb  Pc d c  Pt d t 

Mp 

Mp 

Pc 2t c

Y 2  t  Y c 

2

 

 Ps d s  Prt d rt  Prb d rb  Pw d w  Pt d t 

III- In Slab, Below Prb

C  Pt  Pw  Pc   rb  Ps  Prb  Prt  ts   Pw  Pc  Pt  Prt  Prb  Y  t s   Ps  

 Y 2P  s  Mp     2ts    Prt d rt  Prb d rb  Pc dc  Pwd w  Pt dt 

IV - In Slab, Above Prb Below Prt

C Pt  Pw  Pc  Prb   rt  ts

 Y 2P  s  Mp     2t s    Prt d rt  Prb d rb  Pc dc  Pwd w  Pt dt 

V - In Slab, above Prt

  Ps  Prt 

 P  Pc  Pw  Pt  Prt  Y  t s  rb  Ps  

C Pt  Pw  Pc  Prb  Prt   rt  ts

  Ps 

 P  Pc  Pw  Pt  Prt  Y  t s  rb  Ps  

I - In Web Negative Figure 6.4.3

Pc  Pw  Pt  Prb  Prt   D   P  Pt  Prt  Prb Y    c  1 Pw  2  

II - In Top Flange

Prt  Fyrt Art ; f c = PNA = = Arb, Art Fyrb, Fyrt = = bc, bt, bs tc, tt, tw, ts = Fyt, Fyc, Fyw =

Pc  Pw  Pt  Prb  Prt

 t   P  Pc  Prt  Prb  1 Y   t  w Pt  2  

Ps  0.85 f cbs t s ; Prb  Fyrb Arb ; Pc  Fyc bc t t ;

 Y 2P  s  Mp     2t s    Prt d rt  Prb d rb  Pc dc  Pwd w  Pt dt 





2 Pw  2 Y  D  Y    2D    Prt drt  Prbdrb  Pd t t  Pcdc 

Mp 

Mp 





2 Pt  2 Y  tt  Y    2tt 

Prt drt  Prbdrb  Pwd w  Pc dc  Pw  Fyw Dt w ;

Pt  Fyt bt t t ;

minimum specified 28-day compressive strength of concrete (ksi) plastic neutral axis reinforcement area of bottom and top layer in concrete deck slab (in.2) yield strength of reinforcement of bottom and top layers (ksi) width of compression, tension steel flange and concrete deck slab (in.) thickness of compression, tension steel flange, web and concrete deck slab (in.) yield strength of tension flange, compression flange and web (ksi)

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Case - III Pc1  Y bc Fyc ; Pw1  Y t w Fyw ; Pt1  Y bt Fyt ;

Case - I

Case - II

Case - IV

Case - V

Pc 2  tc  Y bc Fyc

Pw 2  D  Y t w Fyw Pt 2  tt  Y bt Fyt

Figure 6.4-2 Plastic Moment Calculation Cases for Positive Flexure

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Case - I Pw1  Y t w Fyw ;

Case - II

Pw 2  D  Y t w Fyw

Pt1  Y bt Fyt ; Pt 2  tt  Y tt Fyt

Figure 6.4-3 Plastic Moment Calculation Cases for Negative Flexure

6.4.3

Steel Sections The flexural resistance of a steel section (i.e., composite sections in negative flexure and noncomposite sections) is governed by three failure modes or limit states: yielding, flange local buckling and lateral-torsional buckling. The moment capacity depends on the yield strength of steel, the slenderness ratio of the compression flange, f in terms of width-to-thickness ratio (bfc/2tfc) for local buckling and the unbraced length Lb for lateral-torsional buckling. Figure 6.4-4 shows dimensions of a typical I-girder. Figures 6.4-5 and 6.4-6 show graphically the compression flange local and lateral torsional buckling resistances, respectively. Calculations for nominal flexural resistances are illustrated in Table 6.4-4. For sections in straight bridges satisfying the following requirements: 

Fyf  70 ksi



2 Dc E  5. 7 tw Fyc



I yc I yt

 0.3

(AASHTO 6.10.6.2.3-1)

(AASHTO 6.10.6.2.3-2)

The flexural resistance in term of moments may be determined by AASHTO Appendix A6, and may exceed the yield moment.

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bf

tp

tf

bt

Figure 6.4-4 Dimensions of a Typical I-Girder

Figure 6.4-5 I-Section Compression-Flange Flexural Local-Buckling Resistance

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Figure 6.4-6 I-Section Compression-Flange Flexural Torsional Resistance

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Table 6.4-4 Nominal Flexural Resistance for Steel Sections (Composite Sections in Negative Flexure and Noncomposite Section) (Strength Limit State) Flange Nominal Flexural Resistance Fnc  smaller [ Fnc( FLB) , Fnc( LTB) ]

Compression

 Rb Rh Fyc  Fnc  FLB      Fyr  1  1  R F h yc   

(AASHTO 6.10.8.2.1) for  f   pf

  f   pf      pf  rf

   Rb Rh Fyc for  f   pf  

(AASHTO 6.10.8.2.2-1 &2)  Rb Rh Fyc  Fyr    Fnc  LTB   Cb 1   1      Rh Fyc   Fcr  Rb Rh Fyc

Tension

for Lb  L p   Lb  L p       Rb Rh Fyc  Rb Rh Fyc for Lb  L p  Lr   Lr  L p   for Lb  Lr

(AASHTO 6.10.8.2.3-1, 2 &3) (AASHTO 6.10.8.3-1)

Fnt  R h F yt Lb = unbraced length of compression flange (in.)

L p  limiting unbraced length toachieve Rb Rh Fyc  1.0 rt E / Fyc (AASHTO 6.10.8.2.3-4) Lr  limiting unbraced length toachievethe onset of nominalyielding  rt E / Fyr (AASHTO 6.10.8.2.3-5)

 f  slenderess ratio for compression flange 

b fc

(AASHTO 6.10.8.2.2-3)

2t fc

 pf  slenderess ratio for a compact compression flange  0.38

E (AASHTO 6.10.8.2.2-4) Fyc

rf  limiting slenderness ratio for a noncompact flange  0.56

E (AASHTO 6.10.8.2.2-5) Fyc

Fcr  elastic lateral torsional buckling stress (ksi) 

Cb Rb 2 E

 Lb / rt 

2

(AASHTO 6.10.8.2.3-8)





Fyr  smaller 0.7 Fyc , Fyw  0.5Fyc

(AASHTO 6.10.8.2.2)

Cb = moment gradient modifier rt = effective radius of gyration for lateral torsional buckling (in.)

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6.5

SHEAR DESIGN

6.5.1

Design Requirements For I-girder web panels, the following equation shall be satisfied.

Vu  cVn

(AASHTO 6.10.9.1-1)

where Vu is factored shear at the section under consideration (kip); Vn is nominal shear resistance (kip) and c is resistance factor for shear = 1.0.

6.5.2

Nominal Shear Resistance Similar to the flexural resistance, web shear resistance is also dependent on the slenderness ratio in terms of depth-to-thickness ratio (D/tw). For the web without transverse stiffeners, shear resistance is provided by the beam action of shearing yield or elastic shear buckling. For end panels of stiffened webs adjacent to simple support, shear resistance is limited to the beam action only.

Vn  Vcr  CV p

(AASHTO 6.10.9.2-1; 9.3-1)

V p  0.58Fyw Dtw

(AASHTO 6.10.9.2-2)

 D Ek  1.0  1.12 For tw Fyw    1.12 Ek Ek D Ek C  For 1.12   1.40 Fyw t w Fyw  D / t w  Fyw  D Ek  1.57  Ek   D / t 2  F  For t  1.40 F w yw w  yw   (AASHTO 6.10.9.3.2-4,5,6) k 5

5 (d o / D) 2

(AASHTO 6.10.9.3.2-7)

where do is transverse stiffener spacing (in.); C is ratio of the shear-buckling resistance to the shear yield strength; Vcr is shear-buckling resistance (kip) and Vp is plastic shear force (kip). For interior web panels with transverse stiffeners, the shear resistance is provided by both the beam and the tension field actions as shown in Figure 6.5-1.

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Figure 6.5-1 Tension Field Action For

2 Dt w  2 .5 b fc t fc  b ft t ft



(AASHTO 6.10.9.3.2-1)



     0.871  C   Vn  V p C   2   do   1     D  

(AASHTO 6.10.9.3.2-2)

otherwise

   0.871  C  Vn  V p C  2  d  do  1     o  D D 

      

(AASHTO 6.10.9.3.2-8)

where bfc and bft are full width of a compression and tension flange, respectively (in.); tfc and tft are thickness of a compression and tension flange, respectively (in.); tw is web thickness (in.) and do is transverse stiffener spacing (in.).

6.5.3

Transverse Stiffeners Transverse intermediate stiffeners work as anchors for the tension field force so that post-buckling shear resistance can be developed. It should be noted that elastic web shear buckling cannot be prevented by transverse stiffeners. Transverse stiffeners are designed to (1) meet the slenderness requirement of projecting elements to prevent local buckling, (2) provide stiffness to allow the web to develop its postbuckling capacity, and (3) have strength to resist the vertical components of the diagonal stresses in the web.

6.5.4

Shear Connectors To ensure a full composite action, shear connectors must be provided at the interface between the concrete slab and the steel to resist interface shear. Shear connectors are usually provided throughout the length of the bridge. If the longitudinal reinforcement in the deck slab is not considered in the composite

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section, shear connectors are not necessary in negative flexure regions. If the longitudinal reinforcement is included either additional connectors can be placed in the region of dead load contra-flexure points or they can be continued through the negative flexure region at maximum spacing. The fatigue and strength limit states must be considered in the shear connector design.

6.6

COMPRESSION DESIGN

6.6.1

Design Requirements For axially loaded compression members, the following equation shall be satisfied: Pu ≤ Pr = cPn

(6.6-1)

where Pu is factored axial compression load (kip); Pr is factored axial compressive resistance (kip); Pn is nominal compressive resistance (kip) and c is resistance factor for compression = 0.9. For members subjected to combined axial compression and flexure, the following interaction equation shall be satisfied: For

Pu  0 .2 Pr

M M uy  Pu   1.0   ux   M rx M ry  2.0 Pr   For

(AASHTO 6.9.2.2-1)

Pu  0 .2 Pr

Pu 8.0  M ux M uy   1.0   Pr 9.0  M rx M ry 

(AASHTO 6.9.2.2-2)

where Mux and Muy are factored flexural moments (second-order moments) about the x-axis and y-axis, respectively (kip-in.); Mrx and Mry are factored flexural resistance about the x-axis and y-axis, respectively (kip-in.). Compression members shall also meet the slenderness ratio requirements, Kl/r ≤ 120 for primary members, and Kl/r ≤ 140 for secondary members. K is effective length factor; l is unbraced length (in.) and r is minimum radius of gyration (in.).

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

6.6.2

Axial Compressive Resistance For steel compression members with non-slender elements, axial compressive resistance equations specified in the AASHTO (2012) are identical to the column design equations in AISC (2010). For

Pe  0.44 Po

 Po     P    Pn  0.658 e   Po   P For e  0.44 Po

Pn  0.877 Pe Pe 

(AASHTO 6.9.4.1.1-1)

(AASHTO 6.9.4.1.1-2)

 2E

(AASHTO 6.9.4.1.2-1)

2

 Kl     rs  eff

in which

Po  QFy Ag Pe 

 2E 2

 Kl     rs  eff

Ag

(AASHTO 6.9.4.1.2-1)

where Ag is gross cross section area (in.2); K is effective length factor in the plane of buckling; l is unbraced length in the plan of buckling (in.); rs is radius of gyration about the axis normal to the plane of buckling (in.); Q is slender element reduction factor determined as specified in AASHTO Article 6.9.4.2.

6.7

TENSION DESIGN

6.7.1

Design Requirements For axially loaded tension members, the following equation shall be satisfied:

Pu  Pr

(6.7-1)

where Pu is factored axial tension load (kip) and Pr is factored axial tensile resistance (kip).

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For members subjected to combined axial tension and flexure, the following interaction equation shall be satisfied: For

Pu  0 .2 Pr

M M uy  Pu   1.0   ux   M rx M ry  2.0 Pr   For

(AASHTO 6.8.2.3-1)

Pu  0 .2 Pr

Pu 8.0  M ux M uy     1.0 Pr 9.0  M rx M ry 

(AASHTO 6.8.2.3-2)

where Mux and Muy are factored flexural moments about the x-axis and y-axis, respectively (kip-in.); Mrx and Mry are factored flexural resistance about the x-axis and y-axis, respectively (kip-in.). Tension members shall also meet the slenderness ratio requirements, l/r ≤ 140 for primary members subjected to stress reversal, l/r ≤ 200 for primary members not subjected to stress reversal, and l/r ≤ 240 for secondary members.

6.7.2

Axial Tensile Resistance For steel tension members, axial tensile resistance equations are smaller of yielding on the gross section and fracture on the net section as follows: Yielding in gross section:

Pr  y Pny  y Fy Ag

(AASHTO 6.8.2.1-1)

Fracture in net section:

Pr  u Pnu  u Fu AnU

(AASHTO 6.8.2.1-2)

where Pny is nominal tensile for yielding in gross section (kip); Pnu is nominal tensile for fracture in net section (kip); An is net cross section area (in.2); Fu is specified minimum tensile strength (ksi); U is reduction factor to account for shear leg; y is resistance factor for yielding of tension member = 0.95; u is resistance factor for fracture of tension member = 0.8.

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6.8

FATIGUE DESIGN There are two types of fatigue: load and distortion induced fatigue. The basic fatigue design requirement for load-induced fatigue is limiting live load stress range to fatigue resistance for each component and connection detail. Distortion-induced fatigue usually occurs at the web near a flange due to improper detailing. The design requirement for distortion-induced fatigue is to follow proper detailing practice to provide sufficient load paths. For load-induced fatigue consideration, the most common types of components and details in a typical I- girder are (AASHTO Table 6.6.1.2.3-1) listed in Table 6.8-1. Table 6.8-1 I-Section Flexural Design Equations (Strength Limit State) Type of Details

1 2 3 4

Base metal and weld metal at fullpenetration groove-welded splices Base metal at gross section of highstrength bolted slip-critical connections (bolt gusset to flange) Base metal at fillet-welded stud- type shear connectors Base metal at toe of transverse stiffener-to-flange and transverse stiffener-to-web welds

Category (AASHTO Table 6.6.1.2.3-1) B

B C C

Nominal fatigue resistance as shown in Figure 6.8-1 (AASHTO, 2012) is calculated as follows: For infinite fatigue life (N > NTH)

Fn   F TH

(AASHTO 6.1.2.5-1)

For finite fatigue life (N  NTH) 1

3 Fn    A  N

(AASHTO 6.6.1.2.5-2)

in which:

N = (365)(75)n(ADTT)SL

NTH 

A  F   TH  

3

Chapter 6 - Steel Design Theory

(AASHTO 6.6.1.2.5-3) (CA 6.6.1.2.3-2)

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Fatigue Resistance

Fn

Finite life -Fatigue II Lower Traffic Volume P9 Truck

Infinite Life - Fatigue I Higher Traffic Volume 1.75HL93 Truck

Fn   F TH  F n  

 A   N 

1/ 3

N TH 

A

Number of cycle N

F TH 3

Figure 6.8-1 Fatigue Resistance

where A is a constant depending on detail category as specified in AASHTO Table 6.6.1.2.5-1, and (F)TH is the constant-amplitude fatigue threshold taken from AASHTO Table 6.6.1.2.5-3. NTH is minimum number of stress cycles corresponding to constant-amplitude fatigue threshold, (F)TH, as listed in CA Table 6.6.1.2.3-2.

ADTTSL  p(ADTT )

(AASHTO 3.6.1.4.2-1)

where p is fraction of truck traffic in a single lane (AASHTO Table 3.6.1.4.2-1) = 0.8 for three or more lanes traffic, N is the number of stress-range cycles per truck passage = 1.0 for the positive flexure region for span > 40 ft. (CA Table 6.6.1.2.5-2). ADTT is the number of trucks per day in one direction averaged over the design life and is specified in CA 3.6.1.4.2. Fatigue I: ADTT = 2500, N   365 751.0  0.8  2500   0.5475 10   NTH 8

Fatigue II: ADTT = 20, N   365 751.0  0.8 20   438,000  NTH The nominal fatigue resistances for typical Detail Categories in an I-girder are summarized in Table 6.8-2.

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Table 6.8-2 Nominal Fatigue Resistance

Detail Category

Fatigue II

Fn   

120.0 44.0 44.0 11.0

B C C E

6.9

Constant –A ( 108 ) (ksi3)

1 A 3

 N

(ksi) 30.15 21.58 21.58 13.59

Fatigue I

Fn   F TH (ksi) 16.0 10.0 12.0 4.5

SERVICEABILITY STATES The service limit state design is intended to control the elastic and permanent deformations, which would affect riding ability. For steel girder, vehicular live load deflection may be limited to L/800 by AASHTO 2.5.2.6. Based on past successful practice of the overload check in the AASHTO Standard Specifications (AASHTO, 2002) to prevent the permanent deformation due to expected severe traffic loadings, AASHTO 6.10.4 requires that for SERVICE II load combination, flange stresses in positive and negative bending without considering flange lateral bending, ff shall meet the following requirements: For the top steel flange of composite sections

f f  0.95Rh Fyf

(AASHTO 6.10.4.2.2-1)

For the bottom steel flange of composite sections

ff 

fl  0.95Rh Fyf 2

(AASHTO 6.10.4.2.2-2)

For both steel flanges of noncomposite sections ff 

fl  0.8 Rh Fyf 2

(AASHTO 6.10.4.2.2-3)

For compact composite sections in positive flexure in shored construction, longitudinal compressive stress in concrete deck without considering flange lateral bending, fc, shall not exceed 0.6 f c where f c is minimum specified 28-day compressive strength of concrete (ksi). Except for composite sections in positive flexure satisfying D / t w  150 without longitudinal stiffeners, all sections shall satisfy

fc  Fcrw

Chapter 6 - Steel Design Theory

(AASHTO 6.10.4.2.2-4)

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6.10

CONSTRUCTIBILITY An I-girder bridge constructed in unshored conditions shall be investigated for strength and stability for all critical construction stages, using the appropriate strength load combination discussed in Chapter 3. All calculations shall be based on the non-composite steel section only. AASHTO Article 6.10.3 requires checking the following requirements: 

Compression Flange For discretely braced flange (AASHTO 6.10.3.2.1)

fbu  fl   f Rh Fyc fbu 

1 fl   f Fnc 3

f bu   f Fcrw

(AASHTO 6.10.3.2.1-1) (AASHTO 6.10.3.2.1-2) (AASHTO 6.10.3.2.1-3)

where fbu is flange stress calculated without consideration of the flange lateral bending (ksi); Fcrw is nominal bending stress determined by AASHTO 6.10.1.9.1-1 (ksi). For sections with compact and noncompact webs, AASHTO Equation 6.10.3.2.13 shall not be checked. For sections with slender webs, AASHTO Equation 6.10.3.2.1-1 shall not be checked when fl is equal to zero. For continuously braced flanges

fbu   f Rh Fyc 

(AASHTO 6.10.3.2.3-1)

Tension Flange For discretely braced flange

fbu  fl   f Rh Fyc

(AASHTO 6.10.3.2.1-1)

For continuously braced flanges

fbu   f Rh Fyt 

(AASHTO 6.10.3.2.3-1)

Web

Vu  vVcr

(AASHTO 6.10.3.3-1)

where Vu is the sum of factored dead loads and factored construction load applied to the non-composite section (AASHTO 6.10.3.3) and Vcr is shear buckling resistance (AASHTO 6.10.9.3.3-1).

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NOTATION A

=

fatigue detail category constant

ADTT

=

average daily truck traffic in one direction over the design life

ADTTS L

=

single lane ADTT life

Ag

=

gross cross section area (in.2)

An

=

net cross section area (in.2)

Arb

=

reinforcement area of bottom layer in concrete deck slab (in.2)

Art

=

reinforcement area of top layer in concrete deck slab (in.2)

bc

=

width of compression steel flange (in.)

bf

=

full width of the flange (in.)

bfc

=

full width of a compression flange (in.)

bft

=

full width of a tension flange (in.)

bs

=

width of concrete deck slab (in.)

bt

=

width of tension steel flange (in.)

C

=

ratio of the shear-buckling resistance to the shear yield strength

Cb

=

moment gradient modifier

D

=

web depth (in.)

Dcp

=

web depth in compression at the plastic moment (in.)

Dp

=

distance from the top of the concrete deck to the neutral axis of the composite sections at the plastic moment (in.)

Dt

=

total depth of the composite section (in.)

d

=

total depth of the steel section (in.)

do

=

transverse stiffener spacing (in.)

E

=

modulus of elasticity of steel (ksi)

Fcr

=

elastic lateral torisonal buckling stress (ksi)

Fcrw

=

nominal bend-buckling resistance of webs (ksi)

Fexx

=

classification strength specified of the weld metal

Fnc

=

nominal flexural resistance of the compression flange (ksi)

Fnt

=

nominal flexural resistance of the tension flange (ksi)

Fyc

=

specified minimum yield strength of a compression flange (ksi)

Fyf

=

specified minimum yield strength of a flange (ksi)

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Fyr

=

compression-flange stress at the onset of nominal yielding including residual stress effects, taken as the smaller of 0.7Fyc and Fyw but not less than 0.5Fyc (ksi)

Fyrb

=

specified minimum yield strength of reinforcement of bottom layers (ksi)

Fyrt

=

specified minimum yield strength of reinforcement of top layers (ksi)

Fys

=

specified minimum yield strength of a stiffener (ksi)

Fyt

=

specified minimum yield strength of a tension flange (ksi)

Fyw

=

specified minimum yield strength of a web (ksi)

Fyu

=

specified minimum tensile strength of steel (ksi)

fbu

=

flange stress calculated without consideration of the flange lateral bending (ksi)

fc

=

longitudinal compressive stress in concrete deck without considering flange lateral bending (ksi)

ff

=

flange stresses without considering flange lateral bending (ksi)

fs

=

maximum flexural stress due to Service II at the extreme fiber of the flange (ksi)

fsr

=

fatigue stress range (ksi)

=

minimum specified 28-day compressive strength of concrete (ksi)

I

=

moment of inertia of a cross section (in.4)

Iyc

=

moment of inertia of the compression flange about the vertical axis in the plane of web (in.4)

Iyt

=

moment of inertia of the tension flange about the vertical axis in the plane of web (in.4)

K

=

effective length factor of a compression member

L

=

span length (ft)

Lb

=

unbraced length of compression flange (in.)

Lp

=

limiting unbraced length to achieve RbRhFyc (in.)

Lr

=

limiting unbraced length to onset of nominal yielding (in.)

l

=

unbraced length of member (in.)

MAD

=

additional live load moment to cause yielding in either steel flange applied to the short-term composite section and can be obtained from the following equation (kip- in.)

MD1

=

moment due to factored permanent loads applied to the steel section alone (kip-in.)

f c

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MD2

=

moment due to factored permanent loads such as wearing surface and barriers applied to the long-term composite section (kip-in.)

Mp

=

plastic moment (kip-in.)

Mn

=

nominal flexural resistance of the section (kip-in.)

Mrx, Mry

=

factored flexural resistance about the x-axis and y-axis, respectively (kip-in.)

Mu

=

bending moment about the major axis of the cross section (kip-in.)

Mux, Muy

=

factored flexural moments about the x-axis and y-axis, respectively (kip-in.)

My

=

yield moment (kip-in.)

N

=

number of cycles of stress ranges

NTH

=

minimum number of stress cycles corresponding to constant-amplitude fatigue threshold, (F)TH

n

=

number of stress-range cycles per truck passage

Pu

=

factored axial load (kip)

Pr

=

factored axial resistance (kip)

p

=

fraction of truck traffic in a single lane

Q

=

slender element reduction factor

Rh

=

hybrid factor

Rb

=

web load-shedding factor

R

=

radius of gyration

rt

=

effective radius of gyration for lateral torsional buckling (in.)

SLT

=

elastic section modulus for long-term composite sections, respectively (in.3)

SNC

=

elastic section modulus for steel section alone (in.3)

SST

=

elastic section modulus for short-term composite section (in.3)

Sxt

=

elastic section modulus about the major axis of the section to the tension flange taken as Myt/Fyt (in.3)

tc

=

thickness of compression steel flange (in.)

tf

=

thickness of the flange (in.)

tfc

=

thickness of a compression flange (in.)

tft

=

thickness of a tension flange (in.)

tt

=

thickness of tension steel flange (in.)

tw

=

thickness of web (in.)

ts

=

thickness of concrete deck slab (in.)

Vcr

=

shear-buckling resistance (kip)

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Vn

=

nominal shear resistance (kip)

Vp

=

plastic shear force (kip)

Vu

=

factored shear (kip)

f

=

slenderness ratio for compression flange = bfc/2tfc

pf

=

limiting slenderness ratio for a compact flange

rf

=

limiting slenderness ratio for a noncompact flange

(F)TH

=

constant-amplitude fatigue threshold (ksi)

(F)n

=

fatigue resistance (ksi)

f

=

resistance factor for flexure = 1.0

v

=

resistance factor for shear = 1.0

c

=

resistance factor for axial compression = 0.9

u

=

resistance factor for tension, fracture in net section = 0.8

y

=

resistance factor for tension, yielding in gross section = 0.95

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Unit (6th Edition), American Association of State Highway and Transportation Officials, Washington, DC. 2. AASHTO, (2002). Standard Specifications for Highway Bridges, 17th Edition, American Association of State Highway and Transportation Officials, Washington, DC. 3. AISC, (2010). Specifications for Structural Steel Buildings, ANSI/AISC 360-10, American Institute of Steel Construction, Chicago, IL. 4. Caltrans, (2014). California Amendment to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 5. FHWA, (1989). Technical Advisory T5140.22, Federal Highway Administration, Washington, DC. 6. White, D. W., and Grubb, M. A., (2005). “Unified Resistance Equation for Design of Curved and Tangent Steel Bridge I-Girders.” Proceedings of the 2005 TRB Bridge Engineering Conference, Transportation Research Board, Washington, DC.

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CHAPTER 7 POST-TENSIONED CONCRETE GIRDERS TABLE OF CONTENTS 7.1

INTRODUCTION ........................................................................................................... 7-1 7.1.1

General ............................................................................................................ 7-1

7.1.2

Basic Concepts ................................................................................................ 7-2

7.2

MATERIAL PROPERTIES ............................................................................................ 7-4

7.3

GIRDER LAYOUT AND STRUCTURAL SECTION .................................................. 7-6

7.4

PRESTRESSING CABLE LAYOUT ............................................................................. 7-7

7.5

PRESTRESS LOSSES FOR POST-TENSIONING ....................................................... 7-9 7.5.1

Instantaneous Losses ..................................................................................... 7-10

7.5.2

Long Term Loss ............................................................................................ 7-15

7.6

SECONDARY MOMENTS AND RESULTING PRESTRESS LOSS........................ 7-18

7.7

STRESS LIMITATIONS .............................................................................................. 7-18 7.7.1

Prestressing Tendons..................................................................................... 7-18

7.7.2

Concrete ........................................................................................................ 7-19

7.8

STRENGTH DESIGN .................................................................................................. 7-21

7.9

DEFLECTION AND CAMBER ................................................................................... 7-21

7.10

POST-TENSIONING ANCHOR DESIGN .................................................................. 7-24

7.11

DESIGN PROCEDURE................................................................................................ 7-25

7.12

DESIGN EXAMPLE .................................................................................................... 7-31 7.12.1

Prestressed Concrete Girder Bridge Data ..................................................... 7-31

7.12.2

Design Requirements .................................................................................... 7-32

7.12.3

Select Girder Layout and Section ................................................................. 7-32

7.12.4

Determine Basic Design Data ....................................................................... 7-34

7.12.5

Design Deck Slab and Soffit ......................................................................... 7-35

7.12.6

Select Prestressing Cable Path ...................................................................... 7-36

7.12.7

Post Tensioning Losses ................................................................................. 7-46

7.12.8

Cable Path Eccentricities............................................................................... 7-53

7.12.9

Moment Coefficients..................................................................................... 7-56

7.12.10

Gravity Loads................................................................................................ 7-60

7.12.11

Determine the Prestressing Force.................................................................. 7-63

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7.12.12

Determine the Required Concrete Strength .................................................. 7-67

7.12.13

Design of Flexural Resistance ....................................................................... 7-77

7.12.14

Design for Shear............................................................................................ 7-90

7.12.15

Calculate the Prestressing Elongation ......................................................... 7-105

NOTATION ............................................................................................................................. 7-109 REFERENCES ......................................................................................................................... 7-114

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CHAPTER 7 POST-TENSIONED CONCRETE GIRDERS 7.1

INTRODUCTION

7.1.1

General Post-tensioned concrete box girders are widely used in the highway bridges in California. Figure 7.1-1 shows the San Luis Rey River Bridge – a typical cast-in-place post-tensioned (CIP/PT) concrete box girder bridge.

Figure 7.1-1 San Luis Rey River Bridge: A Concrete Box CIP/PT Bridge Basic concepts, definitions and assumptions are first discussed in this Chapter. An example problem with “longhand” solution is then worked through to illustrate typical design procedure.

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7.1.2

Basic Concepts Post tensioning is one of methods of prestressing concrete. The concrete members are cast first. Then after the concrete has gained sufficient strength, tendons (strands of high strength steel wire) are inserted into preformed has ducts and tensioned to induce compressive stresses in the expected tensile stress regions of the member. Concrete must be free to shorten under the precompression. The strands are then anchored and a corrosion protection such as grout or grease, is installed (Gerwick, 1997). Before further discussing prestressing, we should compare it with conventionally reinforced concrete. Prior to gravity loading, the stress level in conventional reinforced concrete is zero. The reinforcing steel is only activated by the placement of the gravity load. The concrete and reinforcing steel act as a composite section. However, once the tensile capacity of the concrete surrounding the longitudinal reinforcement has been surpassed, the concrete cracks. Prestressed concrete activates the steel prior to gravity loading through prestressing the reinforcement. This prevents cracking at service loads in prestressed concrete. Prestressed concrete utilizes high strength materials effectively. Concrete is strong in compression, but weak in tension. High tensile strength of prestressing steel and high compressive strength of concrete can be utilized more efficiently by pretensioning high strength steel so that the concrete remains in compression under service loads activated while the surrounding concrete is compressed. The prestressing operation results in a self-equilibrating internal stress system which accomplishes tensile stress in the steel and compressive stress in the concrete that significantly improves the system response to induced service loads (Collins and Mitchell, 1997). The primary objectives of using prestressing is to produce zero tension in the concrete under dead loads and to have service load stress less than the cracking strength of the concrete along the cross section. Thus the steel is in constant tension. Because of this the concrete remains in compression under service loads throughout the life of the structure. Both materials are being activated and used to their maximum efficiency. Figure 7.1-2 shows elastic stress distribution for a prestressed beam after initial prestressing.

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FC ( Pj )



FC ( Pj e)c1 *

FC ( Pj )

Ig

Ag

Ag

FC ( Pj )

FC ( Pj e)c 2

FC ( Pj )

Ag

Ig

Ag





FC ( Pj e)c1 * Ig

FC ( Pj e)c2 Ig

Note: *Component of Equation is negative because c is on opposite side of center of gravity from the tendon. Tension is denoted as negative (-), compression is denoted as positive (+)

Figure 7.1-2 Elastic Stresses in an Uncracked Prestress Beam. Effects of Initial Prestress by Component (Nilson, 1987) The stress at any point of the cross-section can be expressed as: f pe 

FC ( Pj ) Ag

 FC ( Pj e) y MCs ( Pj ) y     Ig Ig  



(7.1-1)

Where: Ag

=

gross area of section (in.2)

e

=

eccentricity of resultant of prestressing with respect to the centroid of the cross section. Always taken as a positive (ft)

FC

=

force coefficient for loss

fpe

=

effective stress in the prestressing steel after losses (ksi)

Ig

=

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (in.4)

Pj

=

force in prestress strands before losses (ksi)

MCs =

secondary moment force coefficient for loss (ft)

y

distance from the neutral axis to a point on member cross-section (in.)

=

The prestressing force effect is accomplished by two components of the general equation shown above as Equation 7.1-1. The first component is uniform compression stress due to the axial prestressing force. The second component is the bending stress caused by eccentricity of the prestressing steel with respect to the center of gravity of the cross section. This creates a linear change in stress throughout the beam cross

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section (Figure 7.1-2). It is noted that the distance from the neutral axis to the fiber in question y, (y is the general term, c1 and c2 which are more specific terms shown in Figure 7.1-2), may result in a negative value for the bending part of the equation. It is possible the prestressing force will create tension across the center of gravity from the tendon, and therefore part of the beam section may be in tension prior to applying load. The use of prestressed concrete has its advantages and limitations. Some limitations are its low superstructure ductility, the need for higher concrete compressive strengths, and larger member sizes to accommodate ducts inside the girders. Post-tension box girder superstructures are commonly used due to their low costs, their performance throughout the life of the structure, and contractor’s experience with the structure type. Post-tensioning also allows for thinner superstructures. A continuous superstructure increases the stiffness of the bridge frame in the longitudinal direction and gives the designer the option to fix the columns to the superstructure, reducing foundation costs.

7.2

MATERIAL PROPERTIES At first glance, prestressed concrete and reinforced concrete make use of the same two core materials: concrete and steel. However, the behavior of the materials vary due to usage. Conventional concrete structures use deformed bars for reinforcement. Most prestressed applications use tightly wrapped, low-relaxation (lo-lax) seven-wire strand. As shown in Figure 7.2-1, the stress strain curves for those steel are quite different.

Figure 7.2-1 Stress-Strain Curves of Mild Steel (Deformed Reinforcing Bars) and Prestress Steel (7-Wire Strand) (Collins and Mitchell, 1997)

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Table 7.2-1 shows the steel material properties for ASTM A706 Grade 60 and ASTM A416 PS Strand Grade 270. The mild reinforcement steel (ASTM A706 Grade 60) used for reinforced concrete has a much lower yield strength and tensile strength than the prestressing strands (ASTM A416 PS Strand Grade 270). Prestressing steel shall be high strength and possesses superior material properties. This enables a smaller quantity of steel to be used to support the bridge. Higher strength steel is also used because the ratio of effective prestess (prestress force after losses in force) to initial prestress (prestress force before losses in force) of high strength steel is much higher than that of mild steel (Figure 7.2-2). This is because losses, which will be discussed below, consume a large percentage of the strain in the elastic range of the mild steel, but a small portion of the prestressed steel. Table 7.2-1 Steel Material Properties for Reinforced and Prestressed Concrete Grade 60 ASTM A706

Grade 270 ASTM A416 PS Strand

fy

60 ksi

250 ksi

fu

80 ksi

270 ksi

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Jacking Stress – Stress in the strand

Effective Prestressstress remaining in strand after losses

Loss of pre-strain due to instantaneous and long term losses Initial Prestress – Stress in steel before any losses

Figure 7.2-2 Loss Effects Comparison Between Prestressing Strand and Mild Reinforcement (Based inpart on Portland Cement Association 2001 Bridge Professors’ Seminar Chicago, IL)

7.3

GIRDER LAYOUT AND STRUCTURAL SECTION Section design is a very important tenant of structure design. An efficient section maximizes the ability of a structure to carry applied loads while minimizing selfweight. Basic mechanics of materials theory shows that the further away the majority of a material lies from the centroid of the shape, the better that shape is at resisting moment. A shape such as a basic “I” is perfect for maximizing flexural strength and minimizing weight. The placement of I-girders side by side results in a box; which is easier to construct and has seismic advantages over individual I-girders. Determination of the typical section of a bridge has been made a simple process. Creation of a typical section begins with the calculation of structure depth for a given span length. Table 7.3-1 lists (AASHTO, 2012) minimum structural depth for various structural spans.

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Table 7.3-1 Traditional Minimum Depth for Constant Depth Superstructures (AASHTO Table 2.5.2.6.3-1, 2012) Minimum Depth (Including Deck)

Material

Reinforced Concrete

Prestressed Concrete

Steel

Superstructure Type Slabs with main reinforcement parallel to traffic T-Beams Box Beams Pedestrian Structure Beams Slabs CIP Box Beams Precast I-Beams Pedestrian Structure Beams Adjacent Box Beams Overall Depth of Composition I-Beam Depth of I-Beam Portion of Composite I-Beam Trusses

When variable depth members are uses, values may be adjusted to account for changes in relative stiffness of positive and negative moment sections Simple Spans Continuous Spans 1.2 (𝑆 + 10) 30

𝑆 + 10 ≥ 0.54 ft 30

0.070L 0.060L 0.035L

0.065L 0.055L 0.033L

0.030L > 6.5 in. 0.045L 0.045L 0.033L

0.027L > 6.5 in. 0.040L 0.040L 0.030L

0.030L 0.040L

0.025L 0.032L

0.033L

0.027L

0.100L

0.100L

In order to use MTD 10-20 (Caltrans, 2008), a simple rule of thumb is girder spacing for a prestressed box girder should not exceed twice the superstructure depth. A larger girder spacing may require a customized deck and soffit slab design and may result in a larger web thickness. MTD 10-20 (Caltrans, 2008) provides predetermined soffit and deck thickness based on girder to girder spacing as well as overhang length.

7.4

PRESTRESSING CABLE LAYOUT To induce compressive stress along all locations of the bridge girder, the prestressing cable path must be raised and lowered along the length of the girder. A typical continuous girder is subjected to negative moments near fixed supports, and positive moments near mid-span. As Equation 7.1-1 shows, eccentricity determines the stress level at a given location on the cross-section. In order to meet the tension face criteria the location of the prestressing cable path will be high (above the neutral axis) at fixed supports, low (below the neutral axis) at midspans, and at the centroid of the section near simply supported connections (Figure 7.4-1). The shape of the cable path is roughly the same as the opposite sign of the dead load moment diagram shown in Figure 7.4-2.

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Figure 7.4-1 Typical Prestressing Path for a Two-Span Bridge

30000

Opposite sign Service Level Moment (k-ft)

25000 20000 15000 10000 5000 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-5000 -10000 -15000 -20000

Location on Bridge from Abutment 1

Figure 7.4-2 Opposite Sign of Dead Load Moment Diagram Determining the most efficient possible final pre-stress cable layout is an iterative process. This requires using Bridge Design Aids (BDA) Sections 11-13 through 11-18 (Caltrans, 2005) to determine a best “guess” initial prestressing force (Pjack) as a function of deck area, span length, and span configuration. Using tables and charts provided in MTD 11-28 (Caltrans, 2010), the designer can then locate critical points along the cable path. The highest point should occur at the locations of highest negative moments, on our example bridge that would be at the bents. At its highest, the duct will fit in just below the bottom transverse mat of steel in the deck. The lowest point will occur near mid-span, and is limited by the location of the top mat of transverse soffit steel.

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7.5

PRESTRESS LOSSES FOR POST-TENSIONING Throughout the life of a prestressed concrete girder, the initial force applied to the pre-stress tendons significantly decreases. This decrease in the force is called loss. Loss of stress in a girder can revert a location previously in compression to tension, or increase a tension stress. This may be very dangerous when stress in concrete is near its stress limit. Because of the significant impact to the structure of these losses, losses must be quantified and accounted for in design. In Caltrans practice, coefficients are usually used to estimate the reduction factor in initial force to find a final prestressing force. These coefficients are called force (Equation 7.5-1) and moment coefficients (Equation 7.5-2). Both coefficients are used to determine a jacking force, as most losses are both functions of and dependant of jacking force. These force coefficients are given as the sum of lost force of each component of loss divided by the allowable stress in the tendon (modification of AASHTO, 2012, Equations 5.9.5.1-1, 5.9.5.1-2).

FC pT  (1 

 f ) i

f ps

(7.5-1)

where: FCpT

=

force coefficient for loss

fi

=

change in force in prestressing tendon due to an individual loss (ksi)

fps

=

average stress in prestressing steel at the time for which the nominal resistance is required (ksi) (5.7.3.1.1-1)

MCP  ( FC pT )(ex )

(7.5-2)

where: MCp

=

primary moment force coefficient for loss (ft)

FCpT

=

total force coefficient for loss

ex

=

eccentricity as a function of x along parabolic segment (ft)

The force coefficient is defined as one at the jacking location and begins decreasing towards zero to the point of no movement. The point of no movement is a finite point of the strand that does not move when jacked and is defined as the location where internal strand forces are in equilibrium. For single-end post tensioning, the point of no movement is at the opposite anchorage from stressing. For two-end tensioning the location is where the movement in one direction is countered by movement from the other direction, and is generally near the middle of the frame. Force coefficients are determined at each critical point along the girder. The product of the force coefficients and strand eccentricities (e) are called moment coefficients. The coefficients determined from the locked in moments at fixed supports

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are used to convert initial strand moment resistant capacities into capacities after losses, or final capacities.

7.5.1

Instantaneous Losses There are two types of losses: instantaneous and long term. The instantaneous losses are due to anchor set, friction, and elastic shortening. Instantaneous losses are bridge specific, yet still broad enough to be estimated in user friendly equations. Therefore a lump sum value is not used and a bridge specific value is calculated. Given below are three different types of instantaneous losses.

7.5.1.1

Anchor Set Loss Anchor Set is caused by the movement of the tendon prior to seating of the anchorage gripping device. This loss occurs prior to force transfer between wedge (or jaws) and anchor block. Anchor set loss is the reduction in strand force through the loss in stretched length of the strand. Once a force is applied to the strands, the wedges move against the anchor block, until the wedges are “caught”. Because of the elasticity of the strands, this movement will cause a loss in strain, stress, and force. This movement and the resulting loss of force prior to being “caught” is the anchor set loss. The force necessary to pull the movement out, will not be captured as the effective force. Even though the size of the slip is small, it still manifests as a reduction in prestressing force. AASHTO 5.9.5.2.1 suggests a common value for anchor set as 3/8 inch. This anchor set loss represents the amount of slip in Caltrans approved anchorage systems. Equation 7.5.1.1-1 puts the Anchor Set into a more familiar change in force and force coefficient form.

FC pA  x pA 

f pA 2(f L )( x pA )  f ps L( f ps )

E p ( Aset ) L 12f L

(7.5.1.1-1)

(7.5.1.1-2)

where: FCpA

=

force coefficient for loss from anchor set

xpA

=

influence length of anchor set (ft)

Ep

=

modulus of elasticity of prestressing

Aset

=

anchor set length (in.)

L

=

distance to a point of known stress loss (ft)

fL

=

friction loss at the point of known stress loss (ksi)

fpA

=

jacking stress lost in the P/S steel due to anchor set (ksi)

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 7.5.1-1 Anchorage System for Multi-Strand Tendon (Collins and Mitchell, 1997) 7.5.1.2

Friction Loss Friction loss is another type of instantaneous loss, which occurs when the prestressing tendons get physically caught on the ducts. This is a significant loss of force on non-linear prestressing paths because of the angle change of the ducts. Friction loss has two components: curvature and wobble frictional losses. Modified Equation AASHTO 5.9.5.2.2b-1 results in Equation 7.5.1.2-1, the equation used to obtain friction losses. Curvature loss occurs when some fraction of the jacking force is used to maneuver a tendon around an angle change in a duct. An example would be: as a tendon is bending around a duct inflection point near a pier or bent, the bottom of the tendon is touching (and scraping) the bottom of the duct. This scraping of the duct is loss of force via friction.

FC pF 

f pF f pj

 e  ( Kx μα)

(7.5.1.2-1)

where: e

=

e is the base of Napierian logarithms

FCpF

=

force coefficient for loss from friction

fpj

=

stress in the prestressing steel at jacking (ksi)

K

=

wobble friction coefficient (per ft of tendon)

x

=

general distance along tendon (ft)



 total angular change of prestressing steel path from jacking end to a point under investigation (rad)

fpF

=

change in stress due to friction loss



=

coefficient of friction

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Table 7.5-1 Provides Wobble Friction Coefficient and Coefficient of Friction as Specified in the California Amendments (Caltrans, 2014). Table 7.5-1 Friction Coefficient K and Coefficient of Friction  Type of Steel Wire or strand

High-strength bars

Type of Duct

K (1/ft)

μ

Rigid and semi-rigid galvanized metal sheathing Tendon Length: < 600 ft 600 ft < 900 ft 900 ft < 1200 ft > 1200 ft Polyethylene Rigid steel pipe deviators for external tendons Galvanized metal sheathing

0.0002

0.15-0.25

0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

0.15 0.20 0.25 >0.25 0.23 0.25

0.0002

0.30

Figure 7.5.1-2 Wobble Friction Losses (Collins & Mitchell, 1997) Wobble losses result from unintended angle changes of the tendon along the length of the cable path (Figure 7.5.1-2). These losses depend on the properties of the duct such as rigidity, diameter, support spacing, and type of duct. Wobble losses are the accumulation of the wobble coefficient over the length of the cable path.

Chapter 7 - Post-Tensioning Concrete Girders

7-12

Force coefficient (decimal % of Pj)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

FCpF = 1.000 fpF = 202.50 ksi

Point of no Movement (2-end)

FCpF = 1.000 fpF = 202.50 ksi

FCpF = 0.879 fpF = 178.00 ksi 2nd End P/S Steel Stress

L1 = 219.6 ft FCpF = 0.773 fpF = 156.53 ksi

L2 = 192.4 ft

1st End P/S Steel Stress Length Along Girder (ft) Figure 7.5.1-3 Two-End Stressing Versus One-End Stressing Friction losses over a long girder begin to add up to a high percentage of the prestressing force. High friction losses can be counteracted by using two-end stressing. As stated above, two-end stressing moves the point of no movement from the anchored end to a point close to the middle of the frame. By jacking the end that was previously anchored, friction stress that was building up in the tendons between the point of no movement and the second end, is effectively pulled out thus reducing the friction losses between the second end and the point of no movement. Figure 7.5.1-3 shows the difference in stress when using two-end stressing instead of one for the design example in Section 7.12.

7.5.1.3

Elastic Shortening When the pre-stressing force is applied to a concrete section, an elastic shortening of the concrete takes place simultaneously with the application of the pre-stressing force to the pre-stressing steel. It is caused by the compressive force from the tendons pulling both anchors of the concrete towards the center of the frame. Therefore, the distance between restraints has been decreased. Because of the elastic nature of the strand decreasing the distance between restraints after the pre-stressing force has been applied, thus reducing the strain, stress, and force levels in the tendons. The equations for elastic shortening in pre-tensioned (such as precast elements) members are shown in AASHTO Equation 5.9.5.2.3a-1. f pES 

Ep Ect

f cgp

(AASHTO 5.9.5.2.3a-1)

The equations for elastic shortening in post-tensioned members other than slabs are shown in Equation AASHTO 5.9.5.2.3b-1.

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

f pES 

N  1 Ep 2 N Ect

fcgp

(AASHTO 5.9.5.2.3b-1)

where: Ect

=

modulus of elasticity of concrete at transfer or time of load application (ksi)

Ep

=

modulus of elasticity of prestressing tendons (ksi)

N

=

number of identical prestressing tendons

fcgp

=

concrete stress at the center of gravity of prestressing tendons, that results from the prestressing force at either transfer or jacking and the self-weight of the member at sections maximum moment (ksi) 

fpES =

change in stress due to elastic shortening loss

The California Amendments to the AASHTO LRFD specify that as the number of tendons increase, the first fractional term converges to 1/2, and the formula is simplified as follows: f pES  0.5

Ep Ect

f cgp

f cgp  f g  f ps 

(CA Amendments 5.9.5.2.3b-1)

M DL e Ig

(

Pj Ag



Pj e 2 Ig

)

(7.5.1.3-1)

where: Ag

=

gross area of section (in.2)

e

=

eccentricity of the anchorage device or group of devices with respect to the centroid of the cross section. Always taken as a positive (in.)

fg

=

stress in the member from dead load (ksi)

fps

=

average stress in prestressing steel at the time for which the nominal resistance is required (ksi)

Ig

=

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (in.4)

MDL =

dead load moment of structure (kip-in.)

Pj

force in prestress strands before losses (ksi)

=

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Turning this into a force coefficient results in: FC pES 

f pES f pi

 0.5

E p fcgp Ect f pi

where: FCpES = force coefficient for loss from elastic shortening

7.5.2

Long Term Loss Long term, time-dependent losses are the losses of presstress force in the tendon during the life of the structure. When using long term losses on post-tensioned members it is acceptable to use a lump sum value (CA Amendments to AASHTO 5.9.5.3) in lieu of a detailed analysis. In the CA Amendments to AASHTO 2012, this value is 20 ksi. When completing a detailed analysis, long term losses are the combination of the following three losses.

7.5.2.1

Shrinkage of Concrete The evaporation of free water in concrete causing the concrete to lose volume is a process known as shrinkage. The amount of shrinkage, and therefore the amount of loss caused by shrinkage, is dependent on the composition of the concrete and the curing process (Libby, 1990). Empirical equations for calculating shrinkage have been developed which rely on concrete strength, and relative humidity of the region where the bridge will be placed. Combining and modifying AASHTO 2012 Equations 5.9.5.3-1, 5.9.5.3-2, and 5.9.5.3-3 results in Equation 7.5.2.1-1.

f pSR  12.0(1.7  .01H )

5 (1  fci )

(7.5.2.1-1)

where:

7.5.2.2

fpSR

=

change in stress due to shortening loss

H

=

average annual ambient mean relative humidity (percent)

fci

=

specified compressive strength of concrete at time of initial loading or prestressing (ksi); nominal concrete strength at time of application of tendon force (ksi)

Creep Creep is a phenomenon of gradual increase of deformation of concrete under sustained load. There are two types of creep, drying creep and basic creep. Drying creep is affected by moisture loss of the curing concrete and is similar to shrinkage, as it can be controlled by humidity during the curing process. Basic creep is the constant stress of the post-tensioning steel straining the concrete. Creep is determined by relative humidity at the bridge site, concrete strengths, gross area of concrete, area of

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7-15

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

prestressing steel, and initial prestressing steel. Combining and modifying AASHTO 2012 Equations 5.9.5.3-1, 5.9.5.3-2, and 5.9.5.3-3 results in Equation 7.5.2.2-1. f pCR  10.0

f pi Aps Ag

(1.7  .01H )

5 (1  f ci )

(7.5.2.2-1)

where:

7.5.2.3

fpCR

=

change in stress due to creep loss

fpi

=

prestressing steel/stress immediately prior to transfer (ksi)

Aps

=

area of prestressing steel (in.2)

Ag

=

gross area of section (in.2)

H

=

average annual ambient mean relative humidity (percent)

fci

=

specified compressive strength of concrete at time of initial loading or prestressing (ksi); nominal concrete strength at time of application of tendon force (ksi)

Relaxation of Steel The relaxation of steel is a phenomenon of gradual decrease of stress when the strain is held constant over time. As time goes by, force is decreasing in the elongated steel. Relaxation losses are dependent on how the steel was manufactured (Figure 7.5.2.3-1). The manufacturing processes used to create prestressing strands result in significant residual stresses in the strand.

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 7.5.2.3-1 7-Wire Strand Production Method (Collins & Mitchell, 1997) The steel can be manufactured to reduce relaxation as much as possible; this steel is called low relaxation (lo-lax). Lo-lax is generally the type of prestressing steel used in Caltrans post-tensioned girder bridges. A lo-lax strand goes through the production of high strength steel (patenting, cold drawing, stranding) and is then heated and cooled under tension. This process removes residual stresses and reduces the time dependant losses due to the relaxation of the strand. AASHTO 2012 allows for the use of lumpsum values. (AASHTO, 2012. 5.9.5.3-1). These are given as 2.4 ksi for lo-lax and 10.0 ksi for stress relieved steel.

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.6

SECONDARY MOMENTS AND RESULTING PRESTRESS LOSS Another type of loss exists based on the frame configuration and support boundary conditions. A continuous prestressed flexural member which is free to deform (i.e. unrestrained by its supports), will deform axially and deflect from its original shape (Libby, 1990). If the prestress reactions are restrained by the supports, moments and shear forces are created as a result of the restraint. Distortions due to primary prestress moments generate fixed end moments at rigid supports. These fixed end moments are always positive, due to the geometry of the cable path, and always enhance the effects of prestressing. These locked in secondary moments decrease the effect of prestressing by lowering the effective prestressing force. For statically indeterminate concrete flexural members, loss of prestress can be tabulated by using the moment distribution method; accounting for eccentricities, and curvature of tendons. Secondary moments vary linearly between supports.

7.7

STRESS LIMITATIONS

7.7.1

Prestressing Tendons Tensile stress is limited to a portion of the ultimate strength to provide a margin of safety against tendon fracture or end-anchorage failures. Stress limits are also used to avoid inelastic tendon deformation, and to limit relaxation losses. Table 7.7.1-1 provides the stress limitations for prestresseing tendons as specified in the California Amendments (Caltrans, 2014). Those limitations can be increased if necessary in long bridges where losses are high.

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Table 7.7.1-1 Stress Limitations for Prestressing Tendons CA Amendments Table CA5.9.3-1, (Caltrans, 2014) Tendon Type Stress-Relieved Strand and Plain HighStrength Bars Pretensioning

Condition

Low Relaxation Strand

Deformed HighStrength Bars

Prior to Seating : short-term fpbt may be allowed

0.90fpy

0.90 fpy

0.90fpy

Immediately prior to transfer (fpbt)

0.70fpu

0.70fpu



At service limit state after all losses (fpe)

0.80fpy

0.80fpy

0.80fpy

0.90fpy

0.90fpy

0.90fpy

0.75fpu

0.75fpu

0.75fpu

0.70fpu

0.70fpu

0.70fpu

0.70fpu

0.70fpu

0.70fpu

0.80fpy

0.80fpy

0.80fpy

Post-tensioning Prior to Seating short term fpbt may be allowed Maximum Jacking Stress: short-term fpbt may be allowed At anchorages and couplers immediately after anchor set Elsewhere along length of member away from anchorages and complers immediately after anchor set At service limit state after losses (fpe)

7.7.2

Concrete Stress in concrete varies at discrete stages within the life of an element. These discreet stages vary based on how the element is loaded, and how much pre-stress loss the element has experienced. The stages to be examined are the Initial Stage: Temporary Stresses Before Losses, and the Final Stage: Service Limit State after Losses, as defined by AASHTO 2012. The prestress force is designed as the minimum force required to meet the stress limitations in the concrete as specified in AASHTO Article 5.9.4 (AASHTO, 2012). During the time period right after stressing, the concrete in tension is especially susceptible to cracking. This is before losses occur when prestress force is the highest and the concrete is still young. At this point the concrete has not completely gained strength. Caltrans project plans should show an initial strength of concrete that must be met before the stressing operation can begin. This is done to indicate a strength required to resist the post tensioning during the concrete’s vulnerable state. During this initial temporary state, the Table 7.7.2-1 (AASHTO, 2012) allows for a higher tensile stress limit and the concrete is allowed to crack. The concrete is allowed to crack because as the losses reduce the high tension stress and the young concrete strengthens, the crack widths will reduce. Then the high axial force from prestressing will pull the cracks closed.

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Table 7.7.2-1 Temporary Tensile Stress Limits in Prestressed Concrete Before Losses in Non-Segmental Bridges (AASHTO Table 5.9.4.1.2-1, 2012)   



Location In precompressed tensile zone without bonded reinforcement In area other than the precompressed tensile zone and without bonded reinforcement In areas with bonded reinforcement (reinforcing bars or prestressing steel) sufficient to resist the tensile force in the concrete computed assuming an uncracked section, where reinforcement is proportioned using a stress of 0.5 fy, not to exceed 30 ksi. For handling stresses in prestressed piles

Stress Limit N/A

0.0948

0.24 0.158

f ci

0.2(ksi)

f ci f ci

(ksi) (ksi)

The final stage of a bridge’s lifespan is known as the “in place” condition. Stresses resisted by concrete and prestressing steel in this condition are from gravity loads. At the service limit, the bridge superstructure concrete should not crack. The code provides for this by setting the stress limit (Table 7.7.2-2) to be less than the tensile strength of the concrete. Under permanent loads tension is not allowed in any concrete fiber (Caltrans, 2014). Table 7.7.2-2 Tensile Stress Limits on Prestressed Concrete at Service Limit State, After Losses, Fully Prestressed Components (CA Amendements Table 5.9.4.2.2-1, 2014) Location Precompresssed Tensile Zone Bridges, Assuming Uncracked Sections—components with bonded prestressing tendons or reinforcement, subjected to permanent loads only.

Stress Limit

No tension

Tension in the Precompressed Tensile Zone Bridges, Assuming Uncracked Sections  For components with bonded prestressing tendons or reinforcement that are subjected to not worse than moderate corrosion conditions, and/or are located in Caltrans’ Environmental Areas I or II.  For components with bonded prestressing tendons or reinforcement that are subjected to severe corrosive conditions, and/or are located in Caltrans’ Environmental Area III.  For components with unbonded prestressing tendons.

Chapter 7 - Post-Tensioning Concrete Girders

0.19

f c (ksi)

0.0948 f c (ksi)

No tension

7-20

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Note that Caltrans’ Environmental Areas I and II correspond to Non-Freeze Thaw Area, and Environmental Area III corresponds to Freeze Thaw Area, respectively. Because concrete is strong in compression, the code-defined compression limits are much higher than corresponding tension limits. Compression limits as shown in Table 7.7.2-3 are in place to prevent the concrete from crushing. The code establishes a portion of the concrete strength to resist both gravity loads and compression from the prestressing tendons. Because compression limits are dependent on stage, like tension limits, careful consideration must not only be taken for force level in the steel but for the loading conditions as well. Table 7.7.2-3 Compressive Stress Limits in Prestressed Concrete at Service Limit State, Fully Prestressed Components (AASHTO Table 5.9.4.2.1-1, 2012) Location In other than segmentally constructed bridges due to the sum of effective prestress and permanent loads

Stress Limit 0.45 f c (ksi)



In segmentally constructed bridges due to the sum of effective prestress and permanent loads

0.45 f c (ksi)



Due to the sum of effective prestress, permanent loads, and transient loads and during shipping and hanling

0.60 w f c (ksi)



7.8

STRENGTH DESIGN Prestressing force and concrete strength are usually determined for the Service Limit States, while mild steel are determined for Strength Limit States. Flexural and shear design are discussed in detail in Chapter 6.

7.9

DEFLECTION AND CAMBER Deflection is a term that is used to describe the degree to which a structural element is displaced under a load. The California Amendments to the AASHTO 2012 code defines camber as the deflection built into a member, other than prestressing, in order to achieve a desired grade. Camber is the physical manifestation of removing deflection from a bridge by building that deformation into the initial shape. This is done in the long term to give the superstructure a straight appearance, which is more pleasing to the public and also for drainage purposes. There are two types of deflections. Instantaneous deflections consider the appropriate combinations of dead load, live load, prestressing forces, erection loads, as well as instantaneous prestress losses. All deflections are based on the stiffness of the structure versus the stiffness of the supports. This stiffness is a function of the Moment of Inertia (Ig or Ie) and Modulus of Elasticity (E). AASHTO 2012 (Equation 5.7.3.6.21) has developed an equation to define Ie based on Ig. With Ie obtained an instantaneous

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deflection can be determined by using a method such as virtual work, or a design software such as CT-BRIDGE. 3   M 3  M  I e   cr  I g  1   cr   I cr  I g   M a    Ma 

(AASHTO 5.7.3.6.2-1)

where: Ie

=

effective moment of inertia (in.4)

Mcr

=

cracking moment (kip-in.)

Ma

=

maximum moment in a member at the stage which the deformation is computed (kip-in.)

Ig

=

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (in.4)

Icr

=

moment of inertia of the cracked section, transformed to concrete (in.4)

M cr  f y

Ig yt

(AASHTO 5.7.3.6.2-2)

where: fy

=

yield strength of mild steel (ksi)

yt

=

distance from the neutral axis to the extreme tension fiber (in.)

The primary function of calculating long-term deflections is to provide a camber value. Permanent loads will deflect the superstructure down, and give the bridge a sagging appearance. Prestressing force and eccentricity cause upward camber in superstructures. When calculating long-term deflections of a bridge, creep, shrinkage, and relaxation of the steel must be considered. This is done by multiplying the instantaneous deflection (deflection caused by DC and PS case) by code defined factor (CA 5.7.3.6.2). This product of instantaneous deflection and long term coefficient is the long term deflection of the bridge. The opposite sign of these deflections is what is placed on the project plans of new structures as camber. Generally, negative cambers (upward deflection) is ignored.

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 7.9-1 Expressions for Deflections Due to Uniform Load and Camber for Hand Checks (Collins and Mitchell, 1997)

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.10

POST-TENSIONING ANCHOR DESIGN The abrupt termination of high force strands within the girder generates a large stress ahead of the anchorage. Immediately ahead of the girder are bursting stresses, and surrounding the anchorages are spalling stresses. The code specifies that for ease of design, the anchorage zone shall consist of two zones (Figure 7.10-1). One a “Local Zone” consisting of high compression stresses which lead to spalling. Also there is a General Zone consisting of high tensile stresses which lead to bursting (AASHTO, Equation 5.10.9.2.1).

Figure 7.10-1 General and Local Zone (AASHTO, 2012) The local zone of the anchorage system is dependent on the nearby crushing demand. Compression reinforcement is used within the local zone to keep concrete from spalling and eventually crushing. The local zone is more influenced by the characteristics of the anchorage device and its anchorage characteristics than by loading and geometry. Anchorage reinforcement is usually designed by the prestressing contractor and reviewed/approved by the design engineer during the shop drawing process. The general zone is defined by tensile stresses due to spreading of the tendon force into the structure. These areas of large tension stresses occur just ahead of the anchorage and slowly dissipate from there. Tension reinforcement is used in the

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

general zone as a means to manage cracking and bursting. The specifications permit the general anchorage zone to be designed using:   

the Finite Element Method the Approximate Method contained in the specifications the Strut and Tie method, which is the preferred method

The result of this design yields additional stirrups and transverse bars into and near the end diaphragm.

7.11

DESIGN PROCEDURE Start

CIP/PT bridge superstructure selected

Span Lengths, and bridge widths are determined by need, existing surroundings Span Lengths (L) Bridge Width (W)

Material properties  Superstructure concrete:  f ′c = 4.0 ksi min, and 10.0 ksi max (Article 5.4.2.1, AASHTO, 2012)  f ′ci min = 3.5 ksi  Normal weight concrete c = 0.15 kcf  Ec = 33,000 c1.5f ′c = (33,000)(0.15)1.5(4) = 3834 ksi (Article 5.4.2.4, AASHTO, 2012)  Prestressing Steel:  fpu = 270 ksi, fpy = 0.9 fpu = 243 ksi (Table 5.4.4.1-1, AASHTO, 2012)  Maximum jacking stress, fpj = 0.75 fpu = 202.5 ksi (CA Amendements, Caltrans, 2014)  Ep = 28,500 ksi (Article 5.4.4.2, AASHTO, 2012)  Mild Steel  A706 bar reinforcing steel:  fy = 60 ksi  Es = 29,000 ksi

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Span Lengths (L) Bridge Width (W) Material Properties (fpu, fy, Ep, Es, Ec)

Select Typical Section D/S Ratio = 0.40 Continuous, 0.45 Simple Span (AASHTO Table 2.5.2.6.3-1) Deck and Soffit Size, Overhand, Typical Section Rebar (MTD 10-20)

Span Lengths (L) Bridge Width (W) Material Properties (fpu, fy, Ep, Es, Ec) Structure Depth (Ds) Girder Spacing (S, Scir) Deck and Soffit Thickness (t, To)





 

Span Lengths (L) Bridge Width (W) Material Properties (fpu, fy, Ep, Es, Ec) Structure Depth (Ds) Girder Spacing (S, Scir) Deck and Soffit Thickness (t, To)

Span Lengths (L) Bridge Width (W) Material Properties (fpu, fy, Ep, Es, Ec) Structure Depth (Ds) Girder Spacing (S, Scir) Deck and Soffit Thickness (t, To) Typical Section Rebar

Seismic Design (see other chapters)

Detail Typical Section

Loads DC = Dead load of structural components and nonstructural attachments (Article 3.3.2, AASHTO, 2012)  Unit weight of concrete (c) (Table 3.5.1-1, AASHTO, 2012)  Includes the weight of the box girder structural section  Barrier rail where appropriate DW = Dead load of wearing surfaces and utilities (Article 3.3.2, AASHTO, 2012)  3 in. Asphault Concrete (A.C.) overlay (3 in. thick of 0.140 kcf A.C.) = 0.035 ksf HL93, which includes the design truck plus the design lane load (Article 3.6.1.2, AASHTO, 2012) California long-deck P15 (CA Amendments to AASHTO LRFD Article 3.6.1.8, Caltrans, 2012)

Calculate Section Properties (Ig, Ag, yt, yb) Span Lengths (L) Bridge Width (W) Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb )

1

MDC, VDC, MDC w/o Barrier MDW, VDW M+HL93, M-HL93, assoc V+HL93, V-HL93 V+HL93, V-HL93, assoc M+HL93, M-HL93 M+P-15, M-P-15, assoc V+P-15, V-P-15 V+P-15, V-P-15, assoc M+P-15, M-P-15 2

Chapter 7 - Post-Tensioning Concrete Girders

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1

Estimate Pj BDA 11-16, and 11-66

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est))

 

2

Prestressing Path Minimum distance to soffit and deck (MTD 11-28) One-end versus two-end stressing

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est)) Eccentricities (e) Calculate Losses Instantaneous Losses  Anchor Set Loss (AASHTO 5.9.5.2.1)  Friction Losses (CA Amendments Table 5.9.5.2.2b-1)  Elastic Shortening (AASHTO 5.9.5.2.3) Long Term Losses  Shrinkage of Concrete (AASHTO 5.9.5.3)  Creep (AASHTO 5.9.5.3)  Relaxation of Steel (AASHTO 5.9.5.3) Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est)) Eccentricities (e) Anchor Set Loss (fpA) Friction Loss (fpF) Elastic Shortening Loss (fpES) Shrinkage Loss (fpSH) Creep (fpCR) Relaxation of Steel (fpR) Secondary Moments  Moment Distribution 4 1

5

3

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

3

4

5

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est)) Eccentricities (e) Anchor Set Loss (fpA) Friction Loss (fpF) Elastic Shortening Loss (fpES) Shrinkage Loss (fpSH) Creep (fpCR) Relaxation of Steel (fpR) Secondary Moment Coefficent (MCs)

Coefficients  Force Coefficients  Moment Coefficients Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est)) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT)

2

Enviromental Area Areas I and II (Non-Freeze-Thaw) Calculate Pj – Service III No tension allowed under dead load (CA Amendments Table 5.9.4.2.2-1) M DC  DW y FC pT ( Pj ) MC pT ( Pj ) y   0 I A I 0.19f ′c (ksi) allowed under SERVICE III (CA Amendments Table 5.9.4.2.2-1) M DC  DW 0.8 HL93 yt (bent ) FC pT ( Pj ) MC pT ( Pj ) y    0.19 f c I A I

Area III (Freeze-Thaw)

Calculate Pj – Service III No tension allowed under dead load (CA Amendments Table 5.9.4.2.2-1) M DC  DW y FC pT ( Pj ) MC pT ( Pj ) y   0 I A I 0.0948f ′c (ksi) allowed under SERVICE III (CA Amendments Table 5.9.4.2.2-1) M DC  DW  0.8 HL93 yt (bent ) FC pT ( Pj ) MC pT ( Pj ) y    0.0948 f c I A I

Prestress Force (Pj)

Prestress Force (Pj) 6

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Estimated Prestress Force (Pj(est)) Prestress Force (Pj) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT)

Chapter 7 - Post-Tensioning Concrete Girders

2

7-28

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

4

6

5

2

No Was Pj closely approximated? i.e. is Pj ~ Pj(est) Yes

No

Do Elastic Shortening Losses converge on assumed? Now that Pj is known, does the assumed Elastic Shortening value approximate the actual? Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Prestress Force (Pj) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT)

Yes

Calculate fpi and fpe Effective stress in concrete after initial losses but before long term losses:

f pi 

Pj FC F Ag



Pj (e)( FC F ) yb Ig

Effective stress in steel after all losses

f pe 

Pj FC pT Ag



Pj ( MC pT ) yb Ig

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Prestress Force (Pj) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT) Design f ′c and f ′ci (AASHTO Initial effective steel stress (fpi) Table 5.9.4.2.1-1) Final effective steel stress (fpe) Dead Load Only f pe  f DC  DW f c  0.45 Live Load + Dead Load f pe  f DC  DW  f HL 93 f c  0.60 Dead Load w/o barrier

f ci 

f pi  f DC w/ o

b

0.60 w

fDC+DW fHL93

Convert Moments into Stresses Dead Load Only My f  I

7 2

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7

2

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Prestress Force (Pj) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT) Compressive Strength of Concrete (f’c) Initial Compressive Strength of Concrete (f’ci)

Flexural Reinforcement Design – Strength I and II AASHTO 5.7.3.1.1-1:  kc  f ps = f pu 1 -   d  p   AASHTO 5.7.3.1.1-4 (for rectangular section, see AASHTO for T-sections):

c

Aps f pu  As f s  As f s f 0.85 f cβ1b  kAps ps dp

AASHTO 5.7.3.2.2-1 a a  a hf    a  M n  Aps f ps  d p    As f s  d s   As f s d s    0.85 f c  b  bw  h f    2 2   2  2 2  Prestressing Steel only

Additional Flexural Steel

Compression Steel

Flanged Section Component a  1 c  h f

Minimum Requirement M n shall be larger than or equal to the lesser of   1.33 Mu as defined in Article 5.7.3.3.2 (AASHTO, 2012) Or AASHTO 5.7.3.3.2-1  S  M cr   3 ( 1 f r   2 f cpe ) Sc  M dnc  c  1  S  nc    Modified AASHTO 5.7.3.1.1-4 (for rectangular section, see AASHTO for T-sections) Pj f ps  As f s  As f s 0.75 f pu a Pj f ps k 0.85 f c b  β1 0.75 f pu d p Use AASHTO 5.7.3.2.2-1 to solve for As once a is known. Check AASHTO C5.7.3.3.1 to make sure that Mild Steel yields

8

Chapter 7 - Post-Tensioning Concrete Girders

2

7-30

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Continued from previous page 8

2

Material Properties (fpu, fy, Ep, Es, Ec) Section Properties (Ig, Ag, yt, yb ) Prestress Force (Pj) Eccentricities (e) Friction Force Coefficient (FCpF) Final Force Coefficient (FCpT) Final Moment Coefficient (MCpT) Compressive Strength of Concrete (f’c) Initial Compressive Strength of Concrete (f’ci) Area of Additional Mild Steel (As)

Shear Reinforcement Design – Strength I and II See AASHTO Figure C5.8.3.4.2-5

Prestress Force (Pj) Area of Additional Mild Steel (As) Area of Shear Stirrup (Av) Shear Stirrup Spacing (s) Final Force Coefficient (FCpT) Compressive Strength of Concrete (f’c) Initial Compressive Strength of Concrete (f’ci)

7.12

DESIGN EXAMPLE

7.12.1

Prestressed Concrete Girder Bridge Data

End

412’-0” 168’-0”

118’-0”

Abut 1

Bent 2

47’-0”

44’-0”

126’-0”

Abut 4

Bent 3

Figure 7.12.1-1 Elevation View of the Example Bridge

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

It is important, in a two-span configuration, to try and achieve equal, or nearly equal spans. In frames consisting of three or more spans, the designer should strive for 75% end spans, with nearly equal interior spans whenever possible. The overall width of the bridge in this example is based on the following traffic requirements: 3 – 12-foot lanes with traffic flow in same direction 2 – 10-foot shoulders 2 – Type 732 barriers supporting Type 7 chain link railing Overall bridge width, W = 58 ft – 10 in. (See Figure 7.12.3-1) Materials Superstructure Concrete: fc = 4.0 ksi min, and 10.0 ksi max (AASHTO Article 5.4.2.1) fci min = 3.5 ksi Normal weight concrete c = 0.15 kcf Ec = 33,000 c1.5fc = (33,000)(0.15)1.5(4) = 3834 ksi (AASHTO Article 5.4.2.4) Prestressing Steel: fpu = 270 ksi, fpy = 0.9 fpu = 243 ksi (AASHTO Table 5.4.4.1-1) Maximum jacking stress, fpj = 0.75 fpu = 202.5 ksi (Caltrans, 2014) Ep = 28,500 ksi (AASHTO Article 5.4.4.2) Mild Steel: A706 bar reinforcing steel fy = 60 ksi, Es = 29,000 ksi

7.12.2

Design Requirements Perform the following design portions for the box girder in accordance with the AASHTO LRFD Bridge Design Specifications, 6th Edition (2012) with California Amendments 2014 (Caltrans, 2014).

7.12.3

Select Girder Layout and Section Table 2.5.2.6.3-1 (AASHTO, 2012) states that the traditional minimum depth for a continuous CIP box girder shall be calculated using 0.040 L, where L is the length of the longest span within the frame. Structure depth, d ~ (0.040)(168) = 6.72 ft

Chapter 7 - Post-Tensioning Concrete Girders

use: d = 6.75 ft = 81 in.

7-32

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Assuming an overhang width that is about 40 – 50% of the clear spacing between girders, and that the maximum girder spacing, Smax, should not exceed (2)(d), where d = structure depth, find the number and spacing of the girders. Overhang width should be limited to 6′-0″ max. When span lengths are of similar length on the same structure, it’s generally a good idea to use the same depth for the entire frame. Maximum girder spacing, Smax = (2)(6.75) = 13.50 ft 

Try 4 girders: As an estimate, assuming the combined width of the overhangs is approximately equal to a bay width, S~W/4. Therefore S4 = 58.83/4 = 14.71 ft. Since S4 = 14.71 > Smax = 13.50 ft, an extra girder should be added to the typical section.



Try 5 girders: S5 = 58.83/5 = 11.77 ft. Since S5 = 11.77 < Smax = 13.50 ft, 5 girders should be used to develop the typical section. Using 5 girders will improve shear resistance, provide one more girder stem for placing P/S ducts, and keep the overhang width less than 6 feet. With 5 girders use an exterior girder spacing of 11 ft -11 in. and an interior girder spacing of 12 feet.

Figure 7.12.3-1 Typical Section View of Example Bridge

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.12.4



The column diameter Dc = 6 ft



2:1 Sloped exterior girders used for aesthetic purposes



Assuming no other issues, the distance as measured perpendicular to “A” line from “A” line to centerline of column is 16 ft-7 in.



Bent cap width is Dc+2 ft = 8 ft



Overhang thickness varies from 12 in. at exterior girder to 8 in. at Edge of Deck (EOD)



Girders are 12 in. wide to accommodate concrete vibration (Standard Plans – Sheet B8-5)



Exterior girders flared to 18 in. minimum at abutment diaphragms to accommodate prestressing hardware (Standard Plans – Sheet B8-5)



Soffit flares to 12 in. at face of bent caps for seismic detailing and to optimize prestress design. Length of flares approximately 1/10 Span Length



All supports are skewed 20 relative to centerline of bridge. Based on guidance material in MTD 11-28 (Caltrans, 2010), use abutment diaphragm thickness of 3 ft -3 in.



Four inches fillets are to be located between perpedicular surfaces except for those adjoining the soffit

Determine Basic Design Data Section Properties Prismatic Section (midspan):  Area (Ag) = 103 ft2  Moment of Inertia (Ig) = 729 ft4  Bottom fiber to C.G. (yb) = 3.80 ft Flared Section (bent):  Area (Ag) = 115 ft2  Moment of Inertia (Ig) = 824 ft4  Bottom fiber to C.G. (yb) = 3.50 ft Loads DC = Dead load of structural components and non-structural attachments (AASHTO Article 3.3.2)  Unit weight of concrete (c) = 0.15 kcf (AASHTO Article 3.5.1)  Includes the weight of the box girder structural section  Type 732 Barrier rail on both sides (0.4 klf each)

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

DW = Dead load of wearing surfaces and utilities (AASHTO Article 3.3.2)  3 in. Asphault Concrete (A.C.) overlay (3 in. thick of 0.140 kcf A.C.) = 0.035 ksf HL93, which includes the design truck plus the design lane load (AASHTO Article 3.6.1.2) California long-deck P15 (CA Amendements to AASHTO LRFD Section 3.6.1.8, Caltrans, 2014) Slab design: Design is based on the approximate method of analysis – strip method requirements from Article 4.6.2.1 (AASHTO, 2012), and slab is designed for strength, service, and extreme event Limit Statess Article 9.5 (AASHTO, 2012). Caltrans Memo to Designers 10-20 (Caltrans, 2008b) provides deck thickness and reinforcement.

7.12.5

Design Deck Slab and Soffit Deck Slab: Refer to MTD 10-20 Attachment 2. Enter centerline girder spacing into design chart and read the required slab thickness and steel requirements. In this example, the centerline spacing of girders is a maximum of 12 feet. Using the chart, the deck slab thickness required is 9 inches.

Table 7.12.5-1 LRFD Deck Design Chart, taken from MTD 10-20, Attachment 2 (Caltrans, 2008) CIP PRESTRESSED BOX, PRECAST-I, & STEEL GIRDERS w/ flange width  24″ "S" "t" Girder CL to CL Top Slab Spacing Thickness

Dimension

Transverse Bars

"D" Bars

"G" Bars

"F"

Size

Spacing

#5 Bars

#4 Bars

11′-9″

8 7/8″

1′-5″

#6

11″

12

5

12′-0″

9″

1′-5″

#6

11″

12

5

12′-3″

9 1/8″

1′-6″

#6

11″

12

5

Soffit Slab: Refer to MTD 10-20 Attachment 3 (formerly BDD 8-30.1). Enter effective girder spacing into the design chart. Read the required slab thickness and steel requirements. In this example, the effective spacings for interior and exterior bays are 11 ft and 8 ft  4 in., respectively. Use a constant soffit thickness of 8.25 in. and “E” bar spacing based on 11 ft, and “H” bar spacing based on individual bay widths.

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 7.12.5-2 LRFD Soffit Design Chart, taken from MTD 10-20, Attachment 3 (Caltrans, 2008)

7.12.6

Select Prestressing Cable Path In general, maximum eccentricities (vertical distance between the C.G.s of the superstructure concrete and the P/S steel) should occur at the points of maximum gravity moment. These points usually occur at the maximum negative moment near the bent cap, or at the maximum positive moment regions near midspan. In order to define the prestressing path for this frame, we need to get an estimate of the jacking force, Pj. An estimate of Pj will aid us in determining how much vertical room is needed to physically fit the strands/ducts in the girders, and will help optimize the prestress design. Estimate pounds of P/S steel per square meter of deck area using the chart found on the next page.

Chapter 7 - Post-Tensioning Concrete Girders

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Chapter 7 - Post-Tensioning Concrete Girders

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0

50

3 Span Box Girder w/ 3/4 L end spans d/s = 0.030 - 0.050 =0.15; f's = 270 ksi

L

150

3/4 L

Span Length (ft)

200

250

300

Figure 7.12.6-1 Estimate of Required Pj from the Modified for use with English Units BDA Chapter 11-15, (Caltrans, 2005)

d/s = 0.050

d/s = 0.040

100

d/s = 0.030

d/s = 0.035

3/4 L

350

400

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-37

Prestressing Steel (lbs per sq.ft. deck area)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

With a length of span = 168 ft, and a depth of span ratio of 0.040, read 2.75 lb per ft2 deck area off the chart shown below: Estimate Pj using the Equation on BDA, page 11-66 (Caltrans, 2005). Total weight of prestressing steel = (2.80 lb/ft2)(deck area) = (2.80 lb/ft2) (412.0 ft)(58.83 ft) = 67,870 lb Re-arranging the English equivalent of the equation found on BDA, page 11-66, (Caltrans, 2005) to solve for Pj results in the following:

Pj 

W  202.5

(7.12.6-1)

L frame  3.4

where: Pj

=

force in prestress strands before losses (ksi)

Lframe

=

length of frame to be post-tensioned (ft)

W

=

weight of prestressing steel established by BDA, page 11-66 (lb)

Pj 

67, 870 lb  202.5 412 ft  3.4

 9, 810 kips

Develop preliminary maximum eccentricities at midspan and bent cap using MTD 11-28 (Caltrans, 2010) found on the next page. Determine “D” value based on estimate of Pj : 

Pj /girder = 9,810 kips/5 girders = 1,962 kips/girder.



Enter “D” chart for cast-in-place girders (MTD 11-28 Attachment 2, Caltrans 2010), and record a value of “D” as 5 inches. This chart accounts for the “Z” factor, which considers the vertical shift of the tendon within the duct, depending on whether you are at midspan, or at the centerline of bent. The “D” values produced in this chart are conservative, and the designer may choose to optimize the prestressing path by using an actual shop drawing to compute a “D” value.

Chapter 7 - Post-Tensioning Concrete Girders

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Chapter 7 - Post-Tensioning Concrete Girders

0

3

6

9

12

15

18

0

500

1500

2000

2500

3000

3500

4000

4500

5000

Figure 7.12.6-2 MTD 11-28 Attachment 2 “D” Chart used to Optimize Prestressing Tendon Profile (Caltrans, 2010)

1000

Detail “A” C.G of Prestressing Steel See Memo to Designer 11-31

“D” Chart for Cast-in-Place P/S Concrete Girders

5500

6000

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-39

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

21/2″clr C.G Prestressing Force

“D”

“ts”

“K”

Figure 7.12.6-3

1″ clr

MTD 11-28 Attachment 1 Tendon Configuration at Low Point of Tendon Profile (Caltrans, 2013)

The value K for a bridge with a skew of less than 20 is the distance from the top mat of steel in the soffit to the bottom mat of steel in the soffit. Therefore, for this bridge, K is:

K  t  clrint

(7.12.6-2)

where: K

=

distance to the closest duct to the bottom of the soffit or top of the deck (in.)

t

=

thickness of soffit or deck (in.)

clrint =

clearance from interior face of bay to the first mat of steel in the soffit or deck (usually taken as 1 in.) (in.)

δlp  ts  1  " D "

(7.12.6-3)

where: lp

=

offset from soffit to centroid of duct (in.)

ts

=

thickness of soffit (in.)

Using Figure 7.12.6-2 and Equation 7.12.6-3, determine offset from bottom fiber to the C.G. of the P/S path at the low point:

δlp  tsoffit  1  " D "  8.25  1  5  12.25 in.

Chapter 7 - Post-Tensioning Concrete Girders

7-40

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

* “tD”

“K”

C.G. Prestressing Force

1″ clr

“D” * - An increase in the “K” value will usually be required when main bent cap reinforcement falls below the limits shown here, particularly when the skew angle of the bent cap exceeds 20 degrees, and a thickened top slab is required.

21/2″ clr

Figure 7.12.6-4 MTD 11-28 Attachment 1 Tendon Configuration at High Point of Tendon Profile (Caltrans, 2013)

δ hp  td  1  " D "

(7.12.6-4)

where: hp =

offset from deck to centroid of duct (in.)

td

thickness of deck (in.)

=

Using Figure 7.12.6-4 and Equation 7.12.6-4, determine offset from top fiber to the C.G. of the P/S path at the high point: δlp  tdeck  1  " D "  9  1  5  13 in.

Another method to optimize the prestressing path is to use an actual set of P/S shop drawings to find “D”. Both post-tensioning subcontractors as of the publication of this material use 27 tendon - 0.6 in. diameter strand systems, with a maximum capacity of 27 strands @ 44 kips/strand = 1188 kips. The calculation of “D” is as follows: The equation for Pj in BDA, page 11-66 gave us an estimate of 1962 kips/girder. Assuming 0.6 in. diameter strands, with Pj per strand = 44 kips, the number of strands per girder is as given in Equation 7.12.6-5: strands

=

girder strands girder

force per girder

(7.12.6-5)

force per strand 

force per girder force per strand



1, 927 kips/girder

 43.79

44 kips/strand

Chapter 7 - Post-Tensioning Concrete Girders

7-41

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Use: 44 Strands 

Assume 22 strands in duct A, and 22 strands in duct B.



Find “D” based on the Equation 7.12.6-6

Figure 7.12.6-4 Example of Sub-contractor Tendon Layout for a Two-Duct per Girder Configuration n

 (n  d ) i

"D" 

i

i 1

n

n

Z

(7.12.6-6)

i

i 1

where: ni =

number of strands in the i th duct

di =

distance between C.G. of i th duct and the i th duct LOL (See Figure 7.12.6-4) (in.)

Z

C.G. tendon shift within duct (in.)

=

Chapter 7 - Post-Tensioning Concrete Girders

7-42

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

n

"D" 

 (n  d ) i

i

i 1 n

n

Z 

( na  d a )  ( nb  d b ) na  nb

Z

i

i 1



(22  2.19)  [22  (2.19  4.13)] 22  22

 0.75 in.

"D" = 5.0 in. Final design offsets using Equations 7.12.6-3 and 7.12.6-4: Values from Tendon Layout:

δlp  tsoffit  1  " D "  8.25  1  5.0  12.25 in.

δhp  tdeck  1  " D "  9.0  1  5.0  13.0 in. Values using MTD 11-28 “D”:

δlp  12.25 in. δ hp  13.0 in. It is noted that for each change in Pjack there is an accompanying change in “D” value. Changes in “D” values results in rerunning a model, and can change other portions of the design. Because of this, it is recommended to use a conservative value that will result in stable “D” values. However, this is an area where Pj can be decreased through iteration, if necessary. In this case, these values are equal, use a minimum value of 12.5 in. at the soffit and of 13.5 in. at the deck. Once the high and low points of the tendon path are established, the locations of the inflection points can be obtained. Locating the inflection points at the 10% span length locations on either side of the bent cap not only delivers adequate moment resistance to this zone, but provides a smooth path that allows for easy tendon installation. The vertical position of these inflection points lie on a straight line between the high and low points of the tendon path, at the 0.1L locations. Similar triangles can be used to find the vertical location of the inflection point: (See Figures 7.12.6-5 and 7.12.6-6) Spans 1 and 3: 

yBD = 81– 13.5– 12.5 = 55 in.



Similar Triangles:



Rearranging yields: yBC 

55



yBC

0.5  0.1 0.5 55 in.  0.5

Chapter 7 - Post-Tensioning Concrete Girders

0.6

 45.83 in.

7-43

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 7.12.6-5 Spans 1 and 3 Inflection Point Sketch Span 2: 

yFH = 81– 13.5– 12.5 = 55 in.



Similar Triangles:



Rearranging yields: yFG 

55 0.4  0.1



yFG 0.4

55  0.4 0.5

 44 in.

Figure 7.12.6-6 Span 2 Inflection Point Sketch. Middle Spans of Frames Typically have Inflection Points at Mid-Span The final cable path used for design is shown in Figure 7.12.6-7 on the following page. The yb = 45.625 in. values at the abutment diaphragms is the distance to the C.G. of the Concrete Box Section, with 6 in. of tolerance (up or down) to allow for constructability issues.

Chapter 7 - Post-Tensioning Concrete Girders

7-44

Figure 7.12.6-7 Final Cable Path as it would Appear on the Plans

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Chapter 7 - Post-Tensioning Concrete Girders

7-45

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

One vs. Two-End Stressing: According to MTD 11-3, “One-end stressing is considered economical when the increase in Pj does not exceed 3%” when compared to two end stressing. An increase in Pj corresponds to an equivalent increase in materials, and 3% is the breakeven point between cost of materials vs. the cost of time and labor in moving the stressing operation to the opposite end of the frame. Two-end stressing controls the design of most 3+ span frames, and we will assume it controls in this example problem. Two-end stressing is used as a means to control friction loss.

7.12.7

Post Tensioning Losses

7.12.7.1

Friction Loss Angle change of P/S path: The cumulative angle change of the P/S path must be found in order to find the friction loss. Each individual parabola (10 in. this example) must be isolated so that angle change (yij) can be calculated for each segment. Friction losses have a cumulative effect, and increase as you get further away from the jacking end. Use the following formula to solve for angle change in each parabolic segment: 2 yij

αij 

lij

(rad )

(7.12.7.1-1)

where: yij =

height of individual parabola (in.)

lij =

length of individual parabola (in.)

Find the angle change in segment BC in span 1: y BC = 4.861 – 1.042 = 3.82 ft lBC = 0.5L1 = (0.5)(126) = 63 ft

αij 

2 yBC lBC



2  3.82 63

 0.121(rad )

The table shown on the next page includes a summary of values that will be used to calculate initial friction losses: Note in Table 7.12.7.1-1 that the prestressing cable is located at the exact neutral axis of the member (3.813 in.). However, for all future equations the rounded number 3.8 in. will be used.

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Table 7.12.7.1-1 Summary of P/S Path Angle Changes used in Friction Loss Calculations Left-end Right-end

Left-end Right-end

yij Calculation

yij

xij Calculation

xij

xAK

xKA

ij

AK

KA

Segment

(ft)

(ft)

(ft)

(ft)

(ft)

(ft)

(rad)

(rad)

(rad)

AB

3.813 - 1.042

2.77

(0.4)(126)

50.4

50.4

412.0

0.110

0.110

1.166

BC

4.861 - 1.042

3.82

(0.5)(126)

63.0

113.4

361.6

0.121

0.231

1.056

CD

5.625 - 4.861

0.76

(0.1)(126)

12.6

126.0

298.6

0.121

0.352

0.934

DE

5.625 - 4.708

0.92

(0.1)(168)

16.8

142.8

286.0

0.109

0.462

0.813

EF

4.708 - 1.042

3.67

(0.4)(168)

67.2

210.0

269.2

0.109

0.571

0.704

FG

4.708 - 1.042

3.67

(0.4)(168)

67.2

277.2

202.0

0.109

0.680

0.595

GH

5.625 - 4.708

0.92

(0.1)(168)

16.8

294.0

134.8

0.109

0.789

0.486

HI

5.625 - 4.861

0.76

(0.1)(118)

11.8

305.8

118.0

0.130

0.919

0.376

IJ

4.861 - 1.042

3.82

(0.5)(118)

59.0

364.8

106.2

0.130

1.048

0.247

JK

3.813 - 1.042

2.77

(0.4)(118)

47.2

412.0

47.2

0.117

1.166

0.117

412.0

1.166

Initial Friction Coefficient: The percent in decimal form of jacking stress remaining in the P/S steel after losses due to friction, FCpF , can now be calculated based on the cumulative lij and ij computed above. Using Equation 7.5.1.2-1, modified Equation 5.9.5.2.2b-1 (AASHTO, 2012) from Article 7.5.1.2 above: FC pF 

f pF f pj

 e  ( Kx μα )

(7.5.1.2-1) Inputting values from CA Amendments to AASHTO LRFD Table 5.9.5.2.2b-1 (Caltrans, 2014)  

K = 0.0002 per ft  = 0.15 (Lframe < 600 ft)

Find the percent in decimal form of Pj remaining after the effects of friction loss at the 0.1 point in span 2. 

 ( Kx  μα) At pt. E: FC pF  e

  

xE (left-end) = 142.8 ft E (left-end) = 0.4616 rad FCpF (@E) = e – [(0.0002)(142.8) + (0.15)(0.462)] = 0.907

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Table 7.12.7.1-2 includes a summary of values of initial friction losses: Table 7.12.7.1-2 Summary of Cumulative Angle Change for Two-End Stressing Left-end Stressing

7.12.7.2

Location

xAK (ft)

AK (rad)

A B C D E F G H I J K

0.0 50.4 113.4 126.0 142.8 210.0 277.2 294.0 305.8 364.8 412.0

0.000 0.110 0.231 0.352 0.462 0.571 0.680 0.789 0.919 1.048 1.166

e

 KxAK  AK 

=FCpF (decimal %) 1.000 0.974 0.944 0.925 0.907 0.880 0.854 0.837 0.819 0.794 0.773

Right-end Stressing xKA (ft)

KA (rad)

e KxKA KA  =FCpF

412.0 361.6 298.6 286.0 269.2 202.0 134.8 118.0 106.2 47.2 0.0

1.166 1.056 0.934 0.813 0.704 0.594 0.486 0.376 0.247 0.117 0.000

0.773 0.794 0.819 0.836 0.853 0.878 0.905 0.923 0.943 0.973 1.000

(decimal %)

Anchor Set Losses The method used for determining losses due to anchor set is based on “similar triangles”. The assumption is that the effects of friction is the same whether a tendon is being stressed, or released back into the duct to seat the wedges. Most of the time, the end of the influence length of anchor set, xpA, lies between the high and low inflection points from the jacking end in a multi-span frame. The equations that solve for FCpA and xpA are shown on the next page:

Figure 7.12.7.2-1 Typical Anchor Set Loss

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FC pA 

f pA f ps



2(f L ) x pA

(7.5.1.1-1)

L( f ps )

E p ( Aset ) L

x pA 

(7.5.1.1-2)

12f L

Define the anchor set loss diagram by finding xpA and FCpA in span 1 due to the left-end stressing operation: Given: Ep = Aset = L = fpj fpu

= = = = =

fL

28,500 ksi 0.375 in. Distance from point A to point C = 0.9 L1 = (0.9) (126 ft) = 113.4 ft 0.75 (0.75)(270 ksi) 202.5 ksi ( Table 5.9.3-1 AASHTO, 2012) [1 - FCpF (@ point C)](202.5 ksi) [1 – 0.944](202.5) = 11.34 ksi

Solving for xpA and FCpA at the intersection of initial losses and anchor set (the location xpA away from the anchor).

E p (  Aset ) L

x pA 

12f L

FC pA 

f pA f ps





28, 500(0.375)113.4

 94.37ft

12(11.34)

2(f L )( x pA ) L( f ps )



2(11.34)(94.37) 113.4(202.5)

 0.093

Figure 7.12.7.2-2 Bridge Specific Anchor Set Loss

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The anchor set loss diagram is found in a similar manner in Span 3 due to the second (right) end stressing operation. Given: Ep = 28,500 ksi Aset = 0.375 in. L = Distance from point K to point I = 0.9 L3 = (0.9)(118 ft) = 106.2 ft fpj = 0.75fpu = (0.75)(270ksi) = 202.5 ksi (Table found in commentary to AASHTO Article 5.9.3-1) fL = [1 - FCpF (@ point I )] (202.5 ksi) = [1 – 0.943] (202.5) = 11.54 ksi Solving for xpA and FCpA at the intersection of initial losses and anchor set (the location xpA away from the anchor). x pA 

E p ( Aset ) L

FC pA 

7.12.7.3

12f L f pA f ps





28,500(0.375)106.2  90.53 ft 12(11.54)

2(f L )( x pA ) L( f ps )



2(11.54)(90.53) 106.2(202.5)

 0.097

Elastic Shortening Losses due to elastic shortening are usually assumed at the beginning and then checked once more for convergent numbers for MDL, Pj, and e have been found. Therefore, based on experience, we will assume a realistic and typically conservative value of fpES = 3 ksi for this practice problem. To turn this into a Force Coefficient: f pES 3 ksi FC pES    0.015 f ps 202.5 ksi

7.12.7.4

Approximate Estimate of Time Dependent Long-Term Loss The long-term change in prestressing steel stress due to creep of concrete, shrinkage of concrete, and relaxation of P/S steel occur over time and begin immediately after stressing. 5 f pSH  12.0(1.7  .01H ) (7.5.2.1-1) (1  fci ) f pCR  10.0

f pi Aps Ag

(1.7  .01H )

f pR  2.4 ksi for lo-lax strands Chapter 7 - Post-Tensioning Concrete Girders

5 (1  f ci )

(7.5.2.2-1) (AASHTO 5.9.5.3-1)

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Most of the research done on time-dependant losses considered precast concrete girders, without much consideration given to continuous, CIP PT box girder structures. Ongoing research, indicate time-dependant losses as high as 30 ksi may be appropriate for cast-in-place, post-tensioned structures. This portion of the code has undergone revision from 2008 to 2014.

f pLT  20 ksi

(CA Amendments 5.9.5.3)

For this example, let’s use 25 ksi. This is a reasonable value and was the value used in the 2008 CA Amendments. Converting to a Force Coefficient: f pR 25 FC pLT    0.123 f ps 202.5

7.12.7.5

Total Loss of Prestress Force As stated in section 7.5 the loss of force in the prestressing steel is cumulative. In lieu of more detailed analysis, prestress losses in members constructed and prestressed in a single stage, relative to the stress immediately before transfer, may be taken as:

FC  (1 

 f ) i

(7.5-1) f ps A summary of our calculated immediate and total prestress stress remaining along the prestressing path is included in Table 7.12.7.5-1.

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Table 7.12.7.5-1 Summary of Cumulative Prestress Loss Percentage of Cumulative Prestress Stresses Remaining After All Losses Left-end Stressing

Right-end Stressing

Location

FCpF

FCpA

FCpES

FCpLT

FCpF

FCpA

FCpES

FCpLT

A

1.000

0.907

0.893

0.770

0.773

0.773

0.758

0.635

B

0.974

0.933

0.917

0.794

0.794

0.794

0.779

0.656

C

0.944

0.944

0.929

0.806

0.819

0.819

0.804

0.681

D

0.925

0.925

0.910

0.787

0.836

0.836

0.821

0.698

E

0.907

0.907

0.892

0.769

0.853

0.853

0.838

0.715

F

0.880

0.880

0.865

0.742

0.878

0.878

0.863

0.740

G

0.854

0.854

0.839

0.716

0.905

0.905

0.890

0.767

H

0.837

0.837

0.822

0.699

0.923

0.923

0.908

0.785

I

0.819

0.819

0.804

0.681

0.943

0.943

0.928

0.805

J

0.794

0.794

0.779

0.656

0.973

0.930

0.915

0.792

K

0.773

0.773

0.758

0.635

1.000

0.903

0.890

0.767

FCpI

FCpT

FCpI

FCpT

(ft)

Figure 7.12.7.5-1 Summary of Prestress Losses for Two-End Stressing Chapter 7 - Post-Tensioning Concrete Girders

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7.12.8

Cable Path Eccentricities In order to design the jacking force, and track stresses in both top and bottom fibers, cable path eccentricities must be calculated at the 10th points of each span in the frame. Derived from the equation of a parabola, the diagram and formula shown below can be used to compute eccentricities for parabolic P/S paths: 







ex    yij  1  

x2 

 xij 2  

c

(7.12.8-1)

where: ex =

eccentricity as a function of x along parabolic segment (ft)

yij =

height of the individual parabola (ft)

x

=

location along parabolic segment where eccentricity is calculated (percent of span L)

xij =

length of parabolic segment under consideration (must originate at vertex) (percent of span L)

c

shifting term to adjust eccentricities when yij does not coincide with the C.G. of concrete

=

Figure 7.12.8-1 Cable Path Calculation Diagram

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Find the cable path eccentricities in Span 1:

Figure 7.12.8-2 Cable Path Calculation Diagram from Abut 1 to the Low Point of Span 1

  0.42   e0.0 L1    2.771 1   0  0 ft 2   0.4      0.32   e0.1L1    2.771 1   0  1.212 ft 2   0.4      0.22   e0.2 L1    2.771 1   0  2.078 ft 2   0.4      0.12   e0.3 L1    2.771 1   0  2.598 ft 2   0.4      0.02   e0.4 L1    2.771 1   0  2.771 ft 2   0.4   

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Figure 7.12.8-3 Cable Path Calculation Diagram from the Low Point of Span 1 to the Inflection Point Near Bent 2





e0.4 L1   3.819  1 

0.02  

 0.5    0.12   3.819  1  2  0.5 

2

   1.048  2.771 ft 

    1.048  2.618 ft  2   0.2   e0.6 L1   3.819  1   1.048  2.160 ft 2   0.5      0.32   e0.7 L1   3.819  1   1.048  1.396 ft 2   0.5      0.42   e0.8 L1   3.819  1   1.048  0.327 ft 2   0.5      0.52   e0.9 L1   3.819  1   1.048  1.048 ft 2   0.5    e0.5 L1

The eccentricity at the centerline of Bent 2 must be calculated using the section properties that include the soffit flare: e1.0L1 =yCD + c + yb = 0.764 + 1.048 + 0.310 = 2.122 ft

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Figure 7.12.8-4 Cable Path Calculation Diagram from the Inflection Point Near Bent 2 to CL of Bent 2 Cable path eccentricities for all three spans are summarized in the Figure 7.12.8-5 below: Abut 1

Bent 2

Bent 3

2.122’ 1.048’

e 1.212’ x 2.078’ 2.598’ 2.771’

A

2.122’

0.895’

B

CG P/S Steel

0.709’

1.396’ - 2.160’ 2.618’

1.855’ 2.542’

C D

e

CG Concrete

0.327’

e x

2.771’

E

F

1.048’

0.895’

e x

0.000’

Abut 4

x

0.709’ 1.855’

2.542’

G H I

0.000’ 0.327’ e 1.212’ 1.396’ x 2.078’ 2.160’ 2.598’ 2.618’ 2.771’

K

J

Figure 7.12.8-5 Cable Path Eccentricities

7.12.9

Moment Coefficients Now that we’ve identified both the eccentricity and the percentage of prestressing force present at each tenth point along each span in the frame, we can now find the moment coefficients. The moment coefficients will help us solve for Pj, as well as compute flexural stresses in the concrete for determining fc and fci. The total moment coefficient consists of two parts; primary and secondary moment coefficients.

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Primary Moment Coefficient: The primary moment coefficient at any location along the frame is simply defined as the total force coefficient (FCpT) multiplied by the eccentricity (ex). Find the primary moment coefficients at each tenth point in Span 1:

MCP  ( FC pT )(ex )

(7.12.9-1)

where: MCp FCpT ex

= = =

primary moment force coefficient for loss (ft) total force coefficient for loss eccentricity as a function of x along parabolic segment (ft)

@ 0.0 L1: (0.770) (0) @ 0.1 L1: (0.776) (-1.212) @ 0.2 L1: (0.782) (-2.078) @ 0.3 L1: (0.788) (-2.598) @ 0.4 L1: (0.794) (-2.771) @ 0.5 L1: (0.800) (-2.618) @ 0.6 L1: (0.806) (-2.160) @ 0.7 L1: (0.812) (-1.396) @ 0.8 L1: (0.812) (-0.327) @ 0.9 L1: (0.806) (1.048) @ 1.0 L1: (0.787) (2.122)

= = = = = = = = = = =

0 ft -0.941 ft -1.625 ft -2.047 ft -2.200 ft -2.094 ft -1.741 ft -1.134 ft -0.266 ft 0.845 ft 1.670 ft

Secondary Moment Coefficient: Prestress secondary moments occur in multispan post-tensioned concrete frames where the superstructure is “fixed” to the column. The result of this fixity is an indeterminate structure. Prestress secondary moments are made up of two components: 

Distortions due to primary prestress moments, MCP = (FCpT)(ex), generate fixed-end moments at rigid column supports. These fixed-end moments are always positive due the the geometry of the cable path and always enhance the effects of prestressing at the bent caps. On the other hand, this component of prestress secondary moment always reduces the flexural effects of the prestressing force near midspan.



Prestress shortening of superstructure between rigid supports generates moments in the columns, which result in fixed-end moments in the superstructure. This component of secondary prestress moment occurs in frames with three or more spans. Long frames with short columns result in larger secondary prestress moments, which can be a significant factor in the design of the superstructure.



There are several analysis methods a designer can use to find the prestress secondary moments for a given frame. In stiffness based frame analysis software packages, the forces generated by prestressing the concrete are

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replicated with a series of uniform and point loads. In other words, primary “internally applied” moments and axial loads are converted into “externally applied” loads. The drawback of this method is that it is extremely difficult to do by hand, especially in multi-span frames. 

Each span within the frame is transformed into a simple span so that the ends can rotate freely.



Create an MCP /EI diagram, as the applied prestress moments are simply the prestress force times eccentricity.



Using conjugate beam theory, sum moments about one end of the beam to solve for the rotation at the opposite end. The moment needed to rotate the end of the beam back to zero is the “fixed-end” secondary moment due to MCP distortion.



When a frame is three spans or longer, secondary prestress deflections are generated in the column supports. The resulting column moments are a result of prestress shortening of the superstructure between the interior spans of the frame.



The fixed-end moments of the two components of prestress secondary moments are then combined; with the use of moment distribution, these fixedend moments are distributed to both the superstructure and columns based on the relative stiffness of each member.



Because Pj is still unknown, the prestress secondary moments must be solved for in terms of coefficient (MCS).

The secondary prestress moment coefficients used in this example problem are a result of the conjugate beam and moment distribution methods of analysis. The primary (MCP) and secondary (MCS) are then added together algebraically resulting in the total moment coefficient (MCPT). A summary of MCP, MCS, and MCPT are summarized in the following diagram:

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Prestress Moment Coefficient

Chapter 7 - Post-Tensioning Concrete Girders

-3

-2

-1

0

1

2

3

A

0.193

-1.623

0.386

-2.042

B

-2.195

-1.809 -2.081

1.412

1.713

2.342

-1.724

-1.117

0.509

C

-0.263

E

-0.540

-1.401

-0.547

-2.056

F

-1.903

0.852

1.539

-1.902

-1.400

0.313

0.852

G

H

2.0

0.686

1.392

I

-0.262

0.844

-1.115

-1.146

0.476

-1.749

0.381

J

-2.186

-2.034

-1.424

0.286 0.095 0.024 0.000

-1.615

-0.839

K

-0.934

-0.213 3.0 -0.236

0.190

Abutment 4

-1.806

-1.597

2.5

0.571

-2.073

-0.449

0.666

-1.717

0.499

0.762

0.857

1.700

2.241

0.920

1.429

2.285

-0.539

0.853 0.853

-0.546

-1.048

-1.202

-1.049

1.5

0.854 0.854

Length Along Frame (ft)

0.315

0.855 0.855 0.854 0.688

0.856

MCpt

MCs

MCp

Bent 3

Figure 7.12.9-1 Moment Coefficients

D

1.0

0.856

0.934

1.544

2.375

1.432

Bent 2

0.869 0.772 0.845 0.676

-0.441

0.579

-1.145

0.5

0.483

-1.753 -1.599

0.290

-1.430

-0.843

-0.222

-0.939

0.0

-0.201

0.000

0.023

0.097

Abutment 1

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7.12.10

Gravity Loads DC = Dead load of structural components and nonstructural attachments.  Includes the self weight of the box section itself, assuming a unit

weight of concrete (wc) = 0.15 (kcf) (Round up from 0.145) Table 3.5.1-1(AASHTO, 2012)  Type 732 barrier rail: (2 barriers) (0.4 klf ea.) = 0.80 klf DW = Dead load of wearing surface and utilities.  3 in. A.C. Overlay: (56 ft) (0.035 ksf) = 1.96 klf

Vehicular Live load (LL): The application of vehicular live loads on the superstructure shall be calculated separately for flexure and shear design. In each case, live load distribution factors shall be calculated for an interior girder, then multiplied by the total number of girders in the cross-section. Treating all girders as interior is justifiable because exterior girders become interior when bridges are widened. Distribution of live load per lane for moment in interior beams, with two or more design lanes loaded: 0.3

 13   S   1        N c   5.8   L 

0.25

(AASHTO Table 4.6.2.2.2b-1)

where: Nc =

number of cells in a concrete box girder (Nc  3)

S

=

spacing of beams or webs (ft) (6.0  S  13.0)

L

=

individual span length (ft) (60  L  240)

(Note: if L varies from span to span in a multi-span frame, so will the distribution factors) Distribution of live load per lane for shear in interior beams, with two or more design lanes loaded: 0.9

 S   d       7.3   12.0 L 

0.1

(AASHTO Table 4.6.2.2.3a-1)

where: S

=

spacing of beams or webs (ft) (6.0  S  13.0)

L

=

individual span length (ft) (20  L  240)

(Note: if L varies from span to span in a multi-span frame, so will the distribution factors)

Chapter 7 - Post-Tensioning Concrete Girders

7-60

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

d

depth of member (in.) (35  d  110)

=

Calculate the number of live load lanes for both moment and shear design for Span 1: 7.12.10.1

Live Load Lanes for Moment Using an equation from Table 4.6.2.2.2b-1: 0.3

 13   S  1 .        Nc   5.8  L 

0.25

0.3

 13   12  1        4   5.8  126 

0.25

= 0.879 lanes/girder

Number of live load moment lanes (Span 1) = (0.879 lanes/girder)(5 girders) = 4.395 lanes 7.12.10.2

Live Load Lanes for Shear Using an equation from Table 4.6.2.2.3a-1: 0.9

0.1

0.1

0.9  S   d   81  12       =    = 1.167 lanes/girder  7.3   12.0 L   7.3   (12.0)(126) 

Number of live load shear lanes (Span 1)

= (1.167 lanes/girder)(5 girders) = 5.835 lanes

Notes:  Assuming a constant structure width, the only term that will probably vary within a frame is the span length, L. Therefore, the number of live load lanes for both moment and shear will vary from span to span. Article 4.6.2.2.1 (Caltrans, 2014).  In negative moment regions, near interior supports, between points of DC flexural contraflexure, the “L” used to calculate negative moment is the average length of the two adjacent spans.  The Dynamic Load Allowance Factor (IM) (in the LFD code known as Impact) is applied to the design and permit trucks only, not the design lane load. Table 3.6.2.1-1 (Caltrans, 2014) summarizes the values of IM for various components and load cases.  The results of a gravity load analysis, including the unfactored moment envelopes of the (DC), (DC + DW) and (DC + DW+ HL93) load cases, are summarized in the diagram on the next page:

Chapter 7 - Post-Tensioning Concrete Girders

7-61

Chapter 7 - Post-Tensioning Concrete Girders

-60000

-50000

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

A

0.0

15280

23232

3673 2430

7667

9198

2542

-8749

1.0

C

-44609

E

-48134

D

-26395

-19046

-16062

-5127

-30989

-34502 -36714

-29118

-19430

-30484

-23030

-14688

-7377

2143

9552

Bent 2

F

1.5

17445 12659 10690

21175

11538

17869

19838

23511

31050

830

-30012

I

-42003

-32280 -34319

-45552

-8107

-22387 -27244

-18889

-28969

-14366

2.0

G H

-24134

-17034

965

5886

13665

21462

Length Along Frame (ft)

-544 -443

20550

20672

30596

34068

Bent 3

-15818

-9614

7192

447

532

2.5

11079

24791

J

6520

12916

13103 10888

13167

20337

K

3.0

1862

2209

3444

7735

12099

Abutment 4

15621 15545

25643

13144

6839

8115

16799

22948

Figure 7.12.10-1 Gravity and Service Load Moment Envelopes for Design of Prestressing

B

MDC+MDW+0.8HL93

MDC + MDW

MDC (w/o barrier)

0.5

15946 13799

20126

10911

26825

16370

29634

18554 18917

28460

13774 15640 12880 9095

Abutment 1

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-62

Unfactored Gravity Load Moment (kip-ft)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.12.11

Determine the Prestressing Force The design of the prestressing force, Pj , is based on the Service III Limit States. The Service III Limit States is defined as the “Load combination for longitudinal analysis relating to tension in prestressed concrete superstructures with the objective of crack control…” Article 3.4.1 (AASHTO, 2012). The following loads, and corresponding load factors, shall be considered in the design of Pj : DC

= Dead load of structural components and non-structural attachments, ( = 1.0) Article 3.3.2 (AASHTO, 2012)

DW = Dead load of wearing surface and utilities, ( = 1.0) Article 3.3.2 (AASHTO, 2012) HL93 = Service live load, ( = 0.8) Section 3.3.2 (AASHTO, 2012) Service III Limit States load cases: CA Amendments Table 5.9.4.2.2 (Caltrans, 2014). Case 1: No tension allowed for components with bonded prestressing tendons or reinforcement, subjected to permanent loads (DC, DW) only. M DC  DW y FC pT ( Pj ) MC pT ( Pj ) y   0 I A I

(7.12.11-1, modified 7.1-1)

Case 2: Allowable tension 0.19fc ksi are for components subjected to the Service III Limit States (DC, DW, (0.8) HL93), and subjected to not worse than moderate corrosion conditions, located in Environmental Areas I or II. Allowable tension 0.0948fc ksi are for components subjected to severe corrosion condions located in Enviromental Area III. M DC  DW  0.8 HL 93 yt ( bent ) I



FC pT ( Pj )



MC pT ( Pj ) y

A

I

 0.19 fc or 0.0948 fc

(7.12.11-2, modified 7.1-1) The design of the jacking force usually controls at locations with the highest demand moments within a given frame. Upon inspection of the demand moment diagram plotted earlier in this example, the design of Pj will control at one of two locations:  

The right face of the cap at Bent 2 (top fiber) Mid-span of Span 2 (bottom fiber)

Load cases 1 and 2 must be applied at both the right face of the cap at Bent 2, and at midspan of Span 2, with the overall largest Pj controlling the design of the entire frame.

Chapter 7 - Post-Tensioning Concrete Girders

7-63

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Solve for the jacking force based on two-end stressing data gathered earlier in the example problem: Right face of the cap at Bent 2 (top fiber):

C T

C T

0

C T

Case 1

yt(bent)

-0.19f’c

C T

Case 1

Case 2 Neutral Axis Case 2

Case 1:

Case 2:

M DC  DW yt (bent face) I (bent face)

+

FC pT Pj A(bent face)

M DC  DW  0.8 HL93 yt (bent ) I (bent )

+

+

MC pT Pj yt (bent face)

FC pT Pj A(bent face)

I (bent face)

+

=0

MC pT Pj yt (bent face) I (bent face)

= 0.19

fc

Figure 7.12.11-1 Elastic Stresses in an Uncracked Prestress Beam. Effects of Prestress by Component at Top of the Beam Reading Figure 7.12.13-1 MDC+DW = -36,713 kip-ft MDC+DW+0.8HL93 = -48,134 kip-ft Interpolating the Force Coefficent from Table 7.12.7.5-1 between points D and E FC@pt D = 0.787 FC@pt E = 0.769 Span 2 Length = 168 ft Distance from CL of column to face of cap (pt D) = 4 ft Distance from CL of column to location of first inflection point of Span 2 (pt E) = 16.8 ft  0.787  0.769     4  0   0.787  0.783  0  16.8 

Reading Figure 7.12.9-1 for MCPT @ the face of cap at Bent 2: MCPT = 2.375

Chapter 7 - Post-Tensioning Concrete Girders

7-64

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Rearranging terms of 7.12.11-1 and 7.12.11-2 to solve for Pj: M DC  DW yt (bent face )

0 I (bent ) Case 1: Pj   MC pT yt ( bent face ) FC pT  A( bent face ) I ( bent face ) ( 36, 713)(3.25)

0 824 Pj    8, 957 kips 0.783 (2.375)(3.25)  115 824

M DC  DW  0.8 HL 93 yt ( bent face ) I ( bent face )

Case 2: Pj  

FC pT A( bent face ) ( 48,134)(3.25) 824 0.783

Pj  





 (0.19) f c 144 

MC pT yt ( bent ) I ( bent face )

 (0.19) 4(144)  8, 359 kips

(2.375)(3.25)

115

824

Mid-span of Span 2 (bottom fiber): C T

C T

C T

C T Neutral Axis Case 2

yb(mi

Case 2

Case 1

d)

Case 1

0 Case 1: Case 2:

M DC  DW yb ( mid ) I ( mid )

+

FC pT Pj

M DC  DW  0.8 HL93 yb ( mid ) I ( mid )

+

A( mid ) +

MCPT Pj yb ( mid )

FC pT Pj A( mid )

I ( mid ) +

=0

MC pT Pj yt (bent ) I (bent )

-0.19f’c

= 0.19

f c

Figure 7.12.11-2 Elastic Stresses in an Uncracked Prestress Beam. Effects of Prestress by Component at the Bottom of the Beam

Chapter 7 - Post-Tensioning Concrete Girders

7-65

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Reading the Force Coefficent from Table 7.12.7.5-1 at point F FC@pt F = 0.742 Reading Figure 7.12.9-1 for MCPT @ the CL of Span 2: MCPT = -1.202 Reading Figure 7.12.10-1 MDC+DW = 23,511 kip-ft MDC+DW+0.8HL93 = 34,068 kip-ft M DC  DW yb ( mid )

0

I ( mid )

Case 1: Pj  

FC pT A( mid )



( MC pT ) yb ( mid ) I ( mid )

(23, 511)( 3.8)

Pj  

0 729  9,100 kips 0.742 ( 1.202)( 3.8)  103 729

M DC  DW  0.8 HL 93 yb ( mid ) I ( mid )

Case 2: Pj  

FC A( mid )



(34, 068)( 3.8)

Pj  

729 0.742 103



 (0.19) f c

MCyb ( mid ) I ( mid )

 (0.19)( 4) (144)

( 1.202)( 3.8)

 9,124 kips

729

Therefore, Pj = 9,124 kips, round to the nearest 10 kips, Pj = 9,120 kips Notes: The overall largest Pj was calculated at the midspan of Span 2 under the Case 2 load condition. Two observations can be made: Now that we have a Pj we can check our elastic shortening assumption using CA Amendments 5.9.5.2.3b-1 and Equation 7.5.1.3-1:

f cgp  f g  f ps f pES  0.5

Pj Pj e2 M DL e  (  ) Ig Ag Ig

Ep f Ect cgp

Chapter 7 - Post-Tensioning Concrete Girders

(7.5.1.3-1) (CA Amendments 5.9.5.2.3b-1)

7-66

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 7.12.11-1 Values of fcgp and fpES Loc

fcgp

fpES

Loc

fcgp

fpES

Loc

fcgp

fpES

0.024 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.968

-0.615 -0.770 -1.170 -1.579 -1.772 -1.682 -1.357 -0.929 -0.632 -0.797 -1.281

-2.285 -2.863 -4.348 -5.870 -6.585 -6.252 -5.043 -3.453 -2.349 -2.964 -4.761

1.024 1.100 1.200 1.300 1.400 1.500 1.600 1.700 1.800 1.900 1.976

-1.546 -0.847 -0.655 -1.137 -1.677 -1.903 -1.689 -1.155 -0.665 -0.830 -1.504

-5.745 -3.147 -2.433 -4.227 -6.232 -7.071 -6.277 -4.293 -2.471 -3.083 -5.590

2.034 2.100 2.200 2.300 2.400 2.500 2.600 2.700 2.800 2.900 2.975

-1.424 -0.934 -0.594 -0.791 -1.187 -1.538 -1.694 -1.586 -1.246 -0.832 -0.615 Average

-5.292 -3.470 -2.208 -2.940 -4.412 -5.716 -6.297 -5.894 -4.629 -3.091 -2.285 -4.351

At this point, we would rerun our numbers using the 4.4 ksi value for fpES . However, for this example we will choose not to rerun the numbers. The 1.4 ksi difference between assumed and calculated result in about a 3% increase in Pj .

7.12.12

Determine the Required Concrete Strength Now that the jacking force has been calculated for this structure, we can determine the stresses in the concrete due to prestressing. Prestress stresses need to be computed in order to determine the initial (fci) and final (fc) concrete strengths required. The design of fci is based on concrete stresses present at the time of jacking, which includes the initial prestress stress, fpi . The initial prestress stress considers losses due to friction (FCpF) only. The design of fc is based on service level concrete stresses that occur after a period of time, which includes the effective prestress stress, fpe . The effective (total) prestress stress considers the effects of all prestress losses (FCpT). The equations for concrete stresses due to prestressing are as follows:

f pi 

Pj FCF

f pe 

Pj FC pT

Ag

Ag



Pj (e)( FCF ) yb



Pj (MC pT ) yb

Ig

Ig

(7.12.12-1)

(7.12.12-2)

where: Ag

=

gross area of section (in.2)

FC

=

force coefficient for loss

FCpF

=

force coefficient for loss from friction

Chapter 7 - Post-Tensioning Concrete Girders

7-67

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

e

=

eccentricity of the anchorage device or group of devices with respect to the centroid of the cross section. Always taken as a positive (ft)

fpe

=

effective stress in the prestressing steel after losses (ksi)

fpi

=

initial stress in the prestressing steel after losses, considering only the effects of friction loss. No other P/S losses have occurred (ksi)

Ig

=

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (in.4)

MCPT

=

total moment force coefficient for loss (ft)

Pj

=

force in prestress strands before losses (kip)

Both initial and final stresses for the concrete top and bottom fibers due to the effects of prestressing have been calculated, and are shown on the following two diagrams.

Chapter 7 - Post-Tensioning Concrete Girders

7-68

Chapter 7 - Post-Tensioning Concrete Girders

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

A

-0.366

0.081

-0.449

0.023

B

-0.293

0.059

-0.113

0.246

C

0.131

0.549

D

1.0

E

0.081

F

-0.307

-0.084 1.5

-0.269

-0.140

-0.032

0.148

-0.269

-0.140

-0.032

0.080

0.125

1.001

G

H

I

0.436

2.0

0.430

0.548 0.472

0.621

0.436

0.495 0.504

0.833

0.571 0.560

0.764

0.931

0.996

0.921

Bent 3

0.866 0.839

0.464 0.468 0.395 0.389 0.324

0.191

0.456 0.460

fpe

fpi

Length Along Frame (ft)

0.125

0.192

0.473 0.469 0.465 0.439 0.461 0.437 0.431 0.390 0.396 0.325

0.869

Pj(MC)y/Ig

Pj(FC)/Ag

-0.115

0.128

0.246

0.376

0.059

0.078

-0.409

-0.294

2.5 -0.054

0.195

0.491 0.489 0.487

0.493

Figure 7.12.12-1 Top Fiber Concrete Stresses Due to Prestressing

-0.464

-0.410

-0.054

0.5

0.079

0.197

0.379

0.494

0.766

0.585 0.593

0.834

0.935

0.928

1.025

0.916

Bent 2 1.022

0.496 0.505

0.625

0.487 0.489 0.491 0.492

-0.093

0.034

-0.050

0.114

0.480 0.483

0.302

-0.051

-0.216

0.0

0.261

0.422

0.477

0.540

Abutment 1

J

0.472

-0.365

0.081

0.113

K

-0.215

-0.055 3.0

0.259

0.302

0.478 0.474

-0.050

0.022

-0.463-0.448

-0.093

0.033

0.485 0.481

0.535

Abutment 4

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-69

Initial and Final Concrete Stresses due to P/S (ksi)

Chapter 7 - Post-Tensioning Concrete Girders

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

A

0.0

0.095

0.398

0.841

1.062

1.432

0.686 0.638

1.278

B

0.5

0.547

0.744

0.807

-0.208

D E

Length Along Frame (ft)

F

1.5

0.210

G

-0.268

-0.037

0.328

-0.067

I

-0.440

-0.327

0.072

-0.185 -0.167 -0.359

H

0.686 0.495

0.253

0.436

2.0

0.147

0.289

0.430

0.472

0.763

0.468 0.365

0.645

1.094

0.464

0.546

0.744

0.806

1.288

0.456 0.460

0.804

0.852

1.345

Bent 3

0.836 0.685

0.867

1.014

1.430

0.102

2.5

0.491 0.489 0.487 0.440

0.493

0.640

1.027

1.276

Figure 7.12.12-2 Bottom Fiber Concrete Stresses Due to Prestressing

C

1.0

-0.037

0.211 0.079 0.054

0.289

-0.327 -0.268 -0.434-0.453

-0.193

-0.070

0.102

0.645

0.764

1.096

1.288

fpe

fpi

Pj(MC)y/Ig

Pj(FC)/Ag

0.473 0.469 0.465 0.461 0.437 0.431 0.365

Bent 2

0.254

0.686 0.496

1.028

0.487 0.489 0.491 0.492 0.494 0.441 0.325

0.837

0.869

1.017

0.891

1.084

1.490

0.480 0.483

0.677

0.540 0.477

0.708 0.755

1.009 0.952

1.284

1.445

Abutment 1

1.443

J

0.948

1.284

0.472

0.751

0.715

1.008

K

3.0

0.101

0.478 0.474 0.397

0.677

0.841

0.485 0.481

0.890

1.059 1.081

1.488

Abutment 4

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-70

Initial and Final Convtere Stresses due to P/S (ksi)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Design of both initial and final concrete strengths required are governed by the Service I load case. This load case is defined as a load combination relating to the normal operational use of the bridge with a 55 mph wind and all loads taken at their nominal value ( = 1.0) Article 3.4.1(AASHTO, 2012). 7.12.12.1

Design of fc The definition of fc is “the specified strength concrete for use in design” Article 5.3 (AASHTO, 2012). Article 5.4.2.1 (AASHTO, 2012) specifies the compressive strength for prestressed concrete and decks shall not be less than 4.0 ksi. Additionally, Table 5.9.4.2.1-1 (AASHTO, 2012) lists compressive stress limits for prestressed components at the service Limit States after all losses as: 

In other than segmentally constructed bridges due to the sum of the effective prestress and permanent loads, the concrete has a compressive stress limit of 0.45 fc (ksi).

f pe  f DC  DW

fc 

0.45

where: fDC+DW = 

(ksi)

(7.12.12-3)

stress in bridge from DC and DW load cases (ksi)

Due to the sum of effective prestress, permanent loads and transient loads, the concrete has a compressive stress limit of 0.60 fc (ksi).

fc 

f pe  f DC  DW  f HL 93 0.60

(ksi)

(7.12.12-4)

Solving for fc at face of support at Bent 2 (0.0 L2 point). Using Equations 7.12.12-2 and 7.12.12-3 for only the DL case. At the cap face for Bent 2 the deck is in tension under service loads, therefore the prestressing steel is close to the deck to pull the section together, and resist tension. The controlling concrete strength demand will be opposite of the tension where the service loads act in compression, and the prestressing force acts in tension. fpi

 

Pj FC pT Ag



Pj ( MC pT ) yb Ig

 9,120 (0.783)  (9,120 )(2.375)(3.5)   30.0 ksf   115 824

Chapter 7 - Post-Tensioning Concrete Girders

7-71

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Notice that the stress due to the prestressing steel is negative. This means that the prestressing steel is pulling together the face of the concrete that is in tension, causing the side opposite side of the prestressing steel to be in tension.

f pe  30.0 ksf f DC  DW 

1

 0.21ksi

144

M DC  DW ( yb ) Ig

f pe  f DC  DW



36,713(3.5)

1

f DC  DW  156.0 ksf

fc 



824

 156.0 ksf

 1.08 ksi

144

0.21  1.08

0.45

 1.9 ksi

0.45

Using Equation 7.12.12-4 for only the LL case.

f HL 93 

M HL 93 ( yb ) Ig

f HL 93  60.6 ksf



1

14, 275 kip-ft(3.5ft) 824 ft 4

 60.6 ksf

 0.42 ksi

144

Using Equation 7.12.12-5 for only the service load case.

fc 

f pe  f DC  DW  f HL 93 0.60



0.21  1.08  0.42

 2.16 ksi

0.60

Therefore the minimum fc controls from Article 5.4.2.1 (AASHTO, 2012): fc = 4.0 ksi

Chapter 7 - Post-Tensioning Concrete Girders

7-72

Figure 7.12.12-3 Determining Final Concrete Strength Required

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Chapter 7 - Post-Tensioning Concrete Girders

7-73

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.12.12.2

Design of fci The definition of fci is “the specified compressive strength of concrete at time of initial loading of prestressing” (AASHTO Article 5.3). There are two criteria used to design fci, and they are as follows: The compressive stress limit for pretensioned and post-tensioned concrete components, including segmentally constructed bridges, shall be 0.60 fci (ksi). Only the stress components present during the time of prestressing shall be considered (AASHTO Article 5.9.4.1.1). f ci 

f pi  f DC w/ o

where: fDC w/o b =

0.60

b

(ksi)

(7.12.12-6)

stress in concrete due to the Dead Load of the structural section only (ksi)

The specified initial compressive strength of prestressed concrete shall not be less than 3.5 ksi (MTD 11-3). Solving for fci at the cap face Bent 2 (0.02 L2 point). Interpolating the Force Coefficent due to Friction from Table 7.12.7.5-1 between points D and E FCpF@pt D = 0.925 FCpF@pt E = 0.907 Span 2 Length = 168 ft Distance from CL of column to face of cap (pt D) = 4 ft Distance from CL of column to location of first inflection point of Span 2 (pt E) = 16.8 ft

 0.925  0.907     4  0   0.925  0.921  0  16.8  Interpolating the eccentricities from Figure 7.12.8-5 between points D and E e@pt D = 2.122 ft e@pt E = 0.896 Span 2 Length = 168 ft Distance from CL of column to face of cap (pt D) = 4 ft Distance from CL of column to location of first inflection point of Span 2 (pt E) = 16.8 ft

 2.122  0.896     4  0   2.122  1.830  0  16.8 

Chapter 7 - Post-Tensioning Concrete Girders

7-74

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Using Equations 7.12.12-1 and 7.12.12-5 fpi = 

Pj FCF Ag



Pj (e)( FCF ) yb Ig

 9,120 (0.921)  9,120 (1.830 ) (0.921) (  3.5)   7.75 ksf   115 824

f pi  7.5

1

 0.054 ksi

144

Using Equation 7.12.12-6

f DC w/ o B 

M DC w/ o b ( yb ) Ig



30, 990 (3.5) 824

 68.2 ksf

f DC w / o B  68.2 ksf  0.474 ksi

Using Figure 7.12.12-4 the minimum fc controls from MTD 11-3: fci = 3.5 ksi

Chapter 7 - Post-Tensioning Concrete Girders

7-75

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

0.863

A

-1.025-1.057

0.0

0.996

-1.219

0.739

-1.363

0.649

Abutment 1

Top Fibers

Bottom Fibers

Chapter 7 - Post-Tensioning Concrete Girders B

-1.464

1.0

0.197 0.132

0.524

D

E

-1.300

0.629

F

-1.222

-1.669

-1.460 -1.562

2.0

G H

I

J

-1.615 -1.632 -1.685 -1.714 -1.716 -1.686

2.5

-1.483

0.645

3.0

0.979

-1.286

K

-1.079

0.809

Abutment 4

0.530 0.504 0.461 0.430 0.419 0.429 0.462 0.291 0.077

-1.348

0.601

0.688

Specified minimum f'ci = 3.5 ksi

-1.095 -1.044 -1.069 -1.128

1.5

Length Along Frame (ft)

-1.181

0.709

0.763 0.790 0.783 0.748

Figure 7.12.12-4 Determining Initial Concrete Strength Required

C

-1.451 -1.521 -1.554 -1.574 -1.585 -1.589-1.595 -1.613

0.5

0.592 0.556 0.529 0.510 0.496 0.480

Specified minimum f'ci = 3.5 ksi

Bent 3 Reserve Capacity

Bent 2

Reserve Capacity

Final Concrete Strength Required (ksi)

3.5

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-76

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7.12.13

Design of Flexural Resistance For rectangular or flanged sections subjected to flexure about one axis where the approximate stress distribution as specified in Article 5.7.2.2 (AASHTO, 2012) is used and for which fpe is not less than 0.5 fpu, the average stress in the prestressing steel, fps, may be taken as:   

f ps  f pu  1 

kc 

 d p 

 f py k  2  1.04  f pu 

(AASHTO 5.7.3.1.1-1)

  

(AASHTO 5.7.3.1.1-2)

Alternatively, the stress in the prestressing steel may be determined by strain compatibility (see AASHTO 5.7.3.2.5). For rectangular section behavior, the distance between the neutral axis and the compressive face can be represented as:

c

Aps f pu  As f s  As f s 0.85 f c β1b  kAps

(AASHTO 5.7.3.1-4)

f ps dp

Substitute fps for fpu after solving Equation 5.7.3.1.1-1 (AASHTO, 2012) The factored resistance Mr shall be taken as:

M r  M n

(AASHTO 5.7.3.2.1-1)

For flanged sections subjected to flexure about one axis and for biaxial flexure with axial load as specified in Article 5.7.4.5, (AASHTO, 2012), where the approximate stress distribution specified in Article 5.7.2.2 (AASHTO, 2012) is used and the tendons are bonded and where the compression flange depth is less than a=1 c, as determined in accordance with Equation 5.7.3.1.1-3, the nominal flexural resistance may be taken as: a a a h    a  M n  Aps f ps  d p    As f s  d s   As f s d s    0.85 f c(b  bw )h f   f  2 2   2  2 2  Prestressing Steel only

f ps f pu

Additional Flexural Steel

 0.9

k  0.28

Chapter 7 - Post-Tensioning Concrete Girders

Compression Steel

Flanged Section Component a=β1c  h f

(AASHTO C5.7.3.1.1-1)

(AASHTO C5.7.3.1.1-1)

7-77

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Aps 

Pj

(7.12.13-1)

0.75 f pu

where: Aps

=

area of prestressing steel (in.2)

As

=

area of non-prestressed tension reinforcement (in.2)

b

=

width of the compression face of a member (in.)

c

=

distance from extreme compression fiber to the neutral axis (in.)

dp

=

distance from extreme compression fiber to the centroid of the prestressing tendons (in.)

f c

=

specified compressive strength of concrete used in design (ksi)

fps

=

average stress in prestressing steel at the time for which the nominal resistance is required (ksi)

fpu

=

specified tensile strength of prestressing steel (ksi)

Mr

=

factored flexural resistance of a section in bending (kip-in.)

Mn

=

nominal flexure resistance (kip-in.) 

1

=

ratio of the depth of the equivalent uniformly stressed compression zone assumed in the strength Limit States to the depth of the actual compression zone



=

resistance factor

The factored ultimate moment, Mu , shall be taken as the greater of the following two Strength I and II Limit Statess as defined in California Amendments Article 3.4.1 and Table 3.4.1-1 (Caltrans, 2014). Strength I: Mu(HL93) = 1.25 (MDC ) + 1.50 (MDW) + 1.75 (MHL93) + 1.00 (MP/S s) Strength II: Mu(P-15) = 1.25 (MDC ) + 1.50 (MDW) + 1.35 (MP-15) + 1.00 (MP/S s) The largest value of Mu indicated the governing Limit States at a given location. It is possible to have different Limit Statess at different locations. Unless otherwise specified, at any section of a flexural component, the amount of prestressed and nonprestressed tensile reinforcement shall be adequate to develop a factored flexural resistance, Mr (min), at least equal to the lesser of: 1.33 Mu as defined in Section 5.7.3.3.2 (AASHTO, 2012)

f r  0.24 f r   S M cr   3 ( 1 f r   2 f cpe )Sc  M dnc ( c  1)  Snc   Chapter 7 - Post-Tensioning Concrete Girders

(AASHTO 5.4.2.6) (AASHTO 5.7.3.3.2-1)

7-78

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The second part of the equation is crossed out because this is not a composite section. Article 5.7.3.3.2 defines: 1

=

1.6 for super structures that are not precast segmental

2

=

1.1 for bonded tensions

3

=

0.75 if additional mild reinforcement is A 706, grade 60 reinforcement

fr

=

modulus of rupture of concrete (ksi)

fcpe

=

compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi)

Mcr

=

cracking moment (kip-in.)

Sc

=

section modulus for the extreme fiber of the composite sections where tensile stress is caused by externally applied loads (in.3)

1

=

flexural cracking variability factor

2

=

prestress variability factor

3

=

ratio of specified minimum yield strength to ultimate tensile strength of reinforcement

where:

The prestressing steel present in the section by itself may be enough to resist the applied factored moment. However, additional flexural steel may have to be added to provide adequate moment resistance for the Strength I and II Limit Statess. Flexural steel provided for seismic resistance can be relied upon for Strength Limit Statess. AASHTO Article 5.7.3.3, defines a limit on tension steel to prevent overreinforced sections, has been eliminated in the 2006 interims. The current approach involves reducing the flexural resistance factor when the tensile strain in the reinforcement falls below 0.005. In other words, over-reinforced sections are allowed by the code, but a more conservative resistance factor is applied. Conventional designs will likely result in tensile strains greater than 0.005. The tensile strain can be determined using the c/de ratio. From a simple plane strain diagram assuming concrete strain of 0.003, a c/de ratio of 0.375 corresponds to a tensile strain of 0.005. If the c/de ratio exceeds 0.375, then a reduced must be used as defined in Article 5.5.4.2 (Caltrans, 2014).

Chapter 7 - Post-Tensioning Concrete Girders

7-79

Chapter 7 - Post-Tensioning Concrete Girders

-40000

-30000

-20000

-10000

0

10000

20000

30000

A

0.0

1555

2977 2157 5849 1022 273

8074

10491

1716

9941

2084

12383

B

0.5

1838 1225 286

2257

8762

-13958

-7424

-7767

-981

-9187 -13994

C D

-32619

-30651

-23630

-20455 -21153

E

-15703 -12634 -16912 -14275

-9318

-2134 -2575 -4095 -3850 1.0

Bent 2

-5729 -9618

-72 -471

2308

12405

F

1.5

2627

13196

20884

26007

2365

12343

18811

23868

1521

9747

12144

17846

Length Along Frame (ft)

1408

9864

11251

18002 18364

24083

6152 868 97

8646

G

2.0

-2502

H

-30493

-28678

-23466

I

-19885 -20569

-14041

-12155 -16458

-9531

-3826 -3602

-8875 -13774 -15126

-1909

Bent 3

-14629

-7755

-8536

-1077

472 61

8325

13717

912

2.5

1477

J

1755

12254 12527

11667

19991

1746

11558

1451

9275

11465

15932

K

3.0

1545

2869 5456 1960 869 248

6866

9495

Abutment 4

13866 13799

21727 22020

7203

10855

18867

Figure 7.12.13-1 Gravity Load Moment Envelopes for Design of Flexural Resistance

Mp-15

Mhl93

Mdw

2125

14819

11518

20768

9686

13396 13070

14531

24451 24096

16470 16793

22246

Mdc

13564

17837

Abutment 1

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-80

Unfactored Moments (kip-fti)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Find the flexural resistance of the section at the right face of cap at Bent 2 considering the Area of P/S steel only. If required, find the amount of additional flexural steel needed to resist the factored nominal resistance Mn. MP/S s = Pj (MCs)

(7.12.13-2)

where: MP/S s = moment due to the secondary effects of prestressing (k-ft) Step 1: Determine the controlling Strength Limit State used to determine the factored ultimate moment, Mu: Strength I: (MCs)

= 0.856 ft (Figure 9.12.12-1)

MP/S s

= (9,120 kips) (0.856 ft) = 7,810 kip-ft

Mu(HL93)

= 1.25 (MDC ) + 1.50 (MDW) + 1.75 (MHL93) + 1.00 (MP/S s)

Mu(HL93)

= 1.25 (-32,619) + 1.50 (-4,095) + 1.75 (-14,275) + 1.00 (7,810) = -64,090 kip-ft

Strength II: Mu(P-15)

= 1.25 (MDC ) + 1.50 (MDW) + 1.35 (MP-15) + 1.00 (MP/S s)

Mu(P-15)

= 1.25 (-32,619) + 1.50 (-4,095) + 1.35 (-23,630) + 1.00 (7,810) = -71,010 kip-ft

The Strength II Limit State controls, Mu = -71,010 kip-ft Step 2: Compute Mcr to determine which criteria governs the design of the factored resistance, Mr (1.33Mu or using AASHTO 5.7.3.3.2-1). Sc 

I bent face ybent face



(824 ft 4 )124 3.25 ft(12)

 438,100 in.3

f r  0.24 fc  0.37 4  0.48 ksi fcpe = 1.025 ksi (from plot of P/S stresses)

 S  M cr   2 ( 1 f r   2 f cpe ) Sc  M drc  c  1   Snc  





 0.75 1.6(0.48)  1.1(1.024)  438,100  0  622, 700 kip-in.

Chapter 7 - Post-Tensioning Concrete Girders

7-81

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Mcr = 622,700 kip-in. = 51,900 kip-ft Mr (min) = the lesser of: Mcr = (51,900) = 51,900 kip-ft 1.33Mu = 1.33(71,010) = 94,440 kip-ft Therefore, Mr (min) = 51,900 kip-ft Step 3: Compute the nominal moment resistance of the section based on the effects of the prestressing steel using AASHTO 5.7.3.1.1-4 only and substituting out Aps:

Pj

c

Aps f pu  As f s  As f s 0.85 f c β1b  kAps

f pu dp



0.75 f pu

f pu  As f s  As f s

0.85 f c β1b  k

Pj

f pu

0.75 f pu d p

Assuming no compression or tension resisting mild steel, As and As both equal zero. b

=

The soffit width = overall width – overhang width (including slope)

b

=

58(12)+10 – 2[5(12)+10.5] = 517 in.

dp

=

structure depth – prestressing force distance to deck (interpolated between points D and E)

dp

=

81 – 17.0 = 64.0 in.

9,120

c

0.75

00

 9,120  0.85(4)(0.85) (517)  0.28    0.75(64) 

 7.9 in.

7.9 in. < hsoffit = 12 in.; therefore, rectangular section assumption is satisfied Using AASHTO Equation 5.7.3.1.1-1:   

f ps  f pu  1  k

c 

7.9     270 1  0.28   260.7 ksi dp  64.0  

Modifying AASHTO Equation 5.7.3.2.2-1 for rectangular sections produces Equation 7.12.13-3:  

M n  Aps f ps  d p 

a



2

Chapter 7 - Post-Tensioning Concrete Girders

(7.12.13-3)

7-82

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

From Article 5.7.2.2 (AASHTO, 2012):

a  β1c

(7.12.13-4)

a  β1c  0.85(7.9)  6.7 in. Aps 

Pj

9,120



0.75 f pu

 45.0 in.

2

0.75(270)

a 6.7    M n  Aps f ps  d p    45.0(260.7)  64.0    711, 500 kip-in. 2 2    Mn = 711,500 kip-in. = 59,300 kip-ft Calculating Mn = 0.95 (59,300) = 56,350 kip-ft > 751,900 kip-ft shows that no additional flexural steel is required. However, for illustrative purposes, let’s determine As based on Mr = 77,300 kip-ft. Step 4: Compute the area of mild steel required to increase Mn to resist the full factored resistance, Mr: Rearranging AASHTO 5.7.3.1.1-4 and Equation 7.12.13-4 and substituting out the values for fps and Aps, results in Equation 7.12.13-5.

Pj a

0.75 f pu

f pu  As f s  As f s

0.85 f c b 



k Pj f pu

(7.12.13-5)



β1 0.75 f pu d p

9,120  (60) As  0 a 0.75   3.35  0.0166 As 2   0.28(9,120) 2  0.85(4) (517)   (0.85)(0.75) (64.0)  

Note that we will assume fs = fy. This assumption is valid if the reinforcement at the extreme steel tension fiber fails. We can check this by measuring t at the end of the calculation. Modifying AASHTO Equation 5.7.3.2.2-1 for rectangular sections produces Equation 7.12.13-6.  

M n  Aps f ps  d p 

a

a    As f y  d s   2 

2

Chapter 7 - Post-Tensioning Concrete Girders

(7.12.13-6)

7-83

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

where: ds =

distance from extreme compression fiber to the centriod of the nonprestressed tensile reinforcement (in.)

ds =

d – 0.5(hdeck) = 81 – 0.5(9.125) = 76.4 in.

 77,300 12    0.95   (45.0)(260.6) (64.0  (3.35  0.0166 As ))  ...   …+ 60 As (76.4 – (3.5 + 0.0166 As)) 0.99 As2 – 4,190 As + 264,990 = 0 Solving the quadratic equation: As = 64.2 in.2

(65 # 9 bars As = 65 in.2)

Step 5: Verify Assumptions – Two assumptions were made in the determination of As. The first was that the mild steel would yield and we could use fy for fs. The validity of this assumption can be checked by calculating the t. According to the California Amendments (Caltrans, 2014) and Figure 7.12.13-2 if t is > 0.005 then the section is tension controlled and 1.00, which is more conservative than the 0.95 we used initially. These values can be easily obtained with a simple strain diagram setting the concrete strain to 0.003.

Figure 7.12.13-2 Variation of  with Net Tensile Strain t for Grade 60 Reinforcement and Prestressed Members (California Amendments Figure C5.5.4.2.1-1, 2014)

Chapter 7 - Post-Tensioning Concrete Girders

7-84

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

cu = 0.003 c

de - c

Figure 7.12.13-3 Strain Diagram where: c

=

distance from extreme compression fiber to the neutral axis (in.)

de =

effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.)

cu =

failure strain of concrete in compression (in./in.)

t

net tensile strain in extreme tension steel at nominal resistance (in./in.)

=

Having just calculated As, it is possible to calculate both c and de using AASHTO 5.7.3.1.1-4 and substituting out fps and Aps:

Pj 0.75

c

 As f s  As f s

0.85 f s β1b  k

Pj

0.75 d p

9,120



1

0.75

 (64.2)(60)

9,120  1  (0.85)(4)(0.85)(517)  (0.28)   0.75  64.0 

 10.35 in.

Therefore c = 10.35 in.  12 in., the rectangular section assumption satisfied.

de 

Aps f ps d p  As f y d s Aps f ps  As f y

(45.02)(260.6)(64.2)  (64.2)(60)(76.50) (45.02)(260.6)  (64.2)(60) de = 67.1 in. 

Chapter 7 - Post-Tensioning Concrete Girders

7-85

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

By similar triangles: ε c  ε t

de  c

c

εt 

:

εc 0.003 de  c    67.1  10.35  0.016 > 0.005  c 10.35

Therefore, the mild steel yields and = 1.00 Find the flexural resistance of the section at midspan of Span 2 considering the area of P/S steel only. If required, find the amount of additional flexural steel needed to resist the factored nominal resistance, Mn : Step 1: Determine the controlling Strength Limit State used to determine the factored ultimate moment, Mu: Strength I: MP/S s

=

(9,120) (0.854) = 7,790 kip-ft

Mu(HL93) =

1.25 (MDC ) + 1.50 (MDW) + 1.75 (MHL93) + 1.00 (MP/S s)

Mu(HL93) =

1.25 (20,884) + 1.50 (2,627) + 1.75 (13,196) + 1.00 (7,790)

=

60,930 kip-ft

Strength II: Mu(P-15) =

1.25 (MDC ) + 1.50 (MDW) + 1.35 (MP-15) + 1.00 (MP/S s)

Mu(P-15) =

1.25 (20,884) + 1.50 (2,627) + 1.35 (26,007) + 1.00 (7,790)

=

72,940 kip-ft

The Strength II Limit State controls, Mu = 72,940 kip-ft Step 2: Compute Mcr to determine which criteria governs the design of the factored resistance, Mr (1.33Mu or using AASHTO 5.7.3.3.2-1).

Sc 

I mid yb  mid



(728.94)

ft 3  331, 475 in.3

3.8

f r  0.24 fc  0.24 4  0.48 ksi fcpe = 0.851 ksi (from plot of P/S stresses)





M cr   3   1 f r   2 f cpe Sc  M dnc 

 0.75 1.6  0.48   1.1 0.851  331, 500  423,800 kip-in.

Mcr = 423,800 kip-in. = 35,300 kip-ft

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Mr (min) = the lesser of: 1.33Mu = 1.33(72,940) = 97,000 kip-ft Therefore Mr (min) = 35,300 kip-ft Step 3: Compute the nominal moment resistance of the section based on the effects of the prestressing steel using AASHTO 5.7.3.1-4 only and substituting out fps and Aps:

Pj

c

Aps f pu  As f s  As f s 0.85 f c β1b  kAps

f pu dp



0.75 f pu

f pu  As f s  As f s

0.85 f c β1b  k

Pj

f pu

0.75 f pu d p

Assuming no compression or tension resisting mild steel, As and A′s both equal zero. b

= compression (top) flange width = 58 ft 10 in. = 706 in.

dp = structure depth – prestressing force distance to soffit dp = 81– 12.5 = 68.5 in. 9,120 00 0.75 c  5.8in.  9,120  1 0.85(4)(0.85)(706)  0.28    0.75  68.5

5.8 in. < hdeck = 9.0 in., therefore, rectangular section assumption satisfied.   

f ps  f pu 1 

kc  5.8     270 1  0.28   263.6 ksi dp  68.5  

Using Equations 7.12.13-3 and 7.12.13-4: a  β1c  0.85(5.8)  4.9 in. a 4.9    M n  Aps f ps  d p    (45.0)(263.6)  68.5    783,500 kip-in. 2 2   

Mn = 783,500 kip-in. = 65,300 kip-ft Calculating Mn = 0.95 (65,300) = 62,000 kip-ft > 33,500 kip-ft shows that no additional flexural steel is required. At this point, the calculation should stop, and we would not include any mild steel for the bottom of the superstructure at midspan. However, to illustrate this example we will continue by setting Mr = 75,000 kip-ft.

Chapter 7 - Post-Tensioning Concrete Girders

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Step 4: Compute the area of mild steel required to increase Mn to resist the full factored, Mr . Using Equation 7.12.13-5:

Pj a

0.75 f pu

f pu  As f s  As f s

0.85 f c b 

Pj f pu k β1 0.75 f pu d p

9,120  (60) As  0 a 0.75   2.48  0.012 As 2  0.28  9,120  1  2  0.85(4)(706)    0.85  0.75  68.5  

Again assume fs = fy. This assumption is valid if the reinforcement at the extreme steel tension fiber fails. We can check this by measuring t at the end of the calculation. Using Equation 7.12.13-6:  

M n  Aps f ps  d p 

a

a    As f y  d s   2 2 

  (75,000) (12)      (45.0) (263.6) ( 68.6  (2.48  0.012 Ag ))  ... 0.95   ...  60 Ag (76.44  (2.48  0.012 Ag ))

0.73As2  4,292As + 164,257 = 0 Solving the quadratic equation: As = 38.520 in.2 (39 # 9 bars As = 39.00 in.2) Step 5: Verify the two assumptions that were made in the determination of As. The first was that the mild steel would yield and we could use fy for fs. The validity of this assumption can be checked by calculating the t. According to the California Amendments (Caltrans, 2014) and Figure 7.12.13-2 if t is > 0.005 then the section is tension controlled and 1.0. These values can be easily obtained with a simple strain diagram setting the concrete strain to 0.003.

c = 0.003 c

de - c Figure 7.12.13-4 Strain Diagram

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Having just calculated As., it is possible to calculate both c and de using AASHTO 5.7.3.1.1-4 and substituting out fps and Aps: Pj 0.75

c

9,120

 As f s  As f s

0.85 f cβ1b  k

Pj

1

0.75 d p



0.75

 (39.0)(60)

9,120  1  (0.85)(4)(0.85)(706)  (0.28)   0.75  68.5 

 6.9 in.

ts = 8.25 in. c = 6.9 in.  8.25 in., therefore, the rectangular section assumption is satisfied.

de 

Aps f ps d p  As f y d s Aps f ps  As f y



(45.04) (263.6) (68.5)  (39.00) (60) (76.44) (45.04)(263.6)  (39.00) (60)

de = 69.8 in. By similar triangles:

εt 

εc εt  therefore c de  c

εc 0.003  69.8  6.9   0.027 > 0.005  de  c   c 6.9

Therefore, the mild steel yields and = 1.00.

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 7.12.13-1 Additional Mild Steel Required per Location of Bridge Loc

As req'd

Mn

Loc

As req'd

Mn

Loc

As req'd

Mn

0.024

0.0

35492

1.024

65.0

81785

2.034

64.0

80527

0.100

0.0

46543

1.100

0.0

51660

2.100

20.0

60549

0.200

7.0

59535

1.200

0.0

33687

2.200

0.0

38316

0.300

0.0

63394

1.300

0.0

54413

2.300

0.0

48864

0.400

0.0

65317

1.400

0.0

62681

2.400

0.0

58036

0.500

0.0

63592

1.500

0.0

65317

2.500

0.0

63592

0.600

0.0

58036

1.600

0.0

62681

2.600

0.0

65317

0.700

0.0

48864

1.700

0.0

54413

2.700

0.0

63394

0.800

0.0

38316

1.800

0.0

40528

2.800

0.0

57031

0.900

15.0

58814

1.900

0.0

51660

2.900

0.0

46543

0.968

66.0

81518

1.976

62.0

80767

2.975

0.0

35699

7.12.14

Design for Shear Where it is reasonable to assume that plane sections remain plane after loading, regions of components shall be designed for shear using either the sectional method as specified in Article 5.8.3 (AASHTO, 2012), or the strut-and-tie method as specified in Article 5.6.3 (AASHTO, 2012). When designing for nominal shear resistance in boxgirders, it is appropriate to use the sectional method. In the sectional design approach, the component is investigated by comparing the factored shear force and the factored shear resistance at a number of sections along its length. Usually, this check is made at the tenth point of the span and at locations near the supports. Where the reaction force in the direction of the applied shear introduces compression into the end region of a member, the location of the critical section for shear shall be taken as dv from the internal face of support. Figure CB5.2-5 shown on the next page, illustrates the shear design process by means of a flow chart. This Figure is based on the simplified assumption that 0.5 cot  =1.0.

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure CB5.2-5 Flow Chart for Shear Design Containing at Least Minimum Transverse Reinforcement (AASHTO, 2012)

Chapter 7 - Post-Tensioning Concrete Girders

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Chapter 7 - Post-Tensioning Concrete Girders

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

A

0.0

88

682

695

1317

68

596

538

1105

42

333 489

858

Abutment 1

-455 -309

-10 -77

-621 -400

0.5 -282

-36

-831

-487

-88

-1022

-692 -582

-491

-62

-1237

-671 -901

-114

C D

-1690

-1498

-1258

-812

-157

-1113

-756

-139

1.0

166

826

E

140

747

1120

1582

105

639

838

1276

36

287 427

721

F

1.5

318 13 2

485

-427 -730

-260

-33

-994

-532

-534

-68

Length Along Frame (ft)

71

532

560

984

-1287

-811 -639

-102

G

-1592

H

-1821

-1308

-1093 -825

-746

-137 -163

2.0

I

133

747

1061

1433

149

800

1193

1598

Bent 3

109

662

862

1181

84

575

666

1012

60

474

485

821

36

2.5

282

382

599

Figure 7.12.14-1 Gravity Load Shear Envelopes for Design of Shear Resistance

B

Vp-15

Vhl93

Vdw

Vdc

16

390 128

643

1335

1811

Bent 2

J

90 306 11

462

-621 -387

-13 -102

-483 -823

-294

-37

-1057

-588

-486

-62

Abutment 4

K

-1257

-671

-629

3.0

-80

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

7-92

Unfactored Moments (kip-fti)

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Design the interior girders at the right cap face at Bent 2 to resist the ultimate factored shear demand. Use Figure 7.12.14-1 as a guide, and take advantage of the reduction in demands by using the critical section for shear as dv from the internal face of support. Step 1: Determine dv , calculate Vp . Check that bv satisfies. Equation 5.8.3.3-2. (AASHTO, 2012) dv 

Mn

(7.12.14-1)

As f y  Aps f ps

where: Aps =

area of prestressing steel (in.2)

As =

area of non-prestressed tension reinforcement (in.2)

dv = the effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure fps =

average stress in prestressing steel at the time for which the nominal resistance is required (ksi)

fy

yield strength of mild steel (ksi)

=

Mn =

nominal flexure resistance (kip-in.)

At the right cap face of Bent 2, dv becomes: dv 

Mn As f y  Aps f ps

= 5.25 ft



81,785 64.00(60)  45.04(260.7)

= 63.0 in.

Article 5.8.2.9 (AASHTO, 2012) states that dv should not be less than the greater of 0.9de or 0.72h. Therefore: de = effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) de (min) = 0.9(67.0 in.) = 60.3 in. dv (min) = 0.72(81 in.) = 58.32 in. dv = 63.0 in. > 60.3 in. > 58.3 in. Finding Vp after establishing Equation 7.12.14-2: Vp = Pj () kips

Chapter 7 - Post-Tensioning Concrete Girders

(7.12.14-2)

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

where: Pj =

force in prestress strands before losses (kip)

Vp =

the component in the direction of the applied shear of the effective prestressing force; positive if resisting the applied shear (kip).

  total angular change of prestressing steel path from jacking end to a point under investigation (rad) Since the angle that is formed between the tangent of the parabola and the horizontal changes as the location on the parabola changes, we will take two ’s at each point. One looking back to the previous point, and the other looking forward to the next point. Given such a short distance along the larger parabolas, a triangle can be used to approximate angle change of the much smaller parabola segments:



 l

Figure 7.12.14-2 Components of Alpha The general equation of a parabola: y = ax2 For any parabola with side l and : a 

δ l2

The angle change at any given point on the parabola is its first derivative:

(rads) 

dy 2δ  x dx l 2

(7.12.14-3)

 at a distance dv from the face of cap at Bent 2 2  2.12  3.5    0.90  3.8   63.0  (rads)   4    0.06 2 12    0.1(168) 

Vp = Pj () kips = 9,120 (0.06 rad) = 547 kips

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Vn is the lesser of: Vn = Vc + Vs + Vp

(AASHTO 5.8.3.3-1)

Vn = 0.25 fcbvdv + Vp

(AASHTO 5.8.3.3-2)

where: bv

=

effective web width taken as the minimum web width, measured parallel to the neutral axis, between resultants of the tensile and compressive forces due to flexure. This value lies within the depth dv (in.)

dv

= the effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure (in.)

For now we will use AASHTO 5.8.3.3-2, and return to AASHTO 5.8.3.3-1 later. bv (reqd ) 

Vn  Vp

(7.12.14-4)

0.25 f cd v

where: Vn

= the nominal shear resistance of the section considered (kip)

Vp

=

Vn 

the component in the direction of the applied shear of the effective prestressing force; positive if resisting the applied shear (kip)

Vu

(7.12.14-5)



Using Figure 7.12.14-1 to compare Strength I and Strength II results of Vu at the Bent 2 cap face: Vn @ face ( Str I ) 

Vu

Vn @ pt E ( Str I ) 

Vu

Vn @ face ( Str II ) 

Vu

Vn @ pt E ( Str II ) 

Vu











1.25(1, 334)  1.50(166)  1.75(826)

 3, 736 kips

0.90 

1.25(1,120)  1.50(140)  1.75(745)

 3, 238 kips

0.90 

1.25(1, 334)  1.50(166)  1.35(1,810)

 4,844 kips

0.90 

1.25(1,120)  1.50(140)  1.35(1, 582)

Chapter 7 - Post-Tensioning Concrete Girders

 4,146 kips

0.90

7-95

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Strength II controls so now interpolate to find Vn a distance dv from the face of Bent 2: Vn @ dv fromface 

4,162  4,844  63.0   4  4   4,844  4,564 kips  0.1(168)  4  12 

From above fc = 4 ksi and now using Equation 7.14.12-3 to find bv: bv (reqd ) 

Vn  Vp 0.25 fcd v



4,564  547 0.25(4)63.0

 64.0 in.

This results in bv ~12.8 in. per girder. Flare the interior girders at bent faces to 13 in. for added capacity in future calculations. bv (13 in.flare)  (5girders)(13 in.)  65 in.

Step 2: Calculate shear stress ratio vu / fc using Equation 5.8.2.9-1. (AASHTO, 2012) Vu  V p (AASHTO 5.8.2.9-1) vu   bv d v where: vu =

average factored shear stress on the concrete (ksi) (5.8.2.7) (5.8.2.9)

Vu @ dv from face = 0.90 (4564) = 4,108 kips Vp = 0.9(547) = 492 kips

vu 

Vu  V p  bv d v



4,108  492 0.9(65)63.0

 0.981ksi

vu 0.981   0.245 f c 4

Step 3: If section is within the transfer length of any strands, then calculate the effective value of fpo, else assume fpo = 0.7 fpu. This step is necessary for members without anchorages. fpo = 0.7 fpu = 189 ksi

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Step 4: Calculate x using Equations B5.2-1, B5.2-2 or 5.2-3 (AASHTO, 2012) Assuming the section meets the requirements specified in Article 5.8.2.5 (AASHTO, 2012) Mu x 

where: fpo =

dv

 0.5 N u  0.5 Vu  V p cot   A ps f po

(AASHTO B5.2-1)

2( E s As  E p A ps )

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and the surrounding concrete (ksi)

Nu =

applied factored axial force taken as positive if tensile (kip)

x =

longitudinal strain in the web reinforcement on the flexural tension side of the member (in/in)



angle of inclination of diagonal compressive stresses (degrees)

=

Mu @ face = 1.25(-32,619)+1.50(-4,095)+1.35(-23,620)+0.856(9,120) = -70,997 kip – ft Mu @ pt E = 1.25(-16,910)+1.50(-2,130)+1.35(-14,000)+0.856(9,120) = -35,426 kip – ft By interpolation, find Mu a distance dv from the face of Bent 2:

M u @ d v fromface



 35,426  70,997  63.0 0.1(168)  4

= -56,407 kip – ft

  4  4   70,997   12  = 676,884 kip – in.

Now using AASHTO B5.2-1 with Nu = 0, begin with cot  = 1, and values calculated earlier  676,884

x 

63.0

 0.5 (0)  0.5 4,108  547 (1)  45.04 (189) 2 29,000 (64)  28,500 (45.04)

 0.000639

Step 5: Choose values of and corresponding to next-larger x from AASHTO Table B5.2-1 (California Amendments to AASHTO, 2012). Based on calculated values of vu  0.245 ; x = 0.000639, we obtain: f c

34.3  and = 1.58. 

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table B5.2-1 Values of and for Sections with Transverse Reinforcement (AASHTO, 2012)

Step 6: Determine transverse reinforcement, Vs, to ensure: Vu  (Vc + Vs + Vp) Equations 5.8.2.1-2, 5.8.3.3-1 (AASHTO, 2012). First we must introduce the concrete component of shear resistance: Vc = 0.0316  fcbvdv where: Vc =  =

(AASHTO 5.8.3.3-3)

nominal shear resistance provided by tensile stresses in the concrete (kip) factor relating effect of longitudinal strain on the shear capacity of concrete as indicated by the ability of diagonally cracked concrete to transmit tension

Vc = 0.0316  fcbvdv = 0.0316(1.58) 4(65)(63.0) = 409 kips Vp = 547 kips Vu = 4,108 kips

Combining Equations 5.8.2.1-2 and 5.8.3.3-1 (AASHTO, 2012) results in design equation as follows: Vu ≤  (Vc + Vs + Vp) Rearranging and solving for Vs: Vs 

Vu 4,108  Vc  V p   409  547  3,608 kips  0.90

Chapter 7 - Post-Tensioning Concrete Girders

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BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Vs 

where: Av =

Av f y d v (cot   cot ) sin 

(AASHTO 5.8.3.3-4)

s

area of transverse reinforcement within distance s (in2.)

  angle of inclination of transverse reinforcement to longitudinal axis (º) s  spacing of reinforcing bars (in.) Vs  shear resistance provided by the transverse reinforcement at the section under investigation as given by AASHTO 5.8.3.3-4, except Vs shall not be taken greater than Vu/ (kip) Then  = 90º since stirrups in this bridge are perpendicular to the deck. Thus Equation 5.8.3.3-4 reduces to: Vs 

Av f y d v (cot)

(AASHTO C5.8.3.3-1)

s

Rearranging terms to solve for s

s

Av f y d v (cot)

(7.12.14-6)

Vs

Assume 2 - #5 legs per stirrup ×5 girders, Av = 2(5)(0.31) = 3.10 in.2 s

Av f y dv (cot θ) Vs



3.10 (63.0) (cot 34.3)  4.76 in. 3,608

Use # 5 stirrups, s = 4 in. spacing Step 7: Use Equation 5.8.3.5-1 (AASTHO, 2012) to check if the longitudinal reinforcement can resist the required tension.

Aps f ps  As f y 

Mu d v f

 0.5

Nu c

 Vu



 c



 Vp  0.5Vs  cot θ



(AASHTO 5.8.3.5-1)

Breaking into parts and solving both sides of Equation 5.8.3.5-1 results in: 45.0(260.2) + 64.0(60) = 15,549 kips 56, 407(12) 63.0(0.9)

 0   4, 564  547  0.5(3, 520)  cot 36.0  15, 044 kips

Chapter 7 - Post-Tensioning Concrete Girders

7-99

BBRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Since 15,549 is greater than 15,044 therefore the shear design is complete. Had the left side been smaller than the right side we would use the following procedure to determine As. If the conditions of Equation 5.8.3.5-1 (AASHTO, 2012) were met then the shear design process is complete. Step 8: If the right side of AASHTO 5.8.3.5-1 was greater than the left side, we would need to solve Equation 5.8.3.5-1 to increase As to meet the minimum requirements of 5.8.3.5-1. Mu As 

d v f

 0.5

Nu c

 Vu



 c



 Vp  0.5Vs  cot θ  Aps f ps



fy

(AASHTO 5.8.3.5-1)

Design the exterior girders right cap face at Bent 2 to resist the ultimate factored shear demand. Use Figure 7.12.14-1 as a guide, and take advantage of the reduction in demands by using the critical section for shear as dv from the internal face of support. In this example, all values will be per girder, since only the exterior girder is affected by this analysis. Use the modification chart found in BDA 5-32 to amplify values of Vu . Step 1: Determine dv , calculate Vp . Check that bv satisfies Equation 5.8.3.3-2. (AASHTO, 2012) From above: dv = 63.0 in. Vp = 547 kips or 110 kips/girder Vn is the lesser of AASHTO 5.8.3.3-1 and AASHTO 5.8.3.3-2: Vn = Vc + Vs + Vp Vn = 0.25 fcbvdv + Vp For now we will use AASHTO 5.8.3.3-2, and return to AASHTO 5.8.3.3-1 later. Equations 7.12.14-4 and 7.12.14-5 respectively state:

bv (reqd )  Vn 

Vn  Vp 0.25 f cd v

Vu 

Chapter 7 - Post-Tensioning Concrete Girders

(7.12.14-4)

(7.12.14-5)

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From above Strength II controls: Vn @ dv from face = 4,570 kips We can now use BDA 5-32 (Figure 7.12.14-2) to amplify the obtuse exterior girders shear demand. For an exterior girder with a 20º skew the chart reads that the modification factor is 1.4. Now Vn(mod) = Vn (ext )  1.4

4564 5

 1278 kips

From above fc = 4 ksi and now using Equation 7.12.12-3 to find bv:

bv (reqd ) 

Vn  Vp 0.25 f cd v



1, 278  110 0.25(4)63.0

 18.5 in.

This results in bv = 18.5 in. per girder. We will flare exterior girders at bent faces to 19 in. for added capacity in future calculations.

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Figure 7.12.14-3 Shear Modification Factor Found in BDA 5-32 (Caltrans, 1990) Step 2: Calculate shear stress ratio vu / fc using Equation 5.8.2.9-1. (AASHTO, 2012) vu 

Vu  V p bv d v

Vu @ dv from face = 0.90 (1,2780) = 1,150 kips vu 

Vu  V p  bv d v



1,150  100 0.9 (19) 63.0

 0.975 ksi

vu 0.975   0.244 f c 4

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Step 3: If the section is within the transfer length of any strands, then calculate the effective value of fpo, else assume fpo = 0.7 fpu. This step is necessary for members with no anchorage devices. fpo = 0.7 fpu = 189 ksi Step 4: Calculate x using Equations B5.2-1, B5.2-2 or B5.2-3 (AASHTO, 2012) Assuming the section meets the requirements specified in Article 5.8.2.5 (AASHTO, 2012) Mu x 

dv

 0.5 N u  0.5 Vu  V p cot   A ps f po 2( E s As  E p A ps )

From above and in terms of a single girder:

M u @d v from face 

-676,884 kip  in  135,877 kip  in 5

Now using AASHTO 5.8.3.4.2-1 with Nu = 0, begin with cot  = 1, and values calculated earlier…  135,530 45.04  0.5 (0)  0.5 1,150  100 (1)  (189) 63.0 5 x   0.0012 65 45.04   2 29,000 ( )  28,500 ( ) 5 5  

Step 5: Choose values of and corresponding to next-larger x from AASHTO Table B5.2-1 (California Amendments to AASHTO, 2012). Based on calculated values of vu  0.244 and x = 0.0012, we obtain f c

35.8and = 1.50



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Table B5.2-1 Values of and for Sections with Transverse Reinforcement (AASHTO, 2012)

Step 6: Determine transverse reinforcement, Vs, to ensure: Vu  (Vc + Vs + Vp) Equations 5.8.2.1-2, 5.8.3.3-1 (AASHTO, 2012). First use AASHTO 5.8.3.3-3 to determine the concrete component of shear resistance. Vc = 0.0316 fcbvdv Vc = 0.0316 fcbvdv = 0.0316 (1.50)4 (19)(63.0) =113.5 kips Vp = 110 kips Vu = 1,110 kips Combining Equations 5.8.2.1-2 and 5.8.3.3-1 (AASHTO, 2012) results in design equation as follows: Vu ≤ (Vc + Vs +Vp) Rearranging and solving for Vs: Vs 

Vu 1,110  Vc  V p   113.5  110  1,010 kips  0.9

Then using Eq. 7.12.14-6

s

Av f y d v (cot) Vs

Assume 2 - #5 legs per stirrup × 1 girders, Av = 2 (1) (0.31) = 0.62 in.2

s

Av f y d v (cot ) (0.62) (60) (63.0) (cot 35.8)   3.22 in. Vs 1,010

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Use # 5 stirrups, s = 3 in. spacing Step 7: Use Modified Equation 5.8.3.5-1 (AASTHO, 2012) to check if the longitudinal reinforcement per girder can resist the required tension. Aps f ps 5



As f y 5



Mu d v f

 0.5

Nu c

 Vu



 c



 Vp  0.5Vs  cot θ



Breaking into parts and solving both sides of Equation 5.8.3.5-1 results in:

45.04(260.2) 64(60)   3,112 kips 5 5 11, 281(12) 63.0(0.9)

 0   1, 278  110  0.5(1,039)  cot 37.4  3, 236 kips

3,112 is not greater than 3,236. Use the following procedure to determine As. Step 8: Solve Equation 5.8.3.5-1 to increase As to meet the minimum requirements of 5.8.3.5-1: Mu As 

As 

d v f

 0.5

Nu c

 Vu



 c



 Vp  0.5Vs  cot θ  Aps f ps



fy

3,236 

45.04(260.2) 5  14.87 in.2 60

If the right side of Equation 5.8.3.5-1 is equal to 3,290, the exterior 1/2 bay 64 reinforcement should be increased from As   12.8 in.2 to As = 15 in.2 5

7.12.15

Calculate the Prestressing Elongation Tendon elongation calculations are necessary to help ensure the proper jacking force is delivered to the superstructure. Elongation calculations are one way for construction field personnel to check the actual Pj force applied to tendons. Since the structure has been designed for two-end stressing, both first and second end elongations need to be computed. Figure 7.12.15-1 shows the information that should be included on the contract plans.

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Article 7.12.11 Article 5.9.5.2.1 (California Amendments to AASHTO, 2014)

Article 7.12.12

Table 7.12.7.5-1

Figure 7.12.15-1 Decal Shown on Contract Plans

Force coefficient (decimal % of Pj)

Based on the location and magnitude of fpF (stress with friction losses) shown on the contract plans, the post-tensioning fabricator develops a simplified diagram, like the one shown in Figure 7.12.15-2. FCpF = 1.000 fpF = 202.50 ksi

Point of no Movement (2-end)

FCpF = 0.879 fpF = 178 ksi

FCpF = 1.000 fpF = 202.5 ksi

2nd End P/S Steel Stress L1 = 219.6 ft L2 = 192.4 ft

FCpF = 0.773 fpF = 156 ksi

1st End P/S Steel Stress Length Along Girder (ft) Figure 7.12.15-2 Simplified Diagram

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The 1st end elongation calculation (from the 1st end of jacking side to the anchorage side, the entire length of the span): The prestressing elongation is based on the stress-strain relationship and results in Equation 7.12.15-1:

f avg Lx

E 

(7.12.15-1)

Ep

Ep =

modulus of elasticity of prestressing tendon (ksi)

favg =

average stress in the strand from jacking end to point of no movement (ksi)

E =

change in length of prestressing tendons due to jacking (in.)

For one-end stressing, MTD 11-1 (Caltrans, 2013) provides the following formula: T (1  )(L  3.5' ) ΔE  o 2E p

(7.12.15-2)

In Equation 7.12.15-2, the 3.5′ term is the expected length of jack. For two end stressing, MTD 11-1 (Caltrans, 2013) provides the following formula:

1st 

To (1  )L1  (3  1) L2  2E p

T (1  )L2  2nd  o Ep

(7.12.15-3)

(7.12.15-4)

where:

1st

= elongation after stressing the first end (in)

2nd = elongation after stressing the second end (in) To

= steel stress at the jacking end before seating (generally 202.5 ksi) (ksi)



= initial force coefficient at the point of no movement

L

= Length of tendon (ft)

L1

= Length of tendon from first stressing end to the point of no movement (ft)

L2

= Length of tendon from point of no movement to second stressing end (ft)

For our bridge, let’s use the values shown on the graph above and applying the 3 ft length of jack:

1st 

202.5 (2) 28,500

[(1+0.879)(219.6)+((3×0.897)-1)(192.40)]=2.585 ft =31.02 in.

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When applying these measurements in the field, it is necessary to determine the “measureable” elongation. The measureable elongation includes the length of tendon in the contractor’s jack (3 ft). It is been found that by applying 20% of Pj before monitoring elongations, the tendon is allowed to shift from it’s resting place to it’s final position. Therefore, only 80% of the elongation calculated above is measureable. 1st = (0.8)(31.02) = 24.81 in.  The 2nd end elongation calculation (from the 2nd end of jacking side to the point of no movement):  2 nd 

202.5 (1  0.879) (192.4)  0.165 ft  1.98 in. 28,500

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NOTATION Ag

=

gross area of section (in.2)

Aps

=

area of prestressing steel (in.2)

As

=

area of non-prestressed tension reinforcement (in.2)

As

=

area of compression reinforcement (in.2)

Av

=

area of transverse reinforcement within distance s (in.2)

b

=

width of the compression face of a member (in.)

bv

=

effective web width taken as the minimum web width, measured parallel to the neutral axis, between resultants of the tensile and compressive forces due to flexure. bv lies within the depth dv (in.)

bw

=

web width (in.)

c

=

distance from extreme compression fiber to the neutral axis (in.)

clrint

=

clearance from interior face of bay to the first mat of steel in the soffit or deck (Usually taken as 1 in.) (in.)

d

=

depth of member (in.)

de

=

defective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.)

di

=

distance between C.G. of i th duct and the i th duct LOL (See Figure 7.12.7-4) (in.)

dp

=

distance from extreme compression fiber to the centroid of the prestressing tendons (in.)

ds

=

distance from extreme compression fiber to the centriod of the nonprestressed tensile reinforcement (in.)

dv

= the effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure (in.)

Ect

=

modulus of elasticity of concrete at transfer or time of load application (ksi)

Ep

=

modulus of elasticity of prestressing tendons (ksi)

FCpA

=

force coefficient for loss from anchor set

FCpES =

force coefficient for loss from elastic shortening

FCpF

=

force coefficient for loss from friction

FCpT

=

total force coefficient for loss

e

=

eccentricity of resultant of prestressing with respect to the centroid of the cross section. Always taken as a positive. (ft) The base of Napierian logarithms

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ex

=

eccentricity as a function of x along parabolic segment (ft)

favg

=

average stress in the strand from jacking end to point of no movement (ksi)

f c

=

specified compressive strength of concrete used in design (ksi)

fci

=

specified compressive strength of concrete at time of initial loading or prestressing (ksi); nominal concrete strength at time of application of tendon force (ksi)

fpe

=

effective stress in the prestressing steel after losses (ksi)

fpi

=

initial stress in the prestressing steel after losses, considering only the effects of friction loss. No other P/S losses have occurred (ksi)

fcgp

=

concrete stress at the center of gravity of prestressing tendons, that results from the prestressing force at either transfer or jacking and the self-weight of the member at maximum moment sections (ksi)

fcpe

=

compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi)

fDC+DW =

stress in concrete from DC and DW load cases (ksi)

fDCw/o b =

stress in concrete due to the Dead Load of the structural section only (ksi)

fpi

=

prestressing steel stress immediately prior to transfer (ksi)

fg

=

stress in the member from dead load (ksi)

fHL93

=

stress in concrete from HL93 load cases (ksi)

fpo

=

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and the surrounding concrete (ksi)

fps

=

average stress in prestressing steel at the time for which the nominal resistance is required (ksi)

fpu

=

specified tensile strength of prestressing steel (ksi)

fpy

=

yield strength of prestressing steel (ksi)

fr

=

modulus of rupture of concrete (ksi)

fy

=

yield strength of mild steel (ksi)

f’y

=

specified minimum yield strength of compression reinforcment (ksi) 

H

=

average annual ambient mean relative humidity (percent)

hf

=

compression flange depth (in.)

Icr

=

moment of inertia of the cracked section, transformed to concrete (in.4)

Ie

=

effective moment of inertia (in.4)

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Ig

=

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (in.4)



=

distance to the closest duct to the bottom of the soffit or top of the deck (in.)

k

=

wobble friction coefficient (per ft of tendon)

L

=

distance to a point of known stress loss (ft), individual span length (ft) (60  L  240)

Lframe

=

length of frame to be post-tensioned (ft)

Ma

=

maximum moment in a member at the stage which the deformation is computed (kip-in.)

Mcr

=

cracking moment (kip-in.)

Mn

=

nominal flexure resistance (kip-in.)

Mr

=

factored flexural resistance of a section in bending (kip-in.)

MDL

=

dead load moment of structure (kip-in.)

MP/S s

=

moment due to the secondary effects of prestressing (k-ft)

MCp

=

primary moment force coefficient for loss (ft)

MCs

=

secondary moment force coefficient for loss (ft)

MCPT

=

total moment force coefficient for loss (ft)

N

=

number of identical prestressing tendons

Nc

=

number of cells in a concrete box girder (Nc  3)

Nu

=

applied factored axial force taken as positive if tensile (kip)

Pj

=

force in prestress strands before losses (kip)

lij

=

length of individual parabola (in.)

ni

=

number of strands in the i th duct

Sc

=

section modulus for the extreme fiber of the composite sections where tensile stress is caused by externally applied loads (in.3)

s

 spacing of reinforcing bars (in.)

t

=

thickness of soffit or deck (in.)

td

=

thickness of deck (in.)

ts

=

thickness of soffit (in.)

Vc

=

nominal shear resistance provided by tensile stresses in the concrete (kip)

Vn

= the nominal shear resistance of the section considered (kip)

Vp

=

the component in the direction of the applied shear of the effective prestressing force; positive if resisting the applied shear (kip)

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Vs

 shear resistance provided by the transverse reinforcement at the section under investigation as given by AASHTO 5.8.3.3-4, except Vs shall not be taken greater than Vu / (kip)

vu

=

average factored shear stress on the concrete (ksi)

W

=

weight of prestressing steel established by BDA page 11-66 (lb)

x

=

general distance along tendon (ft), location along parabolic segment where eccentricity is calculated (% span L)

xpA

=

influence length of anchor set (ft)

y

=

general distance from the neutral axis to a point on member cross-section (in.)

y ij

=

height of individual parabola (in.)

yt

=

distance from the neutral axis to the extreme tension fiber (in.) 

Z

=

C.G. tendon shift within duct (in.)



 angle of inclination of transverse reinforcement to longitudinal axis (º) total angular change of prestressing steel path from jacking end to a point under investigation (rad)



 factor relating effect of longitudinal strain on the shear capacity of concrete, as indicated by the ability of diagonally cracked concrete to transmit tension

1

=

ratio of the depth of the equivalent uniformly stressed compression zone assumed in the strength limit state to the depth of the actual compression zone

hp

=

offset from deck to Centroid of duct (in.)

lp

=

offset from soffit to Centroid of duct (in.)

Aset

=

anchor set length (in.)

E

=

change in length of prestressing tendons due to jacking (in.)

fi

=

change in force in prestressing tendon due to an individual loss (ksi)

fL

=

friction loss at the point of known stress loss (ksi)

fpA

=

jacking stress lost in the P/S steel due to anchor set (ksi)

fpCR

=

change in stress due to creep loss

fpES

=

change in stress due to elastic shortening loss

fpF

=

change in stress due to friction loss

fpSR

=

change in stress due to shrinkage loss 

cu

=

failure strain of concrete in compression (in./in.)

s

=

net longitudinal tensile strain in section at the centroid of the tension reinforcement (in./in.)

t

=

net tensile strain in extreme tension steel at nominal resistance (in./in.)

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

=

angle of inclination of diagonal compressive stresses (degrees)



=

resistance factor 



=

coefficient of friction

1

=

flexural cracking variability factor

2

=

prestress variability factor

3

=

ratio of specified minimum yield strength to ultimate tensile strength of reinforcement

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REFERENCES 1. AASHTO, (2012). LRFD Bridge Design Specifications, American Association of State Highway and Transportation Officials, 6th Edition, Washington, D.C. 2. Collins, M. P. and Mitchell, D., (1997). Prestressed Concrete Structures, Response Publications, Toronto, Canada. 3. Libby, J. R., (1990). Modern Prestressed Concrete: Design Principles and Construction Methods – 4th edition, Van Nostrand Reinhold New York, NY. 4. Nawy, E. G., (2005). Reinforced Concrete: A Fundamental Approach – 5th Edition, Pearson Prentice Hall Upper Saddle River, NY. 5. Nilson, A. H., (1987). Design of Prestressed Concrete, Wiley Hoboken, NJ. 1987. 6. Gerwick, B. C. Jr., (1997). “Chapter 11 – Construction of Prestressed Concrete”, Concrete Construction Engineering Handbook, Editor, Nawy, E.G., CRC Press, Boca Raton, FL. 7. Caltrans, (2014). California Amendments to the AASHTO LRFD Bridge Design Specifications  Sixth Edition, California Department of Transportation, Sacramento, CA. 8. Caltrans, (2013). Memo to Designers 11-11 − Prestressed Concrete Shop Drawing Review, California Department of Transportation, Sacramento, CA. 9. Caltrans, (2005). Bridge Design Aids Chapter 11 – Estimating, California Department of Transportation, Sacramento, CA. 10. Caltrans, (2008b). Memo to Designers 10-20 – Deck and Soffit Slabs, California Department of Transportation, Sacramento, CA. 11. Caltrans, (2010). Memo to Designers 11-3 – Designers Checklist for Prestressed Concrete, California Department of Transportation, Sacramento, CA. 12. Caltrans, (1994). Memo to Designers 11-28 Attachment 1 – Prestress Clearances for CIP P/S Box Girder Structures, California Department of Transportation, Sacramento, CA. 13. Caltrans, (1996). Memo to Designers 11-28 Attachment 2 – Clearance Requirement for Ducts, California Department of Transportation, Sacramento, CA. 14. Caltrans, (1994). Memo to Designers 11-28 Attachment 31 – ''D'' Chart for Cast-inPlace Girders Estimating, California Department of Transportation, Sacramento, CA.

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15. Caltrans, (1989). Memo to Designers 15-2 – Concrete Girder Spacing, California Department of Transportation, Sacramento, CA.

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CHAPTER 8 PRECAST PRETENSIONED CONCRETE GIRDERS TABLE OF CONTENTS 8.1 

INTRODUCTION ............................................................................................... 8-1 

8.2 

PRECAST GIRDER FEATURES ....................................................................... 8-3 

8.3 

8.4 

8.2.1 

Typical Sections and Span Ranges ..................................................................... 8-3 

8.2.2 

Primary Characteristics of Precast Girder Design .............................................. 8-7 

8.2.3 

Methods to Vary Strand Eccentricity and Force .............................................. 8-11 

PRECAST BRIDGE TYPES ............................................................................. 8-15  8.3.1 

Single-Span Bridges ......................................................................................... 8-15 

8.3.2 

Multi-Span Bridges .......................................................................................... 8-16 

8.3.3 

Spliced Girder Bridges ..................................................................................... 8-22 

DESIGN CONSIDERATIONS ......................................................................... 8-26  8.4.1 

Materials ........................................................................................................... 8-26 

8.4.2 

Prestress Losses ................................................................................................ 8-27 

8.4.3 

Flexure .............................................................................................................. 8-30 

8.4.4 

Shear ................................................................................................................. 8-31 

8.4.5 

Deflection and Camber..................................................................................... 8-32 

8.4.6 

Anchorage Zones.............................................................................................. 8-37 

8.4.7 

Diaphragms and End Blocks ............................................................................ 8-37 

8.4.8 

Lateral Stability ................................................................................................ 8-38 

8.5 

DESIGN FLOW CHART .................................................................................. 8-39 

8.6 

DESIGN EXAMPLE ......................................................................................... 8-41  8.6.1 

Problem Statement ........................................................................................... 8-41 

8.6.2 

Select Girder Depth, Type, and Spacing .......................................................... 8-43 

8.6.3 

Establish Loading Sequence ............................................................................. 8-44 

8.6.4 

Select Materials ................................................................................................ 8-45 

8.6.5 

Calculate Section Properties ............................................................................. 8-46 

8.6.6 

Determine Loads .............................................................................................. 8-49 

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 8.6.7 

Perform Structural Analysis ............................................................................. 8-50 

8.6.8 

Estimate Prestressing Force and Area of Strands ............................................. 8-55 

8.6.9 

Estimate Prestress Losses ................................................................................. 8-59 

8.6.10  Design for Service Limit State ......................................................................... 8-62  8.6.11  Design for Strength Limit State........................................................................ 8-75  8.6.12  Check Reinforcement Limits............................................................................ 8-80  8.6.13  Design for Shear ............................................................................................... 8-82  8.6.14  Design for Interface Shear Transfer between Girder and Deck ....................... 8-93  8.6.15  Check Minimum Longitudinal Reinforcement ................................................ 8-95  8.6.16  Pretensioned Anchorage Zone Reinforcement ................................................. 8-96  8.6.17  Deflection and Camber..................................................................................... 8-97 

NOTATION ................................................................................................................. 8-103  REFERENCES ............................................................................................................ 8-111 

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

CHAPTER 8 PRECAST PRETENSIONED CONCRETE GIRDERS 8.1

INTRODUCTION Precast concrete elements such as girders, piles, deck panels, and pavement are being used with increasing frequency in California. This chapter focuses exclusively on precast pretensioned concrete girders, referred to herein as PC girders. PC girders are a type of prestressed concrete girder that facilitates rapid construction of a bridge using girders that are fabricated off-site and then transported and erected into place at the job site. Once the deck is poured, the structural section becomes composite, minimizing deflections. Because PC girders require little to no falsework, they are a preferred solution for jobs where Accelerated Bridge Construction (ABC) is sought, where speed of construction, minimal traffic disruption, and/or environmental impact is required, and where temporary construction clearance is limited. PC girders employ high performance concrete for strength, durability, and/or constructability and tend to be more economical and competitive when significant repeatability exists on a job (i.e., economy of scale). The use of PC girders in California highway bridge system has increased rapidly in recent years (Figure 8.1-1).

A) Pretensioned bulb-tee girders

B) Pretensioned wide flange girder

Figure 8.1-1 Example of Precast Pretensioned Concrete Girder Sections Similar to cast-in-place (CIP) post-tensioned (PT) girders, PC girders are prestressed to produce a tailored stress distribution along the member at service level to help prevent flexural cracking. For member efficiency, the girders have precompressed tensile zones-regions such as the bottom face of the girder at midspan where compression is induced to counteract tension due to expected gravity loads (e.g., self-weight, superimposed dead loads such as deck weight, barrier weight, and overlay, as well as live loads). To achieve this, PC girders employ prestressing

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strands that are stressed before the concrete hardens. This is in contrast to PT girders, in which the tendons are stressed after the concrete hardens. However, PC girders may also be pretensioned, then post-tensioned, and are sometimes spliced together to form a single span or continuous superstructure. As shown in Figure 8.1-2, pretensioning requires the use of a stressing bed, often several hundred feet long for efficient casting of a series of members in a long line, and using abutments, stressing stands, jacks, and hold-downs/hold-ups to produce the desired prestressing profile. The transfer of strand force to the concrete members by bond is typically evident by the upward deflection (camber) of members when the strands are detensioned (cut or burned) at the member ends. Steam curing of members allows for a rapid turnover of forms (typically one-day cycle or less) and cost efficiency. Control during fabrication of PC girders also permits the use of quality materials and provides many benefits compared to CIP PT girders, such as higher strength materials (e.g., f´ci, f´c) and modulus of elasticity, as well as reduced creep, shrinkage, and permeability. Article 5.5.4.2.1 of CA Amendments to AASHTO LRFD Bridge Design Specifications (Caltrans, 2014) takes advantage of this higher quality control and thus increases the resistance factor, , for tensioned-controlled sections from 0.95 for CIP PT members to 1.0 for PC girders.

Stressing jack

Removable abutment Formwork

Original length, L

End abutment

Precasting bed A) Strands tensioned

L - ES*

ES = elastic shortening B) Strands detensioned

Figure 8.1-2 Pretensioning of Members with Straight Strands on Stressing Bed

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8.2

PRECAST GIRDER FEATURES

8.2.1

Typical Sections and Span Ranges The designer may select from a wide variety of standard sections, as described in Chapter 6 of the Bridge Design Aids (BDA). Girder sections not covered in this section are considered non-standard and must be approved by the Type Selection Meeting. Figure 8.2-1 shows representative PC girder sections, and Table 8.2-1 lists typical and preferred span lengths for eight common PC girder types, including four standard California girders (I, bulb-tee, bath-tub, and wide-flange) and the California voided slab, as well as three other PC girders (box, delta, and double-tee). Table 8.2-1: PC Girder Types and Span Lengths (Caltrans, 2012) Girder Type

Possible Span Length(ft)

California I-girder California bulb-tee girder California bath-tub girder California wide-flange girder California voided slab Precast box girder Precast delta girder Precast double-tee girder

50 to 125 80 to 150 80 to 150 80 to 200 20 to 70 40 to 120 60 to 120 30 to 100

Preferred Span Length(ft) 50 to 95 95 to 150 80 to 120 80 to 180 20 to 50 40 to 100 60 to 100 30 to 60

Among these girders, the I-girder is most commonly used and has been in use in California for nearly 60 years. With bridge span lengths normally ranging from 50 ft to 125 ft, the I-girder typically uses a depth-to-span ratio of approximately 0.05 to 0.055 for simple spans and approximately 0.045 to 0.05 for multi-span structures made continuous for live load. The bulb-tee and bath-tub (or U-shape) girders are targeted for bridge spans up to 150 ft. The depth-to-span ratio is slightly smaller than that for I-girders: 0.045 to 0.05 for simple spans and 0.04 to 0.045 for continuous structures, respectively. However, due to the weight limits for economical hauling, the length of bath-tub girders is usually restricted to a range of 100 ft to 120 ft. The California wide-flange girder (Figure 8.2-2) was recently developed in coordination with California precasters to produce more efficient bottom and top flange areas that permit design for spans up to 200 ft, with a depth-span ratio of 0.045 (simple) and 0.04 (continuous). The larger bottom bulb accommodates nearly 20% more strands than the standard California bulb tee and, due to its shape, provides enhanced handling and erection stability at longer spans. Greater economy is also anticipated due to larger girder spacing and reduction in girder lines. Standard sections have been developed for both pretensioning alone, as well as combined preand post-tensioned sections. For longer span lengths, special permits for hauling, trucking routes, and erection must be verified.

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Other girders that are less commonly used include girders with trapezoidal, double-tee, and rectangular cross sections as well as box girders. These are sometimes used for cost effectiveness and aesthetics. Precast box girders are often used for railway systems and relatively short span lengths ranging from 40 ft to 100 ft. It should be noted that using the given bridge depth-to-span ratios to determine the girder section is approximate but is usually a reasonable starting point for initial design and cost estimates. Normally, girder spacing is set at approximately 1.25 to 1.75 times the bridge superstructure depth. When a shallow girder depth is required, girder spacing may have to be reduced to satisfy all design criteria, which may result in increased cost.

A) I girder

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B) Bulb-tee

C) Bath-tub

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D) Wide-flange

Figure 8.2-1 Example PC Girder Sections (Caltrans, 2012)

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Figure 8.2-2 California Wide-Flange Girders

8.2.2

Primary Characteristics of Precast Girder Design At the heart of the prestressed concrete design philosophy is the positioning of the prestressing strands within the PC girder: the center of gravity of the strands (CGS) is deliberately offset from the center of gravity of the concrete section (CGC) to establish an eccentricity, defined as the distance between the CGS and CGC at a section. This eccentricity produces a beneficial tailored flexural stress distribution along the length of the member to counteract the flexural tension expected from gravity loads. The largest eccentricity is provided at locations where tension is expected to be the greatest (e.g., at midspan of simple span girder). For PC girder design, the following three basic stages are addressed: Transfer, service, and ultimate. 

Transfer refers to the stage at which the tensile force in the strands is transferred to the PC girder, by cutting or detensioning the strands after a minimum girder concrete strength has been verified. Because the girder is simply supported and only self-weight acts with the prestressing at this stage, the most critical stresses typically occur at the ends of the girder or harping points (also known as drape points). Both tensile and compressive stresses should be checked at these locations against AASHTO LRFD stress limits.



Service refers to the stage at which girder and deck self-weight act on the non-composite girder, together with additional dead loads (e.g., barrier and wearing surface) and live load on the composite section. This stage is checked using the AASHTO LRFD Service I and III load combinations (AASHTO, 2012). Per Caltrans Amendments Table 5.9.4.2.2.-1 (Caltrans, 2014), the girder must also be designed to prevent tension in the precompressed tensile zones (“zero tension”) due to permanent loads.

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

Ultimate refers to the Strength Limit State. Flexural and shear strengths are provided to meet all factored load demands, including the Caltrans P15 design truck (Strength II load combination).

In service limit state design, the concrete stresses change at various loading stages. In general, there are three major stages that need to be considered in the design, and these stages are described in the following sections. 

Stage I: Cast and stress girder (transfer) (Fig. 8.2-3): o

Strands are stressed to jacking force within form. Girder concrete is cast. Once concrete gains sufficient strength, strands are cut, transferring prestressing force to the girder.

o

Girder self-weight is supported by the PC girder alone.

o

This transfer stage is a temporary condition. Tensile stresses are limited to

0.0948 f ci'  0.2 ksi

for section without bonded

reinforcement or 0.24 f ci' for section with reinforcement sufficient to resists the tensile force in the concrete per Table 5.9.4.1.2-1 (AASHTO, 2012). The compressive stresses are governed by limits in Article 5.9.4.1.1 of LRFD Specifications (AASHTO, 2012).

C

C

T (Mg/S) - Self wt.

Girder

C

(P/A) Prestress

T

T*

C (Pe/S) Prestress

C  0 . 6 f c' Stage I Concrete Stresses

* T  0.0948 f ci' or 0.2 ksi for section without bonded reinforcement * T  0 .24

f ci' for section with reinforcement sufficient to resist concrete tensile force

Figure 8.2-3 Representative Concrete Flexural Stress Distribution at Stage I (Transfer)

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

Stage IIA: Erect girder and cast deck slab (Fig. 8.2-4): o

Girders are transported to job site and erected on structure supports. Diaphragms and concrete deck are cast.

o

When deck concrete is wet, deck slab does not contribute to section modulus for flexural resistance.

o

Temporary construction loads for machinery (e.g., Bidwell) need to be accounted for.

o

Girder self-weight plus weight of diaphragms and deck are supported by the PC girder alone.

o

This stage is a temporary condition. Tensile and compressive stresses are governed by the limits in Article 5.9.4.1 of LRFD Specifications (AASHTO, 2012).

C

C

T

Neutral Axis

C

T

M slab Stage I concrete stresses

S (Slab DL)

C  0.6 f c' Stage IIA concrete stresses

Figure 8.2-4 Representative Concrete Flexural Stress Distribution at Stage IIA (Erection and Deck Pour) 

Stage IIB: Construct barrier rails (Fig. 8.2-5) o

Deck concrete hardens and barrier rails are constructed. The girder and deck act together as a composite section.

o

Girder self-weight plus weight of diaphragms and deck are supported by the PC girder alone and additional dead load (haunch and barrier rails) is supported by the composite section.

o

Tensile and compressive stresses are governed by the limits in Article 5.9.4.1 of LRFD Specifications (AASHTO, 2012).

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M slab C S

C C Neutral Axis (new) Neutral Axis

C Composite Section of Girder and Deck

C DL ADL 0 (No Tension)

T

Stage IIA concrete stresses

M ADL S

ADL on Composite Section

Stage IIB Concrete Stresses

Figure 8.2-5 Representative Concrete Flexural Stress Distribution at Stage IIB (Barrier Rail Construction). 

Stage III: Open to traffic (Fig. 8.2-6): o Girder and deck continue to act as a composite section. o Girder self-weight plus weight of diaphragms and deck are supported by the PC girder alone. Additional dead load (haunch and barrier rails) and live loads are supported by the composite section. o This stage is a permanent condition. Compressive and tensile stresses are governed by the limits in LRFD Specifications Table 5.9.4.2.1-1 and Table 5.9.4.2.2-1 (AASHTO, 2012), respectively. C

C

C C

C

Neutral AxisNeut

C

DC+DW Composite Section of Girder and DeckGirder with Wet Deck

Stage IIB Stresses Adjusted for Stage III

T

M LL 1 S HL-93

T  0.19 f c'

Service Level Stage III Concrete Stresses

Figure 8.2-6 Representative Concrete Flexural Stress Distribution at Stage III (Open to Traffic). Chapter 8 – Precast Pretensioned Concrete Girders

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8.2.3

Methods to Vary Strand Eccentricity and Force Efficient design of PC girders typically requires varying the strand eccentricity along the length of the member and/or limiting the strand force at transfer. PC girders are fabricated, transported, and initially installed as simply-supported segments. For a simply-supported girder with straight strands, the large eccentricity between the CGS and the CGC section helps reduce tension and possible cracking at midspan at service level. However, excessive flexural tensile stresses may develop at the top of the girder segments near the ends, where counteracting flexural stresses due to selfweight are minimal. Excessive flexural compressive stresses may similarly develop. The critical location near the ends is at the transfer length, the distance from the end of the girder at which the strand force is fully developed. For this temporary condition, Table 5.9.4.1.2-1 of LRFD Specifications (AASHTO, 2012) specifies appropriate stress limits to mitigate cracking and compression failure.

Figure 8.2-7 Draped Strand Profile (Pritchard, 1992)

Figure 8.2-8 Hold-Down Assembly in Stressing Bed (Ma and Schendel, 2009)

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To reduce the tensile and compressive stresses at the ends of girders, the designer normally considers two primary methods, both of which are used in California: 

Harping (or draping) strands to reduce the strand eccentricity (Figures 8.2-7 and 8.2-8):. o

o



Advantages of harping include: 

Flexural design efficiencies due to the strand CGS achieving a profile corresponding to the moment envelope



Reduction of eccentricity at member ends to control concrete stresses at these critical regions at transfer



Additional shear capacity due to the contribution of the vertical component of the prestress force in the harped strands

Disadvantages of harping include: 

Safety issues and precaster ability to economically deflect and anchor harped strands



Slightly higher cost for fabrication and embedded hold-down devices



Beam form patching to accommodate variable hold down locations

Debonding (or shielding) select strands at the member ends to reduce the transfer prestress force (Figure 8.2-11): o

o

Advantages of debonding include: 

Reduction in concrete stresses at member ends



Simpler fabrication by the use of straight strands in the stressing bed



Elimination of hold-down devices

Disadvantages of debonding include: 

Potential increase in design compressive strength of concrete



Increased design effort to determine debonding patterns, shear reinforcement, and camber

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Figure 8.2-9 Bottom Fiber Stress Distribution at Transfer: Harping vs. Debonding (PCI Bridge Design Manual 2011)

Figure 8.2-10 Top Fiber Stress Distribution at Transfer: Harping vs. Debonding (PCI Bridge Design Manual 2011) By draping the strands in a PC girder, the eccentricity can be varied in linear segments along the length of the girder by mechanically deflecting some of the stressed strands in the casting beds prior to casting using hold-downs and hold-ups, as shown in Figures 8.2-7 and 8.2-8. Although draping is limited to strands within the web, only a portion of the strands typically needs to be draped to achieve the required eccentricity at girder ends. Typically, the drape points are located between approximately 0.33L and 0.4L. Some fabricators may not have suitable equipment for all drape profiles. In addition, the drape angle must be limited to ensure that jacking requirements and hold-down forces do not exceed available capacity. The patterns in Figures 8.2-9 and 8.2-10 provide a comparison of the bottom and top fiber stresses associated with draped and debonded strands.

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A) Single strand sheathing

B) Debonded strands in PC girder

Figure 8.2-11 Plastic Sheathing Used for Debonding Strand Alternatively, the designer may choose to limit transfer stresses by reducing the prestress force through debonding strands along a portion of the girder length at member ends. This is known as partial debonding. Figure 8.2-11 shows debonding of a strand by encasing the strand in a plastic sheathing. Debonding strand prevents the prestressing force from developing in the debonded region and causes the critical section for stresses to shift a transfer length (i.e., 60 strand diameters, per LRFD Specifications) beyond the end of debonding. Caltrans Amendments (Caltrans, 2014) limit the number of partially debonded strands to 33% of the total number of strands and the number of debonded strands in any horizontal row to 50% of the strands in that row. Increases in development length at ultimate are also addressed in Article 5.11.4.3 of LRFD Specifications (AASHTO, 2012). Due to the limitations in number of debonded strands at the girder bottom, the temporary stress at girder top at the ends may still exceed the allowable stress limits, especially for longer span girders. One solution is to use temporary strands at the girder tops that are shielded along the member length except at the girder ends. These strands can be cut at a later stage such as erection, when they are no longer needed, by providing an access pocket formed in the girder top.

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8.3

PRECAST BRIDGE TYPES There are three main PC bridge types: i) precast pretensioned girders, ii) precast post-tensioned spliced girders, and iii) precast segmental girders. Table 8.3-1 summarizes the typical span lengths for these bridge types. Table 8.3-1 Precast Bridge Types and Span Lengths (Caltrans, 2012) Bridge Type Precast pretensioned girder Post-tensioned spliced girder Precast segmental girder

Possible Span Length (ft) 30 to 200 100 to 325 200 to 450

Preferred Span Length (ft) 30 to 180 120 to 250 250 to 400

The selection among these three bridge types is normally decided by span length requirements. As shown in Table 8.3-1, a single precast, pretensioned girder could be designed to span from 20 ft to 200 ft. Trucking length, crane capacity, and transporting routes may limit the girder length (and weight) that could be delivered. Therefore, a girder may need to be manufactured in two or more segments and shipped before being spliced together on-site to its full span length. Such splicing techniques can be applied by using post-tensioning systems for both single-span and multiple-span bridges, which span up to 325 ft. For span lengths over approximately 250 ft, precast segmental girder bridges may be considered, which is beyond the scope of this document. Section 8.3.3 further addresses spliced girder bridges.

8.3.1

Single-Span Bridges As the simplest application of PC girders, single-span bridges normally consist of single girders. As shown in Figure 8.3-1, girders are set onto bearing pads at seattype abutments. Dead and live load effects are based on a simply supported condition. PC girders obviously lend themselves to being single-span elements because they are fabricated as single elements. Abutments can be seat-type or end diaphragm-type.

Figure 8.3-1 Single-Span I Beam Lowered onto Abutments at Mustang Wash Bridge (Bridge No. 54-1279L, Caltrans)

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8.3.2

Multi-Span Bridges Many design considerations for single-span bridges apply to multi-span bridges because girders or girder segments exist as single-span elements for several stages, namely, fabrication, transportation, erection, and deck pour. In addition, some multispan bridges or portions thereof are constructed using expansion joints that can produce a simply supported condition for a span. Most multi-span bridges are constructed with simple-span girders made continuous for live load to increase efficiency and redundancy. Limiting expansion joints, designing deck reinforcement to serve as negative moment reinforcement at interior bents, and providing girder continuity at bents by using a continuous CIP deck and/or CIP diaphragms accomplishes this. In addition, some bridges are detailed to provide an integral connection with full moment transfer between the superstructure and substructure. To achieve this, use CIP diaphragms at bent caps; reinforcing bars between the bent cap, diaphragm, and girders; and/or longitudinal post-tensioning. An integral connection provides not only longitudinal continuity for live load but also longitudinal continuity for seismic loading. Due to moment continuity between the superstructure and substructure, columns in multi-column bents may be designed to be pinned at their base, thus reducing foundation cost. The following sections summarize three typical bent cap configurations for achieving continuity in multi-span bridges:

8.3.2.1



Drop caps



Inverted-tee caps



Integral caps with precast post-tensioned girders

Drop Caps

Figure 8.3-2 Drop Cap at Chuckwalla Wash Bridge (Bridge No. 54-1278L, Caltrans)

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Drop caps are bent caps that provide intermediate supports for girders together with live-load continuity (Figure 8.3-2). Drop caps are commonly detailed to provide a non-integral connection-without moment continuity to the substructure but with moment continuity in the superstructure through negative moment reinforcement in the deck. Simple-span girders are placed on bearing pads at the top of drop caps. Girders at the top of drop caps are normally tied together with a CIP diaphragm and dowels placed through the webs at the ends of the girders. As shown in Figure 8.3-3, steel pipe shear keys may extend from the top of the drop cap into the CIP diaphragms at bent caps. With pipe shear keys, moment transfer is prevented between the superstructure and substructure, and the bearing can more easily be replaced if needed.

Figure 8.3-3 Nonintegral Drop Cap Detail Using Pipe Shear Key

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With proper design and detailing of the diaphragm and bent cap, an integral connection can be developed between the superstructure and substructure, as shown in Figure 8.3-4. For example, the system can be designed to emulate seismic performance of a continuous CIP PT concrete bridge if the joint between girder and cap (due to positive moment during a seismic event) is prevented from opening. One method is to extend pretensioning strands through the joint for development within the cap, in accordance with the requirements of MTD 20-6 (Caltrans, 2001). As mentioned in the subsequent section on integral caps with post-tensioned precast girders, post-tensioning of the girders to the cap at intermediate supports can also be used. The designer is encouraged to clearly detail the reinforcement between the superstructure, diaphragm, and bent cap so that conditions assumed in design realistically match field conditions.

Figure 8.3-4 Integral Drop Cap Detail Adequate seat width must be provided for drop caps to prevent unseating due to longitudinal displacement in a seismic event. Aesthetics should also be considered in the use of drop caps, as they lack the clean lines of inverted-tee caps or CIP PT box girders with integral caps.

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8.3.2.2

Inverted-Tee Caps Using an upside down “T” shaped cross section with a ledge, inverted-tee caps combine the ability to place precast girders directly on the bent and the aesthetic appeal of the flush bottom of cap with the precast girders. Hooked reinforcement extending from side faces of the cap is placed between girders, and a diaphragm is cast to tie the girders and cap together. A deck is later cast for live-load continuity. This is shown in Figures 8.3-5 and 8.3-6.

Figure 8.3-5 Dapped End Girder with Inverted-Tee Cap (Snyder, 2010)

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Figure 8.3-6 Existing Inverted-Tee to Dapped End Girder Connection Detail Designers have commonly modeled this connection as a pin (i.e., non-integral connection between the superstructure and substructure) due to the assumption that the connection would degrade to a pin in a seismic event. However, recent research demonstrated that plastic hinges do indeed form at the column top, confirming that moment continuity develops due to the use of CIP diaphragm and dowel bars through the girder webs (Snyder, 2010). For this connection type, continuity at the column top may be assumed, and joints may be designed for the force transfer associated with plastic hinging. Confining reinforcement at the column top is required. Designers should consult with the Caltrans Earthquake Committee for further Seismic Design Criteria (SDC) updates and instructions for seismic design of inverttee cap-girder connections. 8.3.2.3

Integral Caps with Precast Post-Tensioned Girders Post-tensioning PC girders through a CIP bent creates an integral connection between the superstructure and substructure as well as a frame that is continuous for service, strength, and extreme event limit states (Figure 8.3-7). In addition, such a connection provides a means for bridge widening using PC girders to match the performance and appearance of an existing CIP PT bridge. Without an integral connection, continuity is not effectively developed at the bent cap, which would require columns and foundations to be designed to provide the necessary fixity at the base of the structure. If the connection between post-tensioned PC girders and the bent cap is designed and detailed properly, the system can emulate the seismic performance of a continuous CIP PT concrete bridge (Holombo et al., 2000; Castrodale and White, 2004). Post-tensioning of the girders to the cap and intermediate supports is intended

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to prevent joint opening due to positive moment during a seismic event. Extending bottom pretensioning strands into the cap for development provides positive moment capacity.

Figure 8.3-7 Integral Bent Cap Connection Using Longitudinal Post-tensioning of PC Girders

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8.3.3

Spliced Girder Bridges Due to limitations in transportation length and member weight, as well as stressing bed size, a girder may need to be fabricated in two or more segments and shipped before being spliced together on-site to its full span length. Such splicing techniques can be applied to both single-span and multiple-span bridges. By using this approach, the designer has significant flexibility in selecting the span length, number and location of intermediate supports, segment lengths and splice locations. Splicing is more commonly used for multi-span bridge construction. However, spliced girders have also been used successfully in the construction of several singlespan bridges in California such as the Angeles Crest Bridge (208 ft). Splicing of girders is typically conducted on-site, either on the ground adjacent to or nearby the bridge location, or in place using temporary supports. Figure 8.3-8 shows two precast bathtub girder segments being placed on temporary supports in preparation for field splicing at midspan.

Figure 8.3-8 Precast Bathtub Girder Segments Spliced Near Midspan Using Temporary Supports at Harbor Blvd. Overcrossing (Bridge No. 22-0108, Caltrans) Full continuity needs to be developed between spliced girder segments. This is commonly achieved using post-tensioning tendons between segments and mechanical coupling of reinforcement that is extended from the ends of the girder segments within a CIP closure pour. Figure 8.3-9 shows these details at the closure pour, including the use of couplers for PT ducts and ultimate splice couplers for reinforcement.

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Figure 8.3-9 Details of Spliced Girder Closure Pour Using Mechanical Splices and PT Duct Couplers (Bridge No. 22-0108, Caltrans) Post-tensioning spliced girders not only provides continuity but also enhances structural efficiency. Post-tensioning enhances interface shear capacity across the splice joint (closure pour), which normally includes roughened surfaces or shear keys (Figure 8.3-9). When splicing together multiple spans of PC girders, it is critical that the precast girder placement, post-tensioning sequence, and material properties be properly defined. Figure 8.3-10 shows the construction sequence of a typical two-span (or multi-span) spliced girder bridge. At each stage, the following must be checked: concrete compressive strength and stiffness, creep and shrinkage of concrete, and tension force in the prestressing steel (and debonded length, if needed). The designer must consider each stage as the design of an individual bridge with given constraints and properties defined by the previous stage.

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Figure 8.3-10 Spliced Bridge Construction Sequence (Bridge No. 22-0108, Caltrans)

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The simplest multi-span precast spliced girder system includes consideration of a minimum of four stages or steps after fabrication and before service loads, as follows: 

Transportation: The girder acts as a simply supported beam, with supports defined by the locations used by the trucking company. Typically, the manufacturer or trucking company is responsible for design and check of loads, stability, and bracing during transportation and erection of the girder.



Erection: The girder initially acts as a simply supported beam, with supports defined by the abutments, bents or temporary falsework locations. A CIP closure pour is placed after coupling of PT tendons and reinforcing bars in the splice joint. Optionally, a first stage of posttensioning may be applied before the deck pour instead of after the deck pour (not shown in Figure 8.3-10).



Deck pour: The deck is poured but not composite with the girders until attaining full strength. Therefore, the girders alone carry girder selfweight and the wet deck weight.



Post-tensioning: The hardened deck and girder act compositely, and the girders are spliced together longitudinally using post-tensioning. As the number of girders that are spliced and the stages of post-tensioning increases, so does the complexity of design.

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8.4

DESIGN CONSIDERATIONS PC girder design must address three basic stages—transfer, service, and ultimate—as well as additional stages if post-tensioning is introduced. PC girder design, including section size, prestress force (number and size of strands), strand layout, and material properties, may be governed by any of these stages. Although design for flexure dominates the PC girder design process, other aspects must also be considered, such as prestress losses, girder shear and interface shear strength, deflection and camber, anchorage zones, diaphragms, and end blocks. The following sections briefly introduce the various aspects of PC girder design. The designer is encouraged to read the references cited in the following sections, particularly LRFD Specifications (AASHTO, 2012), Caltrans Amendments (Caltrans, 2014), Caltrans Memo To Designers (MTD) 11-8 (Caltrans, 2014), Caltrans Bridge Design Aids 6-1 (Caltrans, 2012), and Chapters 5 and 6.

8.4.1

Materials

8.4.1.1

Concrete Concrete used in PC girders produced under plant-controlled conditions is typically of higher strength and higher quality than for CIP concrete. Per MTD 11-8, the minimum concrete compressive strength at release, f´ci, and minimum 28-day concrete compressive strength, f´c, for PC girders is 4 ksi. In addition, the concrete compressive strength at release, f´ci, may be selected as large as 7 ksi and f´c as large as 10 ksi. However, designers should verify with local fabricators’ economical ranges of f´ci on a project-specific basis, especially for f´ci and f´c exceeding these limits. Minimum concrete compressive strengths may also be specified at girder erection and for post tensioning, when used. In most PC girder design, a relatively large value of f´ci is used in design, which typically controls the overall concrete mix design. If an excessively large value of f´ci is required in design to resist temporary tensile stresses at transfer in areas other than the precompressed tensile zone, such as the top flange at girder ends, then bonded reinforcement or prestress strands may be designed to resist the tensile force in the concrete, per stress limits in LRFD Specifications Table 5.9.4.1.2-1 (AASHTO, 2012). This helps reduce the required f´ci used in design. The relatively large value of f´ci used in design also results in a relatively large value of f´c (e.g., often in excess of 7 ksi), which is normally larger than that required to satisfy the concrete compressive strength requirements at the serviceability and/or ultimate limit state. In cases where a larger f´ci is required to produce an economical design (e.g., girders of long span, shallow depth, or wide spacing), high strength concrete mixes that require longer than the normal 28-day period may be specified. Current Standard Specifications allow 42 days for achieving specified strength and 56 days for low cement mixes. However, designers should verify the impact of such a decision on the overall construction schedule.

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Advantages of the concrete used in PC girders produced under plant-controlled conditions are wide ranging. Higher modulus of elasticity and lower creep, shrinkage, and permeability are by-products of the relatively higher compressive strength and steam curing process used for PC girders. In addition, reduced effects of creep and shrinkage for PC girders occur after installation because most creep and shrinkage occurs prior to erection. Supplementary cementitious materials (SCMs) and regional materials may also be used for benefits in cost, material properties, and environmental impact through the use of in-house batch plants, mix designs, and sustainability practices. Self-consolidating concrete (SCC), a highly flowable yet cohesive concrete that consolidates under its own weight, is becoming more commonly used in precast plants. It provides significant advantages such as elimination of external and internal vibration for consolidation and reduced manual labor and equipment requirements resulting in reduced construction time; excellent consolidation, even in congested regions of reinforcement; higher level of quality control; extremely smooth concrete surfaces, even in negative draft regions; eliminated need for patching; increased safety; and lower noise levels, usually combined with higher strength and improved durability. Some disadvantages of SCC include more costly material, stricter control on selection and measurement of materials, larger number of trial batches, greater sensitivity to water content, more rapid hardening, faster drying, higher formwork design loads (for fluid pressure), as well as greater experience and care in handling and production of SCC. 8.4.1.2

Steel For economy, PC girders commonly use 0.6 in. diameter, 270 ksi (Grade 270), low-relaxation strands. Use of 0.5 in. diameter strands is less common because the 0.6 in. diameter strands provide a significantly higher efficiency due to a 42% increase in capacity. However, 0.375 in. diameter strands are commonly used for stay-in-place, precast deck panels. If epoxy coated prestressing strands are required, a note should be shown on the design plans, and the corresponding section of the Standard Specifications should be used. Deformed welded wire reinforcement (WWR), conforming to ASTM A497 and Caltrans Standard Specifications based on a maximum tensile strength of 60 ksi, is permitted and commonly used as shear reinforcement in PC girder design.

8.4.2

Prestress Losses From the time prestressing strands are initially stressed, they undergo changes in stress that must be accounted for in design. Figure 8.4-1 illustrates the change in strand stress over time for a typical pretensioned girder.

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Figure 8.4-1 Strand Stress vs. Time in Pretensioned Girder (Tadros et al., 2003) Prestress losses in prestressed concrete members consist of instantaneous (or immediate) and time-dependent losses in prestressing strands. Total losses can be estimated using the LRFD Specifications approach: ∆fpT = ∆fpES  ∆fpLT

(AASHTO 5.9.5.1-1)

where: ∆fpT = total change in stress due to losses (ksi) ∆fpES = sum of all losses or gains due to elastic shortening or extension at the time of application of prestress and/or external loads (ksi) ∆fpLT = losses due to long-term shrinkage and creep of concrete, and relaxation of the steel (ksi) Losses are normally defined from the time of initial stress (immediately after seating of strands for PC girders). Time-dependent losses of prestress include concrete creep and shrinkage and steel relaxation. LRFD Specifications (AASHTO, 2012) provides an approximate estimate and refined estimate for determining timedependent losses. The background can be found in the National Cooperative Highway Research Program (NCHRP) Report 496, Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders (Tadros et al., 2003). For PC girders, instantaneous loss refers to loss of prestress due to elastic shortening of the girder at transfer. Elastic gain refers to increase in strand stress due to strand extension related to application of external loads.

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A reasonable estimate of prestress losses is critical to properly estimate the required prestress force (and thus the required number of strands). Overestimating losses leads to a larger than necessary initial prestress force, which results in larger initial tensile and compressive stresses and may, in turn, result in cracking and larger than expected camber. Overestimation of losses tends to reduce design efficiency because of the increase in number of strands, f´ci cost of the concrete mix, and/or curing time. In addition, problems in girder placement and haunch height in the field may result from excessive camber. Although underestimating losses could potentially produce adverse effects such as flexural cracking in the precompressed tensile zone at service level, such problems have rarely been found to occur in practice. 8.4.2.1

Instantaneous Losses In PC girders, the entire prestressing force is applied to the concrete in a single operation. For pretensioned members, the loss due to elastic shortening can be calculated from AASHTO Eq. 5.9.5.2.3a-1, as shown below:

f pES 

Ep Ect

fcgp

(AASHTO 5.9.5.2.3a-1)

where: ∆fpES

= sum of all losses or gains due to elastic shortening or extension at the time of application of prestress and/or external loads (ksi)

fcgp

= the concrete stress at the center of gravity of prestressing tendons due to the prestressing force immediately after transfer and the self-weight of the member at the section of maximum moment (ksi)

Ep

= modulus of elasticity of prestressing steel (ksi)

Ect

= modulus of elasticity of concrete at transfer or time of load application (ksi)

Calculation of ∆fpES requires iteration for fcgp. However, iteration can be avoided by using LRFD Specifications Eq. C5.9.5.2.3a-1 (AASHTO, 2012) for ∆fpES. It is important that LRFD Specifications Articles C5.9.5.2.3a and C5.9.5.3 be consulted when using transformed section properties in the stress analysis. 8.4.2.2

Time-Dependent Losses LRFD Specifications (AASHTO, 2012) provides two methods to estimate the time-dependent prestress losses: approximate method (Article 5.9.5.3) and refined method (Article 5.9.5.4). This chapter introduces a sample calculation using the approximate method. However, for cases in which the refined method is required or preferred, the designer should consult Article 5.9.5.4 of AASHTO LRFD (AASHTO, 2012). Chapter 9 of the PCI Bridge Design Manual (2011) provides useful PC girder design examples with prestress loss calculations using both the refined and approximate methods.

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Per Article 5.9.5.3, the approximate method is applicable to standard precast, pretensioned members subject to normal loading and environmental conditions, where: 

Members are made from normal-weight concrete



Concrete is either steam- or moist-cured



Prestressing strands use low relaxation properties



Average exposure conditions and temperatures characterize the site

In addition, the estimate is intended for sections with composite decks. This method should not be used for uncommon shapes (volume-to-surface ratios, V/S, significantly different than 3.5 in.), unusual level of prestressing, or with complex construction staging. Long-term prestress losses due to creep and shrinkage of concrete and relaxation of steel are estimated using the following formula, in which the three terms corresponds to creep, shrinkage, and relaxation, respectively: f pLT  10

f pi Aps Ag

 h  st  12 h  st  f pR

(AASHTO 5.9.5.3-1)

where: Ag

= gross area of girder section

Aps = area of prestressing steel fpi

= prestressing steel stress immediately prior to transfer (ksi)

H

= average annual ambient mean relative humidity (percent)

h

= correction factor for relative humidity of ambient air = 1.7-0.01H

st

= correction factor for specified concrete strength time at of prestress transfer to concrete member = 5/(1+ f´c)

∆fpR = an estimation of relaxation loss taken as 2.4 ksi for low relaxation strand, 10 ksi for stress relieved strand, and in accordance with manufacturers recommendation for other types of strand (ksi)

8.4.3

Flexure Bridge Design Practice provides a detailed summary of flexural design provisions, with limit states for service (including transfer), strength, and fatigue in accordance with LRFD Specifications (AASHTO, 2012) and Caltrans Amendments (Caltrans, 2014). Figures 8.2-3 through 8.2-6 illustrate the change in flexural stress distribution near midspan for a typical PC girder at transfer, deck pour, and service level.

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MTD 11-8 provides specific guidance for design of PC girders, addressing issues such as: 

Order of design (service limit state followed by strength check)



Live load continuity and negative moment reinforcement over the bents



Determination of Pj and centroid of PS steel (CGS) and their inclusion on plan sheets



Harping versus debonding, including tolerances for harping and debonding provisions



Use of temporary strands and associated blockouts



Positive moment reinforcement for continuous spans



Design modifications for long span girders

In addition, the following practical aspects should also be noted in carrying out flexural design of PC girders: 

The initial girder section size is typically based on the minimum depthto-span ratio required for a given girder type.



The specified concrete compressive strengths (initial and 28-day) are commonly governed by the initial compressive strength, f´c , required to limit stresses at transfer.



The total prestress force (number and size of strands) and strand layout are usually determined to satisfy the service limit state (Service III) but may have to be revised to satisfy flexural strength at ultimate (Strength II, California P-15 permit truck).



Girder design is based on the minimum overall depth when computing capacity of the section.

8.4.4

Shear

8.4.4.1

Shear Design for Girders Per MTD 11-8, shear design of PC girders is performed using the sectional method specified in LRFD Specifications Article 5.8.3 (AASHTO, 2012). The sectional method is based on the Modified Compression Field Theory (MCFT), which provides a unified approach for shear design for both prestressed and reinforced concrete components (Collins and Mitchell, 1991). The MCFT is based on a variable angle truss model in which the diagonal compression field angle varies continuously, rather than being fixed at 45˚ as assumed in prior codes. For prestressed girders, the compression field angle for design is typically in the range of 20˚ to 40˚. Per Article 5.8.3.4.3 of the California Amendments (Caltrans, 2014), the LRFD Specifications (AASHTO, 2012) simplified shear design procedure cannot be used in PC girder design.

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For disturbed regions, such as those occurring at dapped ends, shear provisions using the strut and tie method should be used (AASHTO, 2012). In the sectional method, a component is investigated by comparing the factored shear force and the factored shear resistance at a number of sections along the member length. Usually this check is made at a minimum of tenth points along the span as well as at locations near the supports. Because shear design typically follows flexural design, certain benefits can be realized in shear design. For example, when harped strands are used, the vertical component of the harped strand force contributes to shear resistance. In addition, the higher strength concrete specified for flexure enhances the Vc term for shear design. Because flexure-shear interaction must be checked per Article 5.8.3.5 of LRFD Specifications (AASHTO, 2012), the longitudinal reinforcement—based on flexural design—must be checked after shear design, to ensure that sufficient longitudinal reinforcement is provided to resist not only flexure (and any axial forces along the member), but also the horizontal component of a diagonal compression strut that generates a demand for longitudinal reinforcement. LRFD Specifications (AASHTO, 2012) includes an upper limit on the nominal shear resistance, Vn, that is independent of transverse reinforcement, to prevent web crushing prior to yielding of transverse reinforcement. For skewed bridges, live load shear demand in the exterior girder of an obtuse angle must be magnified in accordance with LRFD Specifications (AASHTO, 2012) Article 4.6.2.2.3c unless a three-dimensional skewed model is used. To accommodate field bending of stirrups, #4 or #5 stirrups are commonly preferred. In most cases, the size of stirrups should not exceed #6. 8.4.4.2

Interface Shear Design Interface shear should be designed based on the shear friction provisions of LRFD Specifications (AASHTO, 2012) Article 5.8.4 and MTD 11-8.

8.4.5

Deflection and Camber

8.4.5.1

Key Aspects for Design Designers must address potentially challenging issues related to downward deflection and upward camber of PC girders. Camber in a PC girder occurs instantaneously at transfer but can increase to much larger values long-term, particularly due to creep and shrinkage of the concrete. Excessive camber at erection may cause potential intrusion of the top flange of the girder into the CIP deck. Although the contractor is responsible for deflection and camber calculations (per Caltrans Standard Specifications and MTD 11-8), the designer is responsible for specifying a midspan haunch thickness and calculating the minimum haunch thickness at supports, which affects the total bridge depth at both mid-span and at supports. In order to calculate the minimum haunch thickness at supports, girder deflections at release and at erection, as well as immediate girder deflection due to

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the deck weight, must be considered. To complete the deflection design and provide better construction support, the following guidelines are recommended: 

Specify unfactored instantaneous girder deflections on plan sheets: Per Caltrans Standard Specifications, the contractor is responsible for deflection and camber calculations and any required adjustments for deck concrete placement to satisfy minimum vertical clearance, deck profile grades, and cross slope requirements. However, the designer must provide, on plan sheets, the unfactored instantaneous girder deflections due to: o

Deck and haunch weight on the non-composite girder

o

Weight of barrier rail and future wearing surface on the composite girder-deck section

These deflection components are used to set screed grades in the field. For spliced girders, instantaneous upward deflections due to posttensioning at different stages should be shown on the design plans. 

Determine minimum haunch thickness and specify on plan sheets: The haunch is the layer of concrete placed between the top flange of the girder and bottom of deck to ensure proper bearing. It accommodates construction tolerances such as unknown camber of the girder at time of erection. Because camber values vary along the span length, the actual haunch thickness varies along the span, too. The designer should specify the haunch thickness at mid-span and then calculate the minimum required haunch thickness at supports. The haunch: o

Accommodates variation in actual camber

o

Allows the contractor to adjust screed grades

o

Eliminates potential intrusion of the top flange of the girder into the CIP deck

o

Establishes the seat elevation at supports

Cross slope and width at the top flange of the girder should be considered in determining the specified midspan haunch thickness. The typical section should show: o

Minimum structure depth at centerline of bearing at the supports, including girder depth, deck thickness, plus calculated haunch thickness

o

Minimum structure depth at mid-span, including girder depth, deck thickness, plus any haunch thickness the designer specifies

It should be noted that for girders with large flange widths, such as the CA wide-flange girder, a larger haunch thickness might add a significant concrete quantity and weight to the design. Chapter 8 – Precast Pretensioned Concrete Girders

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8.4.5.2



Satisfy LRFD Specifications for live load deflection: Service level deflections may be checked per Article 2.5.2.6.2 of LRFD Specifications (AASHTO, 2012), which suggests a limit of L/800 for live load deflection due to HL-93 vehicular loading. This is an optional check and not required per LRFD Specifications. Because this is an instantaneous deflection check, no multipliers for long-term deflection should be used. The modulus of elasticity should be determined based on Eq. 5.4.2.4-1 of LRFD Specifications (AASHTO, 2012) and the effective moment of inertia, Ie, should be used per Article 5.7.3.6.2.



Verify girder camber is controlled at key stages: The designer may work with the construction structure representative to ensure that the estimated PC girder camber and camber growth are controlled throughout all key stages, such as fabrication, erection, deck placement, and service level. Camber should not be excessive (i.e., causing concern over intrusion of the top flange of the girder into the CIP deck) and should be positive (upward) under both short-term and long-term conditions. This requires the designer to be aware of girder deflection due to prestress force and dead loads, as well as the timing of their application. This can be especially important for bridge widenings. When more accurate camber values are required for unusual cases such as widening of a long span bridge, the assumed age of the girder at various stages may need to be shown on plan sheets.

Calculation Approaches Total deflection of a girder at any stage is the sum of the short-term and longterm deflections. Short-term deflections are immediate deflections based on the modulus of elasticity and effective moment of inertia of the appropriate section. Some loads (such as girder and deck self-weight) are carried by precast girder alone, while others loads are carried by the much stiffer composite girder-deck system (such as barriers, overlays, as well as live loads). Long-term deflections consist of longterm deflections at erection and long-term deflection at final stage (may be assumed to be approximately 20 years). Long-term deflections at erection are more coarsely determined because of the highly variable effects of creep and shrinkage. Therefore, although theoretical values and various procedures to determine instantaneous and long-term camber and deflection of PC girders are available, calculated values must be viewed as merely estimates. Table 8.4-1 lists common equations for instantaneous camber of PC girders for different prestress configurations. Long-term deflections at erection and final stage are typically estimated based on one of three approaches: 

Historic multipliers (e.g., Table 8.7.1-1 of PCI Bridge Design Manual (2011) shown as Table 8.4-2 below)



Modified multipliers based on regional industry experience



Detailed time-step analysis accounting for various construction stages and varying material properties

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Table 8.4-1 Camber and Rotation Values for Various Prestress Configurations (Naaman, 2004)

Case

1

Case

2

Case

3

Case

4

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Table 8.4-2 PCI-Recommended Multipliers for Estimating Long-term Camber and Deflection for Typical PC Members (PCI, 2011) At Erection (1) (2)

(3) (4) (5) (6)

Deflection () component: Apply to the elastic deflection due to the member weight at transfer of prestress Camber () component: Apply to the elastic camber due to prestress at the time of transfer of prestress Final Deflection () component: Apply to the elastic deflection due to the member weight at transfer of prestress Camber () component: Apply to the elastic camber due to prestress at the time of transfer of prestress Deflection () component: Apply to the elastic deflection due to superimposed dead load only Deflection () component: Apply to the elastic Deflection caused by the composite topping

Without Composite Topping

With Composite Topping

1.85

1.85

1.8

1.8

2.7

2.4

2.45

2.2

3

3

---

2.3

Use of multipliers (either historic or regionally modified) for girders is the most common approach for estimating long-term deflections at erection of routine bridges in California. The design example of Section 8.6 uses the historic multiplier method. Instantaneous deflection due to prestressing force and girder weight is calculated at release. Long-term deflection of precast concrete girders at erection is then calculated as the instantaneous deflection multiplied by a multiplier. In performing calculations, camber due to prestressing force and the self-weight of girder, as well as deflections due to the weight of deck and haunch are calculated using the initial modulus of elasticity of concrete and section properties for the non-composite girder. Then, deflections due to the concrete barrier and future-wearing surface are calculated using gross composite section properties. The historic multiplier method is a simple and straightforward method. Even though it is found to give reasonably accurate prediction of the deflection at time of erection, it, however, is not recommended for estimating long-term deflection of bridges comprise of beams that are made composite with cast-in-place deck slab. This method does not account for the relatively significant effects of cast-in-place concrete deck, as described here. Once the deck is hardened, it restrains the beam from creeping upward (due to prestressing). In addition, the differential creep and shrinkage between girders and cast-in-place concrete deck results in changes of the bridge member deformation. The design example in Section 8.6 illustrates the use of Table 8.4-2 to estimate long-term camber and deflection to determine minimum required haunch thickness at supports. Chapter 9 of the PCI Bridge Design Manual (2011) provides additional example calculations for camber and deflection.

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8.4.6

Anchorage Zones

8.4.6.1

Splitting Resistance End splitting can occur along prestressing strands due to local bursting stresses in the pretensioned anchorage zone. To prevent failure, Article 5.10.10 of LRFD Specifications (AASHTO, 2012) requires vertical reinforcement, As, to be provided within a distance h/4 from the end of the girder to provide splitting or bursting resistance given by the following equation: Pr = fs As

(AASHTO 5.10.10.1-1)

where: As = total area of vertical reinforcement located within the distance h/4 from end of beam (in.2) fs = stress in steel not to exceed 20 ksi Pr= factored bursting resistance of pretensioned anchorage zone provided by transverse reinforcement (kip) Per LRFD Specifications (AASHTO, 2012) Article 5.10.10.1, fs should not exceed 20 ksi and Pr should not be taken as less than 4% of the total prestressing force at transfer. For spliced precast girders where post-tensioning is directly applied to the girder end block, general zone reinforcement is required at the end block of the anchorage area based on Article 5.10.9 of LRFD Specifications (AASHTO, 2012). 8.4.6.2

Confinement Reinforcement Article 5.10.10.2 of LRFD Specifications (AASHTO, 2012) requires reinforcement be placed to confine the prestressing steel in the bottom flange, over the distance 1.5d from the end of the girder, using #3 rebar or larger with spacing not to exceed 6 in. and shaped to enclose the strands.

8.4.7

Diaphragms and End Blocks Although intermediate diaphragms may not be required per Article 5.13.2.2 of LRFD Specifications (AASHTO, 2012), Caltrans practice and MTD 11-8 specify the use of one or more intermediate diaphragms for girders longer than 80 ft to improve distribution of loads between girders and to help stabilize the girders during construction. Also, per Article 5.13.2.2 of LRFD Specifications (AASHTO, 2012), end diaphragms are required at abutments, piers, and hinge joints. Due to increase in fabrication inefficiencies, girder weight, and overall cost, end blocks should only be used where essential for shear resistance. For more information, see MTD 11-8.

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8.4.8

Lateral Stability Because PC girders tend to be rather long slender members, they should be checked for lateral stability during all construction stages, including handling, transportation, and erection. Fabricators are normally responsible for all girder stability checks. However, the designer is encouraged to consider and verify lateral stability during design, especially when non-standard girders are selected. Procedures for checking lateral stability were developed by Mast, 1989 and 1993, and recently summarized in Section 8.10 of the PCI Bridge Design Manual. Some commercial software incorporates this method. The designer should verify specific assumed support and stability parameters (e.g., support locations, impact, transport stiffness, super elevation, height of girder center of gravity and roll center above road, and transverse distance between centerline of girder and center of dual tire) with local fabricators, contractors, and other engineers, as appropriate.

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8.5

DESIGN FLOW CHART

The following flow chart shows the typical steps for designing single-span precast, prestressed concrete girders. The example in the next section closely follows this flow chart. START DEVELOP GEOMETRY - Select Girder Type and Spacing - Determine Structure Depth - Check Deck Thickness SELECT MATERIALS - Select Material Properties for Concrete and Steel CALCULATE SECTION PROPERTIES - Calculate Precast Section Properties - Calculate Composite Section Properties DETERMINE LOADS AND PERFORM STRUCTURAL ANALYSIS - Calculate DC, DW, LL - Calculate Distribution Factors - Calculate Unfactored Shear and Moment Envelopes ESTIMATE PRESTRESS FORCE - Estimate of PS Force under Service Limit III - Calculate Required Area of Strands and CGS ESTIMATE PRESTRESS LOSSES - Estimate Elastic Shortening - Estimate Long-Tem Losses (Approximate or Refined Method) DESIGN FOR SERVICE LIMIT STATE - Check Concrete Stress at Release Condition - Check Concrete Stress at Service Condition NO Stress Limits YES DESIGN FOR STRENGTH LIMIT STATE - FLEXURE - Calculate Factored Applied Moment, Mu - Calculate Nominal Flexural Resistance, Mn - Check Reinforcement Limits Determine Additional Required Aps or As

NO Mn ≥Mu? YES

DESIGN FOR STRENGTH LIMIT STATE - SHEAR - Calculate Factored Applied Shear, Vu - Calculate Concrete Shear Resistance, Vc - Calculate Required Shear Reinforcement - Check Spacing and Reinforcement Limits MORE

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CONTINUED

DESIGN FOR INTERFACE SHEAR - Calculate Interface Shear Reinforcement - Check Reinforcement Limits - Check Maximum Nominal Shear Resistance

CHECK MINIMUM LONGITUDINAL REINFORCEMENT - Check Longitudinal Reinforcement for V-M Interaction

DESIGN FOR ANCHORAGE ZONE - Design Pretensioned Anchorage Zone Reinforcement: Vertical and Confinement

DETERMINE CAMBER, DEFLECTION, AND HAUNCH THICKNESS - Calculate Deck and Rail Deflections for Contract Plans - Check Live Load Deflection against AASHTO LRFD criteria - Determine Minimum Haunch Thickness at Supports for Contract Plans

END

Figure 8.5-1 Precast/Prestressed Concrete Girder Design Flow Chart

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8.6

DESIGN EXAMPLE This example illustrates the design procedure for a typical PC girder using the AASHTO Specifications (AASHTO, 2012) and California Amendments (Caltrans, 2014). To demonstrate the process, a typical interior girder of a 70 ft single-span bridge with no skew is designed using a standard California PC I girder with composite CIP deck to resist flexure and shear due to dead and live loads. The design live load used for service limit design (Service I and III) is the HL-93 design truck, and the Caltrans P15 design truck is used for the strength limit design (Strength II). Elastic flexural stresses for initial and final service limit checks are based on transformed sections. The LRFD Specifications Approximate Method is used to estimate long-term, timedependent prestress losses based on gross section properties. Shear design is performed using the sectional method. Major design steps include establishing structural geometry, selecting girder type and spacing, selecting materials, performing structural analysis, estimating prestress force, estimating prestress losses, service limit state design, strength limit state design, shear design, anchorage zone design, determining girder deflections and determining minimum haunch thickness at supports.

8.6.1

Problem Statement A 70 ft simple-span bridge is proposed to carry highway traffic across a river. Preliminary studies have resulted in the selection of a PC concrete bridge based on traffic and environmental constraints at the site. Figures 8.6-1 and 8.6-2 show the elevation and plan views of the bridge, respectively. The span length (from centerline of bearing to centerline of bearing) is 70 ft and the girder length is 71 ft. The required bridge deck width is 35 ft, which includes a 32 ft roadway and two 1.5 ft concrete barriers. Three inches of polyester concrete overlay are assumed to be placed on the bridge as a future-wearing surface (additional dead load on girders). Design of a typical interior girder must satisfy all requirements of LRFD Specifications Bridge Design Specifications (AASHTO, 2012) and California Amendments (Caltrans, 2014) for all limit states.

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Girder Length = 71'-0" Span Length = 70'-0"

Direction of flow

Figure 8.6-1 Elevation View of the Example Bridge

35-0"

shoulder

BB

CL Freeway

EB

12-0" Traffic Lane 12-0" Traffic Lane shoulder

Figure 8.6-2 Plan View of Example Bridge

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8.6.2

Select Girder Depth, Type, and Spacing For a 70 ft span, the standard California I girder section has been found to be an efficient section, with a minimum structure depth-to-span length ratio (D/L) of 0.055 for simple spans, based on Chapter 6 of Caltrans Bridge Design Aids (2012). Also for PC girders, a girder spacing-to-structure depth ratio (S/D) of 1.5 is commonly used. Span length, L = 70 ft Assuming: Structure Depth, Ds = 0.055 Span Length, L The minimum depth is: Ds = 0.055 (70) = 3.85 ft Because the deck thickness is based on girder spacing and girder spacing is based on structure depth, the concrete slab thickness must be initially assumed. Assume a slab thickness of 7 in. and later verify this value using Table 10-20.1(a) Deck Slab Thickness and Reinforcement Schedule in Memo To Designers (Caltrans, 2008b) after the girder spacing has been determined. Therefore, the minimum girder height = 3.85 (12) – 7 = 39.2 in. Select a 42 in. standard California I girder (CA I42) from BDA 6-1, slightly larger than the minimum height. Assuming a haunch thickness, th = 1 in. at midspan: The structure depth, Ds = 42 + 1 + 7 = 50 in. (4.17 ft) Ds 4.17 = =0.060 > 0.055 70 L

OK

The center-center girder spacing is determined as follows: Maximum girder spacing, S = 1.5 Ds = 1.5 (4.17 ft) = 6.26 ft Total bridge width = 35 ft (assumed) Try a girder spacing, S = 6 ft Overhang length=

35 - 6 (5 spacings) =2.5 ft 2 overhangs

According to MTD 10-20, Attachment 1 (Caltrans, 2013), overhangs should be less than half the girder spacing (S/2) or 6 ft maximum. 2.5 = 0.42 ft < 0.50 ft OK 6 Therefore, use 6 ft girder spacing. Determine deck thickness:

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From MTD Table 10-20.1(b) Deck Slab Thickness and Reinforcement Schedule (Caltrans, 2008b), for girder centerline-to-centerline spacing of 6 ft, the required slab thickness is 7 in. Therefore, a 7 in. deck thickness can be used. The established typical cross section of the bridge is presented in Figure 8.6-3. It consists of six standard California 3 ft - 6 in. PC I-girders (CA I42) with a 7 in. CIP composite deck and two Type 736 concrete barriers.

1-6

1-6

0'-7"

4'-2"

3'-6""

Concrete Barrier

PC I-Girder, Typ.

Figure 8.6-3 Typical Bridge Cross Section.

8.6.3

Establish Loading Sequence The loading sequence and corresponding stresses for a single-span PC girder are normally considered at three distinct stages, as summarized in Table 8.6-1. The table also indicates what section (non-composite versus composite) resists the applied loading. Note: Per Caltrans practice, transportation (shipping and handling) is generally the responsibility of the contractor and PC manufacturer.

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Table 8.6-1 Typical Stages of Loading and Resisting Section for Single-Span PC Girder Stage

Location

I

Casting Yard

IIA

On Site

IIB

On Site

Construction Activity Cast and Stress Girder (Transfer) Erect Girder, Cast Deck Slab

Loads

Resisting Section

DC (Girder) DC (Girder, Diaphragm, Slab), Construction Loads DC (Girder, Diaphragm, Slab)

Construct Barrier Rails

DC (Barrier Rails) DC (Girder, Diaphragm, Slab)

Final Location

III

8.6.4

DC (Barrier Rails) DW (Future Wearing Surface) LL (Vehicular Loading, HL-93 or P15)

Open to Traffic

Girder (Non-composite) Girder (Non-composite) Girder (Non-composite) Girder and Deck (Composite) Girder (Non-composite)

Girder and Deck (Composite)

Select Materials The following materials are selected for the bridge components. The concrete strengths for PC girders at transfer and at 28 days are assumed at this stage of design based on common practice in California. However, these values are subsequently verified during service limit state design: 

Concrete compressive strength and modulus of elasticity: o

PC girder Concrete unit weight is assumed herein wc = 0.15 kcf At transfer: f´ci = 4.8 ksi (80% of f´c at 28 days) Eci = 33,000 w1.5 c

′

(AASHTO 5.4.2.4)

= 33,000 (0.15)1.5 √4.8 = 4,200 ksi Eci = modulus of elasticity of concrete at time of transfer At 28 days: f´c = 6 ksi Ec =33,000 (0.15)1.5 √6 = 4,696 ksi o

Cast-in-place deck slab: Concrete unit weight is assumed herein wc = 0.15 kcf f´c = 3.6 ksi

Chapter 8 – Precast Pretensioned Concrete Girders

(Article 5.4.2.1 of CA; MTD 10-20)

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Ec =33,000 (0.15)1.5 √3.6 = 3,637 ksi o

Prestressing steel: 0.6 in. diameter, seven-wire, low-relaxation strands, Area of each strand, Aps = 0.217 in.2 Grade 270, nominal tensile strength, fpu = 270 ksi (AASHTO Tab 5.4.4.1-1) Yield strength, fpy = 0.9 fpu = 243 ksi

(AASHTO Tab 5.4.4.1-1)

Initial jacking stress, fpj = 0.75 fpu = 202.5 ksi (CA Table 5.9.3-1, 2013) Modulus of elasticity of prestressing steel, (AASHTO Article 5.4.4.2) Ep = 28,500 ksi o

Mild steel - A706 reinforcing steel: Nominal yield strength, fy = 60 ksi Modulus of elasticity of steel, Es = 29,000 ksi

8.6.5

Calculate Section Properties In calculating section properties, gross sections are used for estimating the required prestress force (Section 8.6.8) and for estimating prestress losses using the LRFD Specifications Approximate Method (Section 8.6.9). However, girder flexural stresses are checked at the service limit state based on transformed section properties (Section 8.6.10).

8.6.5.1

Precast Section Figure 8.6-4 shows the standard California Standard 3 ft 6 in. I girder (CA I42) and gross section properties of the girder. Section properties are obtained from BDA 6-1 (Caltrans, 2012).

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yt D = 42"

yb

SECTION PROPERTIES Ag = 474 in.2 Icg = 95,400 in.4 yb = 20 in. yt = 22 in. Sb = 4,770 in.3 St = 4,336 in.3 r = 14.2 in.

Figure 8.6-4 Standard CA I42 Girder (BDA 6-1, 2012) Ag =

gross area of girder section (in.2)

Ig =

gross moment of inertia of girder about centroidal axis (in.4)

yb =

distance from neutral axis to extreme bottom fiber of PC girder (in.)

yt

8.6.5.2

=

distance from neutral axis to extreme top fiber of PC girder (in.)

Sb =

section modulus for bottom extreme fiber of section (in.3)

St =

section modulus for top extreme fiber of section (in.3)

r

radius of gyration (in.)

=

Effective Flange Width CA Amendements Article 4.6.2.6 (Caltrans, 2014) state that the effective flange width, beff, may be taken as the full flange width if S  0 . 32 . L

where: S

=

spacing of girders or webs (ft)

L

=

individual span length (ft)

For this example, S 6   0.09  0.32 L 70

Therefore, the effective flange width beff = S = 72 in. Chapter 8 – Precast Pretensioned Concrete Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 8.6-5 Effective Flange Width. 8.6.5.3

Composite Section To compute properties of the composite section, the CIP deck slab and haunch concrete (same material as deck) are transformed to the higher strength girder concrete using the modular ratio, n. n 

EB ED

(AASHTO 4.6.2.2.1-2)

where: n

=

modular ratio between girder and deck

EB =

modulus of elasticity of girder material (ksi)

ED =

modulus of elasticity of deck material (ksi)

Using AASHTO Eq.4.6.2.2.1-2: n

4 ,696 EB   1 .29 3,637 ED

Transformed flange width 

72 72   55.8 in. n 1.29

Transformed deck area = 55.8(7) = 391 in.2 Transformed haunch width =

19 19   14.7 in. n 1.29

Transformed haunch area = 14.7(1) = 14.7 in.2

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Table 8.6-2 Section Properties - Gross Composite Section Section

Ai (in.2)

yi (in.)

Ai (yi ) (in.3)

Io (in.4)

Ai (Y-yi)2 (in.4)

Deck

391

46.5

18,182

1,681

79,956

Haunch

14.7

42.5

625

1

1,560

Girder

474

20

9,480

95,400

70,550

Total

879.7

-

28,287

97,082

152,066

Ac = 879.7 in.2 YBC 

 Ai yi 28,287   32.2 in. 879.7  Ai

YTC

YTC = 50 – 32.2 = 17.8 in. Ic = 97,082 + 152,066 = 249,148 in.4 S BC 

Neutral Axis yi

YBC

Ic 249 ,148   7,735 in.3 32 .2 YBC

where: yi

= distance from centroid of section i to centroid of composite section

Ac = concrete area of composite section YTC = distance from centroid of composite section to extreme top fiber of composite section Ic

= moment of inertia of composite section

SBC = section modulus of the composite section for extreme bottom fiber of PC girder

8.6.6

Determine Loads

8.6.6.1

Dead Load PC Girder: wg =

474 (0.15) = 0.494 klf 144

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Slab (before reaching design strength): 504 (0.15) = 0.525 klf 144

ws =

Haunch: wh =

19 0.15 = 0.020 klf 144

Dead loads on composite section: Type 732 barrier rail on both sides of deck (concrete area = 144 in.2): 444 wbr = 0.15 = 0.463 klf/barrier 144 Dead load of wearing surfaces and utilities - DW (Article 3.3.2, AASHTO, 2012) 3 in. polyester concrete overlay = 0.035 ksf 8.6.6.2

Live Load At the Service Limit State, LRFD Specifications requires design for the HL-93 vehicular live load. At the Strength Limit State, LRFD Specifications (AASHTO, 2012) and California Amendments (Caltrans, 2014) require design for both HL-93 vehicular live load and the California P15 permit truck. 



HL-93 vehicular live load consists of these combinations: o

Design truck or design tandem (AASHTO Art. 3.6.1.2.1)

o

Design lane load of 0.64 klf without dynamic load allowance (IM) (AASHTO Art. 3.6.1.2.4)

California P15 permit truck: The P15 vehicular live load is the California P15 Permit Design Truck defined in Art. 3.6.1.8 of California Amendments (Caltrans, 2014).

8.6.7

Perform Structural Analysis

8.6.7.1

Dead Load Distribution Factor According to LRFD Specifications Art. 4.6.2.2.1 (AASHTO, 2012), permanent dead loads (including concrete barriers and wearing surface) may be distributed uniformly among all girders provided all of the following conditions are met: 

Width of deck is constant. (OK)



Number of girders, Nb, is not less than four; i.e., Nb = 6 (OK)

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

Girders are parallel and have approximately the same stiffness. (OK)



Roadway part of the overhang, de, does not exceed 3 ft de is taken as the distance from the exterior web of exterior girder to interior edge of curb: de = 2.5 - 1.5 - 0.5(7/12) = 0.71 ft ≤ 3 ft (OK)



Bridge is on a tangent line and curvature in plan is zero. (OK)



Cross-section is consistent with one of the cross-sections shown in AASHTO Table 4.6.2.2.1-1 (AASHTO, 2012). The superstructure is type (k). (OK)

Because the design example satisfies the criteria, the concrete barrier and wearing surface loads can be evenly distributed among the six girders based on the dead load distribution factor (DFDL), which is determined as: DFDL =

Tributary Width 6 = =0.171 Bridge Width 35

Using the DFDL: Barrier, wbr = DC3 = (0.463)(2)(0.171) = 0.159 klf/girder DW = dead load of future wearing surface, 0.035 ksf DW = (0.035)(32)(0.171) = 0.192 klf/girder 8.6.7.2

Unfactored Shear Force and Bending Moment due to DC and DW Dead load shear and moment can be obtained from structural analysis software or can be calculated as follows (for simply-supported, single-span bridges): Shear at x, Vx = w (0.5Lx) Moment at x, Mx = 0.5wx (Lx) where: w = uniform dead load, klf x = distance from left end of girder (ft) L = span length = 70 ft

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Table 8.6-3 Unfactored Shear Force and Bending Moment due to DC and DW Girder Weight Slab, Haunch Wt. Barrier Weight Future Wearing (DC1) (DC2) (DC3) Surface (DW) Dist/Span Location Shear Moment Shear Moment Shear Moment Shear Moment (X/L) (ft) (kip) (kip-ft) (kip) (kip-ft) (kip) (kip-ft) (kip) (kip-ft) 0L 0 17.3 0 19.1 0 5.6 0 6.7 0 0.05L* 3.5 15.6 57.5 17.2 63.4 5 18.5 6 22.3 0.1L 7 13.8 108.9 15.3 120.1 4.4 35 5.4 42.3 0.2L 14 10.4 193.6 11.4 213.6 3.3 62.2 4 75.3 0.3L 21 6.9 254 7.6 280.3 2.2 81.6 2.7 98.8 0.4L 28 3.5 290.3 3.8 320.3 1.1 93.2 1.3 112.9 0.5L 35 0 302.4 0 333.7 0 97.4 0 117.6 Location

*Critical shear section

8.6.7.3

Unfactored Shear Force and Bending Moment due to Live Loads Live loads are applied to the bridge deck on one or more design lanes. Therefore, shear forces and bending moments are normally calculated on a per-lane basis. However, shear forces and moments must then be distributed to individual girders for girder design. LRFD Specifications permits governing values of shear force and moment envelopes to be distributed to individual girders using simplified distribution factor formulas, specified separately for moment and shear (AASHTO Art. 4.6.2.2.2 and Art. 4.6.2.2.3, respectively). As shown previously, the conditions of AASHTO Art. 4.6.2.2 are satisfied for this example bridge. Therefore, the simplified distribution factor formulas are applied to the interior girder design in the following sections.

8.6.7.3.1

Live Load Moment Distribution Factor, DFM (for Interior Girders) The live load distribution factor for moment (DFM, lanes/girder), for an interior girder is governed by the larger value for one design lane versus two design lanes loaded, as shown below. 

One design lane loaded: 0.4 0.3  S   S   K g  DFM  0.060        14   L   12Lts 3 

0.1

(AASHTO Table 4.6.2.2.2b-1) Provided the following ranges are met: 3.5  S  16 S = girder spacing = 6 ft (OK) 4.5  ts  12 ts = thickness of concrete slab = 7 in. (OK) 20  L  240 Chapter 8 – Precast Pretensioned Concrete Girders

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L = span length = 70 ft (OK) Nb = number of girders  4 Nb = 6 (OK) 10,000  Kg  7,000,000 Longitudinal stiffness parameter, Kg = 552,464 in.4 (OK) See calculation below: Kg =n(I + Ae2g )

(AASHTO 4.6.2.2.1-1)

n = EB / ED = 1.29

(AASHTO 4.6.2.2.1-2) 4

I = Icg = 95,400 in. A = Ag = 474 in.2

eg = distance between centers of gravity of girder and deck = 46.5 – 20 = 26.5 in. Kg = 1.29 [95,400 + 474 (26.5)2] = 552,464 in.4  6 DFM  0.06     14 

0.4

 6   70.0 

0.3 

  552,464   12 70 7 3   

0.1

  

== 0.06 + (0.713)(0.479)(1.067) = 0.424 lanes / girder 

Two or more design lanes loaded: 0.6

 S  S DFM  0.075       9.5   L 

0.2

 Kg     12 Lt 3  s  

0.1

(AASHTO Table 4.6.2.2.2b-1)  6  DFM  0.075     9.5 

0.6

 6     70 

0.2 

552, 464     12(70)  7 3   

0.1

= 0.075 + (0.759)(0.612)(1.067) = 0.571 lanes / girder Therefore, DFM for two or more lanes loaded is larger and thus controls. Use DFM = 0.571 lanes / girder

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8.6.7.3.2

Live Load Shear Distribution Factor (DFV) for Interior Girders 

One design lane loaded:

(AASHTO Table 4.6.2.2.3a-1)

 S  DFV  0.36     25 

= 0.36 + 0.24 = 0.6 lanes / girder 

Two or more design lanes loaded:

S S DFV  0.2        12   35 

2

= 0.2 + 0.5 – 0.029 = 0.671 lanes / girder Therefore, DFV for two or more lanes loaded is larger and thus controls. Use DFV = 0.671 lanes / girder Note: The dynamic load allowance factor (IM) is applied to the HL-93 design truck and P15 permit truck only, not to the HL-93 design lane load. Table 3.6.2.1-1 of California Amendments (Caltrans, 2014) summarizes the values of IM for various components and load cases. The live load moment and shear are commonly calculated at tenth points and can be obtained from common structure analysis programs. Spreadsheets can also be used for simple-span structures. In this example, structural analysis software was used to determine the live load moments. The results are tabulated in Table 8.6-4 for HL-93 loading and Table 8.6-5 for P15 loading, respectively. These tables list the envelope values for moment and shear per lane, as well as per girder (for design) using the distribution factors.

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Table 8.6-4 Unfactored Live Load Moment and Shear Force Envelope Values due to HL-93 (LL + IM) Location (ft)

Per Lane† Moment Shear (kip-ft) (kip) 0 102.11 348.5 97.9 655.03 91.56 1144.64 78.18 1468.82 65.24 1657.38 52.75 1695.40 -40.87

DFM (Lane per Girder) 0.571 0.571 0.571 0.571 0.571 0.571 0.571

DFV (Lane per Girder) 0.671 0.671 0.671 0.671 0.671 0.671 0.671

0L* 0 0.05L** 3.5 0.1L 7 0.2L 14 0.3L 21 0.4L 28 0.5L 35 *L = Span Length ** Critical section for shear †These values were obtained from CT Bridge (Include IM = 33%)

Per Girder M(LL+IM) V(LL+IM) (kip-ft) (kip) 0 68.5 199 65.7 373.8 61.4 653.2 52.4 838.2 43.8 945.8 35.4 967.5 -27.4

Table 8.6-5 Unfactored Live Load Moment and Shear Force Envelope Values due to P15 Truck (LL + IM) Location (ft)

Per Lane† Moment Shear (kip-ft) (kip) 0 178.5 532.4 152.3 972 138.86 1566 111.86 2025 89.68 2349 69.43 2328.75 -50.14

DFM (Lane per Girder) 0.571 0.571 0.571 0.571 0.571 0.571 0.571

DFV (Lane per Girder) 0.671 0.671 0.671 0.671 0.671 0.671 0.671

0L* 0 0.05L** 3 0.1L 7 0.2L 14 0.3L 21 0.4L 28 0.5L 35 *L = Span Length ** Critical section for shear †These values were obtained from CT Bridge (Include IM = 25%)

8.6.8

Per Girder M(LL+IM) V(LL+IM) (kip-ft) (kip) 0 119.8 304 102.2 554.7 93.1 893.6 75 1,155.6 60.1 1,340.5 46.6 1,328.9 -33.6

Estimate Prestressing Force and Area of Strands The minimum jacking force, Pj and associated area of prestressing strands, Aps, can be reaonably estimated based on satisfying the two tensile stress limits at the bottom fiber of the PC girder at the Service III limit state: 

Case A) No tension under permanent loads



Case B) Tension limited to prevent cracking under total dead and live loads

It should be noted that, for Service III, only the HL-93 vehicular live load applies. P15 applies to Strength II but not Service III. The critical location for bending moment is normally midspan. However, other locations such as 0.4L (P15

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truck) and harp points can govern and must be checked as well. Gross section properties are used. Calculations for these two critical cases are provided below. Note: Compression is taken as positive (+) and tension as negative (-). 

Case A: No tension is allowed for components with bonded prestressing tendons or reinforcement, subjected to permanent loads (DC, DW) only. Set the stress at the bottom fiber equal to zero and solve for the required effective prestress force (at service, i.e., after losses), P, to achieve no tension.

P Pec  M DC1  M DC 2 M DC3  M DW    Ag Sb  Sb SBC

   0 

Rearranging the equation:  M DC 1  M DC 2 M DC 3  M DW   Sb S BC  P e 1  c Ag Sb

  

As shown in Table 8.6-3 (DC and DW) and Table 8.6-4 (HL-93 vehicular live load), the maximum moment due dead load and live load occurs at midspan. Moments on a per girder basis are used for girder design. MDC1

= unfactored moment due to girder self-weight = 302.4 kip-ft

MDC2

= unfactored moment due to slab and haunch weight = 333.7 kip-ft

MDC3

= unfactored moment due to barrier weight = 97.4 kip-ft

MDW

= unfactored moment due to future wearing surface = 117.6 kip-ft

SBC

= section modulus for the bottom extreme fiber of the composite section = 7,735 in.3

To solve for P, the required effective prestressing force, an estimate of the eccentricity of the noncomposite girder, ec, is needed. To determine ec, the centroid of the prestressing force at midspan can be reasonably estimated to be 4 in. from the bottom of the girder.

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Thus, the eccentricity of prestressing force at midspan based on the noncomposite section is taken as: ec = 20 – 4 = 16 in.  302 .4  333 .7 12  97.4  117 .6 12      4,770 7,735   P 1 16  474 4,770

Required effective prestressing force, P = 353.9 kips 

Case B: Allowable tension for components subjected to the Service III limit state (DC, DW, (0.8) HL-93), subjected to not worse than moderate corrosion conditions, and located in Environmental Areas I or II =  0 .19

f c

.

P Pec  M DC1  M DC2 . M DC3  M DW  0.8M HL93     0.19 f 'c    Ag Sb  Sb S BC  where: MHL93 = moment due to HL-93 loading at midspan = 967.5 kip-ft (Table 8.6-4)  M DC1  M DC 2 . M DC 3  M DW  0.8M HL 93      (0.19) f 'c  Sb S BC   P ec 1  Ag S b  302 .4  333 .7 12    97 .4  117 .6  0.8( 967 .5) 12       0.19 6 4,770 7,735    P 1 16  474 4,770

Required effective prestressing force, P = 488.5 kips The minimum required effective prestressing force, P, at service level for an interior girder is the larger value from Case A and Case B. Therefore, P = Pf = 488.5 kips/girder To determine the minimum required jacking force, an estimate of prestress losses is needed. Thus, assuming total (immediate and longterm) prestress losses of 25% (of the jacking force), the required jacking force (i.e., just before transfer, ignoring minor losses from jacking to detensioning) is: Minimum Jacking Force, Pj =

Chapter 8 – Precast Pretensioned Concrete Girders

488.5 = 651.3 kips 0.75 8-57

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The required area of prestressing strands, Aps, jacked to 0.75 fpu is: Required Aps 

651.3  3.22 in.2 0.75(270)

Number of 0.6 in. diameter strands required 3.22 = = 14.8 strands 0.217 14.8 is rounded to 16, an even number provided for symmetry (about a vertical line through the centroid) to produce a uniform stress distribution in the member. Therefore, use sixteen 0.6 in. diameter low relaxation Grade 270 strands. The actual area of strands is thus: Aps = 16 (0.217) = 3.42 in.2 Total prestressing force at jacking, Pj = 0.75(270)(3.472) = 703 kips It is a common practice in Caltrans to provide contractors with the prestressing force and centroid of prestressing path on contract plans, instead of actual strand patterns. This gives the contractors flexibility in choosing the location and number of strands, based on the setup of their casting bed. However, designers are encouraged to layout an actual strand pattern. This helps ensure the design is constructible and avoids the possible use of too many strands in one girder. The strand pattern is shown in Fig. 8.6-6: six strands at 2.5 in., eight at 4.5 in. and two at 6.5 in. The CGS from the bottom of the girder is: 6(2.5) + 8(4.5) + 2(6.5) 16 = 4 in. from bottom of girder.

CGS =

The actual eccentricity, ec, at midspan for the girder = 20 – 4 = 16 in., matching the assumption used in estimating the prestressing force. Normally, the actual value will vary from the assumption and should be used in subsequent design calculations.

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CGS = 4

2 @ 6.5 8 @ 4.5 6 @ 2.5

Figure 8.6-6 Strand Pattern in PC Girder at Midspan Section

8.6.9

Estimate Prestress Losses Prestress losses were previously estimated in a very approximate way to determine area of strands. With a trial number of strands and layout now determined, prestress losses can be more accurately approximated. Per LRFD Specifications, total prestress losses in prestressing strand stress are assumed to be the sum of immediate and long-term losses. Immediate losses for strands in a PC girder are due to elastic shortening. Long-term losses are primarily due to concrete creep and shrinkage as well as steel relaxation. ∆fpT = ∆fpES + ∆fpLT

(AASHTO 5.9.5.1-1)

where: ∆fpES = change in stress due to elastic shortening loss (ksi) ∆fpLT = losses due to long-term shrinkage and creep of concrete and relaxation of prestressing steel (ksi) ∆fpT = total change in stress due to losses (ksi) 8.6.9.1

Elastic Shortening Immediate elastic shortening losses are easily determined for PC girders using a closed form solution based on LRFD Specifications Commentary Eq. C5.9.5.2.3a-1: f pES 





A ps f pbt I g  e m2 Ag  e m M g Ag





A ps I g  e m2 Ag 

Ag I g E ci Ep

where: Aps = area of prestressing steel = 3.472 in.2

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Ag = gross area of girder section = 474 in.2 fpbt = stress in prestressing steel immediately prior to transfer = 0.75(270) = 202.5 ksi, ignoring minor relaxation losses after jacking Eci = 4,200 ksi Ep = 28,500 ksi em = eccentricity at midspan = 16 in. Ig = moment of inertia of gross section = 95,400 in.4 Mg = midspan moment due to self-weight of girder = MDC1 =302.4 k-ft (12 in./ft) = 3,629 k-in.

f pES 

3.472(202.5)[95,400  162 (474)]  16(3,629)(474) 474(95,400)(4,200) 3.472[(95,400  162 (474)]  28,500

f pES  16.84 ksi The initial prestressing stress immediately after transfer = 202.5 – 16.84 = 185.7 ksi. LRFD Specifications C5.9.5.2.3a notes that when transformed section properties are used in calculating concrete stresses, the effects of losses and gains due to elastic deformation are implicitly accounted for. Therefore, fpES should not be used to reduce the stress in the prestressing strands (and force) for concrete stress calculations at transfer and service level. 8.6.9.2

Long Term Losses – Approximate Method

LRFD Specifications provides two methods to estimate the time-dependent prestress losses: Approximate Method (Article 5.9.5.3) and Refined Method (Article 5.9.5.4). This example uses the LRFD Specifications Approximate Method to estimate long-term, time-dependent prestress losses, based on gross section properties. Per Article 5.9.5.3, the approximate method is applicable to standard precast, pretensioned members subject to normal loading and environmental conditions, where: 

Members are made from normal-weight concrete (OK)



Concrete is either steam- or moist-cured (OK)



Prestressing strands use low relaxation properties (OK)



Average exposure conditions and temperatures characterize the site (OK)

Because the girder in this example satisfies all of the criteria, the Approximate Method can be used. Chapter 8 – Precast Pretensioned Concrete Girders

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Long-term prestress losses due to creep and shrinkage of concrete and relaxation of steel are estimated using the following formula, in which the three terms corresponds to creep, shrinkage, and relaxation, respectively: ∆fpLT =10

fpi Aps Ag

γh γst + 12γh γst + ∆fpR (AASHTO 5.9.5.3-1)

where: fpi = prestressing steel stress immediately prior to transfer (ksi) H = the average annual ambient relative humidity (%) h = correction factor for relative humidity of ambient air = 1.7-0.01H st = correction factor for specified concrete strength time at of prestress transfer to concrete member = 5/(1+ fci) ∆fpR an estimation of relaxation loss taken as 2.4 ksi for low relaxation strand, 10 ksi or stress-relieved strand, and in accordance with manufacturers recommendation for other types of strand (ksi) For this calculation: fpi = 202.5 ksi H = Average annual ambient relative humidity = 70% h = 1.7 – 0.01H = 1.7 = 0.01 (70) = 1

(AASHTO 5.9.5.3-2)

5 5   0.862 1  f ci  1  4.8

(AASHTO 5.9.5.3-3)

 st 

∆fpR = 2.4 ksi for low relaxation strands f pR  10 

202.53.472  10.862   1210.862   2.4

474  1.8  10.3  2.4  25.5 ksi

Total prestress losses: ∆fpT = 16.85 + 25.5 = 42.3 ksi

f pT 

42.3 (100%)  24% 202.5

Effective prestress used with gross non-transformed section: fpe = effective stress in prestressing strands (service level) = 202.5 – 42.3 = 160.2 ksi Check prestressing stress limit at service limit state: 0.8 fpy ≥ fpe

Chapter 8 – Precast Pretensioned Concrete Girders

(AASHTO Table 5.9.3-1)

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0.8 (270)(0.9) = 194.4 ksi > 160.2 ksi, (OK) For gross non-transformed sections, the effective prestressing force after all losses, Pf = 3.472 (160.2) = 556.2 kips Regarding transformed sections, note that with transformed sections used in subsequent sections of this design example, the prestressing force to be used in concrete stress calculations at transfer is the jacking force, Pj, and the prestressing force to be used in concrete stress calculations at service level is the final prestressing force, Pf , based on long-term losses only: Effective prestress used with transformed section: fpe = 0.75 fpu – ∆fpLT = 0.75 (270) – 25.5 = 177 ksi Pf = fpe (Aps) = 177(3.472) = 614.5 kips

8.6.10

Design for Service Limit State Design for the Service Limit State addresses the suitability of the previously estimated strand force and profile based on Stages I, IIA, and III. Concrete stresses are checked at transfer, which may lead to design modifications such as adjusting the strand profile or initial concrete compressive strength fci. The most critical check of stresses at the Service Limit State is normally the check of the tensile stress at the bottom of the girder to prevent possible cracking at Service III (HL-93 vehicular live load).

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8.6.10.1

Calculate Transformed Section Properties

The use of transformed concrete section properties generally leads to more accurate calculations than use of gross section properties. As this is recognized by LRFD Specifications C5.9.5.2.3a, calculations use transformed section properties. Three sets of transformed section properties are needed for the service limit state design. These include transformed section properties of the noncomposite girder at transfer (Stage I), erection, and deck casting (Stage IIA, before deck hardening), as well as the composite section of the girder and deck at service (Stage III). The section property calculations for these stages are presented below. A minor difference in final and initial transformed noncomposite properties results from the use of Ec versus Eci. However, the difference between the composite and noncomposite final properties is significant due to the additional deck area. Table 8.6-6 Transformed Section Properties: Girders and Strands (Initial, Transfer) Section Girder

Ai (in.2) 474

yi (in.) 20

Ai ( yi ) (in.3) 9,480

Ii (in.4) 95,400

Ai (Y-yi)2 (in.4) 200

Strands

20.1‡

4

80.4

≈0

4,736

Total

494.1

--

9,560.4

95,400

4,936

n 1 

E ps E ci

1 

28,500  1  5 .8 4,200

Total Ac = 494.1 in.2 Y Bti 

9,560 .4  19 .4 in. 494 .1

YTti = 42 – 19.35 = 22.6 in. Iti = 95,4000 + 4,936 = 100,336 in.4 S Bti 

I ti 100,336   5,185 in.3 19.4 YBti

STti 

I ti 100,336   4,430 in.3 22.6 YTti

YTti

eti = 19.4 - 4 = 15.4 in. ‡

Neutral Axis

yi eti

YBti

CGS

Strands are transformed using (n – 1)

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Table 8.6-7 Transformed Section Properties: Girders and Strands (Final) Section

Ii (in.4) 95,400

Ai (Y-yi)2 (in.4) 154

4

Ai ( yi ) (in.3) 9,480 70.8

≈0

4,190

--

9,550.8

95,400

4,344

Girder

Ai (in.2) 474

yi (in.) 20

Strands

17.7‡

Total

491.7

n 1 

E ps Ec

Y Btf 

1 

28,500  1  5 .1 4,696

9 ,550 . 8  19 . 4 in. 491 . 7

YTtf YTti = 42 – 19.35 = 22.6 in. Neutral Axis

Iti = 95,4000 + 4,344 = 99,744 in.4 S Btf  S Ttf 

I tf YBtf I tf YTtf



99 ,744  5,135 in. 3 19 .4



99 ,744  4,419 in. 3 22 .6

etf

YBtf

CGS et = 19.4 - 4 = 15.4 in. ‡

Strands are transformed using (n – 1)

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Table 8.6-8 Transformed Section Properties: Composite Girder and Deck (Final) Section Girder

Ai (in.2) 474

yi (in.) 20

Ai (yi ) (in.3) 9,480

Ii (in.4) 95,400

Ai (Y-yi)2 (in.4) 63,671

Strands

17.7

4

70.8

≈0

13,473

Deck

391**

46.5

18,182

1,595

86,922

Haunch

14.7**

42.5

624.8

1

1,750

Total

897.4

-

28,357.6

96,996

165,816

n

Edeck 3,637   0.77 E girder 4,696

Ac = 897.4 in.2 Y BCt 

28 ,357 . 6  31 . 6 in. 897 . 4

YTCt

YTGCt = 42 – 31.6 = 10.4 in

YTGCt

YTCt = 50 – 31.6 = 18.4 in ICt = 96,996 + 165,816 = 262,806 in.4 S BCt 

I Ct 262,806   8,317 in.3 31.6 YBCt

STGCt 

I Ct 262,806   25,271 in.3 10.4 YTCt

STDCt 

I Ct 262,806   18,425 in.3 YDCt 18.4(0.77)

Neutral Axis eCt

YBCt

CGS

eCt = 31.6 – 4 = 27.6 in. **Deck and haunch are transformed using (n = 0.77)

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8.6.10.2

Check Concrete Stresses at Transfer Condition

The check of concrete stresses at transfer investigates the suitability of both the prestressing force and the strand profile for the assumed section. Commonly, strands must be harped or debonded to produce an efficient design that does not overstress the section. In addition, the initial concrete compressive strength, fci may be modified. 

Concrete stress limits: o

Compressive stress limit: Stress limit = 0.6fci = 0.6 (4.8) = 2.880 ksi (AASHTO 5.9.4.1.1)

o

Tensile stress limit: (AASHTO Table 5.9.4.1.2-1)  In area other than precompressed tensile zone without bonded auxiliary reinforcement: Stress Limit  0.0948 f c  0.200 ksi Stress Limit = 0.0948√4.8 = 0.208 ksi Since 0.208 ksi is larger than 0.200 ksi, 0.200 ksi is taken as the limit. 

In areas with bonded auxiliary reinforcement sufficient to resist the tensile force Stress Limit=0.24√4.8=0.526 ksi

Per AASHTO Commentary C5.9.5.2.3a, when checking concrete stresses using transformed section properties, the effects of losses and gains due to elastic deformations are implicitly accounted for. Therefore, the elastic shortening loss, ∆fpES, should not be subtracted from the strand stress in calculating the prestressing force at transfer (taken as Pj because relaxation losses between jacking and transfer are ignored). Check concrete stresses at transfer length section: (straight strands) Transfer length = 60(db) = 60(0.6) = 36 in. = 3 ft (LRFD 5.11.4.1) db = nominal strand diameter (in.) Pj = 703 kips Ati = 494 in.2 Where Ati = Gross area of girder concrete at time of force transfer Eccentricity at 3 ft with CGS strands = 4 in. from bottom Eti = 15.4 in. SBti = 5,185 in.3 STti = 4,430 in.3

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MDC1 = moment at transfer length section due to girder weight, based on total girder length = 0.5(0.494)(3)(71 - 3) = 49.8 kip-ft = 598 kip-in. Calculate concrete stress at top of girder at transfer length:

f top 

Pj Ati



P j eti S Tti



M DC1 S Tti

703 703(15.4) 598   494 4,430 4,430  0.886 ksi (Tensile) 

Because the tensile stress exceeds the upper limit of the allowable tensile stress (0.526 ksi, assuming bonded reinforcement), the effect of prestressing must be reduced. Either harping or debonding a portion of strands near the end of the girders can accomplish this. Note: Selection of harping versus debonding should be discussed with fabricators. For this example, two point harping is selected. Harp points are usually located between 0.33L and 0.4L from girder ends. 0.4L is chosen as the harp point. The calculations below investigate stressses along the member by the use of four strands harped with a profile at the girder ends as follows: two strands at 28.5 in. from bottom and two strands 30.5 in. from bottom. A suitable configuation of harping is easily arrived at by iteration using spreadsheets or commercially available software. From midspan to the harped point (0.4L), CGS is 4 in. from the girder bottom, and at the girder ends, the CGS is 10 in. from the bottom. The strand pattern at the harp points and girder ends are shown in Figure 8.6-7. Hold-down forces and harp angle are normally calculated and checked against limits by local precast producers.

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2 at 30.5 in. 2 at 28.5 in.

2 at 6.5 in. 8 at 4.5 in. 6 at 2.5 in. Strand pattern at 0.4L CGS = 4 in.

6 at 4.5 in. 6 at 2.5 in. Strand pattern at ends CGS = 10 in.

Figure 8.6-7 Strand Patterns at Midspan and Ends of Girder



Check concrete stresses at transfer length section (harped strands): The eccentricity at transfer length (3 ft) with harped strands: eti  f top

28  3 (10.0  4.0)  4.0  9.4 in. 28 Pj Pj eti M DC1    Ati STti STti

703 703(9.4) 598   494 4,430 4,430  0.066 ksi (Compressive) 

Pj Pj eti MDC1 - + Ati STti STti 703 703(9.4) 598 = - + 494 4,430 4,430 = 0.66 ksi (Compressive) < 2.880 ksi ftop =

(OK)

Note that the transformed section properties at midspan are used in the calculations above. It is not necessary to check the stresses at transfer length using transformed properties at transfer length since the stress level is very low.

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

Check concrete stresses at harp points: Harped point location = 0.4L =0.4(70) = 28 ft Pj = 703 kips Ati = 494 in.2 Eccentricity at 0.4L, eti = 15.4 in. SBti = 5,158 in.3 STti = 4,430 in.3 MDC1 = moment at 0.4L due to girder weight, based on total girder length MDC1 = 0.5 (0.494) (28) (70 – 28) = 290.3 kip – ft = 3,484 kip – in. (Table 8.6-3) Calculate concrete stress at top of girder at transfer at 0.4L: ftop = =

Pj Pj eti MDC1 - + Ati STti STti

703 703(15.4) 3,484 + 494 4,430 4,430

= -0.227 ksi (Tensile) Since the tensile stress at top of girder exceeded the limit of 0.200 ksi, auxiliary (mild) reinforcement must be provided at the tensile face (top) to resist the total tensile force. 

Determine auxiliary reinforcement: To find the neutral axis of the section, it is necessary to determine: fbot = =

Pj Pj eti MDC1 - + Ati SBti SBti 703 703(15.4) 3,4884 - + 494 5,185 5,185

= 2.832 ksi (compressive) < 2.889 ksi

(OK)

Locate neutral axis, x, from similar triangles.

Chapter 8 – Precast Pretensioned Concrete Girders

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- 0.227 ksi (tension)

x

42"

2.832 ksi (compression)

Figure 8.6-8 Concrete Stress Distribution

x

42(0.227)  3.12 in. 0.227  2.832

The neutral axis is slightly below the 3 in. rectangular section of the top flange. The value 3.12 in. can be conservatively used to calculate the total tension force as the area of concrete times the average tensile stress. Required tension capacity, FT = 0.5 (0.227) (3.12) (19) = 6.73 kips Using non-prestressed reinforcement (mild steel) at a working stress of 24 ksi Area of mild steel required: 6.73 0.28 in. 24 Provide two #4 bars at top flange for entire length of girder,

As,provided = 2 (0.2) = 0.4 in.2 > 0.28 in.2 (OK) 

Check concrete stresses at midspan: Midspan is not expected to govern over the harp points at transfer because of the beneficial effects of self-weight. However, the calculation is demonstrated herein. The values for Pj, eti, Ati, SBti, and Stti at midspan are the same as at 0.4L. MDC1 = moment at midspan due to girder weight, based on total girder length = (0.494)(70)2/8 = 302.4 kip-ft = 3,629 kip-in. (Table 8.6-3)

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ftop =

Pj Pj eti MDC1 + Ati STti STti =

703 703 15.4 3,629 - + 494 4,430 4,430

= -0.194 ksi (Tensile) This is within the tensile limit of 0.2 ksi (OK) f bot =

=

Pj Ati



Pj eti STti



M DC1 STti

3,629 703 703 15.4 - + 5,185 5,185 494

= 2.804 ksi (Compressive) < 2.880 ksi 8.6.10.3

(OK)

Check Concrete Stresses at Service Condition

The check of concrete stresses at service level investigates the suitability of the section to resist service level loads. Of particular importance is the prevention of flexural cracking of the section at midspan for Service Level III due to the LRFD Specifications HL-93 Vehicular Live Load. In addition, per requirements of the California Amendments (Caltrans, 2014), the section must not develop any tension under permanent loads (only). 

Effective prestressing force, Pf : Because transformed section properties are used, the effective prestressing force (Pf) acting on the section is calculated using the force at transfer, Pj, less long-term losses that were estimated using the Approximate Method: fpe = 0.75 fpu – ∆fpLT = 0.75 (270) – 25.5 = 177 ksi Pf = fpe (Aps ) = 177 (3.472) = 614.5 kips



Concrete stress limits: o

Compressive stress limits 

(AASHTO Table 5.9.4.2.1-1)

Compressive stress limits due to unfactored permanent loads (including girder, slab and haunch, barrier, and future wearing surface) and prestressing force. Load combination: PS + Perm PC girder: 0.45 f´c = 0.45(6) = 2.700 ksi CIP deck: 0.45 f´c,deck = 0.45(3.6) = 1.62 ksi



Compression stress limit due to effective prestress, permanent, and transient loads (including all dead and live loads). Load combination: Service I = PS + Perm + (LL + IM)HL-93

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PC girder: 0.6w fci = 0.6(1.0)(6) = 3.600 ksi CIP deck: 0.6 w fc,deck = 0.6(1.0)(3.6) = 2.160 ksi 

Tensile stress limit

(Table 5.9.4.2.2-1, Caltrans 2014)

For components with bonded prestressing tendons or reinforcement subjected to permanent loads only.

o

Load combination: PS + Perm PC girder: 0 ksi (no tension allowed) For components with bonded prestressing tendons or reinforcement.

o

Load combination: Service III = PS + Perm + 0.8(LL+IM)HL-93 PC girder: -0.19 f 'c = -0.19√6 = -0.465 ksi 

Check concrete stresses at midspan: Bending moments at midspan are given in Table 8.6-9.

Table 8.6-9 Unfactored Bending Moments at 0.4L and Midspan (per Girder). Location 0.4L 0.5L

*MDC1 (kip-ft) 290.3 302.4

*MDC2 (kip-ft) 320.3 333.7

*MDC3 (kip-ft) 93.2 97.4

*MDW (kip-ft) 112.9 117.6

**M(LL+IM)HL93 (kip-ft) 945.8 967.5

*From Table 8.6-3 ** From Table 8.6-4



Check compressive stresses at midspan: PC girders are checked for compressive stresses under the following two load combinations: o

Load combination PS + Perm 

Stress at top of PC girder: ftg =



Pf Atf



Petf STtf



( M DC1  M DC 2 ) ( M DC 3  M DW )  STtf STGCt

614.5 614.5(15.4 ) (302.4  333.7)(12) (97.4  117.6)(12)    491.7 4,418 4,418 25,271

= 0.938 ksi (compression) < 2.7 ksi 

(OK)

Stress at bottom of PC girder: fb =

Pf Atf



Petf S Btf



( M DC1  M DC 2 ) ( M DC 3  M DW )  S Btf S BCt

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

614.5 614.5(15.4 ) (302.4  333.7)(12) (97.4  117.6)(12)    491.7 5,137 5,137 8,317

= 1.299 ksi (compressive) < 2.7 ksi

(OK)

Note that both the top and bottom fibers are in compression. This satisfies the requirement of no tension for components subjected to permanent loads only per California Amendments Table 5.9.4.2.2-1 (Caltrans, 2014). o

Load combination PS + Perm + (LL+IM)HL-93 

f tg





Pf Atf

Compressive stress at top of PC girder:



Petf STtf



( M DC1  M DC2 ) ( M DC3  M DW ) M ( LL IM ) HL93   STtf STGCt STGCt

614.5 614.5(15.4 ) (302.4  333.7)(12) (97.1  117.6)(12) 967.5(12)     491.7 4,418 4,418 25,271 25,271

= 1.397 ksi (Compressive) < 3.600 ksi 

(Service I)

(OK)

Compressive stress at top fiber of deck: This check is not normally required. The deck resists all loads compositely, so that even with a lower compressive strength, the concrete deck is rarely subjected to significant compressive stress. However, designers may desire to check Service I (PS + Perm + (LL + IM)HL-93) for completeness, applying wearing surface, barrier loads, and HL-93 truck and lane loads to the composite section.



Check tensile stresses at bottom of girder at midspan: (Service III) This check to prevent cracking at midspan is normally a critical check that can govern the prestressing force and thus area of prestressing strand. Load combination PS + Perm + 0.8L:

fb 



Pf Atf



Petf S Btf



( M DC1  M DC2 ) ( M DC3  M DW ) 0.8( M ( LL IM ) HL93 )   S Btf S BCt S BCt

614.5 614.5(15.4 ) (302.4  333.7)(12) (97.1  117.6)(12) 0.8(967.5) (12)     491.7 5,137 5,137 8,317 8,317

= 0.182 ksi (Compressive) < 3.6 ksi

(OK)

In this case, not only is the check satisfied, but the girder remained in compression at the bottom fiber.

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

Check concrete stress at harped point (0.4L): The concrete stresses at the harp point (0.4L) should also be checked. These can be checked using the same procedures as for midspan. The calculations are not repeated here, but final results are shown.



Check compressive stresses at 0.4L: o

Load combination PS + Perm ftg = 0.861 ksi (compressive) < 2.7 ksi

(OK)

fb = 1.369 ksi (compressive) < 2.7 ksi

(OK)

Thus also satisfies the requirement of no tension. o

8.6.10.4

Load combination PS + Perm + LL ftg = 1.309 ksi (compressive) < 3.6 ksi

(OK)

fb = 0.278 ksi (compressive) < 3.6 ksi

(OK)

Fatigue Stress Limit

Although fatigue-related compression in concrete deck slabs with multiple PC girders rarely governs design (because of internal arching action), a check of the compressive stress in the deck for the Fatigue I load combination is required per LRFD Specifications Article 5.5.3.1. The compression due to the Fatigue I load combination and one half of the sum of effective prestress and permanent loads cannot exceed 0.4 fc. Per C5.5.3.1, the net concrete stress is usually significantly less than the concrete tensile stress limit for cracking, leading to very small steel stress ranges in the prestressing steel less than limiting values. Fatigue is not likely to control the design and therefore is not checked in this example. Readers are referred to the PCI Bridge Design Manual for more information on how to perform fatigue check.

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8.6.10.5

Effect of Deck Shrinkage

Article 5.9.5.4.3d of LRFD Specifications (AASHTO, 2012) shows the procedure to calculate prestress gain due to shrinkage of deck composite section. However, it is Caltrans practice to ignore this prestress gain due to the fact that the deck is designed as reinforced concrete, and cracks are allowed to form in the section. The calculations will not be presented in this example. The readers are referred to the PCI Bridge Design Manual for information and procedure in estimating prestress gain due to deck shrinkage. 8.6.11

Design for Strength Limit State

Design of PC girders for the service limit state may produce an adequate design for the strength limit state. However, this must be checked because of the significant additional live load design requirement for the Strength II load combination—the P-15 permit truck. 8.6.11.1

Determine Factored Moment

The factored moment at ultimate, Mu, is based on the unfactored moments previously given in Table 8.6-9, shown below: Table 8.6-10 Unfactored Bending Moments at 0.4L and Midspan (per Girder). Location 0.4L 0.5L

*MDC1 (kip-ft) 290.3 302.4

*MDC2 (kip-ft) 320.3 333.7

*MDC3 (kip-ft) 93.2 97.4

*MDW (kip-ft) 112.9 117.6

**M(LL+IM)HL93 (kip-ft) 945.8 967.5

†M(LL+IM)P15 (kip-ft) 1,340.5 1,328.9

*Table 8.6-3; **Table 8.6-4; † Table 8.6-5

Mu is taken as the larger of Strength I and II combinations, per Article 3.4.1 of California Amendments (Caltrans, 2014). Strength I uses the LRFD Specifications (AASHTO, 2012) HL-93 vehicular live load, whereas Strength II uses the California P-15 permit truck. Determine the controlling factored ultimate moment, Mu: By inspection, moments at midspan govern for Strength I. 

Strength I: Mu = 1.25 [MDC1 + MDC2 + MDC3] + 1.5 MDW + 1.75 [M (LL+IM)HL93] Mu(LL+IM)HL93 = 1.25 (302.4 + 333.7 + 97.4) + 1.5 (117.6) + 1.75 (967.5) = 2,786.4 kip-ft/girder



Strength II: Mu = 1.25 [MDC1 + MDC2 + MDC3] + 1.5 MDW + 1.35 [M (LL+IM)P15]

Since live load moment is larger at 0.4L, it is necessary to check both 0.4L and 0.5L sections to find the critical moment demand. 

At 0.4L:

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Mu(LL+IM)P15 = 1.25 (290.3 + 320.3 + 93.2) + 1.50 (112.9) + 1.35 (1,340.5) = 2,858.8 kip-ft 

At 0.5L: Mu(LL+IM)P15 = 1.25 (302.4 + 333.7 + 97.4) + 1.50(117.6) +1.35(1,328.9) = 2,887.3 kip-ft / girder  controls

 8.6.11.2

Strength II governs. Mu = 2,887.3 kip-ft/girder

Calculate Nominal Flexural Resistance

Based on AASHTO 5.7.3.1 and 5.7.3.2, compute nominal flexural resistance of the section, as shown in Figure 8.6-9.

a/2

ybtsybts

de

dp 2-#5

@ 5.5”

cg tensile reinf

cg rebars

2-#5 @ 3.5”

CGS

Figure 8.6-9 Bridge Section at Midspan



Determine average prestressing steel stress at ultimate: In most applications, the average prestressing steel stress at ultimate can be easily determined from AASHTO Eq. 5.7.3.1.1-1, applicable to typical PC girder sections that use bonded tendons and have an effective stress with fpe ≥ 0.5 fpu. If more precise calculations are required, conditions of equilibrium and strain compatibility can be used (AASHTO 5.7.3.2.5). For this example, the effective prestress (after all losses), fpe = 160.2 ksi > 0.5 fpu= 135 ksi, thus Eq. 5.7.3.1.1-1 is applicable.

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 c  f ps  f pu 1  k  d p  

(AASHTO 5.7.3.1.1-1)

For low relaxation strand, fpy/fpu = 0.9 (Table C5.7.3.1.1-1), thus the value k is determined as follows: 2 1.04

(AASHTO 5.7.3.1.1-2)

= 2(1.04 – 0.9) = 0.28 Assuming the compressive stress lies solely within the deck (rectangular stress block develops at ultimate), the neutral axis depth, c, is determined from the following:

c

Aps f pu  As f s  A's f ' s f  0.85 f c' 1b  kA ps  pu  d p 

(AASHTO 5.7.3.1.1  4)

where: Aps = area of prestressing steel = 3.472 in.2 fpu = specified tensile strength of prestressing steel = 270 ksi fpy = yield strength of prestressing steel = 243 ksi As = area of mild steel tension reinforcement = four #5 for longitudinal steel requirement = 4 (0.31) = 1.24 in.2 Note that there are four #5 longitudinal bars added to the bottom bulb of the girder as illustrated in Section 8.6.15. As = area of mild steel compression reinforcement = 0 fs = stress in the mild steel tension reinforcement at nominal flexure resistance = 60 ksi fs = stress in the mild steel compression reinforcement at nominal flexure resistance b

= effective width of flange in compression (deck) = 72 in.

fc = compressive strength in deck concrete at midspan = 3.6 ksi dp = distance from extreme compression fiber (deck) to centroid of prestressing tendon = 50 – 4 = 46 in.

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From AASHTO 5.7.2.2, for fc ≤ 4 ksi 1 = 0.85 for fc > 4 ksi  0.85 – 0.05(f’c – 4) 0.65 fc = 3.6 ksi Therefore, 1 = 0.85 c = distance from extreme compression fiber to the neutral axis assuming tendon prestressing steel has yielded (in.) Assuming that mild reinforcement has also yielded, c is calculated as follows: c

3.472 ( 270 )  1.24 (60 )  270  0.85(3.6)( 0.85)( 72 )  0.28( 3.472 )   46 

= 5.24 in. < 7 in. (deck thickness) Because compressive stresses lie entirely within the slab thickness, the assumption of rectangular section behavior is valid. f ps

5 .24    270  1  ( 0 .28 )  46  

= 261.4 ksi Determine factored flexural resistance, Mr Mr = Mn

(AASHTO 5.7.3.2.1-1)

where: Mr = factored flexural resistance  = resistance factor per AASHTO 5.5.4.2 Mn = nominal flexural resistance For rectangular sections:

a a   M n  A ps f ps  d p    A s f s  d s   2 2   where: ds = distance from extreme compression fiber (deck) to centroid of mild steel = 50 – 4.5 = 45.5 in. a = depth of equivalent rectangular stress block, in. Chapter 8 – Precast Pretensioned Concrete Girders

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= 1 (c) = 0.85(5.24) = 4.46 in.

4.46  4.46    M n  3.472( 261.4) 46    1.26(60) 45.5   2 2     = 42,944 kip-in. = 3,578.7 kip.-ft 

Determine resistance factor, : According to the AASHTO C5.7.3.3.1 and Figure C5.5.4.2.1-1 of California Amendments (Caltrans, 2014) as shown in Figure 8.6-10 if the net tensile strain, t ≥ 0.005, then the section is defined as tensioncontrolled and  =1 for flexure.

Figure 8.6-10 Variation of  with Net Tensile Strain, t, for Grade 60 Reinforcement and Prestressed Members (Figure C5.5.4.2.1-1, California Amendments (Caltrans, 2014)

The net tensile strain is calculated using similar triangles based on the assumed strain distribution through the depth of the section at ultimate, as shown in Figure 8.6-11. cu = 0.003 c

de-c t

Figure 8.6-11 Assumed Strain Distribution through Section Depth at Ultimate

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Variables in Figure 8.6-11 are defined as follows: c = distance from extreme compression fiber to the neutral axis (in.) de = effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) (AASHTO 5.7.3.3.1) cu = failure strain of concrete in compression (in./in.) (AASHTO C5.7.3.1.2 and 5.7.4.4) t = net tensile strain in extreme tension steel at nominal resistance (in./in.) (AASHTO C5.5.4.2.1)

de 

Aps f ps d p  As f y d s 0 Aps f ps  As f y

3.472 261.4 (46) + 1.24(60)(45.5) 3.472(261.4) + 1.24(60) = 45.96 in. =

By similar triangles:

εcu ε  t c de  c εt =

εcu 0.003 de - c = 45.96-5.24 = 0.023 ≥ 0.005 5.24 c

Therefore, the section is tension-controlled, and thus,  = 1. Check flexural capacity of section: Mr = Mn = 1 (3,578.7) = 3,578.7 kip-ft > Mu = 2,877.3 kip-ft

8.6.12

Check Reinforcement Limits

8.6.12.1

Maximum Reinforcement

(OK)

AASHTO Art. 5.7.3.3.1, which defined a maximum limit for flexural tension reinforcement to prevent over-reinforced sections, was eliminated in the 2006 interims. As of 2012, the current approach involves reducing the flexural resistance factor when the net tensile strain in the extreme reinforcement is less than 0.005. Although this approach permits sections with less ductility than previous editions if a smaller resistance factor is applied, sections with a net tensile strain less than 0.004 are not recommended because they have a reduced ductility and are generally uneconomical. Rather, superstructure members shoud be designed for a net tensile strain of at least 0.004, preferrably 0.005 (for which the resistance factor is 1). The net tensile strain can alternatively be checked, ensuring that the c/dt (or c/de) ratio for the section is not greater than 0.375 (which corresponds to a net tensile strain of 0.005).

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8.6.12.2

Minimum Reinforcement

To prevent a brittle failure at intial flexural cracking, AASHTO Article 5.7.3.3.2 (2012) requires that all flexural components have sufficient amount of prestressed and non-prestressed tensile reinforcement to develop a factored flexural resistance, Mr, at least equal to the lesser of: (i) 1.33Mu and (ii) Mcr where: Mu = controlling factored moment demand Mcr = cracking moment as defined in Eqn. 5.7.3.3.2-1 For this example, (i) The controlling factored moment occurs at midspan due to Strength II combination. Mu

= 2,887.3 kip-ft

1.33 Mu = 3,849.0 kip-ft    S (ii) M cr  γ 3  γ1 f r  γ 2 f cpe S c  M dnc  c  1     S nc  where: fr

(AASHTO 5.7.3.3.2-1)

= modulus of rupture of concrete specified in Article 5.4.2.6

fcpe = compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi) Mdnc = total unfactored dead load moment acting on the noncomposite section (kip-in.) Sc

= section modulus for extreme the fiber of the composite section where tensile is caused by externally applied loads (in.3)

Snc = section modulus for extreme the fiber of the noncomposite section where tensile is caused by externally applied loads (in.3) 1

= flexural cracking variability factor = 1.6 for other than PC segmental structures

2

= prestress variability factor = 1.1 for bonded tendons

3

= ratio of specified minimum yield strength to ultimate tensile strength of reinforcement = 1 for prestressed concrete structures

f r  0 . 24

f c'  0 . 24 6  0 . 588 ksi

Chapter 8 – Precast Pretensioned Concrete Girders

(AASHTO 5.4.2.6)

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Pf (e ) Pf c + Sb Ag 556.2 556.2 16 474 4,770 = 3.039 ksi cpe

=

Mdnc = MDC1 + MDC2 = 302.4 + 333.7 = 636.1 kip-ft = 7,633 kip-in. Sc = SBC = 7,735 in.3 Snc = Sb = 4,770 in.3 Mcr = 1

1.6 0.558 + 1.1 3.039 7,735 - 7,633

= 28,389 kip-in. = 2,366 kip-ft

7,735 -1 4,770

Since 1.33Mu (= 3,849 kip-ft) > Mcr (= 2,366 kip-ft)  Mcr controls. Mn = 3,578.7 kip-ft ≥ Mcr = 2,366 kip-ft

8.6.13

(OK)

Design for Shear Shear design of PC girders in this example is performed using the sectional design method of Article 5.8.3 (AASHTO 2012). However, the General Procedure for Shear Design with Tables is used to determine β and , per Appendix B5 of LRFD Specifications (AASHTO, 2012) and California Amendment 5.8.3.4.2 (Caltrans, 2014). A design flow chart is provided in Figure CB5.2-5 of Appendix B5 of the LRFD Specifications (AASHTO, 2012). Note: California Amendment 5.8.3.4.3 (Caltrans, 2014) prohibits the use of LRFD Specifications (AASHTO, 2012) Article 5.8.3.4.1, Simplified Procedure for Nonprestressed Sections, for both prestressed and nonprestressed sections. PC girders are designed by comparing the factored shear force (envelope value) and the factored shear resistance at a number of sections along their length, typically at tenth points along the member length and at additional locations near supports. Per LRFD Specifications (AASHTO, 2012), the shear resistance, Vn, may be taken to consist of the sum of three components: 

Concrete component, Vc, that relies on tensile stresses in the concrete



Steel component, Vs, that relies on the tensile stresses in the transverse reinforcement



Prestressing component, Vp, the vertical component of the prestressing force for harped strands

This example illlustrates shear design only at the critical section. 8.6.13.1

Determine Critical Section for Shear Design

For the common situation near supports where the reaction force in the direction of the applied shear introduces compression into the end region of a member, Article

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5.8.3.2 of LRFD Specifications (AASHTO, 2012) allows the location of the critical section for shear to be taken at a distance, dv, the effective shear depth, from the internal face of the support. Determine effective shear depth, dv : dv = distance between resultants of tensile and compressive forces due to flexure = de  a/2, but not less than the greater of (0.9de, 0.72h)(AASHTO 5.8.2.9) where: h = hc, overall depth of composite section de = effective depth from extreme compression fiber to centroid of tensile reinforcement = hc - ybts where: ybts = centroid of all tensile reinforcement a

= depth of compression block (taken at midspan for simplicity)

Because harped strands are used, de varies near the ends. Thus, either initially assume de based on the straight strands or a location equal to 0.05L for dv to calculate d e. Using the latter approach: dv = 0.05 (70) = 3.5 ft from face of internal support. The centroid of the tensile reinforcement from the bottom fiber, including both prestressing steel (straight strands only) and mild reinforcement, is calculated based on the following, Figure 8.6-12 and Table 8.6-11: a/2

de dv hc

Centroid of tensile reinforcement

PS Steel

ybts

2 - #5 @5.5 2 - #5 @ 3.5

Figure 8.6-12 Definitions of ybts, de, and dv at Section Located Near Support

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Table 8.6-11 Centroid of Tensile Reinforcement Layer 1 2 3 Total

As (in.2) Two #5 = 0.62 Two #5 = 0.62 12(0.217)(270/60) = 11.72 12.96

(As)(yi) (in.3) 2.2 3.4 41 46.6

yi (in.) 3.5 5.5 3.5* -

*Centroid of 12 straight strands on the tension side.

ybts 

46.6  3.6 in. 12.96

de = hc - ybts = 50 – 3.6 = 46.4 in. The depth of compression block, a, at dv can be computed using the procedure presented in the flexure design section. It is found that a = 4.46 in.

a 4.46   2.23 in. 2 2  d e  a / 2  46.4  2.2  44.2 in.  d v  max 0.9d e  0.946.4   41.8 in.  0 .7 h  0.7250.0   36.0 in. 

dv = 44.2 in.= 3.7 ft Because 3.7 ft is larger than the initially assumed value of 3.5 ft, it is conservative to simply use the smaller value for dv (and larger shear) rather than continue iteration. Also, because the bearing pad size has not yet been determined at this stage, it is conservative to assume that the support width equals zero. Therefore, use dv = 3.5 ft from centerline of the support. 8.6.13.2

Determine Shear Force Demand

Determine factored shear force demand and corresponding factored moment demand at dv away from the face of the support, which is taken as 3.5 ft from the centerline of the support. Table 8.6-12 Unfactored Shear Forces and Associated Moments at dv from Face of Support Shear (kip) Associated Moment (kip-ft)

VDC1 15.6 MDC1 57.5

VDC2 VDC3 17.2 5 MDC2 MDC3 63.4 18.5 Table 8.6-3

Chapter 8 – Precast Pretensioned Concrete Girders

VDW 6 MDW 22.3

V(LL+IM)HL93 65.7 M(LL+IM)HL93 199 Table 8.6-4

V(LL+IM)P15 102.2 M(LL+IM)P15 304 Table 8.6-5

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Apply Strength I and Strength II Load Combinations to determine which governs for Vu: Strength I: Vu

= 1.25(VDC1 + VDC2 + VDC3) + 1.5 VDW + 1.75 V(LL+IM)HL93 = 1.25(15.6 + 17.2 + 5) + 1.5(6) +1.75(75.7) = 171.2 kips

Strength II: Vu

= 1.25(VDC1 + VDC2 + VDC3) + 1.5 VDW + 1.35 V(LL+IM)P15 = 1.25(15.6 + 17.2 + 5) + 1.5(6) +1.35(102.2) = 194.2 kips

 controls

Corresponding Strength II factored moment: Mu

= 1.25 [57.5 + 63.4 +18.5] + 1.5[22.3] + 1.35[304] = 618.1 kip-ft

8.6.13.3

Determine Contribution of Concrete

The concrete contribution to shear resistance is determined from the following equation: Vc =0.0316 β

bv dv

(AASHTO 5.8.3.3-3)

where:  = factor indicating the ability of diagonally cracked concrete to transmit tension and shear bv = effective web width taken as the minimum web width within the depth, dv (in.) dv = effective shear depth (in.) For the General Procedure of LRFD Specifications Appendix B5 (AASHTO 2012), the value of  is based on the net longitudinal tensile strain, x, at the middepth of the section for the normal case in which code-minimum transverse reinforcement is provided. This is because such members have the capacity to redistribute shear stresses.

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Figure 8.6-13 Shear Parameters for Section Containing at Least Minimum Amount of Transverse Reinforcement, Vp = 0 (Figure B5.2-1 (AASHTO, 2012)



Determine x: For the General Procedure, the longitudinal strain, , at middepth of the section is typically determined using one of the two equations: o

Eq. B5.2-1 when the strain is tensile (positive)

o

Eq. B5.2-3 when the strain is compressive (negative)

The value of 0.5 cot may be taken equal to 1 (i.e.,  may be taken as 26.6) initially during iterations for  and β, and may also be assumed constant to avoid iterations, without significant loss of accuracy (AASHTO, 2012). Note that for some situations (e.g., PC girders made continuous for live load), using Eq. B5.2-1 may be overly conservative when applied near supports because the prestressing strands will be located on the flexural compression side. In such cases, Eq. CB5.2-1 may be used for greater accuracy (AASHTO, 2012). |Mu | + 0.5Nu + 0.5 Vu - Vp cot θ - Aps fpo dv εx = 2 Es As + Ep Aps

where: | Mu | = absolute value of factored moment corresponding to the factored shear force, not to be taken less than | Vu – Vp | dv = maximum of (618.1 k-ft, | Vu – Vp | dv) Vu

= factored shear force =194.2 kips

Vp

= component in the direction of the applied shear of the effective prestressing force; positive if resisting the applied shear (kip)

Pf

= total strand force = 556.2 kips (Section 8.6.9.2, Pfg, nontransformed section)

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

= angle of harped strands, from Figure 8.6-14 = tan-1

10 - 4 =1.02o 28 x 12

Critical Section dv

Centroid of prestressing steel

 CG = 10 28'-0"

CG = 4

Figure 8.6-14 Girder Elevation Near Support and Critical Section for Shear (Ignoring Bearing Pad)

Vp = 556.2 sin (1.02º) = 9.9 kips | Vu – Vp | dv = |194.2 – 9.9 | 3.5 = 645.2 kip-ft > 618.1 kip-ft Therefore, | Mu | = 645.1 kip-ft = 7,741 kip-in. Nu = factored axial force, taken as positive if tensile and negative is compressive = 0 kips  = angle of inclination of diagonal compressive stresses = 26.5 initially assumed, based on taking 0.5 (cot ) = 1 fpo = a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and the surrounding concrete. For the usual levels of prestressing, a value of 0.7fpu is appropriate for pretensioned members. = 0.7 (270) = 189 ksi Aps = area of prestressing strands on flexural tension side at section = 3.472 in.2 As = 1.24 in.2

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εx

7,741  0.5 (0)  0.5 194.2  9.9 cot( 26.5)  3.472 (189 )  42 2 [29,000 (1.24)  28,500 (3.472 )] 

 288    1.07 x 10  3

269 ,820

Since x is negative at this location at middepth, the section is in compression. Therefore, x must be calculated using Eq. B5.2-3, which accounts for the presence of concrete in compression.

εx

 Mu    0.5N u  0.5 Vu  V p cot θ  Aps f po  d   v 2( Ec Ac  Es As  E p Aps )

Ac = area of concrete on the flexural tension side = 6 (19) + 0.5 (2) (6) (6) + 7 (50/2-6) = 283 in.2

h = 50" h/2 = 25"

= Ac

Figure 8.6-15 Definition of Ac

εx

  7,741   0.5(0)  0.5 194.2  9.9 cot( 26.6)  3.472(189 )  42   2[4,696( 283)  29,000(1.24 )  28,500(3.472 )] 

 288    0.098 x 10  3 2,927 ,760

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

Determine  and  For sections with transverse reinforcement equal to or larger than minimum transverse reinforcement, the value of  (factor for concrete shear contribution) and  (angle of inclination of diagonal compressive stresses) are estimated through iteration from Table B5.2-1 of LRFD Specifications (AASHTO, 2012), shown as Table 8.6-13. To use this, the ratio (vu / fc) is required in addition to x. Using  = 0.9 for shear per LRFD Specifications, vu

vu f 'c



Vu  V p  bv d v



194.2  0.9 (9.9)  0.700 ksi 0.9 (7) (42)



0.700  0.117 6

Table 8.6-13 Values of and  for Sections with Transverse Reinforcement (Table B5.2-1 (AASHTO 2012) vu f c′ ≤0.075 ≤0.1 ≤0.125 ≤0.15 ≤0.175 ≤0.2 ≤0.225 ≤0.25

εx × 1,000 ≤−0.2

≤−0.1

≤−0.05

≤0

≤0.125

≤0.25

22.3 6.32 18.1 3.79 19.9 3.18 21.6 2.88 23.2 2.73 24.7 2.63 26.1 2.53 27.5 2.39

20.4 4.75 20.4 3.38 21.9 2.99 23.3 2.79 24.7 2.66 26.1 2.59 27.3 2.45 28.6 2.39

21 4.1 21.4 3.24 22.8 2.94 24.2 2.78 25.5 2.65 26.7 2.52 27.9 2.42 29.1 2.33

21.8 3.75 22.5 3.14 23.7 2.87 25 2.72 26.2 2.6 27.4 2.51 28.5 2.4 29.7 2.33

24.3 3.24 24.9 2.91 25.9 2.74 26.9 2.6 28 2.52 29 2.43 30 2.34 30.6 2.12

26.6 2.94 27.1 2.75 27.9 2.62 28.8 2.52 29.7 2.44 30.6 2.37 30.8 2.14 31.3 1.93

≤0.5 30.5 2.59 30.8 2.5 31.4 2.42 32.1 2.36 32.7 2.28 32.8 2.14 32.3 1.86 32.8 1.7

≤0.75 33.7 2.38 34 2.32 34.4 2.26 34.9 2.21 35.2 2.14 34.5 1.94 34 1.73 34.3 1.58

≤1 36.4 2.23 36.7 2.18 37 2.13 37.3 2.08 36.8 1.96 36.1 1.79 35.7 1.64 35.8 1.5

From Table 8.6-13 with x = -0.098  10-3 and vu / fc = 0.117, the values of and  could be determined. Although the values to be selected fall between two choices (boxes) in the table, for hand calculations, it is

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normally simpler and conservative to use the value of  in the lower row (larger vu / fc ) and value of in column to the right (larger x) of the computed value in the table. For this design example, first iteration yields:  = 22.8  = 2.94 The angle  was initially assumed to be 26.5, significantly larger than 22.8. Therefore, another iteration is performed using the angle of 22.8.

εx

 7,741    0.5(0)  0.5 194.2  9.9 cot( 22.8)  3.472(189 )  42   2[4,696( 283)  29,000(1.24 )  28,500(3.472 )] 

 253  2,927 ,760

  0.086 x 10  3

From Table 8.6-13, Iteration 2 yields the same values for and . Therefore, no further iteration is required and the following values are used in design at this section:  = 22.8  = 2.94 

Compute concrete contribution to shear resistance, Vc : Vc

 0.0316 β f ' c bv d v

(AASHTO 5.8.3.3-3)

 0.0316 ( 2.94 ) 6 (7) ( 42 ) = 66.9 kips 8.6.13.4

Requirement for Transverse Reinforcement

Check if shear reinforcement is required, i.e., when Vc  0.5 (Vc – Vp) Vu = 194.2 kips > 0.5 (0.9) (66.9 + 9.9) = 34.6 kips Therefore, transverse shear reinforcement is required at the critical section. 8.6.13.4.1 Required Area of Transverse Reinforcement The required area of transverse reinforcement is based on satisfying the following design relationship: 

≤ Vn = Vc + Vp + Vs

(AASHTO 5.8.3.3-1)

Solving this equation for Vs leads to:

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Vs =

Vu 

Vc

Vp

Therefore, the required contribution of the transverse reinforcement, Vs, is: 194.2 - 66.9 - 9.9 = 139 kips 0.9 The required area of transverse reinforcement can conveniently be expressed in design as an area per length, i.e., (Av/s) based on rearrangement of AASHTO Eq. 5.8.3.3-4: Vs =

Vs =

Av fy dv cot  + cot  sin 

s Av Vs = s fy dv cot  + cot  sin  where: s

= spacing of transverse reinforcement measured in a direction parallel to the longitudinal reinforcement

Av = area of shear reinforcement within a distance s 

= angle of inclination of diagonal compressive stresses



= angle of inclination of transverse reinforcement to longitudinal axis = 90 for vertical stirrups

fy

= yield strength of transverse reinforcement

Av 139 = 60(42) cot 22.8o + cot 90o sin 90o s in.2 = 0.023 in. Using #5 double-leg stirrups for transverse reinforcement, Av = 0.31 (2) = 0.62 in.2 0.62 = 27 in. 0.023 Using # 5 double-leg stirrups at 12 in. on center (s = 12 in.) near supports. Spacing, s =

Note: Larger spacing, up to the maximum permitted by LRFD Specifications, may be selected for section beyond 4 ft at the discretion of the designer. This corresponds to a contribution of transverse reinforcement, Vs, to nominal shear resistance: 0.62(60)(42) cot 22.8o = 309.7 kips 12 Vn = Vc + Vp + Vs Vs,provided =

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Vn = 66.9 + 9.9 + 309.7 = 386.5 kips > 8.6.13.5

Vu = 215.8 kips (OK) 

Determine Maximum Spacing for Transverse Reinforcement

Per Article 5.8.2.7 of LRFD Specifications (AASHTO, 2012), the spacing of transverse reinforcement, s, cannot exceed the maximum permissible spacing, smax (i.e., s ≤ smax). The maximum spacing, smax, depends on the level of shear stress, vu. From previous calculation, Vu - Vp = 0.700 ksi  bv dv If vu < 0.125 f´c, smax = 0.8 dv ≤ 18 in. vu =

(CA 5.8.2.7-1)

0.125 f´c = 0.126 (6) = 0.75 ksi > vu = 0.7 ksi smax = 0.8 dv = 33.6 in. smax = 18 in.  controls Therefore, smax = 18 in. Spacing provided s = 6 in. < 18 in. (OK) Note that tighter spacing per Eq. 5.8.2.7-2 applies for cases in which vu  0.125fc. 8.6.13.6

Check Minimum Transverse Reinforcement

The area of transverse reinforcement, Av, provided cannot be less than that required by Eq. 5.8.2.5-1:

Av 

0.0316 fcbv s fy

(AASHTO 5.8.2.5-1)

For s = 12 in. as provided: 7(12) = 0.108 in.2 60 Therefore, #5 double-leg stirrups at 12 in. on center satisfy the minimum transverse reinforcement requirement. Av = 0.62 in.2 > 0.0316√6

8.6.13.7

Check Maximum Nominal Shear Resistance

To ensure that the web concrete will not crush prior to yielding of transverse reinforcement, LRFD Specifications (AASHTO, 2012) requires that the nominal shear resistance, Vn, be limited to the smaller of Eq. 5.8.3.3-1 and Eq. 5.8.3.3-2: Vn = Vc + Vs + Vp = 386.5 kips

(AASHTO 5.8.3.3-1)

Vn = 0.25 f´cbvdv + Vp = 0.25 (6) (7) (42) + 9.9 = 450.9 kips (AASHTO 5.8.3.3-2) Therefore, the nominal shear resistance is 386.5 kips. Chapter 8 – Precast Pretensioned Concrete Girders

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Using the above procedure, the transverse reinforcement along the entire girder can be similarly determined.

8.6.14

Design for Interface Shear Transfer between Girder and Deck The interface shear transfer between precast concrete girder and cast-in-place deck shall be designed according to AASHTO Article 5.8.4. The interface resistance shall satisfy Vri =  Vni ≥ Vui

(AASHTO 5.8.4.1-1 & 5.8.4.1-2)

where Vri = factored interface shear resistance (kip) Vni = nominal interface shear resistance (kip) Vui = factored interface shear (kip)  = resistance factor = 0.9 Using AASHTO Eq. (C5.8.4.2-7), the factored interface shear can be taken as:

Vhi  Vui 

Vu dv

where Vu = factored vertical shear under Strength Limit State dv = effective depth for shear All sections along the entire length of the girder are required to satisfy the Article 5.8.4 requirement. For this example, the interface shear design is only demonstrated at the dv (= 42 in.) location from face of support. From Section 8.6.13.2, the factored shear Vu = 194.2 kips at dv. The factored interface shear, V ui 

194 .2  4 .62 kip/in. 42



Vni  cAcv   Avf  Pc



Acv = bvi Lvi

(AASHTO 5.8.4.1-3) (AASHTO 5.8.4.1-6)

where c

= cohesion factor from AASHTO Art 5.8.4.3 (ksi)

Acv = area of concrete engaged in interface shear transfer (in.2) Avf = area of interface shear reinforcing crossing the shear plane within Acv (in.2) fy

= yield stress of interface shear reinforcement (ksi)

Pc = permanent compressive force (kip) bvi = interface width (in.) Chapter 8 – Precast Pretensioned Concrete Girders

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Lvi = interface length (in.) For CIP concrete slab on clean concrete girder surfaces, free of laitance and with surface roughened to an amplitude of 0.25 in.: c

= 0.28 ksi

(AASHTO Art 5.8.4.3)

µ

=1

(AASHTO Art 5.8.4.3)

Acv = 19 (1) = 19 in.2 The amount of vertical shear reinforcement provided at dv is #5 bar, double legs, at 6 in. spacing (Sect. 8.6.13), acting as interface shear reinforcement extended into the deck. 0.31 = 0.052 in.2 ⁄in. 6 = 0.28(19) + 1[(0.052)(60) + 0] = 8.44 kip/in.

Avf = Vni

Vni= 0.9 (8.44) = 7.6 kip/in. > Vui = 4.62 kip/in. (OK) 8.6.14.1

Check Minimum Interface Shear Reinforcement

The minimum interface shear reinforcement required is Avf, min =

0.05Acv 0.05(19) = 0.016 in.2 ⁄in. 60 fy

Provided Avf = 0.052 in.2 ⁄in. > 0.016 in.2 ⁄in.

Therefore minimum interface shear reinforcement requirement is met. 8.6.14.2

Check Maximum Nominal Shear Resistance

The maximum nominal shear resistance used in design shall be the lesser of the two: 

Vni ≤ K1 f´c Acv

(AASHTO 5.8.4.1-4)



Vni ≤ K2 Acv

(AASHTO 5.8.4.1-5)

where: fc = concrete compressive strength of deck slab (ksi) K1 = fraction of concrete strength available to resist shear K2 = limiting interface shear resistance For CIP concrete slab on clean concrete girder surfaces, free of laitance with surface roughened to an amplitude of 0.25 in.: K1 = 0.3

(AASHTO Article 5.8.4.3)

K2 = 1.8 ksi for normal weight concrete

(AASHTO Article 5.8.4.3)



Vni ≤ 0.3 (3.6) (19) = 20.5 kip/in.  controls

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

Vni ≤ 1.8 (19) = 34.2 kip/in.

The Vni provided = 8.44 kip/in. < 20.5 kip/in. (OK).

8.6.15

Check Minimum Longitudinal Reinforcement The minimum longitudinal reinforcement (including both prestressed and nonprestressed reinforcement on the flexural tension side) at all locations along the girder shall be proportioned to satisfy: A ps f ps  As f y 

Mu d v f

 0 .5

 V   u  V p  0.5V s  cot      c

Nu

c

(AASHTO 5.8.3.5-1) At dv from the face of the support: Mu = 618.1 kip-ft Vu = 194.2 kips Vs = 309.7 kips, but not to exceed

Vu  v

where: Vu 194.2 = = 218.5 kips 0.9 v Therefore, Vs = 215.8 kips Vp = 9.9 kips Nu = 0 kips dv = 42 in.  = 22.8 fps = fpe = 160.2 ksi (from Section 8.6.9.2) The determination of minimum longitudinal reinforcement at dv from face of support is illustrated below. |Mu | dv f 

0.5

Nu c

Vu - Vp -0.5Vs cot  v

618.2(12)

  0.0   194.2  0.5  9.9  0.5218.5 cot 22.8 o  409.7 kips    (42)(1.0)  1. 0   0. 9 

Transfer length, Lt, from girder end is taken as 30(strand diameters) = 30(0.6) = 18 in. This is less than the distance dv from the end. Therefore, the strands have developed the full prestressing force and the effective prestress fps = 160.2 ksi is used. Aps = area of 12 straight strands = 12(0.217) = 2.604 in.2

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As = area of four #5 rebar = 4(0.31) = 1.24 in.2 Aps fps + As fy = 2.604(160.2) + 1.24(60) = 491.6 kips > 409.7 kips Therefore, the minimum reinforcement requirement is satisfied. Note that the minimum reinforcement requirement needs to be satisfied in all locations along the girder. Eq. 5.8.3.5-2 of LRFD Specifications (AASHTO, 2012) must also be satisfied at the inside edge of bearing of simple end supports to the section of critical shear

8.6.16

Pretensioned Anchorage Zone Reinforcement 

Splitting resistance: Article 5.10.10.1 of LRFD Specifications (AASHTO, 2012) requires the following vertical reinforcement be provided within the distance h/4 from the end of the girder to provide splitting resistance to bursting stresses.

Pr  f s As

(AASHTO 5.10.10.1-1)

where: fs = stress in steel not to exceed 20 ksi As = total area of vertical reinforcement located within the distance h/4 from end of beam (in.2) h = overall dimension (height) of the precast I girder in the direction in which splitting resistance is being evaluated (in.) Pr = factored bursting resistance of pretensioned anchorage zone provided by transverse reinforcement (kip), not less than 4% of prestressing force at transfer, Pi Pr = 0.4 (Pi) = 0.04 (0.75) (270) (3.472) = 28.1 kips

As 

28.1  1.41 in.2 20

Using #5 bars with 2 vertical legs, Number of bars required =

1.41  2.3 1.31(2)

Therefore, use three #5 double legs within h/4 (42/4 = 10.5 in.) from end of girder. 

Confinement reinforcement: Article 5.10.10.2 of LRFD Specifications (AASHTO, 2012) requires reinforcement be placed to confine the prestressing steel in the bottom flange, over the distance 1.5d from the end of the girder, using #4 rebar with spacing not to exceed 6 in. and shaped to enclose the strands.

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Place #4 rebar at no more than 6 in. on center over the distance 1.5h = 1.5(42) = 63 in. (conservative) from the end of the girder to confine the prestressing strands in the bottom flange.

#5 stirrup #4 confinement reinforcement

4-#5 cont. (longitudinal reinforcement) Note: See XS Sheet XS1-120 for standard reinforcement (not shown).

Figure 8.6-16 Transverse, Longitudinal, and Confinement Reinforcement Details

8.6.17

Deflection and Camber The following three aspects of deflection and camber are addressed in this design example: 

Determine and specify unfactored instantaneous girder deflections due to deck and rail for plan sheets



Check live load deflection against AASHTO LRFD deflection criteria



Determine and specify minimum haunch thickness at supports for plan sheets

Total deflection of the girder is estimated as the sum of the short-term and longterm deflections. Short-term deflections are immediate and are based on an estimate of the modulus of elasticity and the effective moment of inertia. Girder and deck slab self-weight are carried non-compositely by the girder alone, while dead loads such as barriers and overlays as well as live loads are carried by the composite girder-deck system. Long-term deflections consist of long-term deflections at erection and longterm deflection at final stage (may be assumed to be approximately 20 years).

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8.6.17.1

Calculate Girder Deflections due to Deck and Rail

In this section, the instantaneous, unfactored girder camber and deflections due to prestressing force and self-weight of the deck, haunch, barrier, and future wearing surface are calculated for the contract plans. 

Initial camber due to prestressing force at midspan can be estimated using case (4) of Table 8.4-1. After simplifying, the deflection, ∆p, is expressed as,

Pi  ec L2 e' (bL)2  p   EciI  8 6  where: Pi = total prestressing force immediately in prestress strands after transfer (kips) = Pj – ES = (202.5 – 16.84) (3.472) = 644.6 kips

Eci = modulus of elasticity of concrete at initial (ksi) = 4,200 ksi I = initial gross (non-transformed) moment of inertia of the girder (in.4) = 95,400 in.4 ec = eccentricity of prestressing strands at midspan (in.) = 20 – 4 = 16 in. e' = difference between eccentricity of prestressing strands at midspan and at end of girder (in.) = 10 – 4 = 6 in. L = girder length = 71(12) = 852 in. bL = distance from end of girder to harped point (in.) = 28(12) = 336 in. p 



 16.0(852) 2 6.0(336) 2  644.6    2.15 in. (upward)   4,200(95,400)  9 6 

Immediate deflection due to girder self-weight at midspan: The equation for deflection of a simply supported girder with a distributed load: g 

4 5  wg L  384  Eci I   

where: wg = distributed weight of the girder= 0.494 kip/ft = 0.041 kip/in. Deflection due to beam self weight to be used in computing deflection at erection (with span = 70 ft = 840 in.),

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g = 

5 0.041 840 4 384 4,200 95,400

= 0.66 in. (downward)

Immediate deflection due to weight of deck and haunch at midspan (noncomposite section): ws = 0.525 + 0.02 = 0.545 kip-ft = 0.045 kip-in. Ec = 4,696 ksi s =



5 0.045 840 4 384 4,696 95,400

= 0.65 in. (downward)

Immediate deflection due to barrier weight at midspan (composite section): wbr = 0.159 kip-ft = 0.013 kip-in. Ec = 4,696 ksi, and I = 249,148 in.4 (gross composite section) br =



5 0.013 840 4 384 4,696 249,184

= 0.07 in. (downward)

Immediate deflection due to future wearing surface at midspan (composite section): wfw = 0.195 kip-ft = 0.016 kip-in. Ec = 4,696 ksi, and I = 249,148 in.4 (gross composite section) fw =



5 0.016 840 4 384 4,696 249,184

= 0.09 in. (downward)

Immediate deflections for plans (deck and haunch; barrier rails and future wearing surface): The following unfactored instantaneous girder deflection values at midspan should be shown on the contract plans.

8.6.17.2

o

Deck: Unfactored instantaneous girder deflection due to deck and haunch = ∆s = 0.65 in.

o

Rail: Unfactored instantaneous girder deflection due to barrier rail and future wearing surface = ∆br∆fw= 0.07 + 0.09 = 0.16 in.

Compare Live Load Deflection to AASHTO LRFD Limit

Girder live load deflection check is estimated using composite section properties and concrete strength at service, and compared to the AASHTO LRFD limit per Section 2.5.2.6.2. It should be noted that the deflection criteria in LRFD Section 2.5.2.6.2 is optional for California bridges. However, for specific situations, such as bridge widening where the deflection may impair the minimum vertical clearance, the deflection must be accounted for in the design. Chapter 8 – Precast Pretensioned Concrete Girders

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The instantaneous live load deflection for a simple span bridge occurs at midspan due to the HL-93 truck axles placed in the location shown below in Figure 8.6-17, together with the HL-93 lane load (not shown).

32 kips

32 kips

14- 0"

8 kips 14'-0"

2- 4"

Midspan

Figure 8.6-17 Position of Truck to Produce Maximum Moment

The live load for each lane can be obtained from any structural analysis software program, such as CT Bridge. The deflection for each girder is calculated by multiplying the deflection per lane by the ratio of (number of lanes/number of girders). This ratio can also be estimated using the moment distribution factor (DFM) shown in Sec. 8.6.7.3, which is the simpler way. Since the deck concrete and girder concrete differ in strength, transformed properties are used to calculate the live load deflection. As shown in Sec. 8.6.7.3, the DFM for this bridge is 0.571. ∆LL = DFM (∆LL per lane) = 0.571 (∆LL per lane) From structural analysis software, ∆LL per lane = 1.01 in. (downward) ∆LL = 0.571 (1.01) = 0.58 in. (downward) This instantaneous live load deflection is compared to the AASHTO LRFD recommended limitation of L/800 for general vehicular loading. 70(12) 1.05 in. 800 800 The live load deflection is less than the AASHTO limit and therefore acceptable. 8.6.17.3

Determine Minimum Haunch Thickness

The minimum haunch thickness at supports is intended to help ensure that the specified haunch at midspan is achieved in the field. (See Section 8.4.5.1 for discussion on importance of the haunch.)

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The midspan haunch thickness is first specified by the designer (note that this is no recommended minimum value). Based on this, the designer would calculate the minimum haunch thickness at the supports. Both the minimum haunch thickness at the supports and the designer-specified midspan haunch thickness should be shown on the plans. At erection, the net upward camber of the girder at midspan due to prestressing force plus self-weight, ∆g,erect, is calculated based on the instantaneous deflection components multiplied by the PCI factors. These factors account for the timedependent effects between girder casting and erection. ∆g,erect = (MPCIerect ∆p – MPCIerect ∆g) where: ∆g,erect

= camber at midspan at erection, due to long-term effects of prestressing force and girder self-weight (in.)

MPCIerect = PCI multiplier for long-term effects at erection (Table 8.4-2) ∆p

= immediate camber of girder at transfer due to prestressing force (in.)

∆g

= immediate deflection of girder at midspan at transfer due to self weight (in.)

As shown in Figure 8.6-18, the minimum haunch thickness at the supports, THsup, is then calculated as the difference (at the centerline of the girder) between the longterm camber at midspan at erection, ∆g,erect and the instantaneous deflection of the girder at midspan due to the weight of the deck and haunch, plus the designerspecified haunch at midspan, THmid. TH sup   g , erect   s  TH mid

where: THsup

= haunch thickness at supports

∆g,erect = camber at midspan at erection, due to long-term effects of prestressing force and girder self-weight (in.) ∆s

= immediate deflection at midspan due to deck weight (in.)

THmid

= designer-specified haunch thickness at midspan (in.)

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Bottom of Deck Top of Girder

s

THmid

THsup g,erect

Midspan

Camber after deck placement

Legends: Un-deformed shape Deformed shape (before deck placement) Deformed shape (after deck placement)

Figure 8.6-18 Relationship between Specified Haunch Thickness at Midspan and Minimum Haunch Thickness at Supports

For this example: ∆g,erect = 1.8∆p – 1.85∆g = 1.8 (2.15) – 1.85 (0.66) = 2.65 in. (upward) Specifying a haunch thickness of 1 in. at midspan, the minimum haunch thickness required at the supports is determined as follows: THsup = ∆g,erect + ∆s + THmid = 2.65 + (-0.65) + 1 = 3 in. Therefore, the following should be specified on contract plans: Haunch thickness at midspan: 1 in. Minimum haunch thickness at supports: 3 in. Note that the deck cross slope and girder top flange width do not affect the specified midspan haunch thickness. For situations where the cross slope is relatively large and/or the top flange is very wide (such as the wide flange girders), thickness of the haunch at midspan on both sides of the girder flange must be carefully considered.

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NOTATION a

=

depth of equivalent rectangular compression stress block (in.) (8.6.11.2)

A

=

area of stringer, beam, or component (in.2) (8.2.2)

Ac

=

concrete area of composite section (8.6.5.3)

Acv

=

area of concrete engaged in interface shear transfer (in.2) (8.6.14)

Ag

=

gross area of girder section (in.2) (8.4.2.2)

Ai

=

area of individual component (Table 8.6-2)

Aps

=

area of prestressing steel (in2) (8.4.2.2)

As

=

area of non-prestressed tension reinforcement (in.2) (8.4.6.1)

A´s

=

area of compression reinforcement (in.2) (8.6.11.2)

Atf

=

area of transformed section, at final (8.6.10.3)

Ati

=

(8.6.10.2)

Av

=

area of transverse reinforcement within distance s (in.2) (8.6.13.5)

Avf

=

area of interface shear reinforcing crossing the shear plane within Acv (in.2) (8.6.14)

ADL =

added dead load (kips) (8.2.2)

b

=

width of the compression face of a member (in.) (8.6.11.2)

beff

=

effective flange width (in.) (8.6.5.2)

bL

=

distance from end of girder to harped point (in.) (8.6.17.1)

bv

=

effective web width taken as the minimum web width, measured parallel to the neutral axis, between resultants of the tensile and compressive forces due to flexure; this value lies within the depth, dv (in.) (8.6.13.3)

bvi

=

interface width (in.) (8.6.14)

c

=

distance from extreme compression fiber to the neutral axis (in.) (8.6.11.2); cohesion factor from AASHTO Art 5.8.4.3 (ksi) (8.6.14)

cg

=

center of gravity

CGC =

center of gravity of the concrete section (8.2.2)

CGS =

center of gravity of the strands (8.2.2)

D

=

structure depth (ft) or height of standard shape of PC girder given in BDA 6-1 (in.) (8.2.1)

DC

=

weight of supported structures (kip) (8.2.2)

db

=

nominal strand diameter (in.) (8.6.10.2)

de

=

effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) (8.6.7.1)

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DFDL =

dead load distribution factor (8.6.7.1)

DFM =

live load moment distribution factor (8.6.7.3.1)

DFV =

live load shear distribution factor (8.6.7.3.2)

dp

=

distance from extreme compression fiber to the centroid of the prestressing tendons (in.) (8.6.11.2)

ds

=

distance from extreme compression fiber to the centriod of the nonprestressed tensile reinforcement (in.) (8.6.11.2)

Ds

=

superstructure depth (ft) (8.6.2)

dt

=

distance from extreme compression fiber to the centroid of tensile reinforcement (8.6.12.1)

dv

=

the effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure (in.) (8.6.13.1)

DW

=

superimposed dead load (kip) (8.2.2)

e

=

eccentricity of the anchorage device or group of devices with respect to the centroid of the cross section; always taken as positive (ft); also the base of Napierian logarithms (8.2.1)

e'

=

difference between eccentricity of prestressing steel at midspan and at end (in.) (8.6.17.1)

EB, Ec =

modulus of elasticity of girder material (ksi) (8.6.5.3)

ec

=

eccentricity of strands measured from center of gravity of girder at midspan (in.) (8.6.8)

Eci

=

modulus of elasticity of concrete at initial time (ksi) (8.6.4)

Ect

=

modulus of elasticity of concrete at transfer or time of load application (ksi) (8.4.2.1)

Ecu

=

failure strain of concrete in compression (in./in.) (8.6.11.2)

ED

=

modulus of elasticity of deck material (ksi) (8.6.5.3)

eg

=

distance between centers of gravity of girder and deck (in.) (8.6.7.3.1)

em

=

eccentricity at midspan (8.6.9.1)

Ep, Eps =

modulus of elasticity of prestressing tendons (ksi) (8.4.2.1)

Es

=

modulus of elasticity of mild reinforcing steel (ksi) (8.6.4)

etf

=

distance between centers of gravity of strands and concrete section at time of service (8.6.10.1)

eti

=

distance between centers of gravity of strands and concrete section at time of transfer (in.) (8.6.10.1)

fb

=

concrete flexural stress at extreme bottom fiber (8.6.10.3)

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f´c

=

specified compressive strength of concrete used in design (ksi) (8.1)

f´ci

=

specified compressive strength of concrete at time of initial loading or prestressing (ksi); nominal concrete strength at time of application of tendon force (ksi) (8.1)

fcgp

=

concrete stress at the center of gravity of prestressing tensonds that results from the prestressing force at either transfer or jacking and the self-weight of the member at sections maximum moment (ksi) (8.4.2.1)

fcpe

=

compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi) (8.6.12.2)

fg

=

stress in the member from dead load (ksi) (8.6.9.2)

fpbt

=

stress in prestressing steel immediately prior to transfer (8.6.9.1)

fpe

=

effective stress in the prestressing steel after losses (8.6.9.2)

fpi

=

prestressing steel stress immediately prior to transfer (ksi) (8.4.2.2)

fpj

=

stress in prestressing steel at initial jacking (ksi) (8.6.4)

fpo

=

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and the surrounding concrete (ksi) (8.6.13.3)

fps

=

average stress in prestressing steel at the time for which the nominal resistance is required (ksi) (8.6.11.2)

fpu

=

specified tensile strength of prestressing steel (8.6.4)

fpy

=

yield strength of prestressing steel (8.6.4)

fr

=

modulus of rupture of concrete (ksi) (8.6.12.2)

fs

=

stress in mild tension reinforcement at nominal flexural resistance (ksi) (8.4.6.1)

f´s

=

stress in the mild steel compression reinforcement at nominal flexure resistance (8.6.11.2)

FT

=

required tension capacity provided by reinforcement (8.6.10.2)

ftg

=

concrete stress at top of precast girder (ksi) (8.6.10.3)

fy

=

yield strength of mild steel (ksi) (8.6.11.2)

f´y

=

specified minimum yield strength of compression reinforcment (ksi) (8.6.4)

H

=

average annual ambient mean relative humidity (percent) (8.4.2.2)

h

=

web dimension of PC girder (in.) (8.2.1)

hc

=

overall depth of composite section (in.) (8.6.13.1)

I

=

initial gross (non-transformed) moment of inertia of the girder (in.4) (8.6.17.1)

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Ic

=

moment of inertia of composite section about centroidal axis, neglecting reinforcement (in.4) (8.6.5.3)

Icg, Ig =

moment of inertia of the girder concrete section about the centroidal axis, neglecting reinforcement (in.4) (8.6.5.1)

ICt

=

moment of inertia of composite transformed section (in.4) (8.6.10.1)

Ie

=

effective moment of inertia (in.4) (8.4.5)

Ig

=

gross moment of inertia of girder (in.4) (8.6.5.1)

Io

=

moment of inertia of individual component (in.4) (Table 8.6-2)

Iti

=

moment of inertia of concrete section at initial stage, transformed (in.4) (8.6.10.1)

K1

=

fraction of concrete strength available to resist shear (8.6.14.2)

K2

=

limiting interface shear resistance (8.6.14.2)

Kg

=

longitudinal stiffness parameter (in.4) (8.6.7.3.1)

L

=

span length or girder length (ft) (8.1)

LL

=

live load (kip) (8.6.7.3.2)

Mcr

=

cracking moment (kip-in.) (8.6.12.2)

MADL =

moment due to added dead loads (kip-ft) (8.2.2)

MDC1 =

moment due to self-weight of beam (kip-ft) (8.6.8)

MDC2 =

moment due to self-weight of deck (kip-ft) (8.6.8)

MDC3 =

moment due to self-weight of barrier (kip-ft) (8.6.8)

Mdnc

=

total unfactored dead load moment action on the monolithic or noncomposite section (kip-ft) (8.6.12.2)

MDW

=

moment due to future wearing surface (kip-ft) (8.6.8)

Mg

=

midspan moment due to self-weight of girder (8.2.2)

MHL93 =

moment due to enveloped HL-93 trucks (kip-ft) (8.6.8)

MLL

=

moment due to live loads (8.2.2)

Mn

=

nominal flexure resistance (kip-in.) (8.6.11.2)

MP15

=

moment due to enveloped P15 truck (8.6.11.1)

MPCIerect Mr

=

Mslab =

=

PCI Multipliers for camber/deflection at time of erection (8.6.17.3)

factored flexural resistance of a section in bending (kip-in.) (8.6.11.2) moment due to weight of deck slab (8.2.2)

Mu

=

controlling factored moment demand (8.6.11.1)

Mx

=

moment at location x (kip-ft) (8.6.7.2)

n

=

modular ratio between beam and deck (8.6.5.3)

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Nb

=

number of beams, stringers, or girders (8.6.7.1)

Nu

=

factored axial force, taken as positive if tensile and negative is compressive (kip) (8.6.13.3)

P, Pe =

Effective force in prestress strands after all losses (kip) (8.2.2)

Pc

=

permanent compressive force (kip) (8.6.14)

Pf

=

force in prestress strands after losses (kip) (8.6.13.3)

Pfg

=

effective force in prestress strands after all losses for gross section design (kip) (8.6.9.2)

Pi

=

force in prestress strands after elastic shortening loss (kip) (8.6.17.1)

Pj

=

force in prestress strands before losses (kip) (8.6.8)

Pr

=

factored bearing resistance of anchorages (kip) (8.4.6.1)

r

=

radius (in.) (8.6.5.1)

S

=

spacing of girders or webs (ft) (8.2.2)

S

=

surface of girder section (8.4.2.2)

s

=

spacing of reinforcing bars (in.)

Sb

=

section modulus for the bottom extreme fiber of the girder where tensile stress is caused by externally applied loads (in.3) (8.6.8)

SBC

=

section modulus for the bottom extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3) (8.6.5.3)

SBCt

=

section modulus for the bottom extreme fiber of the composite section transformed (in. 3) (8.6.10.1)

SBtf

=

section modulus for the bottom extreme fiber of the composite section transformed, at service stage (in. 3) (8.6.10.3)

Sc

=

section modulus for the extreme fiber of the composite sections where tensile stress is caused by externally applied loads (in.3) (8.6.12.2)

Snc

=

section modulus for the extreme fiber of the monolithic or noncomposite sections where tensile stress is caused by externally applied loads (in.3) (8.6.12.2)

St

=

section modulus for the top extreme fiber of the sections where tensile stress is caused by externally applied loads (in.3) (8.6.5.1)

Stc

=

section modulus for the top extreme fiber of the composite sections where tensile stress is caused by externally applied loads (in.3)

STDCt =

section modulus for the top extreme fiber of the composite sections at top of deck level, at service, transformed (in.3) (8.6.10.1)

STGCt =

section modulus for the top extreme fiber of the composite sections at top of girder level, at service, transformed (in.3) (8.6.10.1)

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STtf

=

section modulus for the top extreme fiber of the composite sections at top of girder level, at service, transformed (in.3) (8.6.10.3)

STti

=

section modulus for the top extreme fiber precast girder, at initial, transformed (in.3) (8.6.10.2)

T

=

tensile stress in concrete (ksi) (8.2.2)

ts

=

thickness of concrete deck slab (in.) (8.6.7.3.1)

th

=

haunch thickness at midspan (in.) (8.6.2)

THmid =

haunch thickness at midspan (in.) (8.6.17.3)

THsup =

haunch thickness at support (in.) (8.6.17.3)

w

=

uniform dead load, k/ft (8.6.7.2)

wbr

=

uniform dead load—weight of barrier (klf) (8.6.6.1)

wfw

=

uniform dead load—weight of future wearing surface (klf) (8.6.6.1)

wg

=

uniform dead load—weight of girder (klf) (8.6.6.1)

wh

=

uniform dead load—weight of haunch (klf) (8.6.6.1)

ws

=

uniform dead load—weight of deck slab (klf) (8.6.6.1)

V

=

volume of girder (in.3) (8.4.2.2)

Vc

=

nominal shear resistance provided by tensile stresses in the concrete (kip) (8.6.13)

Vn

=

nominal shear resistance of the section considered (kip) (8.6.13)

Vni

=

nominal interface shear resistance (kip) (8.6.14)

Vri

=

factored interface shear resistance (kip) (8.6.14)

Vp

=

component in the direction of the applied shear of the effective prestressing force; positive if resisting the applied shear (kip) (8.6.13)

Vs

=

shear resistance provided by the transverse reinforcement at the section under investigation as given by AASHTO 5.8.3.3-4, except Vs shall not be less than Vu/ (kip) (8.6.13)

Vu

=

factored shear force (kip) (8.6.13.3)

vu

=

average factored shear stress on the concrete (ksi) (8.6.13.3)

Vui

=

factored interface shear resistance (kip/in.) (8.6.14)

x

=

distance from left end of girder (ft) (8.2.1)

Y

=

distance from the neutral axis to a point on individual component (in.) (Table 8.6-2)

yb

=

distance from the neutral axis to the extreme bottom fiber of PC girder (in.) (8.2.1)

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YBC

=

distance from the centroid to extreme bottom fiber of composite section (in.) (8.6.5.3)

YBCt

=

section modulus for the bottom extreme fiber of the composite, at service, transformed (in.3) (8.6.10.1)

YBtf

=

section modulus for bottom extreme fiber of PC girder, at service, transformed (in.3) (8.6.10.1)

YBti

=

section modulus for bottom extreme fiber of PC girder, at initial, transformed (in.3) (8.6.10.1)

ybts

=

centroid of all tensile reinforcement (in.) (8.6.13.1)

yi

=

distance from centroid of section i to centroid of composite section (8.6.5.3)

yt

=

distance from the neutral axis to the extreme top fiber of PC girder (in.) (8.2.1)

YTC

=

distance from the centroid to extreme top fiber of composite section (in.) (8.6.5.3)

Ytg

=

distance from centroid of the composite section to the extreme top fiber of the PC girder (in.) (8.6.10.1)

YTGCt =

distance from centroid of the composite section to the extreme top fiber of the PC girder (in.) (8.6.10.1)

YTti

=

distance neutral axis to the extreme top fiber of the PC girder, transformed (in.) (8.6.10.1)



=

angle of inclination of transverse reinforcement to longitudinal axis (º) total angular change of prestressing steel path from jacking end to a point under investigation (rad) (8.6.13.4.2)



=

factor relating effect of longitudinal strain on the shear capacity of concrete, as indicated by the ability of diagonally cracked concrete to transmit tension (unitless) (8.6.13.3)

1

=

ratio of the depth of the equivalent uniformly stressed compression zone assumed in the strength limit state to the depth of the actual compression zone (5.7.2.2) (8.6.11.2)

∆br

=

deflection due to barrier weight (in.) (8.6.17.1)

∆g

=

camber at midspan at erection due to girder self-weight (in.) (8.6.17.3)

∆g,erect =

camber at midspan at erection due to long-term effects of prestressing force and girder self-weight (in.) (8.6.17.3)

∆ES

change in length due to elastic shortening (8.1)

=

∆fpES =

sum of all losses or gains due to elastic shortening or extension at the time of application of prestress and/or external loads (ksi) (8.4.2)

∆fpLT =

losses due to long-term shrinkage and creep of concrete and relaxation of prestressing steel (ksi) (8.4.2)

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∆fpR

=

an estimation of relaxation loss taken as 2.4 ksi for low relaxation strand, 10 ksi for stress relieved strand, and in accordance with manufacturers recommendation for other types of strand (ksi) (8.4.2.2)

∆fpT

=

total change in stress due to losses (ksi) (8.4.2)

∆fw

=

deflection due to future wearing surface (in.) (8.6.17.1)

∆p

=

camber at midspan due to prestressing force at release (in.) (8.6.17.3)

∆s

=

instantaneous deflection due to weight of deck slab (in.) (8.6.17.3)

cu

=

failure strain of concrete in compression (in./in.) (8.6.11.2)

t

=

net tensile strain in extreme tension steel at nominal resistance (in./in.) (8.6.11.2)

x

=

longitudinal strain in the web reinforcement on the flexural tension side of the member (in./in.) (8.6.13.3)



=

angle of inclination of diagonal compressive stresses (8.6.13.3)

1

=

flexural cracking variability factor (8.6.12.2)

2

=

prestress variability factor (8.6.12.2)

3

=

ratio of specified minimum yield strength to ultimate tensile strength of reinforcement (8.6.12.2)

h

=

correction factor for relative humidity of ambient air. (8.4.2.2)

st

=

correction factor for specified concrete strength time at of prestress transfer to concrete member (8.4.2.2)



=

resistance factor (8.6.11.2)



=

coefficient of friction (unitless) (8.6.14)



=

angle of harped strands (8.6.13.3)

c

=

unit weight of concrete (kcf) (8.6.4)

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REFERENCES 1.

AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, American Association of State Highway and Transportation Officials, 6th Edition, Washington, D.C.

2.

Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications — Sixth Edition 2012, California Department of Transportation, Sacramento, CA.

3.

Caltrans, (2014). Memos to Designers 11-8 Design of Precast, Prestressed Girders, California Department of Transportation, Sacramento, CA.

4.

Caltrans, (2013). Memos to Designers 10-20 Attachment 1: Deck Slab Reinforcement Details, California Department of Transportation, Sacramento, CA.

5.

Caltrans, (2012). Bridge Design Aids Chapter 6-1: Precast Prestressed Concrete Girders, California Department of Transportation, Sacramento, CA.

6.

Caltrans, (2010). Memos to Designers 11-3 Designer’s Checklist for Prestressed Concrete, California Department of Transportation, Sacramento, CA.

7.

Caltrans, (2008a). Memos to Designers 10-20 Deck and Soffit Slabs, California Department of Transportation, Sacramento, CA.

8.

Caltrans, (2008b). Memos to Designers 10-20 Attachment 2: Deck Slab Thickness and Reinforcement Schedule, California Department of Transportation, Sacramento, CA.

9.

Caltrans, (2008c). Memos to Designers 10-20 Attachment 3: Soffit Slab Reinforcement Details, California Department of Transportation, Sacramento, CA.

10.

Caltrans, (2001). Memos to Designers 20-6 Attachment 2: Capacity Procedure (Metric), California Department of Transportation, Sacramento, CA.

11.

Caltrans, (1990). Bridge Design Aids Chapter 10: Type Selection, California Department of Transportation, Sacramento, CA.

12.

Castrodale, R. W. and White, C. D. (2004). National Cooperative Highway Research Program (NCHRP) Report 517: Extending Span Ranges of Precast Prestressed Concrete Girders, Transportation Research Board, Washington, D.C.

13.

Collins, M. P. and Mitchell, D. (1991). Prestressed Concrete Structures, Response Publications, Toronto, Canada

14.

Holombo, J., Priestley, M. J. N., and Seible, F. (2000). “Continuity of Precast, Prestressed Spliced-Girder Bridges Under Seismic Loads.” PCI Journal, 45(2), 40-63.

15.

Ma, J. and Schendel, R. (2009). New California Precast LRFD Standard Drawings.

16.

Mast, R. F. (1993). “Lateral Stability of Long Prestressed Concrete Beams—Part 2.” PCI Journal, 38(1), 70-88.

17.

Naaman, A. E. (2004). Prestressed Concrete Analysis and Design: Fundamentals—Second Edition, Techno Press 3000, Ann Arbor, MI.

18.

Precast/Prestressed Concrete Institute, (2011). PCI Bridge Design Manual—Third Edition, Precast/Prestressed Concrete Institute, Chicago, IL.

Chapter 8 – Precast Pretensioned Concrete Girders

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19.

Precast/Prestressed Concrete Institute, (2010). PCI Design Handbook: Precast and Prestressed Concrete—Seventh Edition, Precast/Prestressed Concrete Institute, Chicago, IL.

20.

Pritchard, B. (1992). Bridge Design for Economy and Durability, Thomas Telford House, London, England.

21.

Snyder, R. (2010). Seismic Performance of an I-girder to Inverted-T Bent Cap Bridge Connection, Iowa State University, Ames, IA.

22.

Tadros, M. K., Al-Omaishi, N., Seguirant, S. J., and Galt, J. G. (2003). National Cooperative Highway Research Program (NCHRP) Report 496: Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders, Transportation Research Board, Washington, D.C.

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CHAPTER 9 STEEL PLATE GIRDERS TABLE OF CONTENTS 9.1 

INTRODUCTION ............................................................................................................... 1 

9.2  

STRUCTURAL MATERIALS ........................................................................................... 1 9.2.1 

Structural Steel ...................................................................................................... 1 

9.2.2   Concrete ................................................................................................................ 2  9.3 

SPAN AND FRAMING ARRANGEMENT ...................................................................... 2 9.3.1   Span Configuration ............................................................................................... 2  9.3.2   Girder Spacing....................................................................................................... 2  9.3.3   Diaphragms and Cross Frames .............................................................................. 3 9.3.4   Lateral Bracing ...................................................................................................... 5 9.3.5   Field Splice Locations ........................................................................................... 5  9.3.6   Expansion Joints and Hinges ................................................................................. 5 

9.4 

SECTION PROPORTION .................................................................................................. 6  9.4.1 

Depth to Span Ratios ............................................................................................. 6 

9.4.2   Webs ...................................................................................................................... 6  9.4.3   Flanges .................................................................................................................. 7 9.4.4   Stiffeners ............................................................................................................... 8  9.5 

STRUCTURAL MODELING AND ANALYSIS .............................................................. 8 

9.6 

DESIGN LIMIT STATES AND PROCEDURES .............................................................. 9  9.6.1   Design Limit States ............................................................................................... 9  9.6.2   Design Procedure .................................................................................................. 9 

9.7 

 DESIGN EXAMPLE - THREE-SPAN CONTINUOUS COMPOSITE PLATE GIRDER BRIDGE ........................................................................................................... 11  9.7.1   Steel Girder Bridge Data ..................................................................................... 11  9.7.2   Design Requirements .......................................................................................... 12 9.7.3   Select Girder Layout and Sections ...................................................................... 13  9.7.4   Perform Load and Structural Analysis ................................................................ 18  9.7.5   Calculate Live Load Distribution Factors ........................................................... 22 9.7.6   Determine Load and Resistance Factors and Load Combinations ...................... 25 9.7.7   Calculate Factored Moments and Shears – Strength Limit States....................... 26

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9.7.8   Calculate Factored Moments and Shears – Fatigue Limit States ........................ 28  9.7.9   Calculate Factored Moments – Service Limit State II ........................................ 30  9.7.10   Design Composite Section in Positive Moment Region at 0.5 Point of Span 2 .. 31 9.7.11   Design Noncomposite Section in Negative Moment Region at Bent 3 .............. 45 9.7.12   Design Shear Connectors for Span 2 ................................................................... 64  9.7.13   Design Bearing Stiffeners at Bent 3 .................................................................... 67  9.7.14   Design Intermediate Cross Frames...................................................................... 70  9.7.15  Design Bolted Field Splices ................................................................................ 77 9.7.16   Calculate Deflection and Camber........................................................................ 98 9.7.17   Identify and Designate Steel Bridge Members and Components ...................... 100  NOTATION ................................................................................................................................ 101  REFERENCES ............................................................................................................................ 105 

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CHAPTER 9 STEEL PLATE GIRDERS 9.1

INTRODUCTION Girder bridges are structurally the simplest and the most commonly used on short to medium span bridges. Figure 9.1-1 shows the Central Viaduct in San Francisco. Steel I-section is the simplest and most effective solid section for resisting bending and shear. In this chapter straight composite steel-concrete plate girder bridges are discussed. Design considerations for span and framing arrangement, and section proportion are presented. A design example of the three span continuous composite plate girder bridge is given to illustrate the design procedure. For a more detailed discussion, reference may be made to texts by Chen and Duan (2014), Baker and Puckett (2013), FHWA (2012), and Taly (2014).

Figure 9.1-1 Central Viaduct in San Francisco

9.2

STRUCTURAL MATERIALS

9.2.1

Structural Steel ASTM A 709 or AASHTO M 270 (Grades 36, 50, 50S, 50W, HPS 50W, HPS 70W and 100/100W) structural steels are commonly used for bridge structures. Chapter 6 provides a more detailed discussion.

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9.2.2

Concrete

9.3

SPAN AND FRAMING ARRANGEMENT

9.3.1

Span Configuration

Concrete with 28-day compressive strength f c = 3.6 ksi is commonly used in concrete deck slab construction. Caltrans MTD 10-20 (Caltrans, 2008) provides concrete deck slab thickness and reinforcement. The transformed area of concrete is used to calculate the composite section properties. For normal weight concrete of f c = 3.6 ksi, the ratio of the modulus of elasticity of steel to that of concrete, n = E/Ec = 8 is recommended by AASHTO (2012). For unshored construction, the modular ratio n is used for transient loads applied to the short-term composite sections, and the modular ratio 3n is used for permanent loads applied to the long-term composite sections.

Span configuration plays an important role in the efficient and cost-effective use of steel. For cases where pier locations are flexible, designers should optimize the span arrangement. Two-span continuous girders/beams are not the most efficient system because of high negative moments. Three- and four-span continuous girders are preferable, but may not always be possible. For multi-span continuous girders, a good span arrangement is to have the end span lengths approximately 70 to 80 percent of the interior span lengths. Equal interior span arrangements are also relatively economical. A span configuration with uplift due to live load plus impact should be avoided. The use of simply supported girders under construction load and continuous girders through steel reinforcement for live load can be an economical framing method (Azizinamini, 2007). This type of framing presents possible advantages over continuous beam designs by eliminating costly splices and heavy lifts during girder erection. The potential drawbacks are that more section depth may be required and the weight of steel per unit deck area may be higher. This framing method needs to be investigated on a case-by-case basis to determine whether it can be economically advantageous. When simply supported span configurations are used, special attention should be given to seismic performance detailing.

9.3.2

Girder Spacing As a general rule, the most economical superstructure design can be achieved using girder spacing within an 11 ft. to 14 ft. range. For spans less than 140 ft., 10 ft. to 12 ft. spacing is preferred. For spans greater than 140 ft., 11 ft. to 14 ft. spacing is recommended. The use of metal deck form panels will limit the spacing to about 16 ft. Girder spacings over 16 ft. may require a transversely post-tensioned deck system. Parallel girder layout should be used wherever possible.

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9.3.3

Diaphragms and Cross Frames The terms diaphragm and cross frame are synonymous. Figure 9.3-1 shows typical types of diaphragms and cross frames used in I-shaped plate girder and rolled beam spans. The K-frames and X-frames usually include a top strut as shown in Figure 9.3-1. Intermediate cross frames provide bracing against lateral torsional buckling of compression flanges during erection and deck concrete placement, and for all loading stages in negative flexure regions. They also provide lateral bracing for wind loads. End cross frames or diaphragms at piers and abutments are provided to transmit lateral wind loads and seismic loads to the bearings.

9.3.3.1

Spacing Arbitrary 25 ft. spacing limit for diaphragms and cross frames was specified in the AASHTO Standard Design Specifications (AASHTO, 2002) and the Caltrans Bridge Design Specifications (Caltrans, 2000). The AASHTO LRFD Bridge Design Specifications (AASHTO, 2012), however, no longer specify a limit on the cross frame spacing, but instead require rational analysis to investigate needs for all stages of assumed construction procedures and the final conditions. Spacing should be compatible with the transverse stiffeners.

Figure 9.3-1 Typical Diaphragms and Cross Frames

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9.3.3.2

Orientation Intermediate cross frames shall be placed parallel to the skew up to a 20o skew and normal to the girders for a skew angle larger than 20o. On skewed bridges with cross frames placed normal to the girders, there may be situations where the cross frames are staggered or discontinuous across the width of the bridge. At these discontinuous cross frames, lateral flange bending stresses may be introduced into the girder flanges and should be considered. Install stiffeners on the back side of connection plates if staggered cross frames are used. Horizontally curved girders should always have the cross frames placed on radial lines. A good economical design will minimize the number of diaphragms with varying geometries. Superelevation changes, vertical curves, different connection plate widths, and flaring girders all work against this goal.

9.3.3.3

Connections Cross frames are typically connected to transverse stiffeners. The stiffeners shall have a positive connection to the girder flange and may either be bolted or welded, although welding is preferred. For bridges built in stages or with larger skew angles, differential deflections between girders due to slab placement can be significant. If differential deflections are significant, slotted holes and hand tight erection bolts with jamb nuts shall be provided during concrete placement, and permanent bolts fully tensioned or field welded connections shall be installed after the barriers are placed. The bolt holes can be field drilled to insure proper fit. Intermediate cross frames between stages shall be eliminated if possible.

9.3.3.4

Design Guidelines 

The diaphragm or cross frame shall be as deep as practicable to transfer lateral load and to provide lateral stability. They shall be at least 0.5 of the beam depth for rolled beams and 0.75 of the girder depth for plate girders (AASHTO 6.7.4.2).



Cross frames should be designed and detailed such that they can be erected as a single unit, and all welding during fabrication should be done from one side to minimize handling costs. As a minimum, cross frames shall be designed to resist lateral wind loads. A rational analysis is preferred to determine actual lateral forces.

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9.3.4



End diaphragms and cross frames at bearings shall be designed to resist all lateral forces transmitted to the substructure. Unless they are detailed as ductile elements, the end diaphragms or cross frames shall be designed to resist the overstrength shear capacity of the substructures. Shear connectors should be provided to transfer lateral loads from the deck to the end diaphragm in accordance with the Caltrans Guide Specifications for Seismic Design of Steel Bridges (Caltrans 2014). When an expansion joint occurs at a support, the end diaphragm shall be designed to resist truck wheel and impact loads.



Effective slenderness ratios (KL/r) for compression diagonals shall be less than or equal to 120 and 140 for horizontally curved girders and straight girders, respectively (AASHTO 6.9.3); and for tension members (L/r) less than or equal to 240 (AASHTO 6.8.4).



Cross frame members and gussets consisting of single angle or WT shapes should be designed for the eccentricity inherent at the gusset connections. Use rectangular gusset plates in lieu of multi-sided polygons.



Steel plate, I girder, and concrete diaphragms may be used at abutments and piers. The use of integral abutments, piers, and bents is encouraged.

Lateral Bracing Bottom chord lateral bracing should be avoided because the bracing creates fatigue-sensitive details and is costly to fabricate, install, and maintain. Flange sizes should be sufficient to preclude the need for bottom flange lateral bracing.

9.3.5

Field Splice Locations Field splices shall preferably be located at points of dead load contraflexure and at points of section change and spaced more than 50 ft. apart. The splice locations are also dependent on shipping and fabrication limits. The length of shipping piece is usually less than 125 ft. and weight less than 40 tons. It is not necessary to locate the splices at the exact contraflexure point, but they should be reasonably close. Field splices are sometimes required to be placed near points of maximum moment in longer spans in order to meet erection requirements. Field splices should always be bolted. Welded field splices shall not be used (CA 6.13.6.2). Adjacent girders should be spliced in approximately the same location.

9.3.6

Expansion Joints and Hinges In-span hinges are generally not recommended for steel bridges since there are not many acceptable solutions for the design of hinges to resist seismic loads. Steel bridges have been designed without expansion joints and hinges at lengths up to 1200 ft. When dropped cap bents are utilized, the superstructure may be separated from the

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substructure with expansion bearings to prevent undue temperature effects on the substructure.

9.4

SECTION PROPORTION

9.4.1

Depth to Span Ratios Figure 9.4-1 shows a typical portion of a composite I-girder bridge consisting of a concrete deck and built-up plate girder I-section with stiffeners and cross frames. The first step in the structural design of a plate girder bridge is to initially size the web and flanges.

Figure 9.4-1 Components of Typical I-Girder Bridge For straight girders, AASHTO Table 2.5.2.6.3-1 specifies the minimum ratio of the depth of steel girder portion to the span length is 0.033 for simply supported spans and 0.027 for continuous spans; the minimum ratio of the overall depth (concrete slab plus steel girder) to span length is 0.04 for simply supported spans and 0.032 for continuous spans. Caltrans traditionally prefers that the minimum ratio of overall depth to span length is 0.045 for simply supported spans and 0.04 for continuous spans. For horizontally curved girders, the minimum depth will more than likely need to be increased by 10 to 20%.

9.4.2

Webs The web mainly provides shear strength for the girder. Since the web contributes little to the bending resistance, its thickness should be as small as practical to meet the web depth to thickness ratio limits D/tw  150 for webs without longitudinal stiffeners, and D/tw  300 for webs with longitudinal stiffeners, respectively (AASHTO 6.10.2.1). It is preferable to have web depths in increments of 2 or 3 in.

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for convenience. Web depths greater than 120 in. will require both longitudinal and vertical splices. The web thickness is preferred to be not less than ½ inch. A thinner plate is subject to excessive distortion from welding. The thickness should be sufficient to preclude the need for longitudinal stiffeners. Web thickness should be constant or with a limited number of changes. A reasonable target would be one or two web sizes for a continuous girder and one web size for a simple span. Web thickness increments should be 1/l6 in. or 1/8 in. for plate thicknesses up to 1 inch, and ¼ inch increments for plates greater than 1 inch.

9.4.3

Flanges The flanges provide bending strength. Flanges should be at least l2 in. wide. A constant flange width for the entire length of the girder is preferred. If the flange area needs to be increased, it is preferable to change the flange thickness. If flange widths need to be changed, it is best to change the width at field splices only. Width increments should be in multiples of 2 or 3 inches. For horizontally curved girders, the flange width should be about one-fourth of the web depth. For straight girders, a flange width of approximately one-fifth to one-sixth of the web depth should be sufficient. For straight girders, the minimum flange thickness should be 3/4 inch. For curved girders, l in. thickness is a practical minimum. The desirable maximum flange thickness is 3 inches. Grade 50 and HPS 70W steels are not available in thicknesses greater than 4 inches. Flange thickness increments should be 1/8 in. for thicknesses up to 1 in., 1/4 in. from 1 to 3 in., and 1/2 in. from 3 to 4 inches. At the locations where the flange thickness is changed, the thicker flange should provide about 25 percent more area than the thinner flange. In addition, the thicker flange should be not greater than twice the thickness of the thinner flange. Both the compression and tension flanges shall meet the following proportion requirements (AASHTO 6.10.2.2) as follows: bf 2t f

bf 

 12

(AASHTO 6.10.2.2-1)

D 6

(AASHTO 6.10.2.2-2)

t f  1.1tw 0.1 

I yc I yt

 10

(AASHTO 6.10.2.2-3) (AASHTO 6.10.2.2-4)

where bf and tf are full width and thickness of the flange (in.); tw is web thickness (in.); Iyc and Iyt are the moment of inertia of the compression flange and the tension flange about the vertical axis in the plane of web, respectively (in.4); D is web depth

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(in.). Equation AASHTO 6.10.2.2-1 ensures the flange will not distort excessively when welded to the web. Equation AASHTO 6.10.2.2-2 ensures that stiffened interior web panels can develop post-elastic buckling shear resistance by the tension field action. Equation AASHTO 6.10.2.2-3 ensures that flanges can provide some restraint and proper boundary conditions to resist web shear buckling. Equation AASHTO 6.10.2.2-4 ensures more efficient flange proportions and prevents the use of sections that may be difficult to handle during construction. It also ensures that the lateral torsional buckling formulas used in AASHTO are valid.

9.4.4

Stiffeners Intermediate transverse stiffeners together with the web are used to provide postelastic shear buckling resistance by the tension field action and are usually placed near the supports and large concentrated loads. Stiffeners without connecting cross frames/diaphragms are typically welded to the girder web and shall be welded to the compression flange and fitted tightly to the tension flange (CA 6.10.11.1.1). Stiffener plates are preferred to have even inch widths from the flat bar stock sizes. Bearing stiffeners are required at all bearing locations. Bearing stiffeners shall be welded or bolted to both sides of the web. Bearing stiffeners should be thick enough to preclude the need for multiple pairs of bearing stiffeners to avoid multiple-stiffener fabrication difficulties. AASHTO 6.10.11.2 requires that the stiffeners shall extend the full depth of the web and as close as practical to the edge of the flanges. Longitudinal stiffeners are required to increase flexure resistance of the web by controlling lateral web deflection and preventing the web bending buckling. They are, therefore, attached to the compression portion of the web. It is recommended that sufficient web thickness be used to eliminate the need for longitudinal stiffeners as they can cause difficulty in fabrication and create fatigue-prone details.

9.5

STRUCTURAL MODELING AND ANALYSIS Steel girder bridges are commonly modeled as beam elements and analyzed as unshored construction. Flexural stiffness of composite section is assumed over the entire bridge length even though the negative moment regions may be designed as non-composite for the section capacity. Longitudinal reinforcing steel in the top mat of the concrete deck within the effective deck width is generally not included in calculating section properties. In the preliminary analysis, a constant flexural stiffness may be assumed. In the final analysis of composite flexural members, the stiffness properties of the steel section alone for the loads applied to noncomposite sections, the stiffness properties of the long-term composite section for permanent loads applied to composite sections, and the stiffness properties of the short-term composite section properties for transient loads, shall be used over the entire bridge length (AASHTO 6.10.1.5), respectively.

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Dead loads are usually distributed to the girders based on tributary area. Live loads distribution is dependent on the girder spacing S, span length L, concrete slab depth ts, longitudinal stiffness parameter Kg, and number of girders Nb (AASHTO 4.6.2.2.1). The more refined analysis using the finite element method may be used in analyzing complex bridge systems such as skewed and horizontally curved bridges.

9.6

DESIGN LIMIT STATES AND PROCEDURES

9.6.1

Design Limit States Steel girder bridges shall be designed to meet the requirements for all applicable limit states specified by AASHTO (2012) and the California Amendments (Caltrans 2014) such as Strength I, Strength II, Service II, Fatigue I and II, and extreme events. Constructability (AASHTO 6.10.3) must be considered. See Chapters 3, 4, and 6 for more detailed discussion.

9.6.2

Design Procedure The steel girder design may follow the flowchart as shown in Figure 9.6-1.

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Start

Select girder layout, framing system, and sections

Perform load and structural analysis Determine LRFD load combinations (CA Table 3.4.1-1)

Perform Flexure Design for the Limit States - Strength (AASHTO 6.10.6.2), Service (AASHTO 6.10.4), Fatigue (AASHTO 6.10.5.1), and Constructibility (AASHTO 6.10.3.2)

Perform Shear Design for the Limit States Strength (AASHTO 6.10.6.3), Fatigue (AASHTO 6.10.5.1), and Constructibility (AASHTO 6.10.3.3)

Perform Shear Connector Design (AASHTO 6.10.10)

Perform Bearing Stiffener Design (AASHTO 6.10.11.2)

Perform Cross Frame Design

Perform Bolted Field Splices Design (AASHTO and CA 6.13.6)

Calculate Deflection and Camber

Girder Design Completed

Figure 9.6-1 Chapter 9 - Steel Plate Girders

Steel I-Girder Design Flowchart 9-10

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7

DESIGN EXAMPLE – THREE-SPAN CONTINUOUS COMPOSITE PLATE GIRDER BRIDGE

9.7.1

Steel Girder Bridge Data A three-span continuous composite plate girder bridge has spans of 110 ft – 165 ft – 125 ft. The superstructure is 58 ft wide. The elevation and plan are shown in Figure 9.7-1. Structural steel: Concrete: Deck: Construction:

A 709 Grade 50 for web, flanges and splice plates Fy = 50 ksi A 709 Grade 36 for cross frames and stiffeners, etc. Fy = 36 ksi f c = 3600 psi; Ec = 3,640 ksi; modular ratio n = 8 Concrete deck slab thickness = 9.125 in. Unshored construction

(a) Elevation

(b) Plan Figure 9.7-1 Three-span Continuous Steel Plate Girder Bridge

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9.7.2

Design Requirements Perform the following design portions for an interior plate girder in accordance with the AASHTO LRFD Bridge Design Specifications, 6th Edition (AASHTO 2012) with the California Amendments (Caltrans 2014). 

Select Girder Layout and Sections



Perform Load and Structural Analysis



Calculate Live Load Distribution Factors



Determine Load and Resistance Factors and Load Combinations



Calculate Factored Moments and Shears – Strength Limit States



Calculate Factored Moments and Shears – Fatigue Limit States



Calculate Factored Moments – Service Limit State II



Design Composite Section in Positive Moment Region at 0.5 Point of Span 2



Design Noncomposite Section in Negative Moment Region at Bent 3



Design Shear Connectors for Span 2



Design Bearing Stiffeners at Bent 3



Design Intermediate Cross Frames



Design Bolted Field Splices



Calculate Camber and Plot Camber Diagram



Identify and Designate Steel Bridge Members and Components

The following notation is used in this example: 

“AASHTO xxx” denotes “AASHTO Article xxx”



“AASHTO xxx-x” denotes “AASHTO Equation/Table xxx-x”



“CA xxx” denotes “California Amendment Article xxx”



“CA xxx-x” denotes “California Amendment Equation/Table xxx-x”



“MTD xxx” denotes “Caltrans Bridge Memo to Designers Article xxx”

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9.7.3

Select Girder Layout and Sections

9.7.3.1

Select Girder Spacing A girder spacing of 12 ft is selected as shown in Figure 9.7-2a.

9.7.3.2

Select Intermediate Cross Frame Spacing Cross frames at spacing of 27.5 ft and 25 ft are selected as shown in Figure 9.7-3 to accommodate transverse stiffener spacing for web design and to facilitate a reduction in required flange thickness of the girder section at the bent.

9.7.3.3

Select Steel Girder Section for Positive Flexure Regions The cross section is usually proportioned based on past practice and proportion limits specified in AASHTO 6.10.2. Interior girder section is shown in Figures 9.7-2b and 9.7-4. Haunch depth shall be carefully selected by considering road slope, top flange thickness, correction of sagging and cambers, embedment of shear connectors as discussed in MTD 12-4 (Caltrans, 2004a). Top Compression Flange The maximum transported length of a steel plate girder is generally limited to a length of about 120 ft and a weight of about 180 kips and may vary due to the locations. It is common practice that the unsupported length of each shipping piece divided by the minimum width of compression flange should be less than or equal to about 85 (AASHTO C 6.10.3.4). For a length of 120 ft, the width of compression flange is preferably larger than (12012)/85 = 17 in. Try top compression flange bfc  tfc = 18  1 (in.  in.). Web AASHTO Table 2.5.2.6.3-1 specifies that for composite girders, the minimum ratio of the depth of steel girder portion to the length of span is 0.033 for simple span and 0.027 for continuous spans. For this design example, the depth of steel girder shall be larger than 0.027(165) = 4.46 ft. = 53.5 in. Try web D  tw = 78  0.625 (in.  in.). Bottom Tension Flange Try bottom tension flange bft  tft = 18  1.75 (in.  in.).

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(a) Bridge Cross Section

1’-11/4”

9 1/8”

(a) Bridge Cross Section

(b) Interior Girder Section

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Figure 9.7-2 Typical Cross Sections

Figure 9.7-3 Framing Plan (Skew not Shown)

For convenience in this example, the ends of the girder have been assumed to match the BB and EB locations. Figure 9.7-4 Elevation of Interior Girder

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Check Section Proportion Limits 



Web without longitudinal stiffeners D 78   124.8  150 O.K. tw 0.625

(AASHTO 6.10.2.1.1-1)

Compression flange b fc 18   9  12 2t fc 2 1.0 

O.K.

(AASHTO 6.10.2.2-1)

D 78   13 6 6

O.K.

(AASHTO 6.10.2.2-2)

b fc  18 

t fc  1.0 in.  1.1tw  1.1 0.625  0.69 in.

OK. (AASHTO 6.10.2.2-3)



Tension flange b ft 18   5.14  12 2t ft 2 1.75  b ft  18 

D 78   13 6 6

O.K.

(AASHTO 6.10.2.2-1)

O.K.

(AASHTO 6.10.2.2-2)

t ft  1.75 in.  1.1tw  1.1 0.625  0.69 in.

OK. (AASHTO 6.10.2.2-3)



Flanges Ratio The flange shall meet the requirement of 0.1  I yc / I yt   10 , where Iyc and Iyt are the moment of inertia of the compression flange and the tension flange about the vertical axis in the plane of web, respectively. This limit ensures more efficient flange proportions and prevents the use of sections that may be difficult to handle during construction. It also ensures that the lateral torsional buckling formulas are valid. 0.1 

I yc I yt



1183 / 12 1.75183 / 12

 0.57  10

O.K. (AASHTO 6.10.2.2-4)

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9.7.3.4

Select Steel Girder Section for Negative Flexure Regions Flanges In the negative moment region, non-composite symmetric steel section is generally used. Try flange plates bf  tf = 18  2 (in.  in.). Web It is more cost effective to use one thickness plate for the web through whole bridge. Try web D  tw = 78  0.625 (in.  in.). Check Section Proportion Limits 



Web without longitudinal stiffeners: D 78   124.8  150 tw 0.625

O.K.

(AASHTO 6.10.2.1.1-1)

Compression and tension flanges bf 18   4.5  12 2t f 2  2.0 

O.K.

(AASHTO 6.10.2.2-1)

O.K.

(AASHTO 6.10.2.2-2)

b f  18 

D 78   13 6 6

t f  2.0 in.  1.1tw  1.1 0.625  0.69 in. O.K. 

Flange ratio 0.1 

I yc I yt



 2 18 3 / 12  2 18 3 / 12

 1.0  10

O.K. 9.7.3.5

(AASHTO 6.10.2.2-3)

(AASHTO 6.10.2.2-4)

Select Transverse Stiffeners It is normal to use stiffener width of 7.5 in. to provide allowances for gusset plate connections of cross frames. Try a pair of stiffeners bt  tp = 7.5  0.5 (in.  in.)

9.7.3.6

Select Bolted Splice Locations For flexural members, splices shall preferably be made at or near points of dead load contraflexure in continuous spans and at points of the section change. As shown in Figure 9.7-3, splices locations for Spans 1 and 3 are selected approximately at 0.7 and 0.3 points, respectively, and for Span 2 are selected approximately at 0.3 and 0.7 points.

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9.7.4

Perform Load and Structural Analysis

9.7.4.1

Calculate Permanent Loads for an Interior Girder The permanent load or dead load of an interior girder includes DC and DW. DC is dead load of structural components and nonstructural attachments. DW is dead load of wearing surface. For design purposes, the two parts of DC are defined as, DC1, structural dead load, acting on the non-composite section, and DC2, nonstructural dead load, acting on the long-term composite section. DC1 usually consists of deck slab concrete (unit weight 150 lbs/ft3), steel girder including bracing system and details (estimated weight 460 lbs/ft for each girder), and an additional 10 percent of deck weight between girders to compensate for the use of permanent steel deck forms as specified in MTD 8-7 (Caltrans, 2015) for bridges designed that are over vehicular or rail traffic in Climate Areas I and II. DC1 is assumed to be distributed to each girder by the tributary area. The tributary width for the interior girder is 12 feet. DC1  1.19.125 / 1212  1.5  1.513.25 1.5 / 120.15  0.46  2.0 kip/ft DC2 usually consists of the barrier rails and specified utility. Type 732 Concrete Barriers (0.41 kips/ft and bottom width = 1.43 ft) are used and no utility is considered for this bridge. DC2 is assumed to be distributed equally to each girder. DC2 = 20.41 / 5  0.164 kip/ft A future wearing surface 35 psf as specified in MTD 15-17 (Caltrans 1988) is assumed. DW is assumed to be distributed equally to each girder. DW = (deck width - barrier width) (wearing surface pressure)/5 = [58 - 2(1.43)] (0.035)/5 = 0.386 kip/ft

9.7.4.2

Determine Live Load and Dynamic Load Allowance The design live load LL is the AASHTO HL-93 (AASHTO 3.6.1.2) and Caltrans P15 vehicular live loads (CA 3.6.1.8). To consider the wheel-load impact from moving vehicles, the dynamic load allowance IM = 33% for the Strength I Limit State, 25% for the Strength II Limit State, and 15% for the Fatigue Limit States are used (CA Table 3.6.2.1-1).

9.7.4.3

Perform Structural Analysis A structural analysis for three-span continuous beams shall be performed to obtain moments and shear effects due to dead loads and live loads including impact. In the preliminary analysis, a constant flexural stiffness may be assumed. In the final analysis of composite flexural members, the stiffness properties of the steel section alone for the loads applied to noncomposite sections, the stiffness properties of the long-term composite section for permanent loads applied to composite sections and

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

the stiffness properties of the short-term composite section properties for transient loads, shall be used over the entire bridge length (AASHTO 6.10.1.5), respectively. In this design example, the analysis is performed by the CT-Bridge computer program and checked by the CSiBridge program. A constant flexural stiffness is assumed for simplicity. Unfactored dead load for an interior girder and live load moments, shears, and support forces for one lane loaded are listed in Tables 9.7-1, 9.7-2, and 9.7-3, respectively. Unfactored moment and shear envelopes for one lane loaded in Span 2 are plotted in Figures 9.7-5 and 9.7-6, respectively. Table 9.7-1 Unfactored Dead and Live Load Moments Span Point x/L

1

2

3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Chapter 9 - Steel Plate Girders

Dead Load Live Load (Interior Girder) (One Lane) DC1 DC2 DW (LL+IM)HL-93 (LL+IM)P15 M_dc1 M_dc2 M_dw +M -M +M -M (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) 0 0 0 0 0 0 0 693 57 135 1143 -249 1937 -576 1144 94 223 1960 -498 3236 -1152 1353 111 264 2465 -747 4118 -1728 1329 108 258 2713 -996 4564 -2304 1045 86 204 2702 -1245 4473 -2880 528 43 103 2470 -1495 4092 -3457 -231 -19 -45 1995 -2104 3136 -4033 -1232 -101 -240 1317 -2404 1758 -4609 -2474 -203 -483 585 -2796 931 -5185 -3959 -325 -772 489 -3426 1035 -5782 -3959 -325 -772 489 -3426 1035 -5782 -1555 -128 -303 638 -1802 623 -2877 304 25 59 1585 -961 2442 -1857 1619 133 316 2586 -838 4883 -1537 2389 196 466 3236 -751 6316 -1216 2615 215 510 3455 -727 6897 -1218 2297 188 448 3274 -886 6410 -1669 1434 118 280 2660 -1045 5021 -2119 26 2 5 1663 -1220 2661 -2570 -1926 -158 -376 643 -1998 687 -3075 -4422 -363 -862 360 -3563 705 -5981 -4422 -363 -862 360 -3563 705 -5981 -2574 -211 -502 574 -2663 635 -4821 -1038 -85 -202 1417 -2237 2058 -4285 186 15 36 2245 -1623 3814 -3750 1097 90 214 2832 -1391 5068 -3214 1695 139 331 3136 -1159 5723 -2678 1981 162 386 3168 -927 5737 -2143 1955 160 381 2893 -696 5167 -1607 1616 133 315 2303 -464 4112 -1071 964 79 188 1344 -232 2419 -536 0 0 0 0 0 0 0

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-2 Unfactored Dead and Live Load Shears Span Point x/L

1

2

3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dead Load Live Load (Interior Girder) (One Lane) (LL+IM)HL-93 (LL+IM)P15 DC1 DC2 DW V_DC1 V_DC2 V_DW +V -V +V -V (kip) (kip) (kip) (kip) (kip) (kip) (kip) 74.0 6.1 14.4 116.2 -22.6 210.2 -52.4 52.0 4.3 10.1 100.8 -23.1 176.1 -52.4 30.0 2.5 5.9 83.8 -25.6 137.8 -52.4 8.0 0.7 1.6 67.9 -37.9 105.3 -62.9 -14.0 -1.1 -2.7 53.4 -52.0 77.4 -83.7 -36.0 -3.0 -7.0 38.2 -66.7 50.0 -109.2 -58.0 -4.8 -11.3 28.6 -81.2 35.7 -138.5 -80.0 -6.6 -15.6 18.5 -96.0 21.0 -170.9 -102.0 -8.4 -19.9 10.7 -110.6 10.8 -200.1 -124.0 -10.2 -24.2 5.2 -124.9 9.4 -227.7 -146.0 -12.0 -28.5 4.5 -136.8 9.4 -262.5 162.2 13.3 31.6 147.1 -12.9 325.5 -27.3 129.2 10.6 25.2 130.7 -13.3 279.2 -27.4 96.2 7.9 18.8 112.0 -16.4 225.5 -27.3 63.2 5.2 12.3 93.2 -27.7 173.9 -33.5 30.2 2.5 5.9 74.8 -41.4 127.8 -57.2 -2.8 -0.2 -0.5 55.5 -57.2 86.5 -88.2 -35.8 -2.9 -7.0 41.4 -74.4 58.5 -124.0 -68.8 -5.6 -13.4 27.4 -92.7 34.2 -169.5 -101.8 -8.3 -19.9 15.8 -111.7 19.4 -220.6 -134.8 -11.1 -26.3 10.3 -130.5 19.4 -274.3 -167.8 -13.8 -32.7 9.9 -147.2 19.4 -321.0 160.4 13.2 31.3 141.9 -2.9 290.1 -5.6 135.4 11.1 26.4 129.3 -4.5 249.7 -5.6 110.4 9.1 21.5 114.2 -10.0 205.7 -11.1 85.4 7.0 16.6 98.9 -18.3 175.3 -21.8 60.4 5.0 11.8 83.4 -28.6 142.1 -37.5 35.4 2.9 6.9 65.9 -40.7 103.5 -58.1 10.4 0.9 2.0 52.9 -54.3 78.9 -83.4 -14.6 -1.2 -2.9 38.3 -69.5 56.4 -114.4 -39.6 -3.2 -7.7 24.6 -86.1 42.9 -151.3 -64.6 -5.3 -12.6 19.0 -104.1 42.9 -193.5 -89.6 -7.3 -17.5 18.6 -120.8 42.9 -233.7

Table 9.7-3 Unfactored Support Forces Location

Abutment 1 Bent 2 Bent 3 Abutment 4

Chapter 9 - Steel Plate Girders

Dead Load Live Load (Interior Girder) (One Lane) DC1 DC2 DW (LL+IM)HL-93 (LL+IM)P15 R_DC1 R_DC2 R_DW +R +R (kip) (kip) (kip) (kip) (kip) 74.0 6.1 14.4 116.2 210.2 308.2 25.3 60.1 244.5 445.3 328.2 27.0 64.0 249.2 447.0 89.6 7.3 17.5 120.8 233.7

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

8000

6316

6000

6897 6410 5021

4883 4000

3455

3236

Moments (kip-ft)

2586

2442 1585

2000

0

-325 -772

638 -128 -303

2615

2389

2661

2297

2660

1619

623

1035 489

3274

1434 466 196 -751

316 133

304 59 25 -961

-838

510 215 -727

448 188

280 118

-886

-1045

643 5

-158 -376

-363 -862

-1220

-1802

-1216

-1537

-1857

-1218

-1926 -1998

-1669 -2119

-2570

-2877

-3426 -3959

705 360

687

26 2

-1555

-2000

-4000

1663

-3075

-3563 -4422

-5782

-6000

DC1

DW

+ (LL+IM)HL-93

- (LL+IM)HL-93

+ (LL+IM)P15

- (LL+IM)P15

-5981

DC2

-8000 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x/L

Figure 9.7-5 Unfactored Moment Envelopes for Span 2 400 325.5

300

279.2 225.5

200 147.1 162.2 129.2 130.7

Shears (kip)

100

0 -12.9

31.6 13.3

173.9 112.0 96.2

127.8 86.5 74.8 63.2 93.2 55.5

25.2 18.8 -13.3 10.6 -16.4 7.9

-27.3 -27.4

12.3 30.2 2.5 5.9 5.2 -27.7 -41.4

-27.3

-33.5

-100

-57.2

58.5

41.4-7.0 -0.5 -0.2 -2.9 -2.8 -57.2 -35.8-74.4

34.2 27.4 -13.4 -5.6

-101.8 -134.8 -111.7 -130.5

-124.0

-300

19.4 19.4 10.3 9.9 -19.9 -8.3 -26.3 -11.1 -13.8 -32.7

-68.8 -92.7

-88.2

-200

19.4 15.8

-147.2 -167.8

-169.5

DC1

DW

+ (LL+IM)HL-93

- (LL+IM)HL-93

+ (LL+IM)P15

- (LL+IM)P15

-220.6 -274.3

-321.0

DC2

-400 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x/L

Figure 9.7-6 Unfactored Shear Envelopes for Span 2

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.5

Calculate Live Load Distribution Factors

9.7.5.1

Check Ranges of Applicability of Live Load Distribution Factors For beam-slab bridges, the distribution of live load is dependent on the girder spacing S, span length L, concrete slab depth ts, longitudinal stiffness parameter Kg, and number of girders Nb. This example is categorized Type “a” (AASHTO Table 4.6.2.2.1-1). The preliminary section shown in Table 9.7-4 is assumed to estimate the longitudinal stiffness parameter, Kg (AASHTO 4.6.2.2.1-1) for the positive moment region in Span 2. Table 9.7-4 Preliminary Section Properties Ai

yi

Aiyi

yi - yNCb

Ai(yi – yNCb)2

Io

(in.2) 18.00 48.75 31.50

(in.) 80.25 40.75 0.875

(in.3) 1,444.5 1,986.6 27.6

(in.) 45.05 5.55 -34.325

(in.4) 36,531 1,502 37,113

(in.4) 1.5 24,716 8.04

98.25

-

3,458.7

-

75,146

24,726

Component Top flange 18  1 Web 78  0.625 Bottom flange 18  1.75 

y NCb 

eg

yNCt

 Ai yi  Ai



3, 458.7  35.2 in. 98.25

y NCt 1.75  78 1  35.2  45.55 in.

C.GNC I NC 

 Io

  Ai  yi  yNCb 

2

 24,726  75,146  99,872 in.4

yNCb

9.125  53.24 in. 2 2  8 99,872   98.25 53.24    3,026,891 in.4   eg  45.55 1.0 13.25 



K g  n I NC  Aeg2



Check ranges of applicability of AASHTO Tables 4.6.2.2.2b-1 and 4.6.2.2.3a-1 for Type “a” structure. Girder spacing: 3.5 ft < S = 12 ft < 16 ft Span length: 20 ft < L = (110, 165 and 125 ) ft < 240 ft Concrete deck: 4.5 in. < ts = 9.125 in. < 12.0 in. Number of girders: Nb = 5 > 4 Stiffness parameter: 10,000 in.4 < Kg = 3,026,891 in.4 < 7,000,000 in.4

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

It is seen that the girder satisfies the limitation of ranges of applicability of the approximate live load distribution factors specified in AASHTO Tables 4.6.2.2.2b-1 and 4.6.2.2.3a-1. Section type “a” (AASHTO Table 4.6.2.2.1-1) will be used. For preliminary design, the term K g /(12 Lt s3 ) may be taken as 1.0. Although the Kg term varies slightly along the span and between spans, the distribution factor is typically not sensitive to the value of Kg. For simplicity, the Kg of Span 2 is used for all spans of this example. 9.7.5.2

Determine Span Length for Use in Live Load Distribution Equations AASHTO Table C4.6.2.2.1-1 recommends the L for use in live load distribution equations as shown in Table 9.7-5. Table 9.7- 5 Span Length for Use in Live Load Distribution Equations     



9.7.5.3

Force Effects Positive Moment Negative Moment—Other than near interior supports of continuous spans Shear Exterior Reaction Negative Moment—Near interior supports of continuous spans from point of contraflexure to point of contraflexure under a uniform load on all spans Interior Reaction of Continuous Span

L (ft) The length of the span for which moment/shear/reaction is being calculated

The average length of the two adjacent spans

Calculate Live Load Distribution Factors Live load distribution factors are calculated and listed in Tables 9.7-6 and 9.7-7 in accordance with AASHTO Tables 4.6.2.2.2b-1 and 4.6.2.2.3a-1. One design lane loaded Kg    3  12 Lts  Two or more design lanes loaded S DFm  0.06     14 

0.4

 S  DFm  0.075     9.5 

S   L

0.6

0.3 

S   L

0.2 

0.1

Kg    3  12 Lts 

DFv  0.36 

;

0.1

;

DFv  0.2 

S 25

S  S    12  35 

2

Kg = 3,026,891 in.4 S = 12 ft ts = 9.125 in.

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-6 Live Load Distribution Factors for Interior Girder for Strength Limit State Span 1* 1 & 2** 2* 2 & 3** 3*

Lane loaded L = 110 ft L = 137.5 ft L = 165 ft L = 145 ft L = 125 ft

Moment DFm (Lane) One Two or More 0.600 0.900 0.554 0.846 0.519 0.805 0.544 0.834 0.573 0.869

Shear DFv (Lane) One Two or More 0.840 1.082 0.840 1.082 0.840 1.082 0.840 1.082 0.840 1.082

Note: * The span length for which moment is being calculated for positive moment, negative moment—other than near interior supports of continuous spans, shear, and exterior reaction. ** Average span length for negative moment—near interior supports of continuous spans from point of contraflexure to point of contraflexure under a uniform load on all spans, and interior reaction of continuous span. Multiple lane presence factors have been included in the above live load distribution factors.

It is seen that live load distribution factors for the case of two or more lanes loaded control the strength and service limit states. For the fatigue limit states, since live load is one HL-93 truck or one P9 truck as specified CA 3.6.1.4.1, multiple lane presence factor of 1.2 should be removed from above factors for the case of one lane loaded (AASHTO 3.6.1.1.2). Table 9.7-7 Live Load Distribution Factors for Interior Girder for Fatigue Limit State Shear DFv (Lane) Moment DFm (Lane) Lane loaded One One Span L = 110 ft 0.500 0.700 1* L = 137.5 ft 0.462 0.700 1 & 2** L = 165 ft 0.433 0.700 2* L = 145 ft 0.453 0.700 2 & 3** L = 125 ft 0.478 0.700 3* Note: * The span length for which moment is being calculated for positive moment, negative moment—other than near interior supports of continuous spans, shear, and exterior reaction. ** Average span length for negative moment—near interior supports of continuous spans from point of contraflexure to point of contraflexure under a uniform load on all spans, and interior reaction of continuous span.

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.6

Determine Load and Resistance Factors and Load Combinations A steel girder bridge is usually designed for the Strength Limit State, and checked for the Fatigue Limit State, Service Limit State II, and Constructibility.

9.7.6.1

Determine Design Equation AASHTO 1.3.2.1 requires that following design equation shall be satisfied for all limit states: (AASHTO 1.3.2.1-1) ii Qi   Rn  Rr

where i is load factor and  is resistance factor; Qi represents force effect; Rn is nominal resistance; i is load modifier factor related to ductility, redundancy, and operational importance and is defined as follows when a maximum value of i is used:  i  D R I  0.95 (AASHTO 1.3.2.1-2)

where D, R , and I are ductility and redundancy and operational factors, respectively. CA 1.3.3, 1.3.4 and 1.3.5 specify that they are all taken to 1.0 for all limit states. Therefore, i = 1.0. For this example, the design equation becomes:

i Qi 9.7.6.2

  Rn  Rr

Determine Applicable Load Factors and Load Combinations According to CA Table 3.4.1-1, considering live load distribution factors for the interior girder and denoting (LL+IM) as unfactored force effect due to one design lane loaded, the following load combinations are obtained as: Strength I: 1.25(DC) + 1.5(DW) + 1.75(DF)(LL+IM)HL-93 Strength II 1.25(DC) + 1.5(DW) + 1.35(DF)(LL+IM)P15 Service II: 1.0(DC) + 1.0(DW) + 1.30(DF)(LL+IM)HL-93 Fatigue I: 1.75(DF)(LL+IM)HL-93 Fatigue II: 1.0(DF)(LL+IM)P9 where DF is the live load distribution factor.

9.7.6.3

Determine Applicable Resistance Factors According to AASHTO 6.5.4.2, the strength limit states in this example. For flexure For shear For axial compression For tension, fracture in net section For tension, yielding in gross section For bearing on milled surfaces For bolts bearing on material For shear connector

Chapter 9 - Steel Plate Girders

following resistance factors are used for the

f v c u y b bb sc

= = = = = = = =

1.00 1.00 0.90 0.80 0.95 1.00 0.80 0.85 9-25

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

For block shear For A325 bolts in shear For weld metal in fillet weld – shear in throat of weld metal

9.7.7

bs = 0.8 s = 0.8 e2 = 0.8

Calculate Factored Moments and Shears – Strength Limit States Using live load distribution factors in Table 9.7-6, factored moments, shears, and support forces for strength limit states I and II are calculated and listed in Tables 9.78, 9.7-9 and 9.7-10, respectively. Strength I: Strength II:

1.25(DC) + 1.5(DW) + 1.75(DF)(LL+IM)HL-93 1.25(DC) + 1.5(DW) + 1.35(DF)(LL+IM)P15

Table 9.7-8 Factored Moment Envelopes for Interior Girder Dead Load Live Load Load Combination DC2 DW (LL+IM)HL-93 (LL+IM)P15 Strength I Strength II Span Point DC1 x/L M_DC1 M_DC2 M_DW +M -M +M -M +M -M +M -M (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) 0.0 0 0 0 0 0 0 0 0 0 0 0 0.1 866 71 203 1800 -392 2354 -700 2940 748 440 3494 0.2 1430 118 335 3087 -784 3932 -1400 4969 1098 482 5814 0.3 1691 139 396 3882 -1177 5004 -2100 6108 1049 126 7230 0.4 1661 135 387 4273 -1569 5545 -2800 6456 615 -617 7728 0.5 1306 108 306 4256 -1961 5434 -3500 5975 -241 -1780 1 7154 0.6 660 54 155 3890 -2355 4972 -4200 4759 -1486 -3331 5840 0.7 -289 -24 -68 3142 -3115 3811 -4606 2762 -3495 -4986 3431 0.8 -1540 -126 -360 2074 -3559 2136 -5264 48 -5585 110 -7290 0.9 -3093 -254 -725 921 -4139 1131 -5922 -3149 -8210 -2939 -9992 1.0 -4949 -406 -1158 770 -5072 1257 -6604 -5743 -11585 -5256 -13117 0.0 -4949 -406 -1158 689 -5072 1124 -6604 -5824 -11585 -5389 -13117 0.1 -1944 -160 -455 899 -2668 677 -3286 -1659 -5226 -1882 -5844 0.2 380 31 89 2233 -1354 2654 -2018 2733 -854 3154 -1518 0.3 2024 166 474 3643 -1181 5307 -1670 6307 1483 994 7971 0.4 2986 245 699 4559 -1058 6864 -1322 8489 2872 10794 2608 0.5 3269 269 765 4867 -1024 7495 -1324 9170 3278 11797 2979 2 0.6 2871 235 672 4612 -1248 6966 -1814 8390 2530 10744 1965 0.7 1793 148 420 3747 -1472 5456 -2303 6107 888 57 7816 0.8 33 3 8 2343 -1719 2891 -2793 2385 -1676 2934 -2751 0.9 -2408 -198 -564 906 -2916 747 -3462 -2263 -6085 -2422 -6631 1.0 -5528 -454 -1293 507 -5200 767 -6734 -6767 -12474 -6508 -14008 0.0 -5528 -454 -1293 547 -5200 828 -6734 -6727 -12474 -6447 -14008 0.1 -3218 -264 -753 873 -3887 745 -5428 -3361 -8121 -3489 -9662 0.2 -1298 -106 -303 2155 -3265 2414 -4825 448 -4972 708 -6532 0.3 233 19 54 3414 -2468 4475 -4399 3719 -2163 4780 -4094 0.4 1371 113 321 4307 -2115 5946 -3771 6112 -311 -1966 7751 0.5 2119 174 497 4769 -1763 6714 -3142 7558 1026 -353 3 9503 0.6 2476 203 579 4818 -1410 6730 -2514 8075 1848 744 9988 0.7 2444 200 572 4400 -1058 6061 -1885 7615 2157 1330 9276 0.8 2020 166 473 3502 -706 4824 -1257 6161 1953 1402 7482 0.9 1205 99 282 2044 -353 2838 -628 3630 1233 957 4424 1.0 0 0 0 0 0 0 0 0 0 0 0

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-9 Factored Shear Envelopes for Interior Girder Dead Load Live Load DC2 DW (LL+IM)HL-93 (LL+IM)P15 Span Point DC1 x/L V_DC1 V_DC2 V_DW +V -V +V -V (kip) (kip) (kip) (kip) (kip) (kip) (kip) 0.0 92.5 7.6 21.6 220.0 -42.8 307.0 -76.5 0.1 65.0 5.4 15.2 190.9 -43.7 257.2 -76.5 0.2 37.5 3.1 8.9 158.7 -48.5 201.3 -76.5 0.3 10.0 0.9 2.4 128.6 -71.8 153.8 -91.9 0.4 -17.5 -1.4 -4.1 101.1 -98.5 113.1 -122.2 0.5 -45.0 -3.8 -10.5 72.3 -126.3 73.1 -159.5 1 0.6 -72.5 -6.0 -17.0 54.2 -153.8 52.1 -202.3 0.7 -100.0 -8.3 -23.4 35.0 -181.8 30.7 -249.7 0.8 -127.5 -10.5 -29.9 20.3 -209.4 15.7 -292.3 0.9 -155.0 -12.8 -36.3 9.8 -236.5 13.7 -332.6 1.0 -182.5 -15.0 -42.8 8.5 -259.0 13.7 -383.5 0.0 202.8 16.6 47.4 278.5 -24.4 475.5 -39.9 0.1 161.5 13.3 37.8 247.5 -25.2 407.8 -40.0 0.2 120.3 9.9 28.2 212.1 -31.1 329.3 -39.9 0.3 79.0 6.5 18.5 176.5 -52.4 254.0 -48.9 0.4 37.8 3.1 8.9 141.6 -78.4 186.7 -83.5 0.5 -3.5 -0.3 -0.8 105.1 -108.3 126.4 -128.9 2 0.6 -44.8 -3.6 -10.5 78.4 -140.9 85.5 -181.2 0.7 -86.0 -7.0 -20.1 51.9 -175.5 49.9 -247.5 0.8 -127.3 -10.4 -29.9 29.9 -211.5 28.4 -322.2 0.9 -168.5 -13.9 -39.5 19.5 -247.1 28.4 -400.7 1.0 -209.8 -17.3 -49.1 18.7 -278.7 28.4 -468.9 0.0 200.5 16.5 47.0 268.7 -5.5 423.8 -8.2 0.1 169.3 13.9 39.6 244.8 -8.5 364.7 -8.2 0.2 138.0 11.4 32.3 216.2 -18.9 300.4 -16.3 0.3 106.8 8.8 24.9 187.3 -34.7 256.0 -31.8 0.4 75.5 6.3 17.7 157.9 -54.2 207.5 -54.8 0.5 44.3 3.6 10.4 124.8 -77.1 151.1 -84.9 3 0.6 13.0 1.1 3.0 100.2 -102.8 115.3 -121.8 0.7 -18.3 -1.5 -4.4 72.5 -131.6 82.4 -167.1 0.8 -49.5 -4.0 -11.6 46.6 -163.0 62.6 -221.1 0.9 -80.8 -6.6 -18.9 36.0 -197.1 62.6 -282.7 1.0 -112.0 -9.1 -26.3 35.2 -228.7 62.6 -341.3

Load Combination Strength I Strength II +V -V +V -V (kip) (kip) (kip) (kip) 341.7 78.9 45.2 428.7 276.4 41.8 9.0 342.8 208.2 1.0 -27.0 250.8 141.8 -58.5 -78.6 167.1 78.2 -121.4 90.2 -145.1 13.1 -185.5 13.8 -218.8 -41.3 -249.2 -43.3 -297.8 -96.6 -313.4 -101.0 -381.3 -147.6 -377.3 -152.1 -460.2 -194.2 -440.5 -190.3 -536.6 -231.7 -499.3 -226.5 -623.7 545.3 242.3 226.9 742.3 460.0 187.4 172.6 620.4 370.4 127.3 118.4 487.6 280.4 51.5 55.1 357.9 191.4 -28.7 -33.8 236.4 100.6 -112.8 121.9 -133.4 19.5 -199.8 26.6 -240.1 -61.2 -288.6 -63.2 -360.6 -137.6 -379.0 -139.1 -489.7 -202.3 -468.9 -193.5 -622.5 -257.3 -554.8 -247.7 -745.0 532.6 258.5 255.7 687.7 467.6 214.2 214.5 587.4 397.9 162.7 165.4 482.1 327.7 105.7 108.6 396.4 257.4 45.3 44.6 306.9 183.0 -18.8 -26.7 209.3 117.3 -85.7 -104.7 132.4 48.4 -155.7 58.3 -191.2 -18.5 -228.1 -2.5 -286.1 -70.3 -303.4 -43.7 -389.0 -112.2 -376.1 -84.8 -488.7

Table 9.7-10 Factored Support Forces for Interior Girder Factored Support Forces Dead Load Live Load (LL+I)HL-93 (LL+I)P15 Location (Interior Girder) R_DC1 R_DC2 R_DW +R +R (kip) (kip) (kip) (kip) (kip) 92.5 7.6 21.6 307.0 220.0 Abutment 1 385.3 31.6 90.2 650.4 463.0 Bent 2 410.3 33.8 96.0 652.9 471.9 Bent 3 112.0 9.1 26.3 341.4 228.7 Abutment 4

Chapter 9 - Steel Plate Girders

Load Combination Strength I Strength II +R +R (kip) (kip) 341.7 428.8 970.0 1157.4 1011.9 1192.9 376.1 488.7

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.8

Calculate Factored Moments and Shears – Fatigue Limit States For load-induced fatigue consideration (CA Table 3.4.1-1), the fatigue moment and shear force ranges are caused by live load only and are calculated by the following equations: Fatigue I (HL-93 Truck): (F) = 1.75 (DF)(LL+IM)HL-93 Fatigue II (P-9 Truck):

(F) = 1.0(DF)(LL+IM)P9

Using live load distribution factors in Table 9.7-7, fatigue limit moment and shear ranges for an interior girder are calculated and listed in Tables 9.7-11 and 9.712. Vu, shear due to the unfactored dead load plus the factored fatigue load (Fatigue I) is also calculated for checking the special fatigue requirement for webs as required by AASHTO 6.10.5.3.

Vu  Vdc1  Vdc2  Vdw 1.75  DFv   LL  IM  HL93

Table 9.7-11 Fatigue I Limit State - Moment and Shears for Interior Girder Fatigue Moment Span Point (LL+IM)HL-93 Factored (LL+IM)HL-93 (LL+IM)HL -93 (One Lane) (Interior Girder) (One Lane) +M -M +M -M M_sr +V -V x/L (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip) (kip) 0.0 0 0 0 0 0 64.6 -11.1 0.1 624 -122 546 -107 653 56.7 -11.1 0.2 1081 -245 946 -214 1160 47.3 -11.1 0.3 1306 -367 1143 -321 1464 38.3 -15.2 0.4 1397 -489 1222 -428 1650 29.8 -23.1 0.5 1358 -611 1189 -535 1724 20.5 -32.6 1 0.6 1264 -734 1106 -642 1748 14.7 -41.8 0.7 1016 -856 889 -692 1581 9.2 -50.5 0.8 640 -978 560 -791 1351 5.5 -58.5 0.9 246 -1101 216 -890 1105 2.3 -66.0 1.0 257 -1223 224 -989 1213 2.3 -71.5 0.0 257 -1223 194 -989 1183 74.8 -6.8 0.1 303 -596 229 -482 712 68.3 -6.8 0.2 843 -509 639 -385 1024 59.9 -6.8 0.3 1338 -421 1014 -319 1333 50.6 -12.8 0.4 1638 -333 1241 -252 1494 40.9 -21.1 0.5 1715 -302 1299 -229 1528 30.0 -30.2 2 0.6 1656 -413 1255 -313 1568 21.9 -39.9 0.7 1375 -525 1042 -398 1440 13.4 -49.6 0.8 891 -637 675 -482 1157 6.9 -59.0 0.9 323 -748 245 -593 838 5.3 -67.7 1.0 193 -1139 147 -903 1049 5.3 -74.4 0.0 193 -1139 162 -903 1065 73.4 -1.6 0.1 267 -1025 223 -813 1036 68.0 -2.2 0.2 719 -911 601 -722 1324 60.7 -5.4 0.3 1156 -797 967 -667 1634 52.8 -9.2 0.4 1456 -683 1218 -572 1790 44.1 -15.4 0.5 1586 -569 1327 -476 1803 33.5 -22.7 3 0.6 1628 -456 1362 -381 1743 25.4 -30.7 0.7 1525 -342 1276 -286 1561 16.1 -39.4 0.8 1215 -228 1017 -191 1207 10.0 -48.6 0.9 728 -114 609 -95 704 9.1 -58.2 1.0 0 0 0 0 0 9.1 -66.6

Chapter 9 - Steel Plate Girders

Fatigue Shear Factored (LL+IM)HL-93 (Interior Girder) +V -V V_sr +V_u -V_u (kip) (kip) (kip) (kip) (kip) 79.2 -13.6 92.8 173.7 80.9 69.4 -13.6 83.1 135.8 52.8 57.9 -13.6 71.5 96.3 24.8 46.9 -18.6 65.4 57.2 -8.3 36.4 -28.3 64.7 18.6 -46.1 25.1 -40.0 65.1 -20.9 -86.0 18.0 -51.2 69.1 -56.1 -125.3 11.3 -61.8 73.1 -90.9 -164.0 6.7 -71.7 78.4 -123.6 -202.0 2.9 -80.8 83.7 -155.5 -239.2 2.9 -87.5 90.4 -183.6 -274.0 91.6 -8.3 99.9 298.7 198.8 83.7 -8.3 92.0 248.7 156.7 73.3 -8.3 81.6 196.2 114.6 62.0 -15.7 77.7 142.7 65.0 50.1 -25.8 75.9 88.7 12.8 36.8 -37.0 73.8 33.3 -40.5 26.8 -48.9 75.7 -18.9 -94.6 16.5 -60.8 77.2 -71.3 -148.6 8.4 -72.3 80.7 -121.6 -202.3 6.5 -82.9 89.4 -165.7 -255.1 6.5 -91.1 97.6 -207.8 -305.4 89.9 -1.9 91.8 294.8 203.0 83.3 -2.7 86.0 256.2 170.2 74.4 -6.6 81.0 215.4 134.4 64.6 -11.3 75.9 173.6 97.7 54.1 -18.9 72.9 131.3 58.3 41.1 -27.8 68.9 86.3 17.4 31.1 -37.7 68.8 44.4 -24.4 19.8 -48.3 68.0 1.1 -67.0 12.3 -59.5 71.8 -38.2 -110.0 11.2 -71.3 82.5 -71.3 -153.8 11.2 -81.5 92.7 -103.2 -195.9

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-12 Fatigue II Limit State - Moment and Shears for Interior Girder

Span Point x/L

1

2

3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fatigue Moment (LL+IM)P9 Factored (LL+IM)P9 (One Lane) (Interior Girder) +M -M +M -M M_sr (kip-ft) (kip-ft) (kip-ft) (kip-ft) (kip-ft) 0 0 0 0 0 1686 -390 843 -195 1038 2749 -779 1375 -390 1764 3579 -1169 1789 -584 2374 3904 -1558 1952 -779 2731 3885 -1948 1942 -974 2916 3476 -2337 1738 -1169 2907 2725 -2727 1363 -1260 2622 1542 -3116 771 -1440 2211 714 -3506 357 -1620 1977 793 -3895 397 -1800 2196 793 -3895 344 -1800 2143 531 -1796 230 -830 1060 2188 -1532 948 -663 1611 3793 -1267 1642 -549 2191 4795 -1003 2076 -434 2511 5144 -933 2228 -404 2632 4857 -1279 2103 -554 2657 3909 -1624 1693 -703 2396 2339 -1970 1013 -853 1866 599 -2315 259 -1049 1308 582 -3626 252 -1643 1895 582 -3626 278 -1643 1921 524 -3264 250 -1478 1729 1792 -2901 857 -1314 2171 3189 -2538 1524 -1213 2738 4119 -2176 1969 -1040 3009 4639 -1813 2217 -867 3084 4689 -1450 2242 -693 2935 4305 -1088 2058 -520 2578 3330 -725 1592 -347 1938 2023 -363 967 -173 1140 0 0 0 0 0

Chapter 9 - Steel Plate Girders

Fatigue Shear (LL+IM)P9 Factored (LL+IM)P9 (One Lane) (Interior Girder) +V -V +V -V V_sr (kip) (kip) (kip) (kip) (kip) 178.6 -35.4 125.0 -24.8 149.8 153.2 -35.4 107.3 -24.8 132.1 123.7 -35.4 86.6 -24.8 111.4 96.0 -35.4 67.2 -24.8 92.0 71.2 -52.6 49.9 -36.8 86.7 46.0 -74.7 32.2 -52.3 84.5 32.8 -102.9 23.0 -72.0 95.0 19.3 -131.6 13.5 -92.1 105.6 9.9 -160.8 6.9 -112.5 119.5 7.2 -188.1 5.0 -131.7 136.7 7.2 -208.9 5.0 -146.2 151.3 229.2 -20.9 160.4 -14.7 175.1 205.0 -20.9 143.5 -14.7 158.2 174.9 -20.9 122.4 -14.7 137.1 143.1 -28.7 100.2 -20.1 120.3 111.1 -50.4 77.8 -35.3 113.1 76.7 -77.5 53.7 -54.3 108.0 52.5 -108.0 36.7 -75.6 112.3 30.1 -139.9 21.1 -97.9 119.0 16.0 -171.8 11.2 -120.3 131.5 16.0 -202.4 11.2 -141.7 152.9 16.0 -227.2 11.2 -159.0 170.2 220.3 -4.7 154.2 -3.3 157.5 199.7 -4.7 139.8 -3.3 143.1 173.0 -10.2 121.1 -7.2 128.3 144.2 -20.0 100.9 -14.0 115.0 113.6 -34.5 79.5 -24.2 103.7 80.0 -53.5 56.0 -37.4 93.4 57.7 -76.3 40.4 -53.4 93.8 35.4 -102.8 24.8 -71.9 96.8 29.0 -131.4 20.3 -92.0 112.3 29.0 -161.9 20.3 -113.3 133.6 29.0 -188.6 20.3 -132.0 152.4

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.9

Calculate Factored Moments – Service Limit State II Using live load distribution factors in Table 9.7-6, factored moments for an interior girder at the Service Limit State II are calculated and listed in Table 9.7-13. Service II: 1.0(DC) + 1.0(DW) + 1.30(DF)(LL+IM)HL-93 Table 9.7-13 Factored Moments for Interior Girder – Service Limit State II

DC1 Span Point M_DC1 (kip-ft) x/L 0.0 0 0.1 693 0.2 1144 0.3 1353 0.4 1329 0.5 1045 1 0.6 528 0.7 -231 0.8 -1232 0.9 -2474 1.0 -3959 0.0 -3959 0.1 -1555 0.2 304 0.3 1619 0.4 2389 0.5 2615 2 0.6 2297 0.7 1434 0.8 26 0.9 -1926 1.0 -4422 0.0 -4422 0.1 -2574 0.2 -1038 0.3 186 0.4 1097 0.5 1695 3 0.6 1981 0.7 1955 0.8 1616 0.9 964 1.0 0

Chapter 9 - Steel Plate Girders

Dead Load DC2 M_DC2 (kip-ft) 0 57 94 111 108 86 43 -19 -101 -203 -325 -325 -128 25 133 196 215 188 118 2 -158 -363 -363 -211 -85 15 90 139 162 160 133 79 0

DW M_DW (kip-ft) 0 135 223 264 258 204 103 -45 -240 -483 -772 -772 -303 59 316 466 510 448 280 5 -376 -862 -862 -502 -202 36 214 331 386 381 315 188 0

Live Load (LL+IM)HL-93 +M -M (kip-ft) (kip-ft) 0 0 1337 -291 2293 -583 2884 -874 3174 -1165 3161 -1457 2890 -1749 2334 -2314 1541 -2644 684 -3075 572 -3768 512 -3768 668 -1982 1659 -1006 2706 -877 3386 -786 3616 -761 3426 -927 2784 -1094 1740 -1277 673 -2166 377 -3863 407 -3863 648 -2887 1601 -2425 2536 -1834 3199 -1571 3543 -1309 3579 -1047 3268 -786 2602 -524 1518 -262 0 0

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.10

Design Composite Section in Positive Moment Region at 0.5 Point of Span 2 For midspan sections, design is normally governed by the bending moments. In following, only flexural design for 0.5 Point Section is illustrated. A similar shear design procedure is shown in Section 9.7.11.

9.7.10.1

Illustrate Calculations of Factored Moments – Strength Limit States Factored force effects are calculated and summarized in Section 9.7.7. Table 9.714 illustrates detailed calculations for factored moments at 0.5 Point of Span 2. Table 9.7-14 Factored Moments at 0.5 Point of Span 2 Load Type DC1

Unfactored Moment (kip-ft) 2,615

DC2

215

DW

510

(LL+IM)HL-93

3,455 (one lane)

(LL+IM)P15

6,897 (one lane)

Controlling DC+DW+(LL+I M) Strength I: Strength II:

9.7.10.2

Factored Moment (kip-ft) MDC1 = 1.25(2,615) = 3,269 (applied to steel section alone) MDC2 = 1.25(215) =269 (applied to long-term composite section 3n = 24) MDW = 1.5(510) = 765 (applied to long-term composite section 3n = 24) M(LL+IM)HL-93 = 1.75(0.805)(3,455) = 4,867 (applied to short-term composite section n = 8) M(LL+IM)P15 = 1.35(0.805)(6,897) = 7,495 (applied to short-term composite section n = 8) Mu = 3,269+ 269 + 765 + 7,495 = 11,798

1.25(DC) + 1.5(DW) + 1.75(DF)(LL+IM)HL-93 1.25(DC) + 1.5(DW) + 1.35(DF)(LL+IM)P15

Calculate Elastic Section Properties Determine Effective Flange Width According to the CA 4.6.2.6, the effective flange width is dependent on the girder spacing to span length ratio (S/L). In this design example, S = 12 ft and L = 165 ft. For the interior girder in Span 2, the effective flange width is as:

 S / L 12 / 165  0.073  0.32 

beff  b  12 ft  144 in.

Chapter 9 - Steel Plate Girders

(CA 4.6.2.6.1-2)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Calculate Elastic Section Properties Elastic section properties for the steel section alone, the steel section and deck slab longitudinal reinforcement, the short-term composite section (n = 8), and the long-term composite section (3n = 24) are calculated and shown from Table 9.7-15 to 9.7-18. Table 9.7-15 Properties of Steel Section Alone Ai

yi

Aiyi

yi - yNCb

Ai(yi – yNCb)2

Io

(in.2) 18.00 48.75 31.50

(in.) 80.25 40.75 0.875

(in.3) 1,444.5 1,986.6 27.6

(in.) 45.05 5.55 -34.325

(in.4) 36,531 1,502 37,113

(in.4) 1.5 24,716 8.04

98.25

-

3,458.7

-

75,146

24,726

Component Top flange 18  1 Web 78  0.625 Bottom flange 18  1.75 

y NCb 

yNCt

yt C.GNC

y NCt 1.75  78 1  35.2  45.55 in. I NC 

yNCb

yw yb

 Io

  Ai  yi  yNCb 

2

 24,726  75,146  99,872 in.4 S NCb 

S NCt 

Chapter 9 - Steel Plate Girders

 Ai yi 3,458.7   35.2 in. 98.25  Ai

I NC 99,872   2,837 in.3 y NCb 35.2 I NC 99,872   2,193 in.3 y NCt 45.55

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Properties of the steel section alone may be conservatively used for calculating stresses under negative moments. In this example, properties of the steel section and deck slab longitudinal reinforcement are used for calculating stresses due to negative moments (AASHTO 6.10.1.1.1c). Assume the total area of longitudinal reinforcement in the deck slab is 1% of concrete deck slab area, we have As as follows: As  0.0112  12 9.125   13 .14 in. 2

Table 9.7-16 Properties of Steel Section and Deck Slab Reinforcement Ai

yi

Aiyi

yi - yNSb

Ai  y i  y NSb 2

Io

(in.2) 13.14

(in.) 88.44

(in.3) 1162.1

(in.) 46.96

(in.4) 28,977

(in.4) 0

98.25 111.39

35.2 -

3,458.4 4,620.5

-6.28 -

3,875 32,852

99,872 99,872

Component Top Reinforcement Steel section 

 

 Ai yi 4,620.5   41.48 in.  Ai 111.39 y NSt 1  78  1.75  41.48  39.27 in.

y NSb 

yNSrb

y NSrb 88.44  41.48  46.96 in.

yNSt yrb

C.GNS C.GNC yNSb

Chapter 9 - Steel Plate Girders

yNCb

I NS   I o   Ai  yi  y NSb 2  99,872  32,852  132,724 in.4 I 132,724 S NSb  NS   3, 200 in.3 41.48 yNSb I 132,724 S NSt  NS   3,380 in.3 39.27 yNSt I 132,724 S NSrb  NS   2,826 in.3 46.96 yNSt

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-17 Properties of Short-term Composite Section (n = 8) Ai

yi

Aiyi

yi - ySTb

Ai  y i  y STb 2

(in.2)

(in.)

(in.3)

(in.)

98.25

35.2

3,458.4

-33.31

(in.4) 109,014

99,872

164.25 262.5

88.44 -

14,526.3 17,984.7

19.93 -

65,241 174,255

1,140 101,012

Component Steel section Concrete Slab 144/8  9.125 

y STb 

ySTb

 Ai



(in.4)

17 ,984.7  68.51 in. 262.5

ySTt  1.75  78  1  68.51 12.24 in.

yc

ySTt C.GST

 Ai y i

Io

I ST   I o   Ai  y i  y STb 

2

101,012  174 ,255  275,267 in.4

S STb 

I ST 275 ,267   4 ,018 in.3 y STb 68.51

S STt 

I ST 275,267   22,489 in.3 y STt 12.24

C.GNC yNCb

Table 9.7-18 Properties of Long-term Composite Section (3n = 24) Ai Component Steel section Concrete Slab 144/24  9.125 

2

yi

Aiyi

yi - yLTb

3

Ai ( y i  y LTb ) 2

Io

(in. ) 98.25

(in.) 35.2

(in. ) 3,458.4

(in.) -19.05

(in. ) 35,655

(in.4) 99,872

54.75 153

88.44 -

4,842.1 8,300.5

34.19 -

64,000 99,655

380 100,252

4

 Ai yi 8,300.5   54.25 in. 153  Ai y LTt 1.75  78 1  54.25  26.5in.

y LTb 

yLTt

yLTb

yc C.GLT C.GNC yNCb

I LT   I o   Ai  yi  y LTb 2 100,252  99,655 199,907 in.4 I 199,907 S LTb  LT   3,685 in.3 yLTb 54.25 199,907 I  7,544 in.3 S LTt  LT  26.5 yLTt

It should be pointed out that the concrete haunch is ignored in calculating composite section properties.

Chapter 9 - Steel Plate Girders

9-34

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.10.3

Design for Flexure – Strength Limit State General Requirement

At the strength limit state, the composite compact section in positive moment regions shall satisfy the requirement as follows: 1 M u  fl S xt   f M n (AASHTO 6.10.7.1.1-1) 3 In this example of the straight bridge, flange lateral bending stress for interior girders fl = 0. The design equation, therefore, is simplified as follows:

Mu   f Mn Check Section Compactness

For composite sections in the positive moment region, it is usually assumed that the top flange is adequately braced by the hardened concrete deck. There is no requirement for the compression flange slenderness and bracing for compact composite sections at the strength limit state. Three requirements (AASHTO 6.10.6.2.2) for a compact composite section in straight bridges are checked as follows: Specified minimum yield strength of flanges:

Web: Section:

Fyf  50 ksi  70 ksi

O.K.

(AASHTO 6.10.6.2.2)

D  124.8  150 tw

O.K.

(AASHTO 6.10.2.1.1-1)

2 Dcp tw

 3.76

E Fyc

(AASHTO 6.10.6.2.2-1)

where Dcp is depth of the web in compression at the plastic moment state and is determined in the following. Compressive force in concrete slab:

Ps  0.85 fc beff ts  0.85 3.6144 9.125  4,021 kips in which ts is thickness of concrete slab Yield force in the top compression flange:

Pc  A fc Fyc  18 150  900 kips Yield force in the web:

Pw  Aw Fyw  78  0.62550  2,438 kips

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Yield force in the bottom tension flange:

Pt  Aft Fyt  18×1.75 50  1,575 kips  Ps  Pc  4,021 + 900  4,921 kips  Pw  Pt  2,438 +1,575  4,013 kips

 Plastic neutral axis is within the top compression flange and Dcp is equal to zero. 2Dcp

 0.0  3.76

tw

E Fyc

O.K.

(AASHTO 6.10.6.2.2-1)

The nominal flexural resistance, Mn, of the composite compact section is, therefore, computed in accordance with AASHTO 6.10.7.1.2. Calculate Plastic Moment Mp

At the plastic moment state, the compressive stress in the concrete slab of a composite section is assumed equal to 0.85 f c , and tensile stress in the concrete slab is neglected. The stress in reinforcement and steel girder section is assumed equal to Fy. The reinforcement in the concrete slab is neglected in this example. The plastic moment Mp is determined using equilibrium equations and is the first moment of all forces about the plastic neutral axis (AASHTO D6.1). Determine Location of Plastic Neutral Axis (PNA) As calculated above, the plastic neutral axis (PNA) is within the top flange of steel girder. Denote that y is the distance from the top of the compression flange to the PNA as shown in Figure 9.7-7, we obtain:

Ps  Pc1  Pc2  Pw  Pt where Pc1  y b fc Fyc Pc2 

t

fc



 y bfc Fyc

in which bfc and tfc are width and thickness of top flange of steel section, respectively; Fyc is yield strength of top compression flange of steel section. Substituting above expressions into the equilibrium equation for y , obtain t fc  Pw  Pt  Ps   1  2 Pc  1  2,438 + 1,575 - 4,021  + 1  0.496 in. < t fc = 1.0 in. y   2 900 

y 

Chapter 9 - Steel Plate Girders

O.K.

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Figure 9.7-7 Plastic Moment Capacity State

Calculate Plastic Moment M p Summing all forces about the PNA, obtain:  tcf  y   y  P d  P d M p  M PNA  Ps d s  Pc1   Pc 2    w w t t 2 2      y2  t  y2 cf    Ps d s b fc Fyc    Pw d w  Pt d t 2    where 9.125 ds   4.125  1  0.496  8.18 in. 2 78 dw  + 1 - 0.496 = 39.50 in. 2 1.75 dt  + 78 + 1 - 0.496 = 79.38 in. 2

 

Mp



0.496  4,0218.18 1850 

2

  2,43839.5  1,57579.38



 1 0.496 2   2 

 254,441 kip - in. = 21,203 kip - ft

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Calculate Yield Moment My

The yield moment M y corresponds to the first yielding of either steel flange. It is obtained by the following formula (AASHTO D6.2):

M y  M D1  M D2  M AD

(AASHTO D6.2.2-2)

From Section 9.7.10.1, factored moments, MD1 and MD2 are as follows:

M D1  M DC1  3,269 kip - ft M D 2  M DC 2  M DW  269  765  1,034 kip - ft Using section moduli SNC, SST and SLT as shown in Section 9.7.10.2 for the noncomposite steel, the short-term and the long-term composite section, we have:

 M M M AD  S ST  Fy  D1  D 2 S NC S LT 

  

For the top flange:

3,26912  1,03412    M AD  22,489  50    2,193 7,544    685,182 kip - in. = 57,099 kip - ft For the bottom flange: 3,26912  1,03412     M AD  4,018  50   2,837 3,685  

(Control)

 131,813 kip - in. = 10,984 kip - ft

 M y  3,269  1,034  10,984  15,287 kip - ft Calculate Flexural Resistance

In this example, it is assumed that the adjacent interior-bent sections are noncompact non-composite sections that do not satisfy requirements of AASHTO B6.2. The nominal flexural resistance of the composite compact section in positive flexure is calculated in accordance with AASHTO and CA 6.10.7.1.2:  M p    My  M n  min  M p 1  1   Mp     1.3 Rh M y

for D p  0.1 Dt   D p / Dt  0.1      for D p  0.1 Dt M p  0.32    for a continous span

(AASHTO and CA 6.10.7.1.2-1,2, 3) where Rh is hybrid factor (AASHTO 6.10.1.10.1) and equal to 1.0 for this example; Dp is the depth from the top of the concrete deck to the PNA; Dt is total depth of the composite section.

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The compact and noncompact sections shall satisfy the following ductility requirement to ensure that the tension flange of the steel section reaches significant yielding before the crushing strain is reached at the top of concrete deck.

Dp  0.42 Dt

(AASHTO 6.10.7.3-1)

Dp  13.25 - 1 + 0.496 = 12.75 in.

Dt  1.75 + 78 +13.25 = 93 in.

Dp  12.75 in.  0.42Dt  0.42  93 = 39.06 in.

O.K.

Dp  12.75 in.  0.1Dt  9.3 in.   M y  D p / Dt  0.1    M p M n  1  1    0.32   M p     15,287  12.75 / 93  0.1   1  1   21,203  20,517 kip - ft  0.32    21,203   1.3Rh M y  1.31.015,287   19,873 kip - ft Use M n = 19,873 kip-ft Check Design Requirement

M u  11,798 kip - ft   f M n  1.019,873  19,873 kip - ft

OK.

(AASHTO 6.10.7.1.1-1) 9.7.10.4

Illustrate Calculations of Fatigue Moment Ranges

Fatigue moment ranges are calculated and summarized in Section 9.7.8. For 0.5 Point of Span 2, using live load distribution factor DFm = 0.433 (Table 9.7.7), fatigue moment ranges are as follows: Fatigue I:

M    DFm  LL  IM  HL 1.75 0.433 LL  IM  HL93 +M = (1.75)(0.433)(1,715) = 1,299 kip-ft -M = (1.75)(0.433)(-302) = -229 kip-ft Fatigue II:

M    DFm  LL  IM  P9 1.0 0.433 LL  IM  P9 +M = (1.0)(0.433)(5,144) = 2,227 kip-ft -M = (1.0)(0.433)(-933) = -404 kip-ft

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9.7.10.5

Check Typical Girder Details – Fatigue Limit States

For load-induced fatigue consideration, the most common types of details in a typical plate girder are (AASHTO Table 6.6.1.2.3-1) listed in Table 6.8-1 and nominal fatigue resistance for those typical details are shown in Table 6.8-2 in Chapter 6. For a section in the positive moment region within mid-span, such as the section at 0.5 Point of Span 2, flexural behavior usually dominates the design. Positive live load moments are applied to the short-term composite section and negative live load moments are applied to the steel section and deck slab longitudinal reinforcement (AASHTO 6.10.1.1.1c). Fatigue stress ranges at the bottom flanges and the top flanges are checked as follows: Fatigue I - HL-93 Truck for Infinite Life:

Flexural fatigue stress ranges at the bottom flange:

  f  

1,299 12  229 12  M M   + S STb S NCb 4,018 3,200

 12.0 ksi O.K. for Category C  16.0 ksi O.K. for Category B Flexural fatigue stress ranges at the top flange:  3.88 + 0.86 = 4.74 ksi

  f  

M S STt



1,299 12  229 12  M  + S NCt 22,489 3,380

 10.0 ksi O.K. for Category C  0.69 + 0.81 = 1.50 ksi  12.0 ksi O.K. for Category C  16.0 ksi O.K. for Category B Fatigue II - P-9 Truck for Finite Life:

Flexural fatigue stress ranges at the bottom flange:

  f  

M S STb



2,227 12  404 12  M  + S NCb 4,018 3,200

 6.65 + 1.52 = 8.17 ksi

 21.58 ksi O.K. for Category C  30.15 ksi O.K. for Category B

Flexural fatigue stress ranges at the top flange: Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

  f  

M S STt



2,227 12  404 12  M  + S NCt 22,489 3,380

 21.58 ksi O.K. for Category C  1.19 + 1.43 = 2.62 ksi  21.58 ksi O.K. for Category C  30.15 ksi O.K. for Category B

9.7.10.6

Check Requirements - Service Limit State General Requirements

Service Limit State II is to control the elastic and permanent deflections under the design live load HL-93 (AASHTO 6.10.4). Live load deflection  may not exceed L/800 (AASHTO 2.5.2.6.2) and is calculated and checked in Section 9.7.16. Illustrate Calculations of Factored Moments - Service Limit State II

It is noted that for unshored construction, DC1, DC2+DW, and live load are applied to the non-composite (steel section alone), long-term and short-term composite sections, respectively. Factored moments at the Service Limit State II are calculated and summarized in Table 9.7.13. The calculation of factored moments for 0.5 Point of Span 2 are illustrated as follows: MDC1 = 2,615 kip-ft

(applied to steel section alone)

MDC2 + MDW = 215 + 510 = 725 kip-ft

(applied to long-term composite section)

M(LL+IM)HL-93 = (1.3)(0.805)(3,455) = 3,616 kip-ft

(applied to short-term composite section)

Check Flange Stresses

In this example, fl = 0 for this interior girder. The requirement becomes:

ff 

M ( LL  IM ) HL 93 M DC 1 M DC 2  M DW    0.95Rh Fyf S NC S LT S ST

 0.95 1.0  50   47.5 ksi



For the top flange

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

ff 



 261512



 72512



 3,61612

2,193 7,544 22,489  14.31  1.15  1.93  17.39 ksi  47.5 ksi O.K. (AASHTO 6.10.4.2.2-1) For the bottom flange ff 

 261512



 72512



 3,61612

2,837 3,685 4,018  11.06  2.36 10.80  24.22 ksi  47.5 ksi O.K. (AASHTO 6.10.4.2.2-2)



For the compression flange AASHTO 6.10.4.2.2 states that for composite sections in positive flexure in which the web satisfies the requirement of AASHTO 6.10.2.1.1, i.e., D/tw  150, satisfying AASHTO 6.10.4.2.2-4 is not required. In this example, D 78  =124.8 < 150 tw 0.625

(AASHTO 6.10.2.1.1-1)

 The compression flange check is not required. 9.7.10.7

Check Requirements - Constructibility General Requirements

At construction stages, steel girders of Span 2 with an unbraced compression flange length Lb = 330 in. carry out the construction load including dead load (selfweight of steel girders and concrete deck slab) and other loads acting on the structure during construction. To prevent nominal yielding or reliance on post-buckling resistance of the steel girder during critical stages of construction, the following AASHTO 6.10.3 requirements for flexural stresses are checked. For 0.5 Point Section, shear effects are very small and shear strength check is not illustrated. A similar design procedure is shown in Section 9.7.11. Calculate Factored Moment – Constructibility

In the constructibility check, all loads shall be factored as specified in AASHTO 3.4.2. In this example, no other construction load is assumed and only factored dead loads are applied on the noncomposite section. The compression flange is discretely braced with an unbraced length Lb = 330 in. within Span 2. The factored moment at 0.5 Point of Span 2 is: Mu = MDC1 = 1.25(2,615) = 3,269 kip-ft

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Check Compression Flange



Web Compactness Limiting slenderness ratio for a noncompact web:

rw  5.7

E 29,000  5.7  137.3 50 Fyc

(AASHTO 6.10.1.10.2-4)

Dc = yNCt - tfc = 45.55 - 1.0 = 44.55 

(See Table 9.7-15)

2Dc 2  44.55    142.6  rw  137.3 tw  0.625

The web is slender, and AASHTO Equations 6.10.3.2.1-2 and 6.10.3.2.1-3 shall be checked. 

Calculate Flange-Strength Reduction Factors Rh and Rb Since homogenous plate girder sections are used for this example, hybrid factor Rh is taken as 1.0 (AASHTO 6.10.1.10.1). When checking constructibility according to AASHTO 6.10.3.2, web loadshedding factor Rb is taken as 1.0 (AASHTO 6.10.1.10.2).



Calculate Flexural Resistance Nominal flexural resistance of the compression flange is the smaller of the local buckling resistance (AASHTO 6.10.8.2.2) and the lateral torsional buckling resistance (AASHTO 6.10.8.2.3) Local buckling resistance  f 

b fc 2t fc



18 =9 2 1

  pf  0.38

Fnc FLB   Rb Rh Fyc  1.01.0 50  50 ksi

E 29,000  0.38  9.15 50 Fyc

(AASHTO 6.10.8.2.2-1)

Lateral torsional buckling resistance Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

rt 

b fc  1 Dc tw  12 1  3 b fct fc   



18  1  44.55 0.625  12 1   3 181.0  

 4.22 in.

(AASHTO 6.10.8.2.3-9) E Fyc

L p  1.0 rt

 1.0  4.22 

29, 000  101.6 in. 50

0.7 Fyc   0.7  50  Fyr  smaller    35 ksi  0.5Fyc  25 ksi  Fyw  50  Use Fyr = 35 ksi Lr   rt

E Fyr

   4.22 

29, 000  381.6 in. 35

(AASHTO 6.10.8.2.3-5)

 Lp  101.6 in.  Lb  330 in.  Lr  381.6 in.   Fyr  Lb  L p    Rb Rh Fyc Fnc  LTB   Cb 1  1    Rh Fyc  Lr  L p    35  330 101.6   1.0 1.050   1.0 1  1      1 . 0 50 381 . 6 101 . 6       37.8 ksi  Rb Rh Fyc  1.0 1.0 50  50 ksi (AASHTO 6.10.8.2.3-2) Use Fnc(LTB) = 37.8 ksi It should be pointed out that Cb factor is taken as 1.0 conservatively for 0.5 Point of Span 2. The nominal flexural resistance of the compression flange is:





Fnc  min Fnc FLB , Fnc LTB  min 50, 37.8  37.8 ksi f bu 

Mu 3,26912    17.9 ksi S NCt 2,193

O.K.

(AASHTO 6.10.3.2.1-2)

  f Fnc  37.8 ksi

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Calculate Web Bend-buckling Resistance 2

2 D  78  k  9   =9   = 27.59  44.55   Dc  0.9 E k 0.9  29,000  27.59  Fcrw    46.2 ksi 2 2 D  78       0.625   tw 

(AASHTO 6.10.1.9.1-2)

 Rh Fyc  1.0  50   50 ksi   smaller    50 ksi  Fyw / 0.7  50 / 0.7  71.4 ksi 

(AASHTO 6.10.1.9.1-1) Use Fcrw = 46.2 ksi

fbu  17.9 ksi   f Fcrw  46.2 ksi

O.K.

(AASHTO 6.10.3.2.1-3)

Check Tension Flange

fbu 

9.7.11

M u 3,269 12   13.8 ksi   f Rh Fyt  50 ksi 2,837 S NCb O.K. (AASHTO 6.10.3.2.2-1)

Design Noncomposite Section in Negative Moment Region at Bent 3 In this example, steel girder sections in negative moment regions are designed as noncomposite sections. When shear connectors are provided in negative moment regions according to AASHTO 6.10.10, sections are considered as composite sections.

9.7.11.1

Illustrate Calculations of Factored Moments and Shears – Strength Limit States

Factored moments and shears are calculated and summarized in Section 9.7.7. Tables 9.7-19 and 9.7-20 illustrate detailed calculations for the section at Bent 3.

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Table 9.7-19 Factored Moments at Section of Bent 3 Load Type

DC1

Unfactored Moment (kip-ft) -4,422

DC2

-363

MDCs = 1.25(-363) = -453.8

DW

-862

MDW = 1.5(-862) = -1,293.0

(LL+IM)HL-93

-3,563 (one lane)

M(LL+IM)HL-93 =1.75(0.834)(-3,563) = -5,200.2

(LL+IM)P15

-5,981 (one lane)

M(LL+IM)P15 = 1.35(0.834)(-5,981) = -6,734.0

MDC1 = 1.25(-4,422) = -5,527.5

Mu = -5,527.5 +(- 453.8)+(-1,293.0)+( -6,734.0) = -14,008

Controlling DC+DW+(LL+IM) Strength I: Strength II:

Factored Moment (kip-ft)

1.25(DC) + 1.5(DW) + 1.75(DF)(LL+IM)HL-93 1.25(DC) + 1.5(DW) + 1.35(DF)(LL+IM)P15

Table 9.7-20 Factored Shears at Section of Bent 3 Load Type DC1

Unfactored Shear (kip) -167.8

DC2

-13.8

VDC2 = 1.25(-13.8) = -17.25

DW

-32.7

VDW = 1.5(-32.7) = -49.05

(LL+IM)HL-93

-147.2 (one lane)

V(LL+IM)HL-93 = 1.75(1.082)(-147.2) = -278.72

(LL+IM)P15

-321.0 (one lane)

V(LL+IM)P15 = 1.35(1.082)(-321.0) = -468.89

Controlling DC+DW+(LL+IM)

9.7.11.2

Factored Shear (kip) VDC1 = 1.25(-167.8) = -209.75

Vu = -209.75 +(-17.25)+( - 49.05)+( - 468.89) = -745.0

Calculate Elastic Section Properties

In order to calculate the stresses, deflection, and camber for this continuous composite girder, the elastic section properties for the steel section alone, the steel section and the deck slab longitudinal reinforcement, the short-term composite section, and the long-term composite section are calculated in Tables 9.7-21, 9.7-22, 9.7-23 and 9.7-24, respectively.

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Table 9.7-21 Properties of Steel Section Alone Ai

yi

Aiyi

yi - yNCb

Ai  y i  y NCb 

(in.2) 36.0 48.75 36.0

(in.) 81.0 41.0 1.0

(in.3) 2,916.0 1,998.75 36.0

(in.) 40.0 0 -40.0

(in.4) 57,600 0 57,600

(in.4) 12 24,716 12

120.75

-

4,950.75

-

115,200

24,740

Component Top flange 18  2 Web 78  0.625 Bottom flange 18  2 

Io

2

 Ai yi  4,950.75  41.0 120.75  Ai yNCt   2 +78 +2 - 41.0 = 41.0

y NCb 

yNCt

yt C.GNC

I NC 

yw

yNCb yb

 Io   Ai  yi  yNCb 

in.

in.

2

 24,740 + 115,200 = 139,940 in.4 I 139,940 S NCb  NC  = 3,413 in.3 yNCb 41.0 S NCt 

I NC 139,940  = 3,413 in.3 yNCt 41.0

Assuming the total area of longitudinal reinforcement is 1% of the concrete area at the interior supports, As = 0.01(12  12)(9.125) = 13.14 in.2, elastic section properties of the steel section and deck slab longitudinal reinforcement are calculated in Table 9.7-22 as follows.

Chapter 9 - Steel Plate Girders

9-47

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-22 Properties of Steel Section and Deck Slab Reinforcement Ai

yi

Aiyi

yi - yNSb

Ai  yi  y NSb 2

Io

(in.2)

(in.)

(in.3)

(in.)

(in.4)

(in.4)

13.14 120.75 133.89

88.69 41.0 -

1,165.4 4,950.8 6,116.15

43.01 -4.68 -

Component Top reinforcement Steel section   

24,307 0 2,645 139,940 26,952 139,940  Ai yi 6,116.15 y NSb    45.68 in. 133.89  Ai

y NSt  2  78  2  45.68  36.32 in. y NSrb 88.69  45.68  43.01 in.

yNSrb yNSt C.GNS

I NS   I o   Ai  yi  y NSb 2

yrb

139,940  26,952  166,892 in.4

I NS 166,892   3,654 in.3 y NSb 45.68 I 166,892 S NSt  NS   4,595 in.3 36.32 y NSt

S NSb 

C.GNC yNSb

yNCb

S NSrb 

I NS 166,892   3,880 in.3 y NSrb 43.01

Table 9.7-23 Properties of Short-term Composite Section (n = 8) Ai

yi

Aiyi

yi - ySTb

Ai  yi  y STb 2

(in.2)

(in.)

(in.3)

(in.)

(in.4)

(in.4)

120.75

41.0

4,950.8

-27.48

91,184

139,940

164.25 285.0

88.69 -

14,567.3 19,518.1

20.21 -

67,087 158,271

1,140 141,080

Component Steel section Concrete slab 144/8  9.125 

19 ,518.1  68.48 in. 285.0  Ai ySTt 2  78 2 68.48 13.52in. y STb 

yc

ySTt

C.GNC

Chapter 9 - Steel Plate Girders



I ST   Io  Ai  yi  ySTb 2

C.GST

ySTb

 Ai y i

Io

yNCb

141,080 158,271  299,351in.4 I 299 ,351 S STb  ST   4 ,371 in.3 y STb 68.48 S STt 

I ST 299,351   22,141 in.3 y STt 13 .52

9-48

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 9.7-24 Properties of Long-term Composite Section (3n = 24) Ai

yi

Aiyi

yi – yLTb

Ai  yi  y LTb 2

Io

(in.2) 120.75

(in.) 41

(in.3) 4,950.8

(in.) -14.88

(in.4) 26,736

(in.4) 139,940

54.75 175.5

88.69 -

4,855.8 9.806.6

32.81 -

Component

Steel section Concrete slab 144/24  9.125 

58,938 380 85,674 140,320  Ai yi 9,806.6 y LTb    55.88 in. 175.5  Ai ySTt  2  78 2  55.88  26.12 in. I ST   I o   Ai  yi  yLTb 2

yLTt

yc

C.GLT yLTb

140,320  85,674  225,994 in.4

I LT 225,994   4,044 in.3 yLTb 55.88 225,994 I  8,652 in.3 S LTt  LT  26.12 y LTt S LTb 

C.GNC yNCb

It should be pointed out that the concrete haunch is ignored in calculating composite section properties. 9.7.11.3

Design for Flexure - Strength Limit States General Requirements

For composite I-sections in negative flexure and non-composite I-sections with compact or non-compact webs in straight bridges, it is strongly recommended to use provisions in AASHTO Appendix A6. In this example of the straight bridge, flange lateral bending stress for interior girders fl = 0. The design equations, therefore, are simplified as follows:

Mu  f Mnc Mu  f Mnt Check Section Compactness

Three requirements for noncompact sections are checked as follows: Specified minimum yield strength of the flanges and web:

Fy  70 ksi

Chapter 9 - Steel Plate Girders

(AASHTO A6.1)

9-49

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

2 Dc 78 E 29, 000   124.8   rw  5.7  5.7  137.3 tw 0.625 Fyc 50

Web:

(AASHTO A6.1-1)

I yc

Flange ratio:

I yt

 218 / 12  2183 / 12 3



 1.0  0.3

(AASHTO A6.1-2)

Since the section at Bent 3 is noncompact, nominal flexural resistance of the Isection is calculated in accordance with AASHTO Appendix A6. It is the smaller of the local buckling resistance (AASHTO A6.3.2) and the lateral torsional buckling resistance (AASHTO A6.3.3). Calculate Flange-Strength Reduction Factors Rh and Rb

Since homogenous plate girder sections are used for this example, hybrid factor Rh is taken as 1.0 (AASHTO 6.10.1.10.1). As shown above, the web is noncompact and web load-shedding factor Rb is taken as 1.0 (AASHTO 6.10.1.10.2). Calculate Flexural Resistance – Based on Compression Flange

Nominal flexural resistance based on the compression flange is the smaller of the local buckling resistance (AASHTO A6.3.2) and the lateral torsional buckling resistance (AASHTO A6.3.3). 

Calculate Local Buckling Resistance f 



b fc 2t fc



18  4 .5 22 

  pf  0.38

M nc FLB  R pc M yc

E 29 ,000  0.38  9.15 F yc 50

(AASHTO A6.3.2-1)

Mp = 2[(18x2)(50)(40)+(39x0.625)(50)(19.5)] = 191,532 kip – in. = 15,961 kip –ft

M yc  S x Fyf  3,41350 170,650 kip - in.  14,221 kip - ft Dcp = Dc = 39 in. (Section is symmetric about neutral axis) 29,000 E rw  5.7  5.7 137.3 50 Fyc

 pw



Dcp





Chapter 9 - Steel Plate Girders

29, 000 50   15,961  0.09   0.54 1.0 14, 221  

2

 Dcp   90.43  rw   137.3  Dc 

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015



2 Dcp tw



 2  39  124.8   0.625



pw Dcp

  90.43

(AASHTO A6.2.1-1)

w  124.8  rw 137.3 and (AASHTO A6.2.2-1) Therefore, for a symmetric section in this example, web is non-compact and web plastification factor is calculated as follows:  D   pw D   pw D  c   90.43   rw 137.3 (AASHTO A6.2.2-6) c  cp   Dcp    Rh M y  w   pw D   M p c   R pc  1  1      My  M     p  rw pw Dc       1.0 14,221  124.8  90.43  15,961   1  1   15,961  137.3  90.43  14,221    1.033

(AASHTO A6.2.2-4)

M nc FLB  R pc M yc  1.03314,221  14,690 kip - ft (AASHTO A6.3.2-1) 

Calculate Lateral Torsional Buckling Resistance In negative moment regions, the bottom compression flange is braced by the cross frame with a spacing of Lb = 330 in. at Span 2 side. h = depth between centerline of flanges = (1.0 + 78 + 1.0) = 80 in. b fc 18 rt    4.7 in.  1 Dctw   1  39  0.625  12 1 12 1   3 b fc t fc   3 18 2.0     L p  1.0 rt

E Fyc

 1.0  4.7 

29,000  113.2 in. 50

Ignoring rebar, from Table 9.7-21 we have Sxc = Sxt = SNCb = SNCt = 3,413 in.3

Fyr

0.7 Fyc   0.7  50   35     smaller  Rh Fyt S xt / S xc  1.0  50 1.0   50  35 ksi  0.5Fyc  25ksi    Fyw  50  (AASHTO A6.3.3)

Chapter 9 - Steel Plate Girders

9-51

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Use Fyr = 35 ksi 3 t fc D tw3 b fc t fc  J    1  0.63  3 3  b fc

 78  0.625

3



3

E Lr  1.95 rt Fyr

  2

 b ft t 3ft   3 

18 2  3

3

 t ft  1 0.63  b ft 

   

2  3 1  0.63   95.6 in. 18  

 Fyr S xc h   1  1  6.76 S xc h  E J  J

29, 000 Lr  1.95  4.7  35

(AASHTO A6.3.3-9)

2

(AASHTO A6.3.3-5)

 35  3, 413 80   95.6 1  1  6.76   29, 000  95.6    3, 413 80   

2

 449.7 in.

(AASHTO A6.3.3-5) It is noted that a conservative Lr can be obtained by AASHTO Equation 6.10.8.2.3-5 as follows: E 29,000 (AASHTO 6.10.8.2.3-5)    4.7   425.0 in. Lr   rt Fyr 35 In this example, Lr = 449.7 in. is used. Since Lb = 330 in. > Lp = 113.2 in., Cb, an equivalent uniform moment factor for lateral torsional buckling which has a minimum value of 1.0 under the uniform moment condition, needs to be calculated. The use of the moment envelope values at both brace points will be conservative for both single and reverse curvature. In this example, the moments M1 and M2 at braced points are estimated from the factored moment envelope shown in Table 9.7-8. At Bent 3, Mu = -14,008 kip-ft and at 0.8 Point with a distance of 33 ft = 396 in. from Bent 3, Mu = -2,751 kip-ft. At braced point with a distance of 27.5 ft. = 330 in. from Bent 3 (Figure 9.7-8), we obtain:

Chapter 9 - Steel Plate Girders

9-52

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Braced Point Bent 3

Braced Point 0.8L

0.9L

-14,008 kip-ft -2,751 kip-ft

-6,631 kip-ft -4,044 kip-ft

66"

132"

198" 396"

Figure 9.7-8 Factored Moment Envelope at Bent 3

 66  M1   2,751    6,631 2,751   4,044 kip - ft  198  2

2

M  M   4,044   4,044  Cb  1.75  1.05  1   0.3 1   1.75 1.05   0.3    14,008   14,008   M2   M2   1.47  2.3 (AASHTO A6.3.3-7) Use Cb = 1.47

 Lp  113.2 in.  Lb  330 in.  Lr  449.7 in.   Fyr S xc  Lb  L p    R pc M yc M nc  LTB   Cb 1  1    Rh M yc  Lr  L p    353,413  330 113.2 1.03314,221  1.47 1  1      1.014,22112  449.7 113.2   17,421 kip - ft  R pc M yc 14,690 kip - ft (AASHTO A6.3.3-2) Use Mnc(LTB) = 14,690 kip-ft 

Determine Flexural Resistance





M nc  min M nc FLB , M nc LTB  min 14,690, 14,690  14,690 kip - ft

Chapter 9 - Steel Plate Girders

9-53

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Calculate Flexural Resistance – Based on Tension Flange

Since the section is symmetric, Rpt = Rpc, Myt = Myc

M nt  R pt M yt  1.03314,221  14,690 kip - ft

(AASHTO A6.4-1)

Check Design Requirement

For both compression and tension flanges:

M u  14,009 kips - ft.   f M nc   f M nt  14,690 kip - ft

O.K.

(AASHTO A6.1.1-1 & A6.1.2-1) 9.7.11.4

Design for Shear – Strength Limit State Select Stiffener Spacing

AASHTO C6.10.2.1.1 states that by limiting the slenderness of transverselystiffened webs to D/tw  150, the maximum transverse stiffener spacing up to 3D is permitted. For end panels adjacent to simple supports, stiffener spacing do shall not exceed 1.5D (AASHTO 6.10.9.3.3). Try interior stiffener spacing do = 165 in. < 3D = 3(78) = 234 in. and end panel stiffener spacing do = 110 in. (for Span 1) and 100 in. (for Span 3) < 1.5D = 1.5(78) = 117 in. Calculate Shear Resistance

Shear resistance for a stiffened interior web is as follows: For do = 165 in. 5 k  5   6.12 165 / 78 2 D  tw

78 Ek  124.8 1.4  1.4  0.625 Fyw

 29,000  6.12 50

 83.41

 Ek  1.57  29,000  6.12         0.358 2F  50 124.82    D / tw   yw  Vp  0.58FywDtw  0.58 50 78 0.625  1,413.8 kips

 C 

1.57

Vcr  CVp  0.3581,413.8  506.1 kips 2 Dtw

 b fct fc  b ft t ft  Chapter 9 - Steel Plate Girders



 2 78 0.625  1.35  18  2  18  2

2.5

9-54

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

    0.871  C  0.8710.358    Vn  Vp C   1,413.8 0.358   843.6 kips   2 2  1 165/ 78  1 d / D    o      Check Design Requirement

Vu  745.1 kips  vVn  1.0 843.6  843.6 kips

O.K

It is noted that for a web end panel adjacent to the simple support, do for the first stiffener adjacent to the simple support shall not exceed 1.5D (AASHTO 6.10.9.3.3). In order to provide an anchor for the tension field in adjacent interior panels, nominal shear strength of a web end panel shall be taken as:

Vn  Vcr  CVp

(AASHTO 6.10.9.3.3-1)

Check Transverse Stiffener

The transverse stiffeners consist of plates welded or bolted to either one or both sides of the web and are required to satisfy the following requirements (AASHTO 6.10.11.1): 

Projecting width D 78 bt  7.5 in.  2.0   2.0   4.6 in. 30 30

O.K. (AASHTO 6.10.11.1.2-1)

16t p  16 0.5  8 in.  bt  7.5 in.  b f / 4 18 / 4  4.5 in.

O.K.

(AASHTO 6.10.11.1.2-2) 

Moment of inertia

For the web panels adjacent to Bent 3, Vu = 745.1 kips > vVcr = (1.0)(506.1) = 506.1 kips, the web tension-field resistance is required in those panels. The moment of inertia of the transverse stiffeners shall satisfy the limit specified in AASHTO 6.10.11.1.3.

It1  b tw3 J

(AASHTO 6.10.11.1.3-3) 1.5

D 4 t1.3  Fyw    It 2  (AASHTO 6.10.11.1.3-4) 40  E  where It is the moment of inertia for the transverse stiffener taken about the edge in contact with the web for single stiffeners and about the mid-thickness of the web for stiffener pairs; b is the smaller of do and D; do is the smaller of the adjacent web panel widths.

Chapter 9 - Steel Plate Girders

9-55

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

J

2 .5

d o / D 2

 2 . 0  0 .5

(AASHTO 6.10.11.1.3-5)

t is the larger of Fyw /Fcrs and 1.0 Fcrs 

0.31E

 bt / t p 

2

 Fys

(AASHTO 6.10.11.1.3-6)

Fys is specified minimum yield strength of the stiffener 2.5 J   2.0   1.441  0.5  Use J  0.5 165 / 782 b = smaller (do = 165 in. and D = 78 in.) = 78 in. 0.31 29, 000   Fcrs   39.96 ksi  Fys  36 ksi  7.5 / 0.5 2

Use Fcrs = 36 ksi

t = larger (Fyw /Fcrs = 50/36 = 1.39; 1.0) = 1.39 It1  btw3 J  780.6253 0.5  9.52 in.4 1.5

It 2

D 4 t1.3  Fyw     40  E 

784 1.391.3   40

1.5

50     29000 

101.6 in.4

2  7.53  0.5  0.625    It  2    7.5  0.5   3.75     158.94  12 2    

in.4

 It 2 101.6 in.4  It1  9.52 in.4  V  V  I t  158.94 in.4  I t1  I t 2  I t1  u v cr    vVn  vVcr   745.1 - 506.1  4  9.52  101.6  9.52   74.73 in.  843.6 - 506.1  O.K. 9.7.11.5

(AASHTO 6.10.11.1.3-9)

Illustrate Calculations of Fatigue Moments and Shears

For bridge details, fatigue moment and shear ranges are calculated and summarized in Section 9.7.8. For the section at Bent 3, live load moments and shears are applied to the steel section only. Fatigue moment and shear ranges are as follows: Fatigue I:

M    DFm   LL  IM  HL 1.75 0.478 LL  IM  HL  0.837 LL  IM  HL

M    DFm   LL  IM  HL 1.75 0.453 LL  IM  HL  0.793  LL  IM  HL V    DFv   LL  IM  HL 1.75 0.7 LL  IM  HL  1.225  LL  IM  HL

Chapter 9 - Steel Plate Girders

9-56

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

+M = (0.837)(193) = 162 kip-ft -M = (0.793)(-1,139) = -903 kip-ft (M) = 162 + 903 = 1065 kip-ft (V) = 1.225(74.4 + 5.3) = 97.6 kips Fatigue II:

M    DFm   LL  IM  P9 1.0 0.478 LL  IM  P9  0.478 LL  IM  P9

M    DFm   LL  IM  P9 1.0 0.453 LL  IM  P9  0.453 LL  IM  P9

V    DFv   LL  IM  P9 1.0 0.7 LL  IM  P9  0.7  LL  IM  P9 +M = (0.478)(582) = 278 kip-ft -M = (0.453)(-3,626) = -1,643 kip-ft (M) = 278 + 1,643 = 1,921 kip-ft (V) = 0.7(16 + 227.2) = 170.2 kips

For special fatigue requirement for the web, factored shear, Vu due to the unfactored dead loads plus the factored fatigue load of Fatigue I for infinite life is calculated as follows: Vu  Vdc1  Vdc 2  Vdw  1.75   DFv   LL  IM  HL

  167.8 13.8  32.7  1.75  0.7  74.4    305.4 kips

9.7.11.6

Check Typical Girder Details and Web - Fatigue Limit States Check Typical Girder Details

From CA Table 6.6.1.2.5-2, the number of stress-range cycles per truck passage for sections near interior support, n = 1.5 for Fatigue I and 1.2 for Fatigue II Limit States. Nominal fatigue resistances are calculated in Table 9.7-25 as follows: Fatigue I: ADTT = 2,500; N = (365)(75)(1.5)(0.8)(2,500) = 0.8213(10)8 > NTH

 Fn    F TH

(AASHTO 6.6.1.2.5-1)

Fatigue II: ADTT = 20, N = (365)(75)(1.2)(0.8)(20) = 525,600 < NTH 1

A 3  Fn     N

Chapter 9 - Steel Plate Girders

(AASHTO 6.6.1.2.5-2)

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Table 9.7-25 Nominal Fatigue Resistance Detail Category

1 3

Fn    A 

(ksi3) B C C E

Fatigue I

Fatigue II

Constant –A ( 108 )

N

120.0 44.0 44.0 11.0

28.37 20.31 20.31 12.79

(ksi)

Fn   F TH (ksi) 16.0 10.0 12.0 4.5

The bending stress ranges for typical girder details Category B (Butt weld for tension flange and bolted gusset plate for lateral bracing) and Category C (Toe of weld for transverse stiffener) are checked as follows: Fatigue I - HL-93 Truck for infinite life:

  f  

  M  S NC



1,065 12  3, 413

 3.74

 12.0 ksi O.K. for Category C  16.0 ksi O.K. for Category B

Fatigue II - P-9 Truck for finite life:   M  1,92112   20.31 ksi O.K. for Category C   f     6.75  28.37 ksi O.K. for Category B S NC 3, 413 It should be pointed out that the above stresses are calculated at the extreme fiber of the top flange for Category B and can be conservatively used for Category C. If the calculation is made at the toe of weld for the transverse stiffeners (Category C), the stress ranges will be smaller than the stress ranges calculated for Category B. AASHTO 6.10.11.1.1 states that the distance between the end of the web-tostiffener weld and the near edge of the adjacent web-to-flange weld or longitudinal stiffener-to-web weld shall not be less than 4tw, but not exceed the lesser of 6tw and 4.0 in. We take this distance = 4tw = 4(0.625) = 2.5 in. and assume web-to-flange weld size of 0.375 inch. The distance between the toe of the weld for the transverse stiffeners to the neutral axis is equal to (41 – 2 – 0.375 – 2.5) = 36.125 in., therefore, stress ranges at the toe of the weld are as follows: Fatigue I - HL-93 Truck for infinite life:

 f  

 M Ctoe I NC

Chapter 9 - Steel Plate Girders



1,065 1236.125  3.30  12.0 ksi O.K. for Category C 139,940

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Fatigue II - P-9 Truck for finite life:

 f  

 M Ctoe I NC



1,9211236.125  5.95  20.31 ksi O.K. for Category C 139,940

Check Special Fatigue Requirement for Web

This requirement is to ensure that significant elastic flexing of the web due to shear is not to occur and the member is able to sustain an infinite number of smaller loadings without fatigue cracking due to the shear.

Vcr  CVp   0.3581,413.8  506.1 kips  Vu  305.4 kips O.K. 9.7.11.7

(AASHTO 6.10.5.3-1)

Design Flange-to-Web Welds

Typical flange-to-web welds shown in Figure 9.7-9 are designed for Strength Limit State. The shear flow at the flange-to-web welds is:

745.118240   7.67 kip/in. Vu Q  I 139,940 Q is the first moment of the steel flange about the neutral axis of the steel girder section. According to AASHTO Table 6.13.3.4-1, the minimum size of fillet weld for plate thickness larger than 3/4 in. is 5/16 in., but need not exceed the thickness of the thinner part jointed. Use two fillet welds tw = 3/8 in. su 

0.625

40

2

18

Neutral Axis

I = 139,940 in.4 Fillet Weld

Figure 9.7-9 Flange-to-Web Welds

Chapter 9 - Steel Plate Girders

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Shear resistance of fillet welds (AASHTO 6.13.3.2.4b) is

Rr  0.6e 2 Fexx

(AASHTO 6.13.3.2.4b-1)

Fexx is classification strength specified of the weld metal. Using E70XX weld metal, Fexx = 70 ksi.

Rr  0.6e 2 Fexx  0.60.870  33.6 ksi For two fillet welds t w = 3/8 in., shear flow resistance is sr  2 t w 0.707  Rr

 2 0.375 0.707 33.6   17.82 kip/in.  su  7.67 kip/in.

O.K.  Use two flange-to-web welds t w = 3/8 in. Shear resistance of the base metal of the web is Rr  0.58v Ag Fy

(AASHTO 6.13.5.3-1)

For web of tw = 0.625 in., shear flow resistance is sr  t w 0.58v Fy

 0.6250.581.0 50   18.13 kip/in.  su  7.67 kip/in.

9.7.11.8

O.K.

Check Requirements – Service Limit State Calculate Moment at Service II

For the section at Bent 3, dead load, DC1, DC2, DW, and live load are applied to the noncomposite section. The factored moments at Service II for Bent 3 are as follows:

M DC1   4,422 kip - ft M DC 2  M DW   363  ( 862)   1,225 kip - ft

M ( LL IM )  1.30.834 3,563   3,863 kip - ft Calculate Web Bend-buckling Resistance 2

2  D  78  k  9   9    36  39   Dc 

Chapter 9 - Steel Plate Girders

(AASHTO 6.10.1.9.1-2)

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Fcrw 

0.9 E k D    tw 

2



0.9  29,000  36   78     0.625 

2

 60.3 ksi

 Rh Fyc  1.0  50   50 ksi   smaller    50 ksi  Fyw / 0.7  50 / 0.7  71.4 ksi 

(AASHTO 6.10.1.9.1-1) Use Fcrw = 50 ksi

Check Flange Stress

In this example, fl = 0 for this interior girder. The requirement becomes:

ff  

M DC1  M DC2  M Dw  M( LLIM ) SNC

 0.80Rh Fyf   0.8 1.0  50  40.0 ksi (AASHTO 6.10.4.2.2-3)

For both compression and tension flanges   4,422  (1,225)  (3,863) f f   3,413 

  12   33.4 ksi  40.0 ksi  

O.K. 

For the compression flange

fc  33.4 ksi  Fcrw  50.0 ksi 9.7.11.9

O.K.

Check Requirements - Constructibility Calculate Factored Moment and Shear - Constructibility

Factored moment and shear at the section of Bent 3 is as: Mu = MDC1 = 1.25(-4,422) = -5,528

kip-ft

Vu = VDC1 = 1.25(-167.8) = -209.8 kips Check Compression Flange

 

Check Web Compactness 2 Dc 78 E 29, 000  124.8   rw  5.7  5.7 137.3 tw 0.625 Fyc 50

(AASHTO 6.10.1.10.2-4)

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The web is noncompact and AASHTO Equations 6.10.3.2.1-1 and 6.10.3.2.1-2 need to be satisfied. Rh = 1.0; 

Rb = 1.0

Calculate Flexural Resistance

Nominal flexural resistance of the compression flange is the smaller of local buckling resistance (AASHTO 6.10.8.2.2) and the lateral torsional buckling resistance (AASHTO 6.10.8.2.3). (1) Local buckling resistance b fc 18 29, 000 E  f    4.5   pf  0.38  0.38  9.15 2t fc 2  2  50 Fyc

Fnc FLB  Rb Rh Fyc  1.01.0 50  50.0 ksi

(AASHTO 6.10.8.2.2-1)

(2) Lateral torsional buckling resistance From Section 9.7.11.3, we have b fc rt   4.7 in.  1 Dc tw  12 1    3 b fc t fc    L p  1.0 rt

E 113.2 in. Fyc

(AASHTO 6.10.8.2.3-9)

(AASHTO 6.10.8.2.3-4)

Fyr  35 ksi Lr   rt

E Fyr

 425.0 in.

(AASHTO 6.10.8.2.3-5)

 L p  113.2 in.  Lb  330 in.  Lr  425.0 in.   Fyr Fnc LTB   Cb 1  1     Rh Fyc

 Lb  L p  R R F R R F   Lr  L p  b h yc b h yc  

(AASHTO 6.10.8.2.3-2)     330 113.2 35 Fnc LTB   1.0  1  1  1.0 1.0 50  1.0  50   425  113.2           39.6 ksi  Rb Rh Fyc  50 ksi Use Fnc(LTB) = 39.6 ksi

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

It should be pointed out that the Cb factor is taken as 1.0 conservatively in the constructibility check. (3) Nominal flexural resistance





Fnc  min Fnc FLB , Fnc LTB  min  50.0, 39.6  39.6 ksi M u 5,528 12    19.4 ksi  f Fnc  39.6 ksi O.K. S NCb 3,413 (AASHTO 6.10.3.2.1-2) Calculate Web Bend-Buckling Resistance fbu 



2

2  D  78  k  9    9    36  39   Dc  0.9 E k 0.9  29, 000  36    60.3 ksi Fcrw  2 2 D  78       0.625   tw 

(AASHTO 6.10.1.9.1-2)

(AASHTO 6.10.1.9.1-1)

 Rh Fyc  1.0  50   50 ksi  smaller   Fyw / 0.7  50 / 0.7  71.4 ksi

Use Fcrw = 50 ksi

fbu  19.4 ksi  f Fcrw  50 ksi

O.K.

(AASHTO 6.10.3.2.1-3)

Check Tension Flange fbu 

M u 5,528 12    19.4 ksi  f Rh Fyt  50 ksi 3, 413 S NCt O.K

(AASHTO

6.10.3.2.2-1) Check for Shear

From Section 9.7.11.4, we obtain: C = 0.358,

Vp = 0.58FywDt = 1,413.8 kips

Vcr  CV p  0.3581,413.8  506.1 kips Vu  209.8 kips  vVcr 1.0506.1  506.1 kips

Chapter 9 - Steel Plate Girders

O.K. (AASHTO 6.10.3.3-1)

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9.7.12

Design Shear Connectors for Span 2 The shear connectors are provided in the positive moment regions and usually designed for fatigue and checked for strength.

9.7.12.1

Design for Fatigue

The range of horizontal shear flow, Vsr, is as follows: Vsr 

Vf Q I ST

where Vf is the factored fatigue vertical shear force range as calculated in Tables 9.711 and 9.7-12, IST is the moment of inertia of the transformed short-term composite section, and Q is the first moment of transformed short-term area of the concrete deck about the neutral axis of the short-term composite section. From Table 9.7-17, IST = 275,267 in.4

Q   Ac / n yc  ySTb   164.2588.44  68.51  3,274 in.3 V f Q 3, 274V f Vsr    0.012V f I ST 275, 267 Try d = 7/8 inch diameter stud, 3 per row, the fatigue shear resistance of an individual stud shear connector, Zr is as follows: Fatigue I: ADTT = 2500 N   365 751.0 0.8 2500  0.5475 10  5.966 10 8

Zr  5.5d 2  5.5 0.875  4.21 kips 2

6

(AASHTO 6.10.10.2-1)

Fatigue II: ADTT = 20 N   365 751.0 0.8 20   438,000  5.966 10

6

  34.5  4.28 log N  34.5  4.28 log 438,000  10.35 (AASHTO 6.10.10.2-3) Zr  d 2  10.35 0.875  7.93kips 2

(AASHTO 6.10.10.2-2)

For 3 - d = 7/8 inch diameter studs, the required pitch of shear connectors, p is obtained as: n Zr 3Z p  r (AASHTO 6.10.10.1.2-1) Vsr Vsr

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

For the positive moment region (0.2L to 0.8L) of Span 2, the detailed calculation is shown in Table 9.7-26. Table 9.7-26 Pitch of Shear Connectors for Span 2 Fatigue I - HL-93 Truck for infinite life

x/L

Fatigue II - P-9 Truck for finite life

Vf (kip)

Vsr = 0.012 Vf (kip/in.)

p (in.)

Vf (kip)

Vsr = 0.012 Vf (kip/in.)

p (in.)

0.2

81.6

0.98

12.9

137.1

1.65

14.4

0.3

77.7

0.93

13.6

120.3

1.44

16.5

0.4

75.9

0.91

13.9

113.1

1.36

17.5

0.5

73.8

0.89

14.1

108.0

1.30

18.3

0.6

75.7

0.91

13.9

112.3

1.35

17.6

0.7

77.2

0.93

13.6

119.0

1.43

16.6

0.8

80.7

0.97

13.0

131.5

1.58

15.1

Select 3–7/8" diameter shear studs with Fu = 60 ksi (AASHTO 6.4.4) at spacing of 12" for the positive moment regions, and 24" for the negative moment regions as shown in Figure 9.7-10. Total number of shear studs from 0.2L to 0.8L Points in Span 2, n = (3)(99+1) = 300 are provided.

3-7/8@12

3-7/8@24 33 Bent 2

3-7/8@24

99 0.2L

33 0.8L

Bent 3

Figure 9.7-10 Pitch of Shear Studs

AASHTO Table 6.6.1.2.3-1 requires that the base metal shall be checked for Category C when the shear studs are attached by fillet welds to the girders. From Section 9.7.10.5, it is seen that this requirement is satisfied.

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.12.2

Check for Strength

In this example of straight bridge, the number of shear connectors between the point of maximum positive moment and each adjacent point of zero moment shall satisfy the following requirement: P n  (AASHTO 6.10.10.4.1-2) Qr

 0.85 fc bts   0.85 3.6 144  9.125   4,021 P  smaller    4,021 kips  As Fy   98.25 50   4,913  (AASHTO 6.10.10.4.2-2) and (AASHTO 6.10.10.4.2-3)

The factored shear resistance of a single d = 7/8 in. shear stud connector is as:

Ec  w3/2  33 fc  150

3/2

Qn  0.5 Asc

f cEc  0.5

 33

 0.875 2 

 34.4 kips  Asc Fu 

4

3,600  3.64 106 psi  3,640 ksi (AASHTO 5.4.2.4-1) 3.6  3, 640 

  0.875 2  60   36.1 kips 4

(AASHTO 6.10.10.4.3-1) Use Qn = 34.4 kips

n 9.7.12.3

P 300 4,021  150    137.5 sc Qn 2 0.85 34.4

O.K.

Determine Shear Connectors at Points of Contraflexure

AASHTO 6.10.10.3 requires that for members that are noncomposite for negative moment regions in the final condition, additional connectors, nac shall be placed within a distance equal to one-third of the effective concrete deck width on each side of the point of permanent load contraflexure.

nac 

As f sr Zr

(AASHTO 6.10.10.3-1)

where fsr is fatigue stress range in the slab reinforcement over the interior support under the Fatigue I load combination for infinite fatigue life. As calculated in Section 9.7.11.5, factored Fatigue I moment range at Bent 3, (M) = 1,065 kips-ft. Using the elastic section property of the steel section and deck slab reinforcement calculated in Table 9.7-22, we obtain:

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

f sr 

  M 

1, 065 12 

 3.3 ksi S NSrb 3,880 It is noted that fsr can be conservatively taken as the fatigue stress range in the top flange as calculated in Section 9.7.11.6. In the past AASHTO Standard Specifications, fsr was assumed as 10 ksi.

nac 

9.7.13



As f sr 13.14 3.3   10.3 studs; 4.21 Zr

Use 12 studs

Design Bearing Stiffeners at Bent 3 The bearing stiffeners consist of one or more plates welded to each side of the web and extend the full height of the web. The purpose of bearing stiffeners is to transmit the full bearing forces from factored loads. The bearing stiffeners shall be designed for axial resistance of a concentrically loaded column (AASHTO 6.10.11.2.4) and for bearing resistance (AASHTO 6.10.11.2.3).

9.7.13.1

Illustrate Calculations of Factored Support Forces at Bent 3

Factored support forces for four supports at the strength limit states are summarized in Table 9.7-10. The calculations of factored support forces at Bent 3 are illustrated as follows: Dead Load

RDC1 = 1.25(328.2) = 410.3 kips RDC2 = 1.25(27) = 33.8 kips RDW = 1.5(64) = 96.0 kips Live Load

R(LL+IM)HL-93 = 1.75(1.082)(249.2) = 471.9 kips R(LL+IM)P15 = 1.35(1.082)(447) = 652.9 kips Controlling Support Force

Ru = 410.3 + 33.8 + 96.0 + 652.9 = 1,193 kips 9.7.13.2

Select Stiffeners

For a short column, assume Pr = 0.85FysAs = 0.85(36)As = 30.6As and we obtain the initial effective column area as: As 

Ru 1,193   38.99 in.2 30.6 30.6

Try two stiffeners, 1.875" 8" PL as shown in Figure 9.7-11.

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.13.3

Check Projecting Width bt  8 in.  0.48t p

E 29,000   0.48 1.875   25.54 in. Fys 36

O.K.

(AASHTO 6.10.11.2.2-1) 9.7.13.4

Check Bearing Resistance

Factored bearing resistance is as:

Rsb r  b Rsb n  1.01.4Apn Fys

(AASHTO 6.10.11.2.3-1 and 6.10.11.2.3-2)

Assuming 1.5 in. cope on bearing stiffeners, the bearing area is: A pn  2 8  1.51.875  24.375 in.2

Rsb r  1.01.424.37536  1,228.5 kips  Ru Center of Bent 3

 1,193 kips

O.K.

Bearing Stiffener

0.625"

16.625"

x-x

18"

8"

y-y

Web

t /2 + 9t w

t /2 + 9t w Flange 13.125"

Figure 9.7.11 Bearing Stiffeners 9.7.13.5

Check Axial Resistance

According to AASHTO 6.10.11.2.4b, for stiffeners welded to the web, the effective column section consists of stiffener plates and a centrally loaded strip of the web extending not more than 9tw on each side of the stiffeners. Stiffener area: Web area:

Chapter 9 - Steel Plate Girders

Ast = (2)(8)(1.875) = 30 in.2 Aweb   18 tw  t p  t w  18  0.625   1.875  0.625 

 13.125  0.625   8.20 in.2

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

As  30  8.20  38.2 in.2

Total effective column area: I x x  rs 

18  0.625  0.6253  1.875   2 8   0.625 12 I x x  As

3

 718 in.4

718  4.34 in. 38.2

Use effective length factor K = 0.75 for the weld end connection (AASHTO 6.10.11.2.4a), unbraced length for the bearing stiffeners l = D = 78 in. Kl 0.75  78  O.K. (AASHTO 6.9.3)  13.5  120 rs 4.34 Axial resistance is calculated in accordance with AASHTO 6.9.4.1 as follows:

 2E

 2 29,000

38.2  59,992.0 kips (AASHTO 6.9.4.1.2-1) 2 13.52  Kl     rs  Po  QFys As  1.03638.2  1,375.2 kips (Q is taken equal to 1.0 for

Pe 

As 

bearing stiffeners in accordance with AASHTO 6.9.4.1.1)



Pe 59,992.0   43.62  0.44 Po 1,375.2

 Po    1,375.2         Pe       59,992.0   1,375.2   1,362.1 kips  Pn  0.658 P  0 . 658  o       (AASHTO 6.9.4.1.1-1)

Pr  c Pn  0.91,362.1  1,225.9 kips  Ru 1,193 kips

O.K.

 Use two 1.875" 8" PL bearing stiffeners 9.7.13.6

Design Bearing Stiffener-to-Web Welds

Fillet welds are usually used for bearing stiffener-to-web connections. According to AASHTO Table 6.13.3.4-1, the minimum size of fillet weld for thicker plate thickness joined larger than 3/4 in. is 5/16 in., but need not exceed the thickness of the thinner part joined. Try two fillet welds tw = 5/16 in. on each stiffener. Shear resistance of fillet welds (AASHTO 6.13.3.2.4b) is

Rr  0.6 e2Fexx

(AASHTO 6.13.3.2.4b-1)

Using E70XX weld metal, Fexx = 70 ksi.

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Rr  0.6 e2Fexx   0.6 0.8 70  33.6 ksi Total length of welds, allowing 2.5 inches for clips at both the top and bottom of the stiffener, is: L = 78 - 2(2.5) = 73 in. Total shear resistance of welds connecting the bearing stiffeners to the web is: Vr   4  0.707  tw L Rr   4  0.707  0.3125  73 33.6   2,168 kips  Vu  1,193 kips

O.K.

 Use two fillet welds tw = 3/8 in. on each stiffener

9.7.14

Design Intermediate Cross Frames An intermediate cross frame consisting of single angles is selected as shown in Figure 9.7-12.

Figure 9.7.12 A Typical Intermediate Cross Frame 9.7.14.1

Calculate Wind Load

Design wind pressure is determined as: 2

V  V2 PD  PB  DZ   PB DZ 10,000  VB 

PB  base wind pressure  0.05 ksf (for beams) V   Z  VDZ  2.5Vo  30  ln    VB   Zo 

(AASHTO 3.8.1.2.1-1) (AASHTO Table 3.8.1.2.1-1) (AASHTO 3.8.1.1-1)

Assume the steel girder is 35 ft. above the low ground, and the bridge is located in the suburban area, Z = 35 ft., Zo = 3.28 ft., Vo = 10.9 mph, V30 = VB = 100 mph (AASHTO Table 3.8.1.1-1). Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

V   Z   100   35  VDZ  2.5Vo  30  ln     2.5 10.9    ln    64.5 mph  100   3.28   VB   Z o  V2 64.52 PD  PB DZ   0.05   0.02 ksf 10, 000 10, 000

In this example, the midspan girder depth d = 80.75 in. = 6.73 ft, depth of deck and barrier = 12.25 + 32 = 44.25 in. = 3.69 ft. Wind load acting on the girder span is as:

WSgirder  PD 3.69  6.73  0.0210.42  0.21 kip/ft  0.3 kip/ft Use WSgirder = 0.3 kip/ft

(AASHTO 3.8.1.2.1)

Wind load acting on the bottom flange is as: WS girder d / 2 

WSbf 



0.36.73 / 2  0.097 kip/ft

3.69  6.73 10.42 Wind force acting on top flange (directly transmitted to the concrete deck) Wtf  0.3 - 0.097  0.203 kip/ft 9.7.14.2

Check Flexural Resistance of Bottom Flange

For cross frame spacing, Lb = 27.5 ft, wind induced moment applied on the bottom flange of the exterior girder is estimated as: M WS 

WSbf L2b 10



0.097 27.52 10

 7.34 kip - ft

(AASHTO C4.6.2.7.1-2)

For the smaller bottom flange, wind induced lateral stress is: fl WS 

M WS t f b 2f

/6



 7.34 12  1.75 18 2 / 6

 0.93 ksi

From CA Table 3.4.1, the load combinations Strength III and V are: For Strength III:

1.25DC + 1.5DW + 1.4WS

For Strength V:

1.25DC + 1.5DW + 1.35DF(LL+IM)HL-93 +0.4WS

It is obvious that Strength V controls design. In this example, the section at 0.5 Point of Span 2 is checked. From Table 9.7-8, Factored moments about major axis of the cross section are as: MDC1 = 3,269 kip-ft;

MDC2 = 269 kip-ft

MDW = 765 kip-ft;

M(LL+IM)HL-93 = 1.35(0.805)(3,455) = 3,755 kip-ft

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Mu = 3269 + 269 + 765 + 3,755 = 8,058 kip-ft The factored lateral bending stress in the bottom flange due to wind load is:

fl   0.4 fl WS   0.4 0.93  0.37 ksi  0.6Fyf  30 ksi

O.K.

(AASHTO 6.10.1.6-1) At the strength limit state, the composite compact section in positive moment regions shall satisfy the requirement as follows: Mu



1 fl S xt  f M n 3

(AASHTO 6.10.7.1.1-1)

where Sxt = Myt/Fyt From Section 9.7.10.3, Myt = 15,287 kip-ft and Mn = 19,873 kip-ft

9.7.14.3

15, 287 12 

S xt 

M yt

Mu 

1 1 3,669 f l S xt  8,058  0.37   8,096 kip - ft 3 3 12   f M n  19,873 kip - ft

Fyt



50

 3, 669 in.3

O.K.

Calculate Forces Acting on the Cross Frame

In order to find forces acting in the cross frame members, a cross frame is treated like a truss with tension diagonals only and solved using statics. The wind force in the top strut is assumed zero because the diagonals will transfer the wind load directly into the deck slab. The horizontal wind forces applied to the brace points are assumed to be carried fully by the bottom strut in the exterior bays. Therefore, the bottom strut in all bays will be conservatively designed for this force. At Strength Limit III, factored wind force acting on the bottom strut is:

Pu  1.4WSbf Lb  1.40.09727.5  3.73 kips Factored force acting on diagonals is:

Pu 

3.73 = cos 6

Chapter 9 - Steel Plate Girders

3.73 5.52  6 2

 5.06 kips

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9.7.14.4

Design Bottom Strut Select Section

Try L 661/2 as shown in Figure 9.7-13. Ag = 5.77 in.2 ; x = y = 1.67 in.;

Ix = Iy = 19.9 in.4; rz = 1.18 in. ;

rx = ry = 1.86 in. Tan  = 1.0

y

x

Figure 9.7-13 Single Angle for Bottom Strut Check Limiting Effective Slenderness Ratio

AASHTO 6.9.3 requires that the effective slenderness ratio, KL/r shall not exceed 140 for compression bracing members. For buckling about minor principal axis (Z-Z), using unbraced length Lz = 6 ft = 72 in. and effective length factor K = 1.0 for single angles regardless of end conditions (AASHTO 4.6.2.5), the effective slenderness ratio is: KLz  rz

1.0  (72) 1.18

 61  140

O.K.

For out-plane buckling about vertical geometric axis (Y-Y), using unbraced length Ly = 12 ft = 144 in. and effective length factor K = 1.0, the effective slenderness ratio is:

KLy ry



1.0 (144) 1.86

 77.4  140

O.K.

Check Member Strength

Since a single angle member is connected through one leg only, the member is subjected to combined flexural moments about principal axes due to eccentrically

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

applied axial load and axial compression. AASHTO Article 6.9.4.4 states that single angles subjected to combined axial compression and flexure may be designed as axially loaded compression members in accordance with AASHTO Articles 6.9.2.1, 6.9.4.1.1, and 6.9.4.1.2, as appropriate, using one of the effective slenderness ratios specified by AASHTO Article 6.9.4.4, provided that: (1) end connections are to a single leg of the angle, and are welded or use minimum two-bolt connections; (2) angles are loaded at the ends in compression through the same one leg; and (3) there are no intermediate transverse loads. It is obvious that the bottom strut meets those three conditions and can be designed in accordance with AASHTO Article 6.9.4.4. 

Determine Effective Slenderness Ratio For equal-leg angles that are individual members,

L (144)   77  80 rx 1.86

144 L  KL   130    72  0.75  72  0.75 1.86 rx  r eff



(AASHTO 6.9.4.4-1)

Determine Slender Element Reduction Factor, Q 

b 6   12  k t 0 .5

E 29,000  0.45  12.8 Fy 36

(AASHTO 6.9.4.2.1-1)

 Q  1.0 

Determine Nominal Axial Compression Strength Axial resistance is calculated in accordance with AASHTO 6.9.4.1 as follows:  2E  2 29,000  5.77   97.72 kips (AASHTO 6.9.4.1.2-1) Pe  A  g 2 1302  KL     rs eff

Po  QFys As  1.0365.77  207.72 kips



Pe 97.72   0.47  0.44 Po 207.72

 Po     207.72        Pe      97.72   207.72   85.33 kips   Pn  0.658 P  0 . 658 o       

(AASHTO 6.9.4.1-1) 

Check Compressive Strength

Pu  3.73 kips  c Pn  0.985.33  76.70 kips

Chapter 9 - Steel Plate Girders

O.K.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.14.5

Design Diagonal Select Section

Try L 445/16, L = 5.5  6 2

2

Ag = 2.4 in.2

rmin = 0.781 in.

= 8.14 ft. = 98 in.

Check Limiting Effective Slenderness Ratio

KL 1.098   123.0  140 rmin 0.791

OK

(AASHTO 6.9.3)

Check Member Strength

A separate calculation similar to the above bottom strut design shows that angle L 445/16 meets specification requirements. 9.7.14.6

Design Top Strut

Since the force in the top strut is assumed zero, we select an angle L 661/2 to provide lateral stability to the top flange during construction and to design for 2 percent of the flange yield strength. Design calculation is similar with the above for the bottom strut and is not illustrated here. 9.7.14.7

Design Connection of Bottom Strut Determine Design Load

For end connections of diaphragms and cross frames in straight girder bridges, AASHTO 6.13.1 requires that it shall be designed for the calculated member forces. In this example, the connection of the bottom strut is designed for the calculated member load, Pu = 3.73 kips. Determine Number of Bolts Required



Select Bolts Try A325 high-strength 3/4 in. diameter bolt with threads excluded from the shear plane, with bolt spacing of 3 in. and edge distance of 1.75 in. For 3/4 in. diameter bolts, minimum spacing of bolts is 3d = 2.25 in. (AASHTO 6.13.2.6.1) and minimum edge distance from center of standard hole to edge of connected part is 1.25 in. for sheared edges (AASHTO Table 6.13.2.6.6-1).



Determine Nominal Resistance per Bolt Calculate nominal shear resistance in single shear

Rn  0.48 Ab Fub Ns

Chapter 9 - Steel Plate Girders

(AASHTO 6.13.2.7-1)

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2

 0.75  2 Ab      0.442 in.  2  Fub  120 ksi

(AASHTO 6.4.3)

Ns = 1 (for single shear)

Rn  0.48Ab Fub Ns  0.48 0.4421201  25.5 kips

(AASHTO 6.13.2.7-1)

Calculate the design bearing strength for each bolt on stiffener material Since the clear edge distance, Lc = 1.75 – (0.75 + 0.0625)/2 = 1.344 in. is less than 2d = 1.5 in. and stiffener material is A709 Grade 36, Fu = 58 ksi.

Rn  1.2 Lc t Fu 1.21.3440.558  46.8 kips

(AASHTO 6.13.2.9-2)

Determine design strength per bolt It is obvious that shear controls and nominal shear resistance per bolt is 25.5 kips. 

Determine Number of Bolts Required The number of bolts required is:

Pu 3.73   0.2 bolts  sRv 0.825.5 Use 2 bolts as shown in Figure 9.7-14. N 

Figure 9.7-14 Bottom Strut Connection

Chapter 9 - Steel Plate Girders

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9.7.15

Design Bolted Field Splices

9.7.15.1

General Design Requirements

For flexural members, splices shall preferably be made at or near points of dead load contraflexure in continuous spans and at points of the section change. AASHTO 6.13.6.1.4a states that bolted splices for flexural members shall be designed using slip-critical connections as specified by AASHTO 6.13.2.1.1. The general design requirements are:   

Factored resistance of splices shall not be less than 100 percent of the smaller factored resistances of the section spliced at the strength limit state (CA 6.13.1). Slip shall be prevented at the service limit state II (AASHTO 6.13.2.1.1) and during erection of the steel and during the casting or placing of the deck (AASHTO 6.13.6.1.4a). Base metal at the gross section shall be checked for Category B at the fatigue limit state (AASHTO Table 6.6.1.2.3-1).

As shown in Figure 9.7-4, bolted field girder splices for Span 2 are located approximately at 0.3 and 0.7 Points. In the following, the design of a bolted splice (Figure 9.7-15) as a slip-critical connection at 0.7 Point will be illustrated. Oversized or slotted holes shall not be used (AASHTO 6.13.6.1.4a). The hole diameter used in calculation shall be 1/16 in. larger than the nominal diameter as shown in AASHTO Table 6.13.2.4.2-1. 9.7.15.2

Design Bottom Flange Splices

Try one outer splice plate 1 in.  18 in., two inner plates 1-1/8 in.  8 in. and one fill plate ¼ in.  18 in. as shown in Figure 9.7-16. Try A325 high-strength d = 7/8 in. bolt threads excluded with bolt spacing of 3 in. and edge distance of 2 in. For 7/8 in. diameter bolts, the minimum spacing of bolts is 3d = 2.625 in. (AASHTO 6.13.2.6.1) and minimum edge distance from center of standard hole to edge of connected part is 1.5 in. for sheared edges (AASHTO Table 6.13.2.6.6-1). The standard hole size for a d = 7/8 in. bolt is 0.9375 in. (AASHTO Table 13.2.4.2-1). Determine Number of Bolts Required – Strength Limit States



Determine Design Forces

Fcf  f Fyf

Pcu  Ae Fcf  Aef Fyf

(CA 6.13.6.1.4c-1) (AASHTO 6.13.6.1.4c)

where Ae is the smaller effective area for the flange on either side of the splice. Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 9.7-15 Bolted Field Girder Splice

F  Ae   u u  An  Ag (AASHTO 6.13.6.1.4c-2)  y Fyt     = 1.0 except that a lower value equal to (Fn/Fyf) may be used. Try 4 – 7/8 in. diameter/row for the flange splices. For smaller flange, we have: An  18  4 0.9375 1.75   24.94 in.2

Ag  181.75  31.5 in.2  F  Ae   u u   f Fyt 

  An   0.865 24.94  27.30 in.2  Ag  31.5 in.2  0.9550     

Use Ae = 27.30 in.2

Pcu  Ae f Fyf  27.301.01.050  1,365.0 kips

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

 = 15/16

Lc = 1.41

Lc = 2.0625

Figure 9.7-16 Bottom Flange Splice



Determine Nominal Resistance per Bolt When the length between the extreme fasteners measured parallel to the line of action of the force is less than 50 in. the nominal shear resistance for an A325 – 7/8 in. diameter bolt is:

Rn  0.48Ab Fub Ns

(AASHTO 6.13.2.7-1)

Ab   0.875 / 2   0.6 in.2 2

Fub 120 ksi Ns = 2 (for double shear) Rn   0.48 0.6120 2  69.1 kips

(AASHTO 6.4.3.1)

Since the clear end distance Lc = 1.875 – 0.469 =1.41 in. < 2d = 1.75 in. (Figure 9.7-16), the nominal bearing resistance for each bolt hole on flange material is:

Rn  1.2Lct Fu

Rn  1 .2 1 .411 .75 65   192 .5 kips

Chapter 9 - Steel Plate Girders

(AASHTO 6.13.2.9-2)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

It is obvious that shear resistance controls and design resistance per bolt is 69.1 kips. 

Evaluate Fill Plate Effects AASHTO 6.13.6.1.5 specifies that fillers ¼ inch and thicker need not be extended and developed provided that the factored shear resistance of the bolts at the strength limit state is reduced by the reduction factor R. R 

  1  2  A f / Ap  1  A f / Ap

(AASHTO 6.13.6.1.5-1)

Af

= sum of area of the fillers on the top and bottom of the connected plate Ap = smaller of either the connected plate area or the sum of the splice plate area on the top and bottom of the connected plate

Af   0.2518  4.5 in.2 1.7518  31.5 in.2  2 Ap  smaller of    31.5 in. 2  2 1.125 8  118  36.0 in.  1  A f / Ap 1  4.5 / 31.5 R    0.889 1   2  4.5 / 31.5  1  2 A f / Ap

 



 

Determine Number of Bolts Required The number of bolts required is:

N 

Pcu 1,365.0   27.78 bolts s R Rn 0.80.88969.1

Use 28 bolts as shown in Figure 9.7-16. Check Slip Resistance of Bolts – Service Limit State II and Constructibility

AASHTO 6.13.2.1.1 and 6.13.6.1.4a require that the bolted connections shall be proportioned to prevent slip at the service limit state II and during erection of the steel and during the casting or placing of the deck. 

Determine Factored Moments For the Service II, factored moment at 0.7 Point of Span 2 is obtained from Table 9.7-13.  M u  1, 434  118  280  2,784  4,616 kip - ft  M u  1, 434  118  280  1,094  738 kip - ft

Chapter 9 - Steel Plate Girders

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For constructibility, factored dead load moment during the casting of the deck at 0.7 Point of Span 2 is obtained as: M DL  1.0  1, 434   1, 434 kip - ft

It is clear that the Service II moment governs the design. 

Check Slip Resistance Assume a non-composite section conservatively at the splice location and use the smaller section property for the bottom flange SNCb = 2,837 in.3 (Table 9.7-15).

Fs 

fs M /S 4,616 12 / 2,837  u NCb   19.28 ksi Rh Rh 1.0

Ru  Fs Ag  19.2831.5  607.3 kips Nominal slip resistance per bolt is:

Rn  Kh Ks Ns Pt

(AASHTO 6.13.2.8-1)

where Kh is hole size factor and is equal to 1.0 for the standard hole (AASHTO Table 6.13.2.8-2); Ks is surface condition factor and is taken 0.5 for Class B surface condition (AASHTO Table 6.13.2.8-3); Pt is minimum required bolt tension and is equal to 39 kips (AASHTO Table 6.13.2.8-1). According to AASHTO 6.13.2.2, factored slip resistance of 28 bolts is: Rr = Rn = (1.0)(0.5)(2)(39)(28)= 1,092 kips > Ru = 607.3 kips

O.K.

Check Tensile Resistance of Splice Plates

Since areas of the inner and outer plates are the same, the flange design force is assumed to be divided equally to the inner and outer plates. In the following, splice plates are checked for yielding on the gross section, fracture on the net section, and block shear rupture (AASHTO 6.13.5.2). 

Yielding on Gross Section

Ag  181.0   2 81.125  36 in.2

Rr   y Ag Fyf  0.95 3650  1,710.0 kips Rr 1,710.0 kips  Pcu  1,365.0 kips 

(AASHTO 6.8.2.1-1) O.K.

Fracture on Net Section Inner plates: An  2 8  2 0.9375 1.125   13.78 in.2 Rr  u Fu AnU  0.8  65 13 .78 1 .0   715 .0 kips 1,365.0 P  682.5 kips Rr  715.0 kips  cu  2 2

Chapter 9 - Steel Plate Girders

(AASHTO 6.8.2.1-2) O.K.

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Outer plate:



An  18  4 0.9375 1.0   14.25 in.2 Rr  u Fu AnU  0 .8  65 14 .25 1.0   741 .0 kips P Rr  741.0 kips  cu  682.5 kips 2 Block Shear Rupture

(AASHTO 6.8.2.1-2) O.K.

Assume bolt holes are drilled full size, reduction factor for hole, Rp is taken equal to 1.0 (AASHTO 6.13.4). For flange splice plates, reduction factor for block shear rupture, Ubs is taken equal to 1.0 (AASHTO 6.13.4). Bolt pattern and block shear rupture failure planes on the inner and outer splice plates are shown in Figure 9.7-17. Inner plates:

Atn  2 6  1.5 1.00.9375  10.34 in.2

Avn  2 20  6.5 0.93751.125  31.29 in.2 Avg  2 20 1.125  45.0 in.2

 Fu Avn  6531.29  2,033.9 kips  Fy Avg  5045.0  2,250.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586531.29  1.06510.34 1,481.4 kips (AASHTO 6.13.4-1) P Rr 1,481.4 kips  cu  682.5 kips O.K. 2 Outer plate: Atn  2 6  1.5 0.9375 1.0   9.19 in.2

Avn  2 20  6.5 0.9375 1.0   27.81 in.2

Avg  2 20 1.0   40.0 in.2

 Fu Avn  6527.81  1,807.7 kips  Fy Avg  5040.0  2,000.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586527.81  1.0659.19 1,316.6 kips (AASHTO 6.13.4-1)

Rr 1,316.6 kips 

Chapter 9 - Steel Plate Girders

Pcu  682.5 kips 2

O.K.

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Figure 9.7-17 Block Shear Rupture - Bottom Flange Splice Plates Check Fracture on Net Section and Block Shear Rupture for Flange



Fracture on Net Section Since the design force is actually based on fracture resistance on the net section, there is no need to check.



Block Shear Rupture Bolt pattern and block shear rupture failure planes on the bottom flange are assumed in Figure 9.7-18.

Figure 9.7-18 Block Shear Rupture – Bottom Flange

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Atn  2 6  1.5 0.9375 1.75   16.08 in.2

Avn  2 19.875  6.5 0.9375 1.75   48.23 in.2

Avg  2 19.8751.75  69.56 in.2

 Fu Avn  6548.23  3,135.0 kips  Fy Avg  5069.56  3,478.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586548.23  1.06516.08  2,290.8 kips

Rr  2,290.8 kips  Pcu  1,365.0 kips

(AASHTO 6.13.4-1) O.K.

Check Fatigue for Splice Plates

Fatigue stress ranges in base metal of the bottom flange splice plates adjacent to the slip-critical connections are checked for Category B (AASHTO Table 6.6.1.2.31). Fatigue normally does not govern the design of splice plates when combined area of inner and outer splice plates is larger than the area of the smaller flange spliced. The fatigue moment ranges at 0.7 Point of Span 2 are obtained from Tables 9.7-11 and 9.7-12. The nominal fatigue resistance is calculated in Table 6.8.2 in Chapter 6. The flexural stresses at the edges of the splice plates are assumed to be the same as the flexural stresses in the girder at those locations. The gross section of the smaller girder section is used to calculate the stresses. Properties of the steel section alone are used conservatively. For the smaller spliced section (Table 9.7-15), SNCb = 2,837 in.3 Fatigue I - HL-93 Truck for infinite life:

 f  

 M  S NCb



1,44012  2,837

 6.09 ksi  16.0 ksi O.K. for Category B

Fatigue II - P-9 Truck for finite life:

 f  

 M  S NCb



2,396 12  2,837

 10.13 ksi  30.15 ksi O.K. for Category B

9.7.15.3

Design Top Flange Splices

Try one outer splice plate 5/8 in.  18 in., two inner plates 3/4 in.  8 in., and one fill plate 1 in.  18 in. as shown in Figure 9.7-19. As the same as the bottom flange, try A325 high-strength d = 7/8 in. bolt threads excluded with bolt spacing of 3 in. and edge distance of 2 inches.

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Figure 9.7-19 Top Flange Splice Determine Number of Bolts Required – Strength Limit State



Determine Design Forces Try 4 – 7/8 in. diameter/row for the flange splices. For the smaller flange, we have: An  18  4 0.9375 1.0   14.25 in.2

Ag  181.0 18 in.2   An   0.865 14.25  15.60 in.2  Ag  18 in.2  0.9550      (AASHTO 6.13.6.1.4c-2) Use Ae = 15.60 in.2  F  Ae   u u   y Fyt 

Pcu  Ae  f Fyf  15.601.01.050  780.0 kips 

Determine Nominal Resistance per Bolt As calculated in Section 9.7.15.2, the nominal shear resistance per A325 – 7/8 in. diameter bolt in double shear is:

Chapter 9 - Steel Plate Girders

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Rn  69.1 kips The nominal bearing resistance for each bolt hole on flange material is:

Rn  1.2Lct Fu

(AASHTO 6.13.2.9-2)

For exterior hole: Lc  1.875  0.469  1.41 in. Rn  1.2 1.411 .0 65   110 .0 kips

It is obvious that shear resistance controls and design resistance per bolt is 69.1 kips. 

Evaluate Fill Plate Effects AASHTO 6.13.6.1.5 specifies that fillers ¼ inch and thicker need not be extended and developed provided that the factored shear resistance of the bolts at the strength limit state is reduced by the reduction factor R: 1  A f / Ap (AASHTO 6.13.6.1.5-1) R  1  2 A f / Ap

 

 

Af  1.018  18 in.2

1.0 18  18 in.2  2 Ap  smaller of    18 in. 2  20.758  0.62518  23.25 in.  R 



   1  18 / 18  0.667 1  2  A f / Ap  1   2 18 / 18  1  A f / Ap

Determine Number of Bolts Required The number of bolts required is

Pcu 780.0   21.5 bolts; s R Rn 0.80.66769.1 N is only 5.8% over 20 bolts shown in Figure 9.7-19, say O.K for this example. However, engineer should redesign the top flange splices for actual projects. N 

Check Slip Resistance of Bolts – Service Limit State II and Constructibility

From Section 9.7.15.2, the moment at Service II is 1,773 kip-ft. Assume noncomposite section conservatively at the splice location and use the smaller section property for the top flange SNCt = 2,193 in.3 (Table 9.7-15). Slip force is calculated as follows: f M /S 4,616 12 / 2,193 Fs  s  u NCb   25.26 ksi (CA 6.13.6.1.4c-5) Rh Rh 1.0

Ru  Fs Ag  25.2618  454.7 kips

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Nominal slip resistance per bolt is:

Rn  Kh Ks Ns Pt

(AASHTO 6.13.2.8-1)

Slip resistance of 20 bolts is: Rr  Rn  1 .0 0 .5 2 39 20   780 .0 kips  Ru  454 .7 kips

O.K.

Check Tensile Resistance of Splice Plates

Since areas of the inner and outer plates differ less than 10%, the flange design force is assumed to be divided equally to the inner and outer plates. In the following, splice plates are checked for yielding on the gross section, fracture on the net section, and block shear rupture (AASHTO 6.13.5.2). 

Yielding on Gross Section Ag  18 0.625   2 8 0.75  23.25 in.2

Rr y Ag Fyf   0.95  23.25 50  1,104.4 kips Rr  1,104 .4 kips  Pcu  780 .0 kips



(AASHTO 6.8.2.1-1) O.K.

Fracture on Net Section Inner plates: An  2 8  2 0.9375 0.75   9.19 in.2 Rr  u Fu AnU  0 .8  65 9.19 1 .0   477 .9 kips 780.0 P  390.0 kips Rr  477.9 kips  cu  2 2 Outer plate: An  18  4 0.9375 0.625   8.91 in.2 Rr  u Fu AnU  0 .8  65 8 .911 .0   463 .3 kips P Rr  463.3 kips  cu  390.0 kips 2



(AASHTO 6.8.2.1-2) O.K.

(AASHTO 6.8.2.1-2) O.K.

Block Shear Rupture The bolt pattern and block shear rupture failure planes on the inner and outer splice plates are assumed in Figure 9.7-20. Inner Plates: Atn  2 6  1.5 0.9375 0.75   6.89 in.2

Avn  2 14  4.5 0.9375 0.75   14.67 in.2

Avg  2 14 0.75  21.0 in.2

Chapter 9 - Steel Plate Girders

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 Fu Avn  6514.67  953.6 kips  Fy Avg  5021.0 1,050.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586514.67  1.0656.89  800.7 kips (AASHTO 6.13.4-1) Pcu Rr  800.7 kips  O.K.  390.0 kips 2

Figure 9.7-20 Block Shear Rupture – Top Flange Splice Plates

Outer plate: Atn  2 6  1.5 0.9375 0.625   5.74 in.2

Avn  2 14  4.5 0.9375 0.625   12.23 in.2

Avg  2 14 0.625  17.5 in.2

 Fu Avn  6512.23  795.0 kips  Fy Avg  5017.5  875.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586512.23  1.0655.74  667.3 kips (AASHTO 6.13.4-1) P Rr  667.3 kips  cu  390.0 kips O.K. 2

Chapter 9 - Steel Plate Girders

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Check Fracture on Net Section and Block Shear Rupture for Flange



Fracture on Net Section Since the design force is actually based on fracture resistance on the net section, there is no need to check.



Block Shear Rupture The bolt pattern and block shear rupture failure planes on the inner and outer splice plates are assumed in Figure 9.7-21.

Figure 9.7-21 Block Shear Rupture – Top Flange Atn  2 6  1.5 0.9375 1.0   9.19 in.2

Avn  2 13.875  4.5 0.9375 1.0   19.31 in.2

Avg  2 14 1.0   28.0 in.2

 Fu Avn  6519.31  1,255.2 kips  Fy Avg  5028.0 1,400.0 kips Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586519.31  1.0659.19 1,060.3 kips (AASHTO 6.13.4-1) Rr 1,060.3 kips  Pcu  780.0 kips O.K. Check Fatigue for Splice Plates

Fatigue stress ranges in base metal of the top flange splice plates adjacent to the slip-critical connections are checked for Category B (AASHTO Table 6.6.1.2.3-1). The fatigue moment ranges at 0.7 Point are obtained from Tables 9.7-11 and 9.7-12. The nominal fatigue resistance is calculated in Table 9.7-25. The flexural stresses at the edges of the splice plates are assumed to be the same as the flexural stresses in the girder at those locations. Gross section of the smaller girder section is used to calculate the stresses. Properties of the steel section alone are used conservatively. For the smaller spliced section (Table 9.7-15), SNCt = 2,193 in.3.

Chapter 9 - Steel Plate Girders

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Fatigue I - HL-93 Truck for infinite life:  M  1,440 12   f    S NCt 2,193  7.88 ksi  16.0 ksi O.K. for Category B

Fatigue II - P-9 Truck for finite life:

 f  

 M  S NCb



2,39612  2,193

 13.11 ksi  30.15 ksi O.K. for Category B

9.7.15.4

Design Web Splices

Try two 7/16 in.  64 in. web splice plates and A325 high-strength d = 7/8 in. bolt threads excluded with bolt spacing of 3 in. as shown in Figure 9.7-15.

Ag   2 0.4375 64  56 in.2 Check Bolt Shear Resistance - Strength Limit States



Calculate Design Forces The design forces for a web splice are shown in Figure 9.7-22.

Vuw   vVn

(1) Shear

(CA 6.13.6.1.4b-1)

From Section 9.7.11.4, Vuw  vVn 843.6 kips (2) Moment induced by eccentrically loaded shear

M vw  Vuw e  843.65.0  4,218 kip - in. (3) Moment resisted by the web Case I – Positive Bending At the strength limit state, the smaller section at the point of splice is a composite compact section. M uw  f

tw Fyw 4

 D 2  4 yo2   

(CA C6.13.6.1.4b-1a)

yo is distance from the mid-depth of the web to the plastic neutral axis. From Section 9.7.10.3, since the plastic neutral axis is located in the top flange (Figure 9.7-7), yo = dw = 39.5 in. > D/2 = 39 in. and all the web is in tension, Equation CA-C6.13.6.1.4b-1a is no longer valid and Muw = 0. Case II – Negative Bending At the strength limit state, the smaller section at the point of splice is a noncompact section.

Chapter 9 - Steel Plate Girders

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Figure 9.7-22 Design Forces for Web Splices

M uw   f



tw D 2 Fnc  Fyw 12



(CA C6.13.6.1.4b-1b)

For bottom flange, assume Fnc = Fyf = Fyw = 50 ksi conservatively

tw D 2 0.625782 50  50 Fnc  Fyw  1.0 12 12  31,687.5 kip - in.



M uw   f



(4) Horizontal force resultant in the web Case I –Positive Bending Since the whole web is in tension, Equation CA C6.13.6.1.4b-2a is no longer valid. Use the following:





Huw   f tw DFyw  1.0 0.625 78 50  2,437.5 kips Case II – Negative Bending Assume Fnc = Fyf = Fyw = 50 ksi, we have:

Chapter 9 - Steel Plate Girders

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H uw  f

tw D 2 Fyw  Fnc  0 2





(CA C6.13.6.1.4b-2b)

(5) Web design forces Case I – Positive Bending Horizontal force

M u  M uw  M vw  0  4,218  4,218 kip - in. Huw  2,437.5 kips

Vertical shear

Vuw  843.6 kips

Total moment

Case II – Negative Bending Horizontal force

M u  M uw  M vw  31,687.5  4,218  35,906 kip - in. Huw  0

Vertical shear

Vuw  843.6 kips

Total moment



Determine Nominal Resistance per Bolt As calculated in Section 9.7.15.2, the nominal shear resistance per A325 – 7/8 in. diameter bolt in double shear is:

Rn  69.1kips The nominal bearing resistance for each bolt hole on web material is:

Rn  1.2Lct Fu For exterior hole: Lc  1.875  0.469  1.41 in.

(AASHTO 6.13.2.9-2)

Rn  1.2 1 .410 .625 65   68 .7 kips

It is obvious that bearing resistance controls and nominal resistance per bolt is 68.7 kips. It is noted that AASHTO 6.13.2.7 specifies that the nominal shear resistance of a fastener in connections greater than 50 in. in length shall be taken 0.8 times the value given by AASHTO 6.13.2.7-1 and 6.13.2.7-2. Although the vertical length of web splices is greater than 50 in., shear resistance of the bolt is not reduced because the resultant shear applied to the bolts is mainly induced by horizontal force. 

Calculate Polar Moment of Inertia Ip of Bolts With Respect to Neutral Axis of Web Section It can be seen that the upper and lower right corner bolts are the most highly stressed and will be investigated. The “Vector” method is used to calculate shear force R on the top right bolt.

Chapter 9 - Steel Plate Girders

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I p   x2   y2



 2 3 30 2  27 2  24 2  212 182 152  12 2  9 2  6 2  32  2 213  21,168 in. 2





2

Check Shear Resistance of Lower Right Corner Bolt Case I - Positive Bending Factored shear force applied on the lower right corner bolt is: Rx 

4, 218  30  Mu y   5.98 kips (  ) Ip 21,168

Ry 

4, 218  3 Mu x   0.60 kips Ip 21,168

()

Rv 

Vuw 843.6   13.39 kips 63  3 21

()

Rh 

H uw 2, 437.5   38.69 kips (  ) 3 21 63   

Rbolt 

Rh



 Rx 2  Rv  R y

2

38.69  5.982  13.39  0.62  46.81 kips  bb Rn  0.868.7   54.96 

kips

O.K.

Case II - Negative Bending Factored shear forces applied on the lower right corner bolt is: 35,906  30  M y Rx  u   50.89 kips (  ) Ip 21,168 Ry 

35,906  3 Mu x   5.09 kips Ip 21,168

()

Rv 

Vuw 843.6   13.39 kips 63  3 21

( )

Rh 

H uw  0  3 21

Rbolt 

Rh



 Rx 2  Rv  R y

2

0  50.892  13.39  5.092  54.14 kips  bb Rn  0.868.7   54.96



kips O.K.

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Check Slip of Bolts – Service Limit State II and Constructability



Determine Factored Moment and Shear From Section 9.7.15.2, factored moment at 0.7 Point of Span 2 at Service II is:  M u  1, 434  118  280  2,784  4,616 kip - ft

From Table 9.7-2, it is obtained Vu  VDC1  VDC 2  VDW  1.3DFv LL  IM HL  93

  68.8   5.6    13.4   1.31.082  92.7    218.2 kips

Moment due to eccentrically loaded shear is:

M vu  Vu e  218.25  1,091.0 kip - in. Moment resisted by the web at Service II M uw 

tw D 2 f s  fos 12

(CA C6.13.6.1.4b-1c)

fs is maximum flexural stress due to Service II at the extreme fiber of the flange and fos is flexural stress due to Service II at the extreme fiber of the other flange concurrent with fs. Assume non-composite section at the splice location, we have

fs 

Mu 4,61612   25.26 ksi 2,193 S NCt

f os 

Mu 4,61612   19.52 ksi 2,837 S NCb

M uw  

(Compression) (Tension)

tw D 2 f s  f os 12

0.625782 12

 25.26 19.52  14,189.7 kips - in.

Horizontal force at web tw D  f s  fos  2 (CA C6.13.6.1.4b-2c) 0.62578  25.26  19.52   139.9 kips  2 Calculate Factored Shear Forces Applied on Upper Right Corner Bolt H uw 



Rx 

M vw  M uw y Ip

Chapter 9 - Steel Plate Girders



1,091.0  14,189.7 30  21.66 21,168

kips ()

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Ry 

M vw  M uw x Ip



1,091.0  14,189.7 3  2.17 kips 21,168

()

Vu 218.2   3.46 kips (  ) 321 63 H uw 139.9 Rh    2.22 kips (  ) 321 63

Rv 

Ru  

Rh



 Rx 2  Rv  R y

2

2.22  21.662  3.46  2.17 2

 24.53 kips 

Check Slip Resistance From Section 9.7.15.2, slip resistance of one bolt is: Rr  Rn  1 .0 0 .5 2 39   39 .0 kips  Ru  24 .53 kips

O.K.

Check Splice Plates



Check Shear Resistance (1)Yielding on the gross section:





Vr  v 0.58 A g Fw  1.0  0.58  2  0.4375  64  50 

O.K.

 1,624 kips  Vuw  843.6 kips (2) Fracture on net section: An  2 64  21 0.9375 0.4375   38.77 in.2 Vr  u 0.58 Fu An   0.80.58 65 38.77 

 1,169 .3 kips  Vuw  843 .6 kips

O.K.

(3) Block shear rupture The bolt pattern and block shear rupture failure planes on the inner and outer splice plates are assumed in Figure 9.7-23. Atn  2 8  2.5 0.9375 0.4375   4.95 in.2 Avn  2 62  20.5 0.9375 0.4375   37.43 in.2

Avg  2 62 0.4375  54.25 in.2

 Fu Avn  6537.43  2,433.0 kips  Fy Avg  5054.25  2,712.5 kips

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Rr  bs R p 0.58Fu Avn  U bs Fu Atn 

 0.8 1.00.586537.43  1.0504.95 1,326.9 kips (AASHTO 6.13.4-1) Rr 1,326.9 kips  Vuw  843.6 kips O.K. 

Check Flexural Resistance Moment of inertia of the splice plates about N.A. of the web section is:   0.4375 64 3    19,115 in.4 I w   2    12   M C 4,218  32  fb  u   7.06 ksi  Fy  50 ksi O.K. 19,115 I sp

Figure 9.7-23 Block Shear Rupture – Web Splice Plates Check Fatigue Stress Ranges

From AASHTO Table 6.6.1.2.3-1, Category B shall be used for base metal at the gross section of the high-strength bolted slip-critical resistant section. Similar assumptions used for flange splices are used for the web splice plates. Flexural stress ranges at the web splice plates are induced by the positive-negative fatigue moments and moments due to the eccentricity of the fatigue shear forces from the centerline of the splices to the center of gravity of the web-splice bolt group. Positive moments are assumed to be applied to the short-term composite section and negative moments are assumed to be applied to the steel section alone in the splice location. Eccentric

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

moments of the fatigue shear forces are assumed to be applied to the gross section of the web splice plates. For the smaller spliced section, INC = 99,872 in.4 (Table 9.715), and IST = 275,267 in.4 (Table 9.7-17). Refer to Tables 9.7-15 and 9.7-17, and Figure 9.7-22, at the locations of the edges of the web splice plates, elastic section modulus for the short-term composite section, the steel section, and the web splice plates are calculated as follows: I ST 275,267 275,267    4,606 in.3 SST wb  Csplicebot  68.511.75  7  ySTb  tbf  7



SST wt  S NC wb  S NC wt 

I ST Csplicetop

Csplicetop

S w splice  2

 ySTt  ttf

7





275,267  64,921 in.3 12.24 1.0  7 









99,872



 yNCb  tbf



99,872

Csplicebot I NC

275,267



I NC



 yNCT  ttf

7 7

99,872  3,776 in.3  35.2 1.75  7  99,872  2,660 in.3 45.55 1.0 7    

0.4375642

 597 in.3 6 It is seen that the bottom edge of the web splice plate obviously controls design and is, therefore, checked. Flexural stress ranges due to fatigue moments (Tables 9.711 and 9.7-12) and eccentric moments of the fatigue shear forces (Tables 9.7-11 and 9.7-12) are: Fatigue I - HL-93 Truck for infinite life:

 f  

M S ST  wb

M



 V    V e S w  splice

39812   16.5  60.85  4,606 3,776 597  2.71  1.26  0.65  4.62 ksi  16.0 ksi O.K. for Category B 

1,042 12

S NT  wb



Fatigue II - P-9 Truck for finite life:

 f   

M S ST  wb



1,69312

M S NT  wb 



 V    V e S w  splice

70312  21.1  97.95

4,606 3,776 597  4.41  2.23  1.00  7.64 ksi  30.15 ksi

Chapter 9 - Steel Plate Girders

O.K. for Category B

9-97

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.16

Calculate Deflection and Camber

9.7.16.1

Determine Stiffness of Girders

AASHTO 2.5.2.6.2 specifies that for composite design, the stiffness of the design cross-section used for the determination of deflection should include the entire width of the roadway and the structurally continuous portions of the railing, sidewalks, and median barriers. AASHTO C6.10.1.5 states that the stiffness properties of the steel section alone for the loads applied to noncomposite sections, the stiffness properties of the long-term composite section for permanent loads applied to composite sections and the stiffness properties of the short-term composite section properties for transient loads, shall be used over the entire bridge length, respectively. In this example, section properties of the steel section alone, the short-term section and the long-term composite sections are calculated in Tables 9.7-15 to 9.718, and Tables 9.7-21 to 9.7-24. 9.7.16.2

Calculate Vehicular Load Deflections

AASHTO 2.5.2.6.2 specifies that the maximum absolute deflection of the straight girder systems should be based on all design lanes loaded by HL93 including dynamic load allowance (Service I load combination) and all supporting components deflected equally. Number of traffic lanes = (deck width - barrier width)/12 = [58 - 2(1.42)]/12 = 4.596  Number of design traffic lanes = 4 For this five-girder bridge, each girder will carry 0.8 design traffic lane equally. Assume that the exterior girders have the same section properties as the interior girders, and use properties of the short-term composite sections of interior girder as shown in Tables 9.7-17 and 9.7-18, vehicular live load deflections are calculated and listed in Table 9.7-27. Comparisons with the AASHTO 2.5.2.6.2 requirement of the vehicular load deflection limit L/800 are also made in Table 9.7-27. Table 9.7-27 Live Load Deflections for Interior Girder Span 1 2 3

L (ft) 110 165 125

Chapter 9 - Steel Plate Girders

Vehicular Load Deflection (in.) 0.211 0.305 0.468

AASHTO Deflection Limit L/800 (in.) 1.650 2.475 1.875

Check O.K. O.K O.K.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.16.3

Calculate Camber

Camber of a structural member is defined as the difference between the shape of the member (the final geometric profile/grade of the member) under full dead load and normal temperature, and its shape at the no-load condition and shop temperature as discussed in MTD 12-3 (Caltrans, 2004b). For a steel-concrete composite girder, camber is the curvature/deformation induced by a fabrication process to compensate for dead load deflections. Camber of a steel-concrete composite girder includes three components: Deck Slab Dead Load, Steel Girder Dead Load, and Added Dead Load. Each camber component has the same magnitude and opposite direction with its respective deflection component. Deck Slab Dead Load and Steel Girder Dead Load deflections are due to the weight of the deck slab, stay-in-place deck form and steel girder, respectively. For unshored construction, it is assumed that all deck concrete is placed at once and deck slab dead load and steel girder dead load are applied to the steel girder section alone (Tables 9.7-15 and 9.7-21). An additional 10% more deflection shall be added to the deck slab dead load deflection to consider deflection induced by concrete shrinkage effects as specified in MTD 12-3 (Caltrans, 2004b). Added Dead Load deflection is due to weight of the curb, railing, utilities, and future AC overlay. Added dead load is applied to the long-term composite section girder (Tables 9.7-18 and 9.7-24). Camber diagram for each girder shall be presented in the design plan. Figure 9.724 shows the camber diagram of the interior girder.

C Bent Bent2 3

C Bent 2

C Abut 1

1/4

1/2 3/4 4 Equal Spaces

Ordinates Camber Component Interior Deck Slab Dead Load (in.) Girder Steel Girder Dead Load (in.) Added Dead Load (in.) Total Camber (in.)

1/4

1/4 0.34 0.12 0.03 0.49

1/2 3/4 4 Equal Spaces

Span 1 1/2 0.33 0.11 0.06 0.50

3/4 0.03 0.02 0.00 0.05

C Abut 34

1/4

1/4 1.12 0.41 0.11 1.64

Span 2 1/2 1.86 0.70 0.42 2.98

1/2 3/4 4 Equal Spaces

3/4 1.05 0.38 0.11 1.54

1/4 0.30 0.10 0.03 0.43

Span 3 1/2 0.83 0.31 0.16 1.30

3/4 0.75 0.28 0.07 1.10

Figure 9.7-24 Camber Diagram of Interior Girder

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

9.7.17

Identify and Designate Steel Bridge Members and Components It is the bridge designer’s responsibility to identify “Fracture Critical Members (FCMs)”, “Main Members”, “Secondary Members” and “Primary Components of Main Members” in designing a new steel bridge and to designate or tabulate them explicitly on the contract documents (plans and/or special provisions). MTD 12-2 (Caltrans 2012) provides guidelines for identification of steel members for steel bridges. Figure 9.7-25 shows the member designations of steel girders for this bridge.

Notes:

T – Denotes Main Tension Member (Non-Fracture Critical Member) C – Denotes Main Compression Member Primary Components of Main Members – flanges, webs, splice plates and cover plates, longitudinal stiffeners, bearing stiffeners and connection stiffeners Secondary Members – All members not designated as T, or C

Figure 9.7-25 Member Designations

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

NOTATION A

=

fatigue detail category constant

ADTT =

average daily truck traffic in one direction over the design life

ADTTS L =

single lane ADTT life

Ae

=

effective area (in.2)

Ag

=

gross cross section area (in.2)

An

=

net cross section area (in.2)

Arb

=

reinforcement area of bottom layer in concrete deck slab (in.2)

Art

=

reinforcement area of top layer in concrete deck slab (in.2)

Atg

=

gross area along the cut carrying tension stress in block shear (in.2)

Atn

=

net area along the cut carrying tension stress in block shear (in.2)

Avg

=

gross area along the cut carrying shear stress in block shear (in.2)

Atn

=

net area along the cut carrying shear stress in block shear (in.2)

bc

=

width of compression steel flange (in.)

bf

=

full width of the flange (in.)

bfc

=

full width of a compression flange (in.)

bft

=

full width of a tension flange (in.)

bs

=

width of concrete deck slab (in.)

bt

=

width of tension steel flange (in.)

C

=

ratio of the shear-buckling resistance to the shear yield strength;

Cb

=

moment gradient modifier

D

=

web depth (in.)

Dcp

=

web depth in compression at the plastic moment (in.)

Dp

=

distance from the top of the concrete deck to the neutral axis of the composite sections at the plastic moment (in.)

Dt

=

total depth of the composite section (in.)

d

=

total depth of the steel section (in.)

do

=

transverse stiffener spacing (in.)

DC

=

dead load of structural components and nonstructural attachments

DC1

=

structural dead load, acting on the non-composite section

DC2

=

nonstructural dead load, acting on the long-term composite section

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

DFm

=

live load distribution factor for moments

DFv

=

live load distribution factor for shears

DW

=

dead load of wearing surface

E

=

modulus of elasticity of steel (ksi)

Ec

=

modulus of elasticity of concrete (ksi)

Fcrw

=

nominal bend-buckling resistance of webs (ksi)

Fexx

=

classification strength specified of the weld metal

Fnc

=

nominal flexural resistance of the compression flange (ksi)

Fnt

=

nominal flexural resistance of the tension flange (ksi)

Fyc

=

specified minimum yield strength of a compression flange (ksi)

Fyf

=

specified minimum yield strength of a flange (ksi)

Fyr

=

compression-flange stress at the onset of nominal yielding including residual stress effects, taken as the smaller of 0.7Fyc and Fyw but not less than 0.5Fyc (ksi)

Fyrb

=

specified minimum yield strength of reinforcement of bottom layers (ksi)

Fyrt

=

specified minimum yield strength of reinforcement of top layers (ksi)

Fys

=

specified minimum yield strength of a stiffener (ksi)

Fyt

=

specified minimum yield strength of a tension flange (ksi)

Fyw

=

specified minimum yield strength of a web (ksi)

f c

=

specified minimum concrete strength (ksi)

fbu

=

flange stress calculated without consideration of the flange lateral bending (ksi)

fos

=

flexural stress due to Service II at the extreme fiber of the other flange concurrent with fs (ksi)

fl

=

flange lateral bending stress (ksi)

fc

=

longitudinal compressive stress in concrete deck without considering flange lateral bending (ksi)

fs

=

maximum flexural stress due to Service II at the extreme fiber of the flange (ksi)

fsr

=

fatigue stress range (ksi)

I

=

moment of inertia of a cross section (in.4)

IST

=

moment of inertia of the transformed short-term composite section (in.4)

It

=

moment of inertia for the transverse stiffener taken about the edge in contact with the web for single stiffeners and about the mid-thickness of the web for stiffener pairs (in.4)

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Iyc

=

moment of inertia of the compression flange about the vertical axis in the plane of web (in.4)

Iyt

=

moment of inertia of the tension flange about the vertical axis in the plane of web (in.4)

K

=

effective length factor of a compression member

Ka

=

surface condition factor

Kg

=

longitudinal stiffness parameter

Kh

=

hole size factor

L

=

span length (ft)

Lb

=

unbraced length of compression flange (in.)

Lp

=

limiting unbraced length to achieve RbRhFyc (in.)

Lr

=

limiting unbraced length to onset of nominal yielding (in.)

MAD

=

additional live load moment to cause yielding in either steel flange applied to the short-term composite section (kip- in.)

MD1

=

moment due to factored permanent loads applied to the steel section alone (kipin.)

MD2

=

moment due to factored permanent loads such as wearing surface and barriers applied to the long-term composite section (kip-in.)

Mn

=

nominal flexural resistance of the section (kip-in.)

Mp

=

plastic moment (kip-in.)

Mu

=

bending moment about the major axis of the cross section (kip-in.)

My

=

yield moment (kip-in.)

N

=

number of cycles of stress ranges; number of bolts

Nb

=

number of girders

NTH

=

minimum number of stress cycles corresponding to constant-amplitude fatigue threshold, (F)TH

n

=

number of stress-range cycles per truck passage

p

=

fraction of truck traffic in a single lane

Q

=

first moment of transformed short-term area of the concrete deck about the neutral axis of the short-term composite section (in.3)

Qi

=

force effect

Rh

=

hybrid factor

Rb

=

web load-shedding factor

Rn

=

nominal resistance

Chapter 9 - Steel Plate Girders

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Rp

=

reduction factor for hole

S

= girder spacing (in.); elastic section modulus (in.3)

SLT

=

elastic section modulus for long-term composite sections, respectively (in.3)

SNC

=

elastic section modulus for steel section alone (in.3)

SST

=

elastic section modulus for short-term composite section (in.3)

Sxt

=

elastic section modulus about the major axis of the section to the tension flange taken as Myt/Fyt (in.3)

tc

=

thickness of compression steel flange (in.)

tf

=

thickness of the flange (in.)

tfc

=

thickness of a compression flange (in.)

tft

=

thickness of a tension flange (in.)

tt

=

thickness of tension steel flange (in.)

tw

=

thickness of web (in.)

ts

=

thickness of concrete deck slab (in.)

Ubs

=

reduction factor for block shear rupture

U

=

reduction factor to account for shear lag

Vcr

=

shear-buckling resistance (kip)

Vn

=

nominal shear resistance (kip)

Vp

=

plastic shear force (kip)

f

=

slenderness ratio for compression flange = bfc/2tfc

pf

=

limiting slenderness ratio for a compact flange

rf

=

limiting slenderness ratio for a noncompact flange

i

=

load factor

D

=

ductility factor

R

=

redundancy factor

I

=

operational factor

(F)TH =

constant-amplitude fatigue threshold (ksi)

(F)n

=

fatigue resistance (ksi)

f

=

resistance factor for flexure

v

=

resistance factor for shear

c

=

resistance factor for axial compression

u

=

resistance factor for tension, fracture in net section

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y

=

resistance factor for tension, yielding in gross section

b

=

resistance factor for bearing on milled surfaces

bb

=

resistance factor for bolt bearing on material

sc

=

resistance factor for shear connector

bs

=

resistance factor for block shear rupture

s

=

resistance factor for bolts in shear

e2

=

resistance factor for shear in throat of weld metal in fillet weld

REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary US Units, (6th Edition), American Association of State Highway and Transportation Officials, Washington, DC. 2. AASHTO, (2002). Standard Specifications for Highway Bridges, 17th Edition, American Association of State Highway and Transportation Officials, Washington, DC. 3. Azizinamini, A., (2007). Development of a Steel Bridge System Simple for Dead Load and Continuous for Live Load, Volume 1: Analysis and Recommendations, National Bridge Research Organization, Lincoln, NE. 4. Barker, R. M. and Puckett, J. A., (2013). Design of Highway Bridges - Based on AASHTO LRFD Bridge Design Specification, 2nd Edition, John Wiley & Sons, Inc., New York, NY. 5. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications – Sixth Edition, California Department of Transportation, Sacramento, CA. 6. Caltrans, (2012). Bridge Memo to Designers 12-2: Guidelines for Identification of Steel Bridge Members, California Department of Transportation, Sacramento, CA. 7. Caltrans, (2008). Bridge Memo to Designers 10-20: Deck and Soffit Slab, California Department of Transportation, Sacramento, CA. 8. Caltrans, (2004a). Bridge Memo to Designers 12-4: Criteria for Control Dimension “Y” on Steel Girders, California Department of Transportation, Sacramento, CA. 9. Caltrans, (2004b). Bridge Memo to Designers 12-3: Camber of Steel-Concrete Composite Girders, California Department of Transportation, Sacramento, CA.

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10. Caltrans, (2000). Bridge Design Specifications, LFD Version, April 2000, California Department of Transportation, Sacramento, CA. 11. Caltrans, (2015). Bridge Memo to Designers 8-7: Stay-in-Place Metal Forms for Cast-inPlace Concrete Decks in Precast Concrete and Steel Superstructures, California Department of Transportation, Sacramento, CA. 12. Caltrans, (1988). Bridge Memo to Designers 15-17: Future Wearing Surface, California Department of Transportation, Sacramento, CA. 13. Chen, W.F., and Duan, L., (2014). Bridge Engineering Handbook, 2nd Edition: Superstructure Design, CRC Press, Boca Raton, FL. 14. FHWA, (2012). Steel Bridge Design Handbook, FHWA NHI-12-052, Federal Highway Administration, Washington, DC. http://www.fhwa.dot.gov/bridge/steel/pubs/if12052/ 15. Taly, N., (2014). Highway Bridge Superstructure Engineering: LRFD Approaches to Design and Analysis, CRC Press, Boca Raton, FL.

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CHAPTER 10 CONCRETE DECKS TABLE OF CONTENTS 10.1

INTRODUCTION ......................................................................................................... 10-1

10.2

CONCRETE DECK TYPES ......................................................................................... 10-1

10.3

10.2.1

Cast-In-Place Concrete Decks....................................................................... 10-1

10.2.2

Precast Concrete Decks ................................................................................. 10-2

DESIGN APPROACH .................................................................................................. 10-2 10.3.1

Structural Behavior of Concrete Decks......................................................... 10-2

10.3.2

Limit State ..................................................................................................... 10-3

10.3.3

Methods of Analysis ..................................................................................... 10-4

10.4

DESIGN CONSIDERATIONS ..................................................................................... 10-6

10.5

DETAILING CONSIDERATIONS .............................................................................. 10-6

10.6

10.5.1

Reinforcement Details................................................................................... 10-6

10.5.2

Skewed Decks ............................................................................................... 10-7

10.5.3

Deck Drains and Access Openings ............................................................... 10-8

DESIGN EXAMPLE – REINFORCED CONCRETE BRIDGE DECK ...................... 10-8 10.6.1

Concrete Deck Data ...................................................................................... 10-8

10.6.2

Design Requirements .................................................................................... 10-9

10.6.3

Determine Minimum Deck Thickness and Cover ......................................... 10-9

10.6.4

Compute Unfactored Dead Load Moments .................................................. 10-9

10.6.5

Compute Unfactored Live Load Moments ................................................. 10-10

10.6.6

Calculate the Factored Design Moments .................................................... 10-11

10.6.7

Positive Flexure Design .............................................................................. 10-12

10.6.8

Negative Flexure Design ............................................................................. 10-14

10.6.9

Check for Crack Control under Service Limit State ................................... 10-16

10.6.10

Minimum Reinforcement ............................................................................ 10-18

NOTATION ............................................................................................................................. 10-21 REFERENCES ......................................................................................................................... 10-23

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CHAPTER 10 CONCRETE DECKS 10.1

INTRODUCTION Bridge decks are an integral part of the bridge structure by providing the direct riding surface for motor vehicles. In addition, bridge decks directly transfer load from the moving traffic to the major load-carrying members. This chapter provides a general description of the various concrete deck types, a discussion of the basic structural behavior of concrete decks, and an overview of major design and detailing considerations. Finally, a design example for a reinforced concrete bridge deck is provided. The example illustrates bridge deck design in accordance with the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) and the California Amendments (Caltrans, 2014).

10.2

CONCRETE DECK TYPES There are two main types of concrete decks, cast-in-place, and precast. The most common type used in Caltrans is the cast-in-place reinforced concrete deck. The other type is used depending on the various conditions like location, traffic, cost, seismicity schedule, and aesthetics (Chen and Duan, 2014).

10.2.1

Cast-In-Place Concrete Decks A cast-in-place concrete deck is a thin concrete slab, either using normal reinforcement or prestressing steel, usually between 7 and 12 inches, with reinforcing steel interspersed transversely and longitudinally throughout the slab. There are several advantages to using a reinforced concrete deck. One of the major advantages is its relatively low cost. Other advantages are ease of construction and extensive industry use. Even though cast-in-place concrete decks have advantages, there are disadvantages using this particular type of deck, such as cracking, rebar corrosion, and tire noise. A large cost of bridge maintenance is in maintaining the riding surface (Fu, et al., 2000). Lack of deck crack control can lead to rebar corrosion and increased life cycle cost, not to mention a poor riding surface for the public.

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10.2.2

Precast Concrete Decks Precast concrete decks consist of either precast reinforced concrete panels or prestressed concrete panels. These panels can either serve as the final deck surface or as a temporary deck to allow placement of a final cast-in-place concrete deck. The advantage of a precast concrete deck is in the acceleration of the construction schedule. Precast panels allow for quicker placement, which, in principle, speeds up overall bridge construction.

10.3

DESIGN APPROACH

10.3.1

Structural Behavior of Concrete Decks It is accepted and widely known that the primary structural behavior of a concrete deck is not pure flexure, but a complex behavior known as internal arching. Concrete slabs behave quite differently than concrete beams under a given load. Research has shown that when a concrete slab starts to crack, the load is initially resisted by a combination of flexure stresses and membrane stresses as shown in Figure 10.3-1 (Csagoly, et al., 1989). The stresses and strain create cracks in three dimensions around the wheel footprint. The way internal arching works is as cracks develop in the bottom of the slab and the slab’s neutral axis shifts upward, compressive stresses develop above the neutral axis to resist further opening of the cracks. The concrete portion above the crack is in a purely elastic state. Therefore, what results is a domed shaped compression zone around the load. The compressive membrane stresses do not resist the loading completely. There is a small flexural component that also resists the loading as well. But the controlling structural mechanic is the membrane compressive stresses created in the upper parts of the slab. For the deck to fail, as the load is increased the deflection also increases. The section around the load becomes overstrained and this results in a cone-shaped section of failed concrete. Therefore, the primary failure mode is punching shear.

Figure 10.3-1 Concrete Deck Showing Flexure and Membrane Forces Chapter 10 – Concrete Decks

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10.3.2

Limit State

10.3.2.1

Service Limit State Concrete decks are designed to meet the requirements for Service I limit state (AASHTO Article 9.5.2). Service limit state is used to control excessive deformation and cracking. According to the California amendment (CA Article 9.5.2), deck slabs shall be designed for Class 2 exposure, therefore,  e  0.75

10.3.2.2

(AASHTO Article 5.7.3.4)

Strength Limit State Concrete decks must be designed for Strength I limit state. Because concrete deck slabs are usually designed as tension-controlled reinforced concrete components, the resistance factor is   0.9 (AASHTO Article 5.5.4.2). Strength II limit state typically is not checked for deck designs. The permit vehicle axle load does not typically control deck design (CA Article C3.6.1.3.3).

10.3.2.3

Extreme Event Limit State Most bridge decks include an overhang with a concrete barrier attached. Therefore, the deck overhang has to be designed to meet the requirements for Extreme Event II. The AASHTO (2012) requires bridge deck overhangs to be designed for the following cases (AASHTO Appendix A13.4): Design Case 1: The transverse and longitudinal forces specified in AASHTO Appendix A13.2 - Extreme Event Load Combination II limit state. Design Case 2: The vertical forces specified in AASHTO Appendix A13.2 Extreme Event Load Combination II limit state. Design Case 3: The loads, specified in AASHTO Article 3.6.1, that occupy the overhang- Load Combination Strength I limit state.

10.3.2.4

Fatigue Limit State Concrete decks supported by multi-girder systems are not required to be investigated for fatigue (AASHTO Article 9.5.3).

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10.3.3

Methods of Analysis

10.3.3.1

Approximate Method of Analysis Caltrans traditionally designs concrete bridge decks as transverse strips as a flexure member. This method is called the Approximate Method of Analysis (AASHTO Article 4.6.2.1). The concrete deck is assumed to be transverse slab strips, which is supported by the girders. To simplify the design, it is assumed that the girders are rigid supports. The AASHTO specifications allow the maximum positive moment and the maximum negative moment to apply for all positive moment regions and all negative moment regions in the slab, respectively. The width of an equivalent strip (interior strip) is dependent on the type of deck used, the primary direction of the strip relative to the direction of traffic, and the sign of the moment. AASHTO Table 4.6.2.1.3-1 only applies for interior strips and not for overhangs.

10.3.3.2

Refined Methods of Analysis The Refined Methods of Analysis (AASHTO Article 4.6.3) as listed in AASHTO 4.4 are acceptable methods for analyzing concrete bridge decks. But these various methods are not typically used to analyze a standard bridge deck. A refined analysis method would be better suited for a more complex deck slab structure, which would require a more detailed analysis.

10.3.3.3

Empirical Method of Analysis Empirical Design (AASHTO Article 9.7.1) is a method of deck slab design based on the concept of internal arching action within concrete slabs. But, until further durability testing of this design method is completed, the empirical design method is not permitted for concrete bridge deck design in California (CA Article 9.7.2.2).

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AASHTO Table 4.6.2.1.3-1 Equivalent Strips Type of Deck

Direction of Primary Strip Relative to Traffic

Concrete:  Cast-in-place

Width of Primary Strip (in.)

Overhang Either Parallel or Perpendicular

45.0 + 10.0X +M: 26.0 + 6.6S -M: 48.0 +3.0S

 Cast-in-place with stay-inplace concrete formwork

Either Parallel or Perpendicular

+M: 26.0 + 6.6S -M: 48.0 +3.0S

 Precast, post-tensioned

Either Parallel or Perpendicular

+M: 26.0 + 6.6S -M: 48.0 +3.0S

Steel:  Open grid  Filled or partially filled grid  Unfilled, composite grids

Main Bars Main Bars Main Bars

1.25 P + 4.0 Sb Article 4.6.2.1.8 applies Article 4.6.2.1.8 applies

Wood:  Prefabricated glulam ○ Non interconnected

Parallel Perpendicular

2.0 h + 30.0 2.0 h + 40.0

○ Interconnected

Parallel Perpendicular

90.0 + 0.84L 4.0 h + 30.0

 Stress-laminated

Parallel Perpendicular

0.8 S + 108.0 10.0 S + 24.0

 Spike-laminated ○ Continuous decks or Interconnected panels

Parallel Perpendicular

2.0 h + 30.0 4.0 h + 40.0

Parallel Perpendicular

2.0 h + 30.0 2.0 h + 40.0

○ Non interconnected panels

S h L P Sb +M -M X

= = = = = = = =

spacing of supporting components (ft) depth of deck (in.) span length of deck (ft) axle load (kip) spacing of gird bars (in.) positive moment negative moment distance from load to point of support (ft)

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10.4

DESIGN CONSIDERATIONS Concrete decks are primarily designed for flexure in the transverse direction; therefore, the spacing of adjacent girders is important. AASHTO Table A4-1 provides a listing of design moments for “S” (girder spacing) values between 4 ft and 15 ft. When “S” exceeds 15 ft, special design is required. The designer needs to consider required deck thickness. According to CA 9.7.1.1 (Caltrans, 2014), the minimum deck thickness is to “conform to the deck design standards developed by Caltrans”. For the typical deck slab, MTD 10-20 (Caltrans, 2008) provides the minimum deck slab thicknesses and reinforcement schedule for various girder types. The typical deck slab thickness varies from 7 in. to 10 3/8 in. depending on the girder type and spacing. The minimum concrete cover is determined in accordance with CA 5.12.3. Extreme environments can have a negative effect on the service life of a concrete deck slab. Corrosion of the reinforcing steel should be a major concern when designing a bridge deck in an extreme environment. There are various ways for the designer to protect against corrosion of the bridge deck. See MTD 10-5 (Caltrans, 2010) for more information on protecting against corrosion. For design purposes, the minimum compressive concrete strength f c' =3.6 ksi shall be used for reinforced concrete (CA Article 5.4.2.1).

10.5

DETAILING CONSIDERATIONS

10.5.1

Reinforcement Details Typical reinforced concrete decks are detailed as part of the superstructure typical section. The designer should use MTD 10-20 (Caltrans, 2008) for required minimum reinforcement and Standard Plan B0-5 for transverse reinforcement spacing diagrams. It is important to check main longitudinal reinforcement spacing and cover to ensure reinforcing steel can fit within slab thickness. For variable width girders it is important for the designer to specify reinforcement spacing and type that differs from the standard superstructure bay. See BDD 8-34 (Caltrans, 1986) for more information on acceptable reinforcement detailing in variable bays. Limits of epoxy-coated reinforcement shall also be specified.

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10.5.2

Skewed Decks Designers must give special consideration to detailing reinforcement for skewed bridge decks. For skews less than 20° the transverse deck reinforcement is parallel to the centerline of the abutment. For skew angles greater than 20° the transverse deck reinforcement is normal to the center line of the girder. Special corner details are usually required to clarify the deck reinforcement and diaphragm connections. BDD 8-36 (Caltrans, 1971) provides examples of the correct way to detail skewed deck corners as shown in Figure 10.5-1.

Figure 10.5-1 BDD 8-36 Skewed Deck Corner Reinforcement

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10.5.3

Deck Drains and Access Openings After determination of the type of deck drain and the deck drain location, it is important to ensure that the drain fits the location specified. Depending on the type of deck drain used it may be necessary to provide additional reinforcement to secure the drain assembly in place. Deck openings are discouraged. Soffit openings are the preferred method to provide access into the bridge bays. If deck openings are used then Standard Plan B7-11 Utility Details provide additional reinforcement for openings.

10.6

DESIGN EXAMPLE – REINFORCED CONCRETE BRIDGE DECK

10.6.1

Concrete Deck Data A typical section of a reinforced concrete box girder bridge is shown in Figure 10.6-1. Max. Girder Spacing: S = 12 ft. Number of Girders: N = 5 Concrete compressive strength: f c' = 3.6 ksi (CA 5.4.2.1) Reinforcement strength: fy = 60 ksi Type 732 Concrete Barrier weight: w732 = 0.410 klf Future wearing surface, wfws = 0.140 kcf (AASHTO Table 3.5.1-1) (Assume 3″ thick asphalt section were, wfws = 0.140 kcf × 0.25ft = 0.035 ksf) Reinforced Concrete unit weight, wrc = 0.150 kcf (AASHTO C3.5.1)

Figure 10.6-1 Reinforced Concrete Box Girder Bridge Typical Section

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10.6.2

Design Requirements Design the concrete bridge deck supported by the girders using the Approximate Method of Analysis (AASHTO Article 4.6.2) in accordance with the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) with the California Amendments (Caltrans, 2014).

10.6.3

Determine Minimum Deck Thickness and Cover According to CA Article 9.7.1.1 (Caltrans, 2014), the minimum deck thickness must conform to the deck design standards specified by MTD 10-20 (Caltrans, 2008). The minimum deck thickness varies depending on the girder type and the centerline to centerline spacing. Therefore, for a reinforced concrete box girder bridge, S = 12.00 ft assumes a minimum thickness t = 9 1/8 in. The minimum concrete cover is determined according to CA Table 5.12.3-1. For the top surface of the bridge deck in a non-corrosive atmosphere the minimum cover is specified as: Deck Top Cover:

Ctop = 2.0 in.

The minimum cover specified for the bottom surface of the deck slab is: Deck Bottom Cover: Cbottom = 1.0 in.

10.6.4

Compute Unfactored Dead Load Moments The dead load moments for the deck slab, barrier and future wearing surface are computed for a one-foot wide section of the bridge deck using any approved structural analysis method. This can include the continuous beam equations, moment distribution, or an acceptable computer analysis program. Table 10.6-1 lists the tabulated unfactored dead load moments for each bay given in tenth points using a finite element analysis.

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Table 10.6-1 Unfactored Dead Load Moments Slab Dead Load-DC1 (kip-ft/ft)

Future Wearing Surface Dead Load-DW (kip-ft/ft)

Bay 1

Bay 2

Bay 3

Bay 4

Bay 1

Bay 2

Bay 3

Bay 4

Bay 1

Bay 2

Bay 3

Bay 4

0

-1.62

-1.30

-1.40

-1.30

-1.81

0.51

-0.25

0.51

-0.26

-0.46

-0.40

-0.46

0.1

-0.84

-0.57

-0.65

-0.60

-1.56

0.43

-0.18

0.28

-0.05

-0.23

-0.18

-0.22

0.2

-0.24

-0.01

-0.07

-0.05

-1.35

0.36

-0.10

0.05

0.11

-0.05

-0.01

-0.02

0.3

0.21

0.40

0.35

0.33

-1.11

0.28

-0.02

-0.18

0.21

0.08

0.11

0.13

0.4

0.05

0.63

0.61

0.55

-0.88

0.21

0.05

-0.42

0.27

0.16

0.18

0.22

0.5

0.60

0.70

0.70

0.60

-0.65

0.13

0.13

-0.65

0.27

0.20

0.20

0.27

0.6

0.55

0.61

0.63

0.48

-0.42

0.05

0.21

-0.88

0.22

0.18

0.16

0.27

0.7

0.33

0.35

0.40

0.21

-0.18

-0.02

0.28

-1.11

0.13

0.11

0.08

0.21

0.8

-0.05

-0.07

-0.01

-0.24

0.05

-0.10

0.36

-1.35

-0.02

-0.01

-0.05

0.11

0.9

-0.60

-0.65

-0.57

-0.84

0.28

-0.18

0.43

-1.56

-0.22

-0.18

-0.23

-0.05

1

-1.30

-1.40

-1.30

-1.62

0.51

-0.25

0.51

-1.81

-0.46

-0.40

-0.46

-0.26

Distance

10.6.5

Barrier Load-DC2 (kip-ft/ft)

Compute Unfactored Live Load Moments The unfactored live load moments are determined from AASHTO Appendix A4, Table A4-1. This table can be used for decks supported on at least three or more girders, in which the minimum superstructure width, between centerline to centerline of exterior girders, cannot be less than 14′-0″. The moments are calculated using the equivalent strip method (AASHTO Article 4.6.2.1.3) for concrete slabs supported on parallel girders. The values given in the table include multiple presence factors and the dynamic load allowance. To be conservative use the largest span length between girders to find the maximum live load force effect. AASHTO Table A4-1 Maximum Live Load Moments Per Unit Width, kip-ft/ft

S 11′ -9″ 12′ -0″ 12′ -3″ 12′ -6″ 12′ -9″

Positive Moment 7.88 8.01 8.15 8.28 8.41

NEGATIVE MOMENT Distance from CL of Girder to Design Section for Negative Moment 0.0 in. 3 in. 6 in. 9 in. 12 in. 18 in. 24 in. 10.01 9.12 8.24 7.36 6.47 5.40 5.05 10.28 9.40 8.51 7.63 6.74 5.56 5.21 10.55 9.67 8.78 7.90 7.02 5.75 5.38 10.81 9.93 9.04 8.16 7.28 5.97 5.54 11.06 10.18 9.30 8.42 7.54 6.18 5.70

From the table, use a maximum spacing of 12′-0″ for the girder spacing. The positive live load moment is 8.01 kip-ft/ft. The maximum negative moment is given in relationship to the distance from the centerline of the girder to the design section. In the design example, the girder width is 8 inches. The maximum negative moment design section is 4 in., which is the distance from the girder centerline to the edge of the girder (AASHTO Article 4.6.2.1.6). The closest value to 4 in. in the Table is 3 in. Therefore, the corresponding maximum negative moment is 9.40 kip-ft/ft. Chapter 10 – Concrete Decks

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10.6.6

Calculate the Factored Design Moments Concrete decks are designed for strength, service and extreme limit states according to AASHTO Article 9.5. Fatigue and fracture limit states need not be investigated for concrete decks supported by multiple girders (AASHTO Article 5.5.3). Therefore, Strength I Load Combination M u   [ DC M DC   DW M DW  (m)(1  IM )( LL )(M LL )]

     

10.6.6.1

  1.0 (Note: Per CA 1.3.4, η = 1.0 until its application is better defined.) For the slab and barrier (DC):  DC  1.25 Maximum factor For future wearing surface (DW):  DW  1.50 Maximum factor Multiple Presence Factor, m =1.20 for one lane of vehicular live load. This value is included in the tabulated moments provided in Table A4-1. Dynamic Load Allowance, IM, is also included in Table A4-1 tabulated moments.  LL  1.75 , for Strength I Load Combinations.

Maximum Positive Factored Moments Based on Table 10.6-1, the maximum unfactored positive moment due to the slab, barrier, and future wearing surface is located in Bay 2 or 3 at a distance of 0.5S. The maximum positive moment equals 8.01 kip-ft/ft, as shown in Section 10.6.5. Therefore, the maximum positive factored moment is Mu = 1.0[(1.25)(0.70+0.13)+(1.5)(0.20)+(1.75)(8.01)] = 15.36 kip-ft/ft

10.6.6.2

Maximum Negative Factored Moments The maximum unfactored negative moment due to the slab, barrier and future wearing surface is located in Bay 1 at a distance of 0 (center of exterior girder). The negative moment can be reduced at the face of the girder. The negative moment can be interpolated between 0.0S and 0.1S. Based on the reduced negative moments at the interior face of the girder, the maximum negative factored moment is Mu = 1.0[(1.25)(1.29+1.71)+(1.5)(0.17)+(1.75)(9.40)] = 20.46 kip-ft/ft

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10.6.7

Positive Flexure Design To design for the maximum positive moment, first, assume an initial bar size. From this initial bar size, the required area of steel can be calculated. Then the reinforcement spacing can be determined. For this example the assumed initial reinforcing steel size is #5. For a #5-bar, Bar area = 0.31 in2 Bar diameter = 0.625 in. Next, determine the effective depth, de , which is equal to the total slab thickness minus the bottom cover, Cbottom , and minus half the bar diameter. Figure 10.6-2 shows the relationship between the effective depth and the slab thickness. d e  t  C bottom 

(bar_ diameter) (0.625)  9.125  1.0   7.813 in. 2 2

Figure 10.6-2 Simple Rectangular Concrete Section with Tension Reinforcement For a rectangular section, assume a  2t , where a is the depth of the equivalent stress block, and t is the deck thickness. Solve for the required amount of reinforcement, as follows:

As 

4M u  z 1  1   2   f y d e z 

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where,

z

1.7 f c' bd e fy

  0.9 (Assume tension control) b = 12 in.

(AASHTO 5.5.4.2.1)

Therefore, z = 9.56 in.2 As = 0.459 in.2 Check a 

a

t , 2

As f y 0.85 f ' c b

 0.75 in  4.56 in.

Next, determine the required bar spacing, bar _ spacing 

12  bar _ area 12  0.31   8.10 in. use, 8 in. spacing for #5 bar. As 0.459

To verify that tension controls the section design and that the proper resistance factor is used to check the strain in the extreme tension reinforcing steel (CA Article 5.7.2.1). The strain, stress and force diagrams for a rectangular concrete section are shown in Figure 10.6-3.

Figure 10.6-3 Development of Bending Strain, Stress, and Force Actions in a Section

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Calculate the tension force and the area of concrete.

12(0.31)  0.465 in.2 8 T  As f y  0.465(60)  27.9 kips #5 @ 8"  As 

T  Ac (0.85) f c'  Ac 

T 27.9   9.12 in.2 ' 0 . 85 ( 3 . 6 ) 0.85 f c

Determine concrete compression block depth, a a

Ac 9.12   0.76 in. b 12

Determine c from the equation a   1 c c

a 0.76   0.89 1 0.85

Check if steel yields,

s 

y (7.813  0.89) (0.003)  (0.003)  0.023  0.004 c 0.89

The steel yields and the section is tension controlled, therefore, the proper resistance factor was used. Finally, check maximum spacing for primary reinforcement (AASHTO Article 5.10.3.2). The spacing of the slab reinforcement shall not exceed 1.5 times the thickness of the slab or 18.0 in. In this case the maximum spacing equals 8.0 in., which is less than either case.

10.6.8

Negative Flexure Design Designing for the maximum negative flexure reinforcement is similar to designing for the maximum positive moment reinforcement. These bars are the primary reinforcing steel over the girders as shown in Figure 10.6-4.

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Figure 10.6-4 Negative Reinforcement over a Typical Girder Assuming #6 bar size. Bar area = 0.44 in.2 Bar diameter = 0.750 in. The effective depth, de , for the negative moment is equal to the total slab thickness minus the top cover, Ctop , and half the bar diameter.

d e  t  Ctop 

(bar _ diameter) (0.750)  9.125  2.0   6.75 in. 2 2

Based on the maximum negative moment z = 8.262 in.2 As = 0.840 in.2 Check a 

a

t , 2

As f y 0.85 f ' c b

 1.209 in.  4.56 in.

Required bar spacing is bar _ spacing 

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12  bar _ area 12  0.44   7.14 in.2 , use 7.0 in. spacing. As 0.740

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Verify that tension controls the section design and that the proper resistance factor is used to check the strain in the extreme tension reinforcing steel (CA Article 5.7.2.1). Finally, check maximum spacing for primary reinforcement per AASHTO Article 5.10.3.2.

10.6.9

Check for Crack Control under Service Limit State Concrete cracking is controlled by the spacing of flexure reinforcement. To improve crack control in the concrete deck, the reinforcement has to be well distributed over the area of maximum tension. Therefore, AASHTO requires steel reinforcement spacing to satisfy the following: s

700 e  2d c  s f ss

(AASHTO 5.7.3.4-1)

In which, s  1

dc 0.7 (h  d c )

where: e =

0.75 for Class 2 exposure conditions

(CA 5.7.3.4 & 9.5.2)

(CA 5.7.3.4) d c = 2 1/2 in. for bridge decks f ss = tensile stress in steel reinforcement at service limit state (ksi) h 10.6.9.1

=

overall thickness of deck (in.)

Determine Maximum Loading under Service Limit State Service I Load Combination M s  1.0 [1.0M DC  1.0M DW  (m)(1  IM )1.0(M LL )]

Positive moment due to service loading: M positive  1.0 [(1.0)(0.70  0.13)  (1.0)(0.20)  (1.0)(8.01)]  9.04 kip-ft/ft

Negative moment due to service loading: M negative  1.0 [(1.0)(1.29  1.71)  (1.0)(.17)  (1.0)(9.40)]  12.57 kip-ft/ft

Negative Service Loading Controls.

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10.6.9.2

Determine Maximum Required Spacing for Crack Control Determine the neutral axis location, y, based on the transformed section properties.

b 2 y  nAs y  nAs d e  0 2 Let

b 2 B  nAs A

C  nAs d e Therefore, y

 B  B 2  4 AC 2A

As = bar area 

12 = 0.754 in.2 7. 0

n=8

A

b 12  6 2 2

B  nAs  8  0.754  6.03 C  nAs d e  (8  0.754  6.75)  40.72

y

 B  B 2  4 AC = 2.151 in. 2A

Calculate the crack moment of inertia, Icr , for the transformed section. by 3 I cr   nAs (d e  y ) 2  167.44 in.4 3 Next, calculate the tensile stress, fss , in the steel reinforcement at service limit state. nM s (d e  y ) f ss   33.15 ksi I cr Finally, determine βs, and then input that value into the formula to calculate the maximum spacing, s, of the reinforcement that would satisfy the LRFD crack control requirement.

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s  1 

s

dc 2.5  1  1.539 0.7(h  d c ) 0.7  (9.125  2.5)

700 e 700  (0.75)  2d c   (2)  (2.5)  5.29 in.  s f ss (1.539)  (33.15)

Figure 10.6-5 shows Caltrans’ Standard Plan for transverse deck reinforcement spacing. To convert the required spacing to Caltrans’ Standard Plan Bridge Details 5-11 or 5-10, you must multiply the spacing by 2. This is due to the fact that the transverse deck reinforcement spacing diagrams add an extra top bar for the given spacing. Therefore, if the required calculated spacing is 5.29 in., then the spacing shown on the typical section would be 10.58 in. In the design example case, we would specify #6 @ S = 10 1/2 in.

Figure 10.6-5 Standard Plan Spacing Details

10.6.10

Minimum Reinforcement Minimum reinforcement is needed in the slab to distribute the load across the slab, for shrinkage, and temperature change. For the typical slab design, AASHTO (2012) requires distribution reinforcement for the top of the slab and the bottom of the slab.

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10.6.10.1

Top of Slab Shrinkage and Temperature Reinforcement The top slab reinforcement is for shrinkage and temperature changes near the surface of the exposed concrete slab. The area of reinforcement has to meet the following requirements: 1.3bh 2( b  h ) f y

(AASHTO 5.10.8-1)

0.11  As  0.60

(AASHTO 5.10.8-2)

As 

where: As b h fy

= = = =

area of reinforcement in each direction and each face (in.2/ft) least width of component section (in.) least thickness of component section (in.) specified yield strength of reinforcing bars  75 ksi

For the design example slab:

As 

2 1.3(12)(9.125) 1.3bh   0.056in. ft 2(b  h) f y 2(12  9.125)(60)

As  0.056  0.11 , therefore, As = 0.11 as a minimum.

Use #4 bars in top slab. Bar area = 0.2 in.2 12(0.2)  21.8 in. use maximum 18.0 in. Bar spacing = 0.11

10.6.10.2

Bottom of Slab Distribution Reinforcement The reinforcement in the bottom of the slab is a percentage of the primary deck reinforcement. The primary deck reinforcement is perpendicular to the direction of traffic, therefore, the requirement is 220 S  67 percent, where S is the effective span length (ft) as specified in Article 9.7.2.3. For the design example, the effective span length is the clear distance from face of girder to face of girder, which is 11 ft. 220

S  220 11  66%  67%

2 Arequired  0.66  1.01  0.67 in.

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To facilitate better lateral load distribution, Caltrans’ Reinforced Concrete Technical Committee recommends that the bottom deck reinforcement be placed within the center half of the deck span. Therefore,

1 11 S   5.5 ft 2 2 2 5.5  0.67 in.  3.685 in.2  0.31 in.2  11.89  12 bars ft Twelve bars are distributed within 1/2 the effective span length. Compared this with the standard Deck Slab Reinforcement Details “G” bars and “D” bars per MTD Table 10-20.1(a): The required “G” bars for 12′-0″ girder spacing are 5 - #4 bars The required “D” bars for 12′-0″ girder spacing are 13 - #5 bars Figure 10.6-6 Shows the detailed deck reinforcement for the design example.

Figure 10.6-6 Bridge Deck Reinforcement Detail

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NOTATION As

=

area of tension reinforcement (in.2)

a

=

depth of equivalent rectangular stress block (in.)

b

=

with of the compression fact of the member (in.)

Cbottom =

required concrete cover over bottom deck reinforcing steel (in.)

Ctop

=

required concrete cover over top deck reinforcing steel (in.)

dc

=

thickness of concrete cover measured from extreme tension fiber to center of bar (in.)

de

=

effective depth from extreme compression fiber to the centroid of the tensile force in the tension reinforcement (in.)

=

specified compressive strength of concrete for use in design (ksi)

fy

=

specified minimum yield strength of reinforcing bars (ksi)

fss

=

tensile stress in mild steel reinforcement at the service limit state (ksi)

h

=

overall thickness of deck (in.)

Icr

=

moment of inertia of the cracked section, transformed to concrete (in.4)

IM

= dynamic load allowance

MDC

=

moment due to dead load (kip-ft)

MDW

=

moment due to dead load wearing surface (kip-ft)

MLL

=

moment due to live load (kip-ft)

Ms

=

moment due to service loads (kip-ft)

Mu

=

factored moment at the section (kip-ft)

m

=

multiple presence factor

N

=

number of girders

n

=

modular ratio = Es/Ec

S

=

center to center spacing of girder (ft.); effective span length (ft)

t

=

thickness of slab (in.)

w

=

uniform load (k/ft)

y

=

distance from neutral axis location to the extreme tension fiber (in.)

e

=

crack control exposure condition factor

DC

=

load factor for permanent dead load

DW

=

load factor for component dead load

f c

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LL

=

load factor for live load



=

load modifier



=

resistance factor

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, 4th Edition, Washington, D.C. 2. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2008). Bridge Memo to Designers (MTD) 10-20: Deck and Soffit Slabs, California Department of Transportation, Sacramento, CA. 4. Caltrans, (2010). Bridge Memo to Designers (MTD) 10-20: Protection of Reinforcement Against Corrosion Due to Chlorides, Acids Sulfates, California Department of Transportation, Sacramento, CA. 5. Caltrans, (2002). Bridge Memo to Designers (MTD) 10-5: Protection of Reinforcement Against Corrosion Due to Chlorides, Acids and Sulfates, California Department of Transportation, Sacramento, CA. 6. Caltrans, (1986). Bridge Design Details (BDD) 8-34: Variable Bay Transverse Reinforcement, California Department of Transportation, Sacramento, CA. 7. Caltrans, (1971). Bridge Design Details (BDD) 8-36: Skewed Deck Corner Reinforcement, California Department of Transportation, Sacramento, CA.

8. Caltrans, (2015). Standard Plans, California Department of Transportation, Sacramento, CA. 9. Chen, W. F., and Duan, L. (2014). Bridge Engineering Handbook, 2nd Edition, CRC Press, Boca Raton, FL. 10. Csagoly, P. and Lybas, J. (1989). “Advance Design Method for Concrete Bridge Deck Slabs,” Concrete International, ACI. Vol.11. 11. Fu, G. K., et al, (2000). “Effect of Truck Weight on Bridge Network Costs”, NCHRP: Report 495, Transportation Research Board, Washington, D.C.

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CHAPTER 12 CONCRETE BENT CAPS TABLE OF CONTENTS 12.1

INTRODUCTION ......................................................................................................... 12-1 12.1.1 Types of Bent Caps .......................................................................................... 12-2 12.1.2 Member Proportioning ................................................................................... 12-11

12.2

LOADS ON BENT CAPS ........................................................................................... 12-12 12.2.1 Permanent Loads ............................................................................................ 12-12 12.2.2 Transient Loads .............................................................................................. 12-13

12.3

DESIGN CONSIDERATIONS ................................................................................... 12-18 12.3.1 Flexural Design .............................................................................................. 12-18 12.3.2 Design for Shear ............................................................................................. 12-30 12.3.3 Check Longitudinal Steel for Tension (Shear-Flexure Interaction) ............... 12-33 12.3.4 Design for Seismic ......................................................................................... 12-35

12.4

DETAILING CONSIDERATIONS ............................................................................ 12-36 12.4.1 Construction Reinforcement........................................................................... 12-36 12.4.2 Side Face Reinforcement................................................................................ 12-37 12.4.3 End Reinforcement ......................................................................................... 12-38 12.4.4 Other Detailing Considerations (Skew).......................................................... 12-38

12.5

DESIGN EXAMPLES ................................................................................................ 12-39 12.5.1 Integral Bent Cap ........................................................................................... 12-39 12.5.2 Drop Bent Cap ................................................................................................ 12-71

NOTATIONS ........................................................................................................................... 12-96 REFERENCES ....................................................................................................................... 12-100

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CHAPTER 12 CONCRETE BENT CAPS 12.1

INTRODUCTION A bent consisting of columns and a bent cap beam is an intermediate support between bridge spans that transfers and resists vertical loads and lateral loads such as earthquake and wind from the superstructure to the foundation. The bent cap beam supports the longitudinal girders and transfers the loads to the bent columns. Concrete bent cap beams may be cast-in-place or precast and may be either conventionally reinforced or prestressed. A typical elevation view of a concrete bent integrally connected with the superstructure is shown in Figure 12.1-1.

Figure 12.1-1 A Typical Integral Concrete Bent Bents can be classified as a single-column, a two-column, or a multicolumn bent as shown in Figure 12.1-2.

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Figure 12.1-2 Typical Bents

12.1.1

Types of Bent Caps The main types of bent caps are: 

Drop bent cap



Integral bent cap



Inverted tee cap

These bent caps may be configured in conventional bent types as shown in Figure 12.1-2, and may possess asymmetric column configurations. Also, they may be utilized in unusual bent types, such as "C" bents, and outrigger bents.

12.1.1.1

Drop Bent Cap A drop bent cap, as shown in Figure 12.1-3, supports the superstructure girders directly on its top. This type of bent cap is generally used when the superstructure consists of precast concrete or steel girders.

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Figure 12.1-3 Overview of Drop Bent Cap Drop bent caps may have different types of connection to the superstructure diaphragm: fixed, pinned, or isolated. Figures 12.1-4 to 12.1-6 show each type of bent cap.

Figure 12.1-4 Drop Cap with Pinned Connection

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Figure 12.1-5 Drop Cap with Isolated Connection

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Figure 12.1-6 Drop Cap with Fixed Connection

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12.1.1.2

Integral Bent Cap An integral bent cap, shown in Figure 12.1-7, is cast monolithically with the superstructure girders, and typically has the same depth as the superstructure. The superstructure girders are framed into the bent cap and are supported indirectly by the bent cap. This type of bent cap is commonly used in cast-in-place concrete box girder construction. The load from the girders is transmitted as point loads along the length of the bent cap.

Figure 12.1-7 Integral Bent Cap As a monolithic connecting element to columns and girders, reinforcement details in integral bent caps can be challenging. Figures 12.1-8 to 12.1-12 show threedimensional schematics of bar reinforcement belonging to components from the superstructure that must be accommodated by the integral bent cap. Engineers must also consider the integration of bar reinforcement from the columns.

Figure 12.1-8 Integral Bent Cap Top Slab Reinforcement

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Figure 12.1-9 Integral Bent Cap Bottom Slab Reinforcement

Figure 12.1-10 Integral Bent Cap Girder Reinforcement Note: For clarity, post-tensioning ducts are not shown.

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Figure 12.1-11 Bent Cap Reinforcement of Integral Bent Cap

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Figure 12.1-12 Integral Bent Cap Cross Section

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12.1.1.3

Inverted Tee Cap Inverted tee cap, as shown in Figure 12.1-13, is typically used with precast concrete girders to increase vertical clearance and to enhance aesthetic appearance. However, from a design standpoint, it is difficult to satisfy seismic demands, and the reinforcement of the ledge of the tee cap presents special challenges in shear, flexure, and bar anchorage.

bledge = ledge width

dledge = ledge depth

bstem = stem width

dstem = stem depth

bf

hcap = bent cap depth

= flange width

Figure 12.1-13 Inverted Tee Bent Cap

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12.1.2

Member Proportioning The bent cap depth should be deep enough to develop the column longitudinal reinforcement without hooks in accordance with SDC 7.3.4 and 8.2.1 (Caltrans, 2013). For integral bent caps, the minimum bent cap width required by SDC 7.4.2.1 (Caltrans, 2013) for adequate joint shear transfer shall be the column sectional width in the direction of interest, plus two feet. Drop caps that support superstructures with expansion joints must have sufficient width to prevent unseating. In accordance to SDC 7.3.2.1 (Caltrans, 2013), the minimum width for non-integrated bent caps is determined by considering displacement of the superstructure due to prestress shortening, creep and shrinkage, thermal expansion and contraction, and earthquake displacement demand. For inverted tee bent caps, the stem must be a minimum three inches wider than the column to allow for the extension of column reinforcement into the cap. Similar to the integral and drop caps, the depth of the inverted tee cap must be adequate to develop the column bar reinforcement without hooks. The ledge width (bledge) must be adequate for the primary flexural reinforcement to develop fully. A bent cap may be evaluated as a “conventional beam” or a “deep beam” in accordance with AASHTO LRFD Article 5.6.3.1 (AASHTO, 2012) to estimate the internal forces of the bent cap. AASHTO Articles 5.6.3.1 and 5.8.1.1 specify that if either of following two cases is satisfied, then a strut-and-tie model may be used: Case 1: Lv,zero < 2d where: Lv,zero = distance from the point of zero shear to the face of the support (in.) d

= distance from the compression face to the centroid of tension reinforcement (in.)

Case 2: A load causing more than ½ of the shear at a support is closer than 2d from the face of the supports. In the past, bent caps were typically designed as "conventional beams" in accordance with the Load Factor Design (LFD) method in Caltrans Bridge Design Specifications (Caltrans, 2000), which was applicable until 2008. The LFD code for flexural design is based on the assumption that plane sections remain plane after loading and that the longitudinal strains vary linearly over the depth of the beam. Furthermore, it assumes that the shear distribution remains uniform. In bent caps, these assumptions may not always be valid. However, the sectional beam method has proved to be acceptable as it generally yields more conservative designs in regions near discontinuities. Furthermore, historical data does not suggest design inadequacies for bent caps. Caltrans will continue to use the sectional method except in very irregular beam geometries. Chapter 12 – Concrete Bent Caps

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12.2

LOADS ON BENT CAPS This section discusses the type of loads that bent caps must be designed to resist and support.

12.2.1

Permanent Loads Permanent loads and forces that are, or are assumed to be, either constant or varying over a long time interval upon completion of construction. For bent cap design, the permanent loads to consider include: 

Dead load of structural components and nonstructural attachments (DC):  Bridge weight: In cast-in-place box girder superstructures, normal weight concrete is 150 pcf, including the weight of bar reinforcing steel and lost formwork  Weight of barrier  Any other type of permanent attachment, such as sound walls or sign structures



Dead load of wearing surfaces and utilities (DW), including all dead loads added to the bridge after it is constructed:  For new bridges, and in accordance with MTD 15-17 (Caltrans, 1988), 35 pounds per square foot must be applied on the bridge deck between the faces of barrier rails to account for three inches of future wearing surface.  Weight from utilities: For example, a 24-inch water line would consist of a uniformly distributed load from the pipe, hardware, and support blocks, as well as the water conveyed in the line.



Force effects due to creep (CR): CR are a time-dependent phenomenon of concrete structures due to sustained compression load. As such, bent caps are generally not affected by the displacement-generated loads unless they are prestressed.



Force effects due to shrinkage (SH): The SH of concrete structures are a time-dependent phenomenon that occurs as the concrete cures. The effects of shrinkage are typically not considered unless the bent cap is unusually long (wide structures). Shrinkage, like creep, also affects prestressed bent caps by creating a loss in prestress force as the structural member shortens beyond the initial elastic shortening.



Secondary forces from post-tensioning (PS): The primary posttensioning forces counteract dead and live load demands. However, PS

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forces introduce load into the members of statically indeterminate bent caps as the cap beams shorten elastically toward the point of no movement. 

Miscellaneous locked-in force effects resulting from construction processes (EL)

Generally, DC and DW are distributed by tributary area (or width) for precast prestressed I-girder, steel girder, and bulb T girder bridges. In other types of structures, DC and DW may be distributed equally to each girder despite varying girder spacing. Those types of structures, such as cast-in-place prestressed concrete box girder sections, are so stiff that dead loads are distributed nearly equally to each girder. The self-weight of the bent cap, DCbent cap, however, is distributed along the length of the bent cap as a tributary load.

12.2.2

Transient Loads Transient Loads are loads and forces that are, or are assumed to be, varying over a short time interval. A transient load is any load that will not remain on the bridge indefinitely. For bend cap design, this includes vehicular live loads (LL) and their secondary effects including dynamic load allowance (IM), braking force (BR), and centrifugal force (CE). Additionally, there may be pedestrian live load (PL), force effects due to uniform temperature (TU), and temperature gradient (TG), force effects due to settlement (SE), water load and stream pressure (WA), wind load on structure (WS), wind on live load (WL), friction force (FR), ice load (IC), vehicular collision force (CT), vessel collision force (CV), and earthquake load (EQ). The primary transient load that the bent cap must support is live load. Force effects from live loads are determined similarly to the methods used for the longitudinal girder analysis—through the use of an analytical process that may involve influence lines. The process of calculating wheel line loads to apply to the bent cap model involves extraction of the unfactored bent reactions for each design vehicle class from the longitudinal analysis model. Note that the reactions are generated for a single truck or lane load for each of the three vehicle classes: LRFD HL-93, Caltrans permit vehicles (P-load) (Caltrans, 2014), and fatigue vehicle.

12.2.2.1

Number of Live Load Lanes Live load lanes (Figure 12.2-1) are not the same as the striped lanes on bridges. For bent design, force effects from a single lane of vehicular live load are acquired from the longitudinal frame analysis. To perform an analysis at the bent, various configurations of a single lane or multiple lanes are considered. Fractional lanes are not allowed for bent cap design, meaning only whole numbers of 12-ft lanes are employed.

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Figure 12.2-1 Number of Live Load Lanes Maximum number of live load lanes following equation:

within a bridge, N, is determined by

N = Integer part of {[width of bridge (ft) − barrier widths (ft)] ∕ 12 ft}

(12.2-1)

Per AASHTO Article 3.6.1.1.1, future changes to clear roadway width should be considered. The lane load is considered uniformly distributed over a 10-ft width. However, designers may simplify the analysis by combining the HL-93 lane load with the HL-93 truck wheel line load. Note that, per AASHTO 3.6.1.2.4, the dynamic load allowance, IM, shall be applied only to the truck load. 12.2.2.2

Multiple Presence Factors, m Multiple presence factors, m, as specified in AASHTO Table 3.6.1.1.2-1, are used to account for the improbability of fully loaded trucks crossing the structure simultaneously and are applied to the vehicular live loading.

12.2.2.3

Vehicular Live Load Positioning An important consideration in the design of bent caps is to determine the maximum or critical force effects by positioning live load lanes. The location of the truck and lane, as shown in Figure 12.2-1, has an important bearing on the force

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effect on the bent cap. Whether the truck is at midspan of a bent cap or at the support, it has an effect on the value of the moment and shear on the bent cap. AASHTO LRFD requires that one truck be placed in each lane transversely: If the bridge can fit four lanes, then up to four trucks can be placed on the bridge, one in each lane. Lanes are placed to produce the maximum force effects in the bent cap. However, one must consider the effect of the multiple presence factor, m, as it is not always evident that placing the maximum number of trucks will garner the maximum force effects in the bent cap. For example, for closely spaced columns within a bent cap, two HL-93 lanes may result in greater shear demand than four HL-93 lanes because the latter case would require a multiple presence factor of 0.65. The designer must also consider that a certain configuration, and number, of live load lane positions may result in maximum shear effects but not necessarily maximum moment effects. For a bent cap supported by multiple columns, it is advisable to use a structural analysis program that is capable of generating combinations of lane configurations, as well as influence lines from moving live loads. CsiBridge, and CTBridge are such programs. 12.2.2.4

HL-93 Design Vehicular Live Load Positioned Transversely HL-93 consists of design truck, or design tandem, and design lane load. Figure 12.2-2 shows one of two alternatives for a design truck, or wheel lines, transversely placed within a 12-ft live load lane. The other alternative is a mirror image of this graphic depiction. The wheel lines may move anywhere within the 12-ft lane as long as AASHTO 3.6.1.3.1 is satisfied. Lanes and wheel lines shall be placed to produce maximum force effects in the bent cap.

Figure 12.2-2 HL-93 Design Truck Positioned Transversely When multiple lanes are applied to the bent cap, the wheel lines may be positioned as shown in Figure 12.2-3:

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Figure 12.2-3 Wheel Line Spacing for Four HL-93 Trucks Figure 12.2-4 shows a 10-ft wide HL-93 lane load placed in 12-ft wide lanes.

Figure 12.2-4 HL-93 Lane Loads Positioned Transversely

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12.2.2.5

Permit Trucks Positioned Transversely Per CA 3.4.1 (Caltrans 2014), for bent cap design, a maximum of two permit trucks shall be placed in lanes that are positioned to create the most severe condition. Figure 12.2-5 shows two permit trucks occupying two adjacent lanes. However, the lanes may be positioned apart if that results in maximum bent cap force effects.

Figure 12.2-5 Permit Trucks Positioned Transversely

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12.3

DESIGN CONSIDERATIONS This section presents the typical design procedure for cast-in-place concrete bent caps. Topics include design for flexure, shear, and shear-flexure interaction.

12.3.1

Flexural Design Reinforced concrete bent caps shall be designed to satisfy the strength, service, and fatigue limit states. The goal of flexural design under the strength limit state is to provide enough resistance to satisfy the strength limit state conditions. This may be achieved by using bar reinforcing steel or prestressing in the cast-in-place concrete bent cap.

12.3.1.1

Flexural Design Process The flexural design process for bent caps consists of 12 primary steps, summarized below: 1)

Calculate factored moments for Strength I and II limit states

2)

Calculate minimum cracking moment, then determine minimum design moment, Mmin

3)

Determine the factored moment demand, Mu

4)

Assume an initial value for area of nonprestressed tension reinforcement, As

5)

Calculate net tensile strain, t, and determine resistance factor, ϕ

6)

Determine whether the section is rectangular or flanged

7)

Calculate the average stress in prestressing steel, ƒps, if the bent cap is posttensioned

8)

Calculate the nominal flexural resistance, Mn

9)

Calculate the factored flexural resistance, Mr

10)

Iterate steps 4 through 9 until Mr ≥ Mu and the design assumptions are verified

11)

Check for serviceability

12)

Check for fatigue

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12.3.1.1.1 Determine the Factored Moment from Strength I and Strength II Limit States Per CA table 3.4.1-1, both Strength I and Strength II limit states are used to calculate Mu for bent caps. For simplicity, Strength III through V, are not being considered as Strength I and II will govern in most bent cap designs. For additional simplicity, moment demand from prestressing, creep, shrinkage, stream pressure, uniform temperature change, temperature gradient, and settlement are not being considered: 

Strength I Mu(HL93)



= 1.25(MDC) + 1.5(MDW) + 1.75(MHL93)

(12.3.1-1)

= 1.25(MDC) + 1.5(MDW) + 1.35(MP‑ 15)

(12.3.1-2)

Strength II Mu(P-15) where: Mu(HL93) =

factored moment demand at the section from HL93 Vehicle

Mu(P-15) =

factored moment demand at the section from the Permit Vehicle

MDC =

unfactored moment demand at the section from dead load of structural components and nonstructural attachments

MDW =

unfactored moment demand at the section from dead load of wearing surfaces and utilities

From the above two limit states, the larger of the two values is the controlling moment. It is possible to have different limit states control at different locations along the bent cap. 12.3.1.1.2 Calculate Minimum Reinforcement Design Moment, Mmin The minimum reinforcement requirement ensures that the flexural design of the bent cap provides either enough post-cracking ductility of the member or a modest margin of safety over Mu:   S  M cr   3   1 f r   2 f cpe S c  M dhc  c  1   S nc 



Sc 

Ig Yt



(AASHTO 5.7.3.3.2-1)

(12.3.1-3)

where: Sc = section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3)

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Ig = moment of inertial of the gross concrete section about the centroidal axis, neglecting the reinforcement (in.4) Yt = distance from the neutral axis to the extreme tension fiber (in.) fr

= modulus of rupture of the concrete (ksi)

fcpe = compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) not including the effects of secondary moment, at extreme fiber of section where tensile stress is caused by externally applied loads (ksi) 1 = flexural cracking variability factor 2 = prestress variability factor 3 = ratio of specified minimum yield strength to ultimate tensile strength of the reinforcement Mcr = cracking moment (kip-in.) Mdnc = total unfactored dead load moment acting on the monolithic or noncomposite section (kip-in.) Snc =

section modulus for the extreme fiber of the monolithic or noncomposite section where tensile stress is caused by externally applied loads (in.3)

Mdnc and Snc typically apply to precast girders before and after the girder become composite with the deck. Therefore, the terms do not typically apply to cast-in-place bent cap design and will be disregarded. Since the design examples are of nonprestressed bent caps, remove fcpe from the equation, and reduce the formula to: Mcr = 3 (1 fr) Sc

(12.3.1-4)

Per AASHTO 5.7.3.3.2, the minimum factored moment demand is: Mmin = min (Mcr, 1.33Mu)

(12.3.1-5)

12.3.1.1.3 Determine the Factored Moment Demand, Mu The maximum value of Mu is: Mu = max(Mu(HL93), Mu(P-15), Mmin)

(12.3.1-6)

12.3.1.1.4 Determine the Resistance Factor,  The AASHTO code requires that factored loads be less than or equal to factored resistances, as shown below:

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Figure 12.3-1 General Equation of LRFD Methodology For flexure design, the AASHTO LRFD code specifies a variable resistance factor. The relationship between  and the steel net tensile strain, εt, is provided in the design specifications. To determine the resistance factor for flexure design of the bent cap, the designer must first calculate the steel net tensile strain. 12.3.1.1.5 Calculate the Section's Net Tensile Strain, εt The net tensile strain, εt, is the tensile strain in the extreme tension steel at nominal flexural strength. The nominal flexural strength is reached when the concrete strain in the extreme compression fiber reaches the assumed ultimate strain of 0.003. The following graphic shows a rectangle section with linear strain distribution along the section.

Figure 12.3-2 Strain Diagram for Rectangular Section where: c = distance from the extreme compression fiber to the neutral axis dt = distance from the extreme compression fiber to the centroid of extreme tension steel

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εcu = failure strain of concrete in compression Based on AASHTO 5.7.2.1, the strain along the section is a linear distribution or directly proportional to the distance from the neutral axis. Using the similar triangle method, the steel net tensile strain can be determined as:  dt   d    0.003 t   c 1  c 1

 t   cu 

(12.3.1-7)

The following graph from CA C5.5.4.2.1-1 shows how the resistance factor, , varies with the section's net tensile strain εt.

Figure 12.3-3 Variation of  with Net Tensile Strain, εt Note that the variation of  has three distinct linear regions. These regions describe ranges of εt that denote whether a section is compression-controlled, tension-controlled, or a transition section. 

Compression-controlled section If εt  0.002, the section is defined as a compression-controlled section. Under external load action, a compression-controlled member fails in a brittle manner with little warning. To avoid this scenario, the design code requires a more conservative resistance factor,  = 0.75.



Tension-controlled section If εt  0.005, the section is defined as a tension-controlled section. Failure of a tension‑ controlled member is more ductile and is considered to have sufficient warning before failure by means of deflection and cracking. Therefore, the design specifications specifies a relatively large resistance factor for tension-controlled members.

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Transition section If εt is located between the above two limits (0.002 < εt < 0.005), the section is defined as a transition section. Under this condition, the resistance factor will vary linearly with εt.

The following figure illustrates the net tensile strain limits:

Figure 12.3-4 Net Tensile Strain Limits. Note: Per CA C5.7.2.1, the net tensile strain for a reinforced (non-prestressed) concrete flexural member, such as a bent cap, shall not be less than 0.004. 12.3.1.1.6 Determine the Factored Flexural Resistance, Mr The factored flexural resistance is: Mr = Mn

(12.3.1-8)

12.3.1.1.7 Determine whether the Section is Rectangular or Flanged For monolithic integral bent caps, it is Caltrans' practice to consider part of the deck and soffit slab to be working with the solid bent cap section as a flanged crosssection as shown in Figure 12.3-5.

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Figure 12.3-5 Flanged Cross Section

Figure 12.3-6 Geometric Model

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where : a = distance from the center of gravity (CG) of the superstructure to the bottom of column = height of the column + depth to CG of the bent cap b = top width of superstructure c = distance from the centerline (CL) of the columns to the edge of deck (e + f) d = distance between the CL of columns e = distance from the CL of exterior girder to the edge of deck (EOD) f = distance from the CL of the column to the CL of the exterior girder In the geometrical model shown in Figure 12.3-6, the flanged cross-section spans from the centerline of the left exterior girder to the centerline of the right exterior girder at the CG of the bent. The rectangular cross-sections span from the centerline of the exterior girder to the edge of deck. These sections are assumed as rectangular in order to simplify the analytical model. For drop bent caps, the section is rectangular for the full length of the bent cap. Although the terms "flanged section" and "rectangular section" describe the geometric section, it is important to note that they are not necessarily accurate depictions of the analytical section. A "flanged section" may be analyzed as a "rectangular section." The scenario exists when the depth of the compression zone, i.e, distance from the extreme compression fiber to the neutral axis, c, is less than the thickness of the compression flange, hƒ (c ≤ hƒ). If the depth of the compression zone is greater than the thickness of the compression flange (c > hƒ), then the section will exhibit "flanged section" behavior. If the section is a rectangular section, then determine c as follows:

c

A ps f pu  As f s  As' f s'  f pu 0.85 f c' 1b  kA ps   dp 

(AASHTO 5.7.3.1.1-4)

   

If the section is a flanged section, then determine c as follows: c

A ps f pu  As f s  As' f s'  0.58 f c' (b  bw )h f 0.85 f c' 1bw

 f pu  kA ps   dp 

   

(AASHTO 5.7.3.1.1-3)

where: Aps = area of prestressing steel (in.2) fpu

= specified tensile strength of prestressing steel (ksi)

c

= distance from extreme compressive fiber to the neutral axis (in.)

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dp

= distance from extreme compression fiber to the centroid of prestressing tendons (in.)

As

= area of mild steel tension reinforcement (in.2)

fs

= stress in the mild steel tension reinforcement at nominal flexural resistance (ksi)

ds

= distance from extreme compression fiber to the centroid of nonprestressed tensile reinforcement (in.)

As = area of compression reinforcement (in.2) fs

= stress in the mild steel compression reinforcement at nominal flexural resistance (ksi)

ds

= distance from extreme compression fiber to the centroid of compression reinforcement (in.)

fc

= specified compressive strength of concrete (ksi) at 28 days, unless another age is specified (ksi)

b

= width of the compression face of the member (in.)

bw

= web width or diameter of a circular section (in.)

hf

= compression flange depth of an I or T member (in.)

1

= stress block factor specified in AASHTO Article 5.7.2.2 For fc  4 ksi, 1 = 0.85 For fc  4 ksi, 1 is reduced at a rate of 0.05 for each 1 ksi of strength in excess of 4 ksi, except 1  0.65

fps

= average stress in prestressing steel; value is often zero for non-prestressed bent caps The average stress in prestressing steel, f ps , may be taken as:  c f ps  f pu 1  k  dp 

k

   

(AASHTO 5.7.3.1.1-1)

= constant that depends on the type of tendon used:  f py   k  21.04    f pu  

(AASHTO 5.7.3.1.1-2)

Alternatively, the following table can be used:

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AASHTO Table C5.7.3.1.1-1 Value of k Type of Tendon

f py f pu

Value of k

Low relaxation strand Stress-relieved strand Type 1 high‑strength bar

0.9 0.85 0.85

0.28 0.38 0.38

Type 2 high‑strength bar

0.8

0.48

Figure 12.3-7 Flexural Resistance Calculation Flowchart Mn is the nominal resistance, which can be calculated by:

æ a hf ö æ æ æ aö aö aö M n = Aps f ps ç d p - ÷ + As f s ç ds - ÷ - As¢ f s¢ç ds¢ - ÷ + 0.85 fc¢ b - bw h f ç - ÷ 2ø 2ø 2ø è è è è2 2 ø

(

)

(AASHTO 5.7.3.2.2-1) where: Aps = area of prestressing steel (in.2) Chapter 12 – Concrete Bent Caps

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fps = average stress in prestressing steel at nominal bending resistance (ksi) dp = distance from extreme compression fiber to the centroid of prestressing tendons (in.) As = area of mild steel tension reinforcement (in.2) fs

= stress in the mild steel tension reinforcement at nominal flexural resistance (ksi)

ds = distance from extreme compression fiber to the centroid of nonprestressed tensile reinforcement (in.) As = area of compression reinforcement (in.2) fs = stress in the mild steel compression reinforcement at nominal flexural resistance (ksi) ds = resistance from extreme compression fiber to the centroid of compression reinforcement (in.) fc = specified compressive strength of concrete (ksi) at 28 days, unless another age is specified (ksi) b

= width of the compression face of the member (in.)

bw = web width or diameter of a circular section (in.) hf = compression flange depth of an I or T member (in.) a

= c1; depth of the equivalent stress block (in.)

1 = stress block factor specified in AASHTO Article 5.7.2.2 The graphic below shows the variables that comprise each component of the formula:

Figure 12.3-8 Components of Flexural Moment For non-prestressed cast-in-place bent caps, in which compression reinforcement is ignored, the term for Mn can be reduced to:

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a  M n  As f s  d s    0.85 f c b  bw h f 2 

12.3.1.2

 a hf     2 2  

(12.3.1-9)

Check for Serviceability Cracks occur in concrete components due to: 

Loading conditions



Thermal effects



Deformations

Cracks occur whenever tension stress in the member exceeds the modulus of rupture of concrete. The severity of flexural cracking in a concrete bent caps can be controlled by providing optimized tension reinforcement layouts, limiting bar sizes, and providing tighter spacing. Per AASHTO 5.7.3.4, the spacing, s, of mild steel reinforcement in the layer closest to the tension face is given by: 700 e  2d c  s f ss

s

(AASHTO 5.7.3.4-1)

where: dc = thickness of concrete cover measured from extreme tension fiber to center of the closest flexural reinforcement (in.) γe = exposure factor = 1.00 for Class 1 exposure condition = 0.75 for Class 2 exposure condition  s 1 

h

dc 0.7h  d c 

(12.3.1-10)

= overall thickness or depth of the component (in.)

fss = tensile stress in steel reinforcement at the service limit state (ksi) Note: In the above equation, the spacing, s, of the bar reinforcing steel is inversely proportional to the stress in the reinforcing steel. Also, per CA 5.7.3.4: 

Class 1 exposure condition applies when cracks can be tolerated due to reduced concerns for appearance and/or corrosion.



Class 2 exposure condition applies when there are increased concerns for appearance and/or corrosion (for example; in areas where de-icing salts are used). Class 2 exposure condition applies to all bridge deck.

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12.3.1.3

Check for Fatigue As per AASHTO 5.5.3.1, the stress range in reinforcing bars due to the fatigue load combination shall satisfy:

(f)  (F)TH

(AASHTO 5.5.3.1-1)

(F)TH = 24 – 0.33 fmin

(AASHTO 5.5.3.2-1)

where:



=

load factor for Fatigue I

f

=

live load stress range (ksi)

fmin

=

minimum live load stress resulting from the Fatigue I load combination, combined with more serve stress from either the permanent loads or the permanent loads, shrinkage, and creep-induced external loads; . positive if tension, negative if compression (ksi)

For the fatigue check:

12.3.2



The fatigue load combination is given in CA Table 3.4.1-1 and features a load factor of 1.75 for the infinite fatigue life.



Apply the IM factor to the fatigue load.



Check both top and bottom reinforcement to ensure that the stress range in the reinforcement under the fatigue load stays within the range specified in the above equation.

Design for Shear The shear design of bent caps involves:   

Determining the stirrup bar size along the length of the bent cap Determining stirrup spacing along the bent cap Checking shear-flexure interaction

The LRFD shear design method is based on the Modified Compression Field Theory. Contrary to the traditional shear design methodology, it assumes a variable angle truss model instead of the 45º truss analogy. The LRFD method also accounts for interaction between shear, torsion, flexure, and axial load, as well as residual tension in concrete after cracking, which was neglected in the traditional method of shear design. California Amendments Article 5.8.3.4.2 specifies that shear resistance of all prestressed and nonprestressed sections shall be determined by AASHTO Appendix B5. The LRFD design code notes that shear design may be considered to the distance, dv, from the face of support (AASHTO 5.8.3.2). However, Caltrans' practice is to evaluate shear to the face of support. See Chapter 5 for detailed discussion.

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Factored shear resistance, Vr, is given by: Vr = Vn

(AASHTO 5.8.2.1-2)

The nominal shear resistance, Vn, is the lesser of: Vn = Vc + Vs + Vp

(AASHTO 5.8.3.3-1)

or Vn = 0.25fcbvdv + Vp

(AASHTO 5.8.3.3-2)

If if the procedures of AASHTO 5.8.3.4.1 or 5.8.3.4.2 are used Vc  0.0316 f c bv d v Vs 

Av f y d v cot   cot   sin  s

(AAAHTO 5.8.3.3-4) (AASHTO 5.8.3.3-4)

f c = specified compressive strength of concrete (ksi)

fy = yield strength of transverse reinforcement (ksi) Vp = component in the direction of the applied shear of the effective prestressing force (kip); positive if resisting the applied shear; typically zero for conventionally reinforced bent caps bv = effective web width (in.) taken as the minimum web width within the depth dv dv = effective shear depth (in.) s = spacing of stirrups (in.)  = factor indicating ability of diagonally cracked concrete to transmit tension and shear  = angle of inclination (°) of diagonal compressive stresses  = angle of inclination (°) of transverse reinforcement to longitudinal axis; typically 90° Av = area of shear reinforcement (in.2) within a distance s The main shear equation may be rearranged and simplified for design purposes, as such: Vs 

Vu  Vc 

(12.3.2-1)

Per AASHTO C5.8.2.4, transverse reinforcement must be provided in all regions where there is a significant chance of diagonal cracking. Transverse reinforcement must be provided where: Vu > 0.5 (Vc + Vp)

(AASHTO 5.8.2.4-1)

Determine the effective shear depth, d. Chapter 12 – Concrete Bent Caps

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As specified in AASHTO 5.8.2.9, the effective shear depth, dv, is taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure. The effective shear depth, dv, is given by: dv  de 

a 2

(12.3.2-2)

where: de = effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) dv needs not be taken to be less than the greater of 0.9de or 0.72h where h is the overall thickness or depth of a member 12.3.2.1

Determine Factor β and Angle θ Using shear stress ratio, vu / fc, and longitudinal strain, x, to find values of β and θ from AASHTO Table B5.2-1. For non-prestressed concrete bent cap, vu 

Vu bv d v

(AASHTO 5.8.2.9-1)

where: Vu = factored shear (kip) bv = effective web width taken as the minimum web width (in.) dv = effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compression force due to flexure, it need not be taken to be less than the greater of 0.9de or 0.72h (in.) The longitudinal strain at the middepth of member x, shall be determined by: | Mu |    0.5 N u  0.5 | Vu  V p | cot   A ps f po  dv  x   (AASHTO B5.2-1) 2 E s As  E p A ps





where:

M u = absolute value of the factored moment, not to be taken less than Vu  V p d v (kip-in.)

Nu

=

factored axial force, taken as positive if tensile and negative if compressive (kip)

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Vu

=

factored shear force (kip)

fpo

=

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and surrounding concrete (ksi)

Es

=

modulus of elasticity of reinforcing bars (ksi)

Ep

=

modulus of elasticity of prestressing tendons (ksi)

Note: The longitudinal strain is a function of the desired angle of compression strut, θ. This necessitates an iterative procedure to solve for the longitudinal strain, x, and θ. Thus, it may be useful to assume that the term (0.5cotθ) equals 1.0 and reduce the number of iterations. 12.3.2.2

Determine the Amount of Shear Stirrups, Av, and Spacing, s Knowing, Vu, Vc, and ϕ, the demand on shear stirrups, Vs , can be determined: Vs 

Vu  Vc 

Knowing Vc, the term

(12.3.2-1)

Av can be determined: s

Av Vs = s f y dv cot q + cot a sina

(

)

(12.3.2-2)

Check for minimum shear reinforcement:

f c bv  Av   0.0316   fy  s  mim

(AASHTO 5.8.2.5-1)

Check for maximum spacing: For vu / fc < 0.125, smax = 0.8dv  18 in.

(CA 5.8.2.7-1)

For vu / fc  0.125, smax = 0.4dv  12 in.

(AASHTO 5.8.2-7-2)

Repeat for other points along bent cap span (typically at tenth point) and obtain Av and s.

12.3.3

Check Longitudinal Steel for Tension (Shear-Flexure Interaction) The effect of shear forces on the longitudinal reinforcement is determined and the adequacy of the reinforcement is checked using the AASHTO interaction equation 5.8.3.5-1. Longitudinal reinforcement along with the vertical steel stirrups and the compression strut in concrete constitute the truss mechanism that carries the applied loads.

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The check of the adequacy of the longitudinal reinforcement may result in added length of the longitudinal bar reinforcing and/or added amount of longitudinal reinforcement. The former occurs if the original bar reinforcement is curtailed, and the latter occurs in case of indirect loading/support such as the case of box girders framing into bent caps. Figure 12.3-9 shows the concept of direct/indirect loading and support, along with the demand on the longitudinal steel from flexure (solid line) and shear (dashed line).

Direct Loading

Indirect Loading

Flexure Shear

Figure 12.3-9 Direct Versus Indirect Loading/Support As can be seen from Figure 12.3-9, the amount of longitudinal steel need not exceed the maximum amount due to flexure demands in the case of direct loading/support. The additional shear demands on the longitudinal steel can be overcome by extending the length of the longitudinal bar reinforcement. However, in the case of girders framing onto other girders at equal depth or height, the shear demand is likely to result in an additional amount of longitudinal steel beyond what is needed to meet the maximum flexure demands.

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A direct loading/support case is typical for drop bent cap beams. Precast and steel girders are applying the load atop the bent cap. Columns are also directly supporting the cap. Since Caltrans practices no curtailment of the longitudinal reinforcement (due to nature of seismic loading), there is no need to check for shear-flexure interaction. Indirect loadings/supports are often encountered in box girder construction. At location of box girders framing onto the cap, the amount of bent cap longitudinal reinforcement needs to be checked using the shear-flexure interaction equation. Similar to drop bent caps, the columns are directly supporting the cap and no added longitudinal reinforcement is required at the face of column support. The tension capacity of the longitudinal reinforcement is determined on the flexural tension side using fully developed steel and at locations of applied concentrated loads (on integral bent caps). At every location, the following three possible load conditions shall be checked:   

Maximum shear and associated moments Maximum positive moments and associated shear Maximum negative moments and associated shear

The longitudinal steel must satisfy: A ps f ps  As f y 

Mu dv f

 0.5

 N u  Vu   V p  0.5Vs  cot    c  v 

(AASHTO 5.8.3.5-1) For the typical case of no prestressing and axial force in bent cap, the above equation reduces to: As f y 

Mu dv f

V    u  0.5Vs  cot     v 

(12.3.3-1)

where: f

= resistance factor for moment

c

= resistance factor for axial load

v

= resistance factor for shear

Note: Vs shall not be taken more than

12.3.4

Vu . 

Design for Seismic For seismic design, resistance factor shall be taken as  = 1.0 (CA 5.5.5).

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12.4

DETAILING CONSIDERATIONS In addition to the main top and bottom longitudinal steel for flexure and vertical stirrups for shear, reinforcement is required for side-face and end reinforcement for crack control, as well as for construction purposes.

12.4.1

Construction Reinforcement This reinforcement is only needed for box girder construction. Concrete for the box girder is usually placed in two stages. The first stage includes placing the soffit slab and the girder stems. It is commonly known as the “stem and soffit” pour since the deck slab is not included. The second stage consists of constructing the top three to four inches of the stem and the deck slab as shown in Figure 12.4-1.

Figure 12.4-1 Concrete Pour Stage The two-stage pouring of concrete results in a construction joint at the bent cap at the top of the girder stem. At this stage (after first pour), the girders are not stressed, longitudinal bottom steel is in place, and the entire bridge is supported on falsework. If falsework underwent any settlement or failed unexpectedly due to impact by an errant vehicle, the bent cap will be subjected to a bending moment similar to that shown in Figure 12.4-2. As shown, there is negative moment at and near the supports (columns) with no top steel to resist this moment.

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As such, it is Caltrans practice to provide additional top longitudinal steel, also commonly referred to as construction reinforcement, underneath the construction joint for the potential loss or settlement of the falsework supporting the bridge. A 10ft length of the exterior stem girder on each side of the bent cap is assumed to act as dead load weight in addition to the bent cap own weight. Detailed weight calculations and design consideration for construction reinforcement are shown in Section 12.5.1, Integral Bent Cap Example. See Figure 12.4-3.

Figure 12.4-2 Potential Moment Diagram after First Pour

Figure 12.4-3 Construction Reinforcement

12.4.2

Side Face Reinforcement Caltrans SDC 7.4.4.3 (Caltrans, 2013) requires side face reinforcement in quantity that amounts to 10 percent of the maximum amount of longitudinal reinforcement of the bent cap. The maximum amount of longitudinal reinforcement will come from either the top or bottom steel. The side face reinforcement, shown in Figure 12.4-4, is placed along the two vertical faces of the bent cap and shall have a maximum spacing of 12 inches. It is permissible to include the construction reinforcement to satisfy part of this requirement.

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Figure 12.4-4 Side-Face Reinforcement

12.4.3

End Reinforcement The end reinforcement is provided along the end face of the bent cap, as shown in Figure 12.4-5, as a crack control measure. There are two basic types of end reinforcement, Z-bars and U-bars as shown in Figure 12.4-5.

Figure 12.4.3-1 in the theBent BentCap Cap Figure 12.4-5 End End Reinforcement Reinforcement in The U-bars (horizontal plane) are typically designed using the shear friction concept to resist the dead and live load of the exterior girder. The number of U-bars per set depends on whether the bent cap width is less than or more than seven feett.

12.4.4

Other Detailing Considerations (Skew) If the skew angle of the bent is 20° or less, the deck and soffit slab reinforcement are placed parallel to the centerline of bent cap. This bar reinforcement configuration allows the bent cap longitudinal reinforcement to be placed as far from the extreme compression fiber as possible to optimize flexural capacity. When the bent cap skew angle is greater than 20°, the deck and soffit slab reinforcement are typically placed perpendicular to the centerline of the bridge and the bent cap top longitudinal steel must be placed below the deck reinforcement or above the soffit reinforcement. Details of bent cap reinforcement for different skew angles can be found in BDD 745 and 7-43 (Caltrans, 1986).

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12.5

DESIGN EXAMPLES The section contains two design examples including a reinforced concrete integral bent cap and a drop bent cap to illustrate the main design process of using the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) and California Amendments (Caltrans, 2014). Figure 12.5-1 shows a general design flowchart for a bent cap. It should be noted that the examples do not constitute a complete bent cap design. Only selected work has been done to demonstrate design methods. For example, tension steel has not been designed for every span of the bent cap as it would be done for an actual bent cap design. Additionally, there are other design considerations not considered in examples. For instance, seismic design will not been addressed. It is hoped, however, that the example will provide good foundation for design of bent caps. The information contained in this section should not be used as a design guide in place of reading the specifications. There are often several ways to solve a design problem. It is recommended that prior to applying any formula or procedure contained within this section, the designer should read the appropriate Articles of the Caltrans currently adopted AASHTO LRFD Bridge Design Specifications and California Amendments to be certain that the described formula or procedure is appropriate for use.

12.5.1

Integral Bent Cap The integral bent cap example shown on the following page is used in a threespan cast-in-place prestressed concrete box girder bridge.

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Figure 12.5-1 Bent Cap Analysis and Design Flowchart

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12.5.1.1

Bent Cap Data Integral bent cap is supported by two column bent as shown in Figure 12.5-2. The box girder has four cells of depth 6.75 ft with a skew angle is 20. Concrete column diameter is 6 ft. diameter columns.

Figure 12.5-2 Two Column Bent Note: These dimensions are perpendicular to the centerline (CL) of the roadway and are not actual dimensions of the bent since the bent is skewed at an angle to the roadway. 12.5.1.2

Design Requirements Perform the structural analysis, flexural, and shear design as shown in Figure 12.5-1 in accordance with the AASHTO LRFD Bridge Design Specifications, 6th Edition (AASHTO, 2012) with California Amendments (Caltrans, 2014).

12.5.1.3

Step 1: Determine Geometry of Bent Bent dimensions along the skew are shown in Figure 12.5-3. Ranges of bent cross section types are shown in Figure 12.5-4.

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Figure 12.5-3 Bent Dimensions along the Skew

Figure 12.5-4 Bent Cross Section Types where : a = distance from the centroid of gravity of the superstructure to the bottom of column b = top width of superstructure c = distance from the CL of the columns to the edge of deck (e + f) Chapter 12 – Concrete Bent Caps

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d = distance between the CL of columns e = distance from the CL of exterior girder to the edge of deck (EOD) f = distance from the CL of the column to the CL of the exterior girder For the example bridge, assume that CG of the bent cap is at mid-depth of the cap beam.

12.5.1.3.1 Rectangular Cross Section The rectangular cross section spans from the CL of the exterior girder to the EOD. These sections are assumed as rectangular in order to simplify the stick model. The dimension e in the geometric model is the distance from the CL of exterior girder to the EOD along the skew, as shown in Figure 12.5-5.

Figure 12.5-5 Rectangular Cross Section ep 

1  28.5 34.5  60  81 in. (13.4)   2  69 

e (alongskew) 

81  86.2 in.  7.18 ft cos(20)

f = c – e = 14.3 - 7.18 = 7.12 ft

12.5.1.3.2 Flanged Cross Section Range of flanged cross section is shown in Figure 12.5-6. Minimum bent cap width = column width + 2 ft

(SDC 7.4.2.1)

Width of bent cap, Bcap = 6 + 2 = 8 ft

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Figure 12.5-6 Range of Flanged Cross Section Effective bent cap flange or overhang width = width of bent cap + 2 (width of overhang at each side), as shown in Figure 12.5-7. Width of overhang is:

(AASHTO 4.6.2.6.5)

Least of -6 (least soffit slab thickness) = 6 (8.25)

= 49.5 in.

0.1(span length of the bent cap)

= 0.1(34)

= 3.4 ft

0.1(2  length of cantilever span)

= 0.1 (214.3) = 2.86 ft = 34.3 in.

= 40.8 in

Total effective flange width = 8 + 2.86 + 2.86 = 13.72 ft = 164.64 in.

Figure 12.5-7 Effective Flange Width

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12.5.1.4

Step 2: Determine Material Property A706 Steel reinforcement with fy = 60 ksi and Es = 29,000 ksi, and concrete with fc = 4 ksi and Ec = 3,645 ksi, are used in design example.

12.5.1.5

Step 3: Determine Bent Reactions due to Permanent and Live Load Bent reactions due to permanent and live load are calculated as follows.

12.5.1.5.1 Reactions due to Permanent Loads The permanent loads are comprised of: 

Dead load of structural components and non-structural attachments (DC)



Dead load of wearing surfaces and utilities (DW)



Self-weight of bent cap

The unfactored bent reactions for DC and DW obtained from a longitudinal analysis are shown in Table 12.5-1. 12.5.1.5.2 Self-Weight of the Bent Cap

The self-weight of the bent cap is modeled as a uniformly distributed load as shown in Figure 12.5-8.

Figure 12.5-8 Permanent Load on Bent Cap

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Self-weight of bent cap = (cross-sectional area of the bent cap solid section)  (average length of bent cap)  (unit weight of concrete) Average length

= 7.12 + 34 + 7.12

Box cell area

= 217.9 ft2

= 48.24 ft

Bent cap self-weigh = (217.90) (8) (0.15) = 261.48 kips Self-weight modeled as uniform load

= 261.48 / 48.24 = 5.42 kips/ft

Note: While calculating the self-weight of the bent cap, be careful not to include the portion of the deck, soffit slab, and girder thicknesses at the bent cap if they have already been included in the longitudinal analysis of the bridge. Table 12.5-1 Unfactored Reactions due to DC, DW, HL-93, P-15, and Fatigue Vehicles Bent Reactions (kip )

Dynamic Allowance Factor

Final Bent Reactions (kip)

DC

2,651.5

NA

2,651.5

DW

325

NA

325

Self-Weight

261.48

NA

261.48

HL-93 Vehicle Truck Lane

114.82 99.34

1.33 1

252

Permit Vehicle

360.77

1.25

451

Fatigue Vehicle

70.66

1.15

81.25

For both DC and DW, these bent reactions obtained from the longitudinal analysis are modeled as concentrated load acting at the CL of each girder framing into the bent cap. 

Reaction due to DC on the bent cap for each girder (kip) = 2651.5 / 5 = 531 kips



Reaction due to DW on the bent cap for each girder (kip) = 325 / 5 = 65 kips

12.5.1.5.3 Live Load as Two Wheels Vehicular live loads as two wheels are applied on the bent caps as shown in Figure 12.5-9. 

HL-93 vehicle: 252 / 2 = 126 kips



Permit vehicle: 451 / 2 = 226 kips



Fatigue vehicle : 82 / 2 = 41 kips

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HL-93 Load Permit Load Fatigue Load

Figure 12.5-9 Bent Reaction due to Live Loads The HL-93 vehicle live loads can be one, two, three, or four lanes. Figure 12.5-10 shows different HL-993 truck placement for transverse analysis using CSiBridge. Figure12.5-11 shows placement of two HL-93 lanes.

Figure 12.5-10 Possible Scenarios for Placement of Live Load

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Figure 12.5-11 Two Scenarios for Two Truck Case Permit vehicle live load may be only one or two lanes as shown in Figure 12.512.

Figure 12.5-12 Permit Trucks Fatigue vehicle live load is only one truck as shown in Figure 12.5-13.

Figure 12.5-13 Fatigue Truck

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12.5.1.6

Step 4: Perform Transverse Analysis The goal of the transverse analysis is to obtain force effect envelopes of all possible live load cases. The example bridge has four design lanes. Each of the different live load trucks can be placed in these design lanes. The number and placement of these trucks depends on the type of load: 

HL-93 vehicle



Permit truck



Fatigue truck

Since there may be many live load cases to consider, computer software is usually used to determine: 

Maximum moment (negative or positive) and associated shear



Maximum shear and associated moment

Available computer programs for transverse analysis are: 

CSiBridge



VBENT



LEAP Bridge - RCPIER



CTBridge

For this example bridge, results are obtained by performing transverse analysis with CSiBrdge. The results from the transverse analysis are obtained at every tenth point or at selected points. These results are shown separately for DC, DW, HL-93, permit, and fatigue loads. Table 12.5-2 and 12.5-3 list the controlling unfactored moments. Note: Bent transverse analysis using CsiBridge is not covered under this chapter. Please refer to the following URL for step-by-step procedures to generate the bent cap model in CSiBridge for transverse analysis: http://onramp.dot.ca.gov/hq/des/sd/SD_training/intro.html

12.5.1.7

Step 5: Perform Flexural Design The design equation is as follows: Mu ≤ Mr =  Mn

(12.5-1)

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12.5.1.7.1 Calculate Controlling Unfactored Moment Table 12.5-2 shows unfactored controlling moments including impact for the bent cap, obtained from the transverse analysis of the bridge using CSiBridge. Table 12.5-2 Unfactored Moments Load and Moment

Moment at Mid-Span (kip-ft)

Moment at Face of Column (kip-ft)

DC, MDC

3,377

-1,760

DW, MDW

339

-217

HL-93 Vehicle, MHL-93

2,683

-1,859

Permit Vehicle, MP-15

4,571

-3,336

12.5.1.7.2 Check Positive Moment at Midspan Factored moments are as follows: 

Strength I M u  1.25M DC  1.5M DW  1.75M HL  93

 1.253,377  1.5339  1.752,683  9,425 kip - ft



Strength II M u  1.25M DC  1.5M DW  1.35M P15

 1.253,377  1.5339  1.354,571  10,901 kip - ft

For the example bent cap, fcpe = 0. The bent cap is designed for the monolithic section to resist all loads. substitute Snc for Sc and cracking moment is calculated by: = 3 ( (1 fr) Sc

Mcr

f r  0.24 f c  0.24 4  0.48 ksi

(12.5-2) (AASHTO 5.4.2.6)

1.60 (for concrete structure, AASHTO 5.7.3.3.2)  (for A706, Grade 60 reinforcement, AASHTO 5.7.3.3.2) Ig (flanged section) = 280.5 ft4; Yt = 40.30 in.; Yb = 40.70 in.

Sc 

Ig Yb



280.5  82.70 kip - ft 40.70 / 12

M cr  Sc f r  0.75 [ (1.60)(0.48) 82.70] 122  6,860 kip  ft

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1.0M cr  M u (min)  Lesser of   1.33M u  1.0 (6,860)  6,860 kip - ft   Lesser of    6,860 kip - ft 1.33 (10,901)  14,498 kip - ft Therefore, the controlling factored moment at mid-span is: Mu = 10,901 kip-ft For the bent section without prestressing steel and with neglecting the effect of compression reinforcement, the Mn equation reduces to:

æ a hƒ ö æ aö M n = As ƒ s ç ds - ÷ + 0.85ƒ¢c b - bw hƒ ç - ÷ 2ø è è2 2ø

(

)

(12.5-2)

Assuming the neutral axis lies in the compression flange (rectangular section behavior), nominal flexural resistance Mn is calculated by: a  M n  As f s  d s   2 

(12.5-3)

where: Mn = nominal flexural resistance (kip-in.) fs

= 60 ksi

ds = 81 – (5.70 + 1.63) = 73.67 in. (assuming vertical bundles of #11) b

= 164.64 in.

bw = 96 in. hf = 9 in. 

= 0.9

a

= c

(AASHTO 5.5.4.2)

where: 











(AASHTO 5.7.2.2)

 As f s   a    0.85 f cb   As f s   c    0.85 f cb 

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Rearrange and substitute design parameters into Eq. (12.5-1) to obtain:   A f  1  Mu  As f s d s   s s   0.9   0.85 f cb  2 

 As 60   1  10,91012    As 6073.67   0.9 . 164.64  2   0.854  Solving for As for positive moment region As

 33.71 in.2

Provide 22- #11 as bottom reinforcement: As = 34.32 in.2

( )

æ As fs ö 34.32 60 c=ç ÷ = 0.85 (4)(0.85)(164.64) = 4.33in. < 9 in. 0.85 f bb ¢ è ø c

(

)

Assumption of the neutral axis within in the flange is correct.

c   M r  0.9 M n  0.9 As f y  d s   2   (4.33)0.85    0.9 34.32(60)  73.67   2    133,120 kip - in.  11,093 kip - ft  M u  10,901 kip - ft





O.K.

Strain diagram is shown in Figure 12.5-14:

cu = 0.003

c

ds

t Figure 12.5-14 Strain Diagram   cu    t   c    d  c     s 

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t  0.003       4 . 33 73 . 67  4 . 33    

Since t = 0.048 > 0.005, assumption of  is correct.

12.5.1.7.3 Check Negative Moment at the Face of Column Factored moments are as follows: 

Strength I M u  1.25M DC  1.5M DW  1.75M HL 93

 1.25 1,760  1.5 217  1.75 1,859   5,779 kip - ft



Strength II M u  1.25M DC  1.5M DW  1.35M HL  93

 1.25 1,760  1.5 217  1.35 3,336   7,029 kip - ft

For the example bent cap, fcpe = 0. Also, the bent cap is designed for the monolithic section to resist all loads, substitute Snc for Sc and cracking moment is calculated by: Mcr

= 3 ( (1 fr) Sc

(12.5-2)

where : f r  0.24 f c  0.24 4  0.48 ksi

(AASHTO 5.4.2.6)

1.60 (for concrete structure, AASHTO 5.7.3.3.2)  (for A706, Grade 60 reinforcement, AASHTO 5.7.3.3.2) Ig (flanged section) = 280.5 ft4; Yt = 40.30 in.; Yb = 40.70 in.

Sc 

Ig Yb



280.5  83.52 ft 2 40.30 / 12

M cr  S c f r  0.75[(1.60)(0.48) 83.52]122  6,928 kip - ft

1.0M cr  M u (min)  Lesser of   1.33M u  

1.0(6,928) 6,928 kip - ft  Lesser of    6,928 kip - ft 1.33 (7,029)  9,349 kip - ft

Therefore, the controlling factored negative moment at the face of column is:

M u  7,029 kip - ft

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Assuming the neutral axis is within the compression flange (rectangular section behavior), nominal flexural resistance Mn is calculated by: a  M n  As ƒ s  d s   2 

where: fs = 60 ksi bw = 96 in. b = 164.64 in. ds = 81 – (5 + 1.63) = 74.37 in. (assuming vertical bundles of #11)  = 0.90 a = c

(AASHTO 5.5.4.2) where 

(AASHTO 5.7.2.2)

Rearrange and substitute design parameters into Eq. (12.5-1) to obtain:   As f y  1   Mu     As f y  d s      0.9  0.85 f c b  2   

( )

7,029 12 0.9

( ) ( )(

æ æ ö æ 1ö ö As 60 £ As 60 ç 74.37 - ç ÷ ç ÷÷ çè è 0.85 4 164.64 ø è 2 ø ÷ø

( )

)

Solving for As for negative moment at the face of the support: As

= 21.33 in.2

Provide 14 - #11 as Top reinforcement (As = 21.84 in.2)

c

As f s 21.8460   2.75 in.  8.25in. 0.85 f cb 0.8540.85161.64

Assumption of the neutral axis within the flange, is correct





c   M r  M n  0.9 As f y  d s   2   2.750.85    0.9 21.84(60) 74.37   2    86,331 kip - in.  7,194 kip - ft

Chapter 12 – Concrete Bent Caps

 M u  7,029 kip - ft

O.K.

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Strain in steel:

  cu    t     c   d s  c 

t  0.003        2.75   74.37  2.75  Therefore assumption of  is correct. 12.5.1.8

Step 6: Check for Serviceability Cracks occur whenever the tension in the gross section exceeds the cracking strength (modulus of rupture) of concrete. One can control or avoid flexural cracking in a concrete component by providing tension reinforcement at certain specified spacing. The spacing, s, of mild steel reinforcement in the layer closest to the tension face:

s 

700 e  2d c  s f ss

(AASHTO 5.7.3.4-1)

in which:

s  1

dc 0.7h  d c 

where: e =

exposure factor taken as 0.75 by considering Class 2 exposure condition (CA 5.7.3.4)

dc =

thickness of concrete cover measured from extreme tension fiber to center of the flexural reinforcement located closest thereto (in.) 1.5 + 0.69 + 0.69 / 2 = 2.54 in.

= h

= overall thickness or depth of the component (in.) = 81 in.

fss = tensile stress in steel reinforcement at the service time limit state (ksi) Tensile stress in steel reinforcement may be calculated based on the transformed section by the following procedure valid for both rectangular and flanged sections (Figure 12.5-15).

Chapter 12 – Concrete Bent Caps

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Figure 12.5-15 A Typical Flanged Section Note: Given b, bw, hf, d, d, As, As, n = Es / Ec, M = applied moment If h  0 and b 

2 h 2f

nd  h f As  n  1h f



 

 d  As , then set bw = b



Set

B

1 h f b  bw   nAs  n  1As bw

Set

C

2 2 h f b  bw  / 2  ndAs  n  1d As bw





x = B2 + C - B (assumes x  d)

)(

(

1 1 I = b x 3 - b - bw x - h f 3 3 f c¢ = f s¢ = fs =

) + nA ( d - x ) + ( n -1) A¢ ( x - d ¢) 3

2

s

2

s

Mx = stress in top fiber of compression flange I

(

nM x - d ¢ I

(

nM d - x I

) = nf ¢ æ 1- d ¢ ö = stress in compression steel cç

è

x ÷ø

) = nf ¢ æ d - 1ö = stress in tension steel

Chapter 12 – Concrete Bent Caps

cç

èx

÷ø

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12.5.1.8.1 Bottom Reinforcement For service load combination, the permit loads are not considered. Mser = 3,377 + 339 + 2,683 = 6,399 kip-ft Bent cap section: be = 13.72 ft = 164.64 in. bw = 96 in. hf = 9.125 in. Es = 29,000 ksi Ec = 3,645 ksi n

= Es / Ec = 7.96

Assume #16-5 as crack control reinforcement. Effective, As1 = (0.31)(16) cos (20) = 4.66 in.2 Tension reinforcement (bottom), As2 = 1.56 (22) = 34.32 in.2 Compression reinforcement (top), As = 1.56 (14) = 21.84 in.2 where: de1 = effective depth from extreme comp fiber to the centroid of the crack control reinforcement = 81 – (1.5 + 0.69 + 0.69/2) = 78.47 in. de2 = effective depth from extreme comp fiber to the centroid of tension reinforcement (bottom) = 81 – (5.7 + 1.63) = 73.67 in. d = effective depth from extreme comp fiber to the centroid of compression reinforcement (top) = 5 + 1.63 = 6.63 in.  be h 2f If    n  1As h f  d   2  lies in the flange.



(

) ( )2

æ 164.64 9 åç 2 ç è



   nAs d  h f  ,

then neutral axis, y,



ö + 7.96 -1 (21.84) 9 - 6.63 ÷ =7,028 ÷ ø

(

)

(

)

< (7.96) (4.66) (78.47 – 9) + (7.96) (34.32) (73.67 – 9) = 20,244

Chapter 12 – Concrete Bent Caps

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Therefore, the neutral axis is within web and the compression block has Tsection shape.

æ ö æ h f ö bw y 2 + n-1 As¢ y - d ¢ ÷ = å nAs d - y å ç be - bw h f ç y - ÷ + 2ø 2 è è ø

(

)

(

) (

)

( )(

)

(164.64 - 96) (9) (y – 9/2) + 96 (y2/2) + (7.96 – 1) (21.84) (y – 6.63) = 7.96 (4.66) (78.47 – y) + 7.96 (34.32) (73.67 - y) Solving for y: y = 14.93 in.



b y 3 be  bw  y  h f I  e  3 3 

3

 n  1As  y  d 2   nAs d  y 2

164.6414.933 164.64  9614.93  9.0   7.96  1(21.84)(14.93  6.63) 2 3 3 2   7.964.6678.47  14.932  7.9634.3273.67  14.93    3

 1,289,972 in.4  62.21 ft 4

f ss 

nM (d  y ) 7.966,3991278.47  14.93   30.6 ksi I 62.21124

bs = 1+ s 

2.54 = 1.046 0.7 81- 2.54

(

)

7000.75  2 2.54  11.32 in. 1.046(30.6)

12.5.1.8.2 Top Reinforcement Calculations for crack reinforcement at top of bent cap are not shown. Designer can use WinCONC or similar computer program to check serviceability of section. Flexural reinforcements are shown in Figure 12.5-16.

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Figure 12.5-16 Bent Cap Cross Section 12.5.1.9

Step 7: Check for Fatigue Unfactored fatigue moments at midspan are listed in Table 12.5-3. Table 12.5-3 Unfactored Fatigue Load Moments at Midspan. Load DC

Max. and Min. Moment at Mid-span (kip-ft) +Positive -Negtive 3,377 0

Max. and Min. Moment at Face of Column (kip-ft) +Positive -Negative 0 -1,760

DW

339

0

0

-217

Fatigue Vehicle I

789

-264

144

-504

For Fatigue I load combination, a load factor of 1.75 shall be used. Mu(max)

= 3,377 + 339 + 1.75(789) = 5,096.8 kip-ft

Mu(min)

= 3,377 + 339 + 1.75(-264) = 3,254 kip-ft

I = 69.5 ft4, nM (d  y ) fs  I

y = 14.96 in.

fs(max) = 7.96 (5,096.8)(12) (78.47 – 14.96)/(69.5)(124) = 21.45 ksi fs(min) = 7.96 (3,254)(12) (78.47 – 14.96)/(69.5)( 124) = 13.70 ksi

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(f) = fs(max) - fs(min) = 21.45 – 13.70 = 7.75 ksi (F)TH = 24 – 0.33 fmin = 24 – 0.33(13.7) = 19.48 ksi

(f) = 7.75 ksi < (F)TH = 19.48 ksi

OK

Fatigue requirement at the midspan is met. The following critical locations shall be checked for fatigue requirement: 

At maximum difference in positive moment region (bottom steel)



At maximum difference in negative moment region (top steel)



Moment reversal (top and bottom steel)

Calculations for fatigue check at top of bent cap not shown. Designer can use WinCONC or similar computer program to check fatigue limit state of the section. 12.5.1.10

Step 8: Perform Shear Design The shear design equation is as follows: Vu ≤ Vr = Vn

(12.5-2)

Shear design is performed for critical section located at the face of support. 12.5.1.10.1

Calculate Factored Shear Table 12.5-4 lists unfactored shears including impact from CSiBridge analysis: Table 12.5-4 Unfactored Shear at Face of Support



Load

Max. Shear (kip)

Assoc. Moment (kip-ft)

Max. Moment (kip-ft)

Assoc Shear (kip)

DC

-888

-1760

-1760

-888

DW

-98

-217

-217

-98

Design Vehicle

-338

588

-1860

-66

Permit Vehicle

-607

1054

-3335

-119

Strength I: Vu(max) Vu

= 1.25(-888) + 1.5(-98) + 1.75(-338) = -1,849 kips

Mu(assoc) = 1.25(-1,760) +1.5 (-217) +1.75(558)

Chapter 12 – Concrete Bent Caps

= -1,549 kip-ft

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12.5.1.10.2

Strength II: Vu(max) Vu

= 1.25(-888) + 1.5(-98) + 1.35(-607)

= -2,076 kips

Mu(assoc)

= 1.25(-1,760) +1.5(-217) +1.35(1,054) = -1,102 kip-ft

Determine  and  Using AASHTO-LRFD Table B5.2-1 for sections with minimum amount of transverse reinforcement. bv = 8 ft = 96 in. dv, using results from flexural analysis: dv  de 

0.852.75  73.20 in. a  74.37  2 2

0.9d e  0.974.37  66.93 in.  d v  73.20 in.  larger  0.72h  0.7281  58.3 in. Use dv = 73.2 in. Shear stress: vu 

Vu 2,076   0.328 ksi  bv d v 0.9 9673.2

Shear stress factor: vu 0.328   0.082 f c 4

Determine x at mid-depth  | Mu |    0.5 N u  0.5 | Vu  V p | cot   Aps f po  d  x   v 2 Es As  E p Aps





(AASHTO 5.8.3.4.2-4)

As there is no prestressing force and axial force in bent cap, the above equation reduces to:

 Mu    0.5Vu cot   dv  x   2E s As 

(12.5-3)

As = area of fully developed steel on flexural tension side of member = 28.08 in.2 Assuming 0.5cot  = 1 and use absolute values for Mu and Vu for strain calculation.

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 1,10212   2,076  73.2   0.001386 1.386103 x   229,00028.08 AASHTO Table B5.2-1 lists values of θ and β as function of shear stress factor vu / fc and strain at mid-depth of the bent cap x: AASHTO Table B5.2-1 Values of θ and β for Sections with Transverse Reinforcement vu / fc ≤0.075 ≤0.1 ≤0.125 ≤0.15 ≤0.175 ≤0.2 ≤0.225 ≤0.25

vu ≤–0.2 f c≤–0.1 22.3 6.32 18.1 3.79 19.9 3.18 21.6 2.88 23.2 2.73 24.7 2.63 26.1 2.53 27.5 2.39

20.4 4.75 20.4 3.38 21.9 2.99 23.3 2.79 24.7 2.66 26.1 2.59 27.3 2.45 28.6 2.39

εx  1,000 ≤–0.05

≤0

≤0.125

≤0.25

≤0.5

≤0.75

≤1

21 4.1 21.4 3.24 22.8 2.94 24.2 2.78 25.5 2.65 26.7 2.52 27.9 2.42 29.1 2.33

21.8 3.75 22.5 3.14 23.7 2.87 25 2.72 26.2 2.6 27.4 2.51 28.5 2.4 29.7 2.33

24.3 3.24 24.9 2.91 25.9 2.74 26.9 2.6 28 2.52 29 2.43 30 2.34 30.6 2.12

26.6 2.94 27.1 2.75 27.9 2.62 28.8 2.52 29.7 2.44 30.6 2.37 30.8 2.14 31.3 1.93

30.5 2.59 30.8 2.5 31.4 2.42 32.1 2.36 32.7 2.28 32.8 2.14 32.3 1.86 32.8 1.7

33.7 2.38 34 2.32 34.4 2.26 34.9 2.21 35.2 2.14 34.5 1.94 34 1.73 34.3 1.58

36.4 2.23 36.7 2.18 37 2.13 37.3 2.08 36.8 1.96 36.1 1.79 35.7 1.64 35.8 1.5

From above table:  = 2.18  = 36.7 12.5.1.10.3

Determine Shear Reinforcement Concrete contribution to shear resistance:

Vc  0.0316 f c bv d v  0.03162.18 4 9673.2  968 kips Demand on shear stirrups: Vs 

Vu 2,076  Vc   968  1,339 kips  0.7

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Required shear stirrups: Vs  1,339 

Av f y d v s

cot  

Av 6073.2 cot 36.7 o s





Av  0.227 in.2 /in. s

Minimum shear reinforcement:

f c¢ æ Av ö 4 2 çè s ÷ø = 0.0316 f bv = 0.0316 60 (96) = 0.1 in. /in. y min Required stirrups spacing: Av = use six legs #5 Av = 0.31 (6) = 1.86 in.2 Required s 

1.86  8.19 in. 0.227

Use s = 6 in. 12.5.1.10.4

Check Maximum Spacing For

vu < 0.125 , Smax = 0.8dv ≤ 18 in. f c¢

For

vu ³ 0.125 , Smax = 0.4dv ≤ 12 in. f c¢

At this particular section: vu 0.328   0.082 , Smax = 0.8(73.2) = 58.56 in. but not greater than 18 in. f c 4

Maximum spacing allowed = 18 in. Six legs #6 at 6 in. at the face of column meet this requirement. Note: Place bent cap stirrups parallel to girders.

12.5.1.10.5

(BDD 7-45.1)

Check Tenth Points along Bent Cap For Vu, Mu, ,  , Vc, Vs, S, Avmin, and Smax, Figure 12.5-17 shows stirrup spacing along the bent cap length.

Chapter 12 – Concrete Bent Caps

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Figure 12.5.1-19 shows stirrup spacing along bet cap length 35

Face of column

Stirrups Spacing, in.

30

25

20

Max Spacing = 18” 15

10

5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

length along beam Figure 12.5.1-15 Stirrup Spacing of Bent Cap

Figure 12.5.1-19 Stirrup Spacing along Bet Cap Length

Figure 12.5-17 Stirrup Spacing along Bent Cap Length 12.5.1.10.6

Check Longitudinal Reinforcement (Shear-Flexural Interaction) For a bent cap without prestressing steel and axial force, the longitudinal steel must satisfy: As f y 

Mu dv f

V    u  0.5Vs  cot     v 

(12.3.3-1)

Note: In practice, after the design is complete, engineer must check seismic design requirements as per SDC and check the longitudinal steel in the bent cap to handle seismic moments. To determine the tension in the longitudinal steel (the shear-flexure interaction), check at every location, usually at the 10th point of the span and at concentrated load: 

Maximum shear and associated moments



Maximum positive moments and associated shear



Maximum negative moments and associated shear

Longitudinal reinforcement at the first interior girder locations is checked in this example. Table 12.5-5 lists unfactored shears including impact from CSiBridge analysis at the location of the first interior girder:

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Table 12.5-5 Unfactored Shears at Location of First Interior Girder

DC

-878

Assoc. Moment (kip-ft) -373

DW Design vehicle Permit vehicle

-98

-64

-64

-98

-315

990

-1756

-66

-565

1,776

-3,149

-119

Max. Shear (kip)



Max. Moment (kip-ft) -373

Assoc. Shear (kip) -878

Strength I Maximum shear and associated moments: Vu = 1.25(-878) + 1.5(-98) + 1.75(-315) = -1,796 kips Mu(assoc) = 1.25(-373) + 1.5(-64) + 1.75(990) = 1,170 kip-ft Maximum moment and associated shear: V u(assoc) = 1.25(-878) + 1.5(-98) + 1.75(-66)

= -1,360 kips

Mu(max) = 1.25(-373) + 1.5(-64) + 1.75(-1,756) = -3,635 kip-ft 

Strength II Maximum shear and associated moments: Vu = 1.25(-878) + 1.5(-98) + 1.35(-565) = -2,007 kips Mu(assoc) = 1.25(-373) + 1.5(-64) + 1.35(1,776) = 1,835 kip-ft Maximum moment and associate shear: Vu(assoc) = 1.25(-878) + 1.5(-98) + 1.35(-119) = -1,405 kips Mu(max) = 1.25(-373) + 1.5(-64) + 1.35(-3,149) = -4,813 kip-ft

•

Check case of maximum shear and associated moment: Vu(max) = -2,007 kips Mu(assoc) = 1,835 kip-ft Compute x and  for this particular location under this loading: As(bot)

= 37.44 in.2

dv

= 73.67 – (0.85)(4.33)/2 = 71.83 in.

nu = 0.0876 ; x = 0.001077;  = 36.55o f c¢

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1.866071.83 cot 36.55o s 6 Vu 2,007  1,802.3 kips    2,230 kips  0.9

Vs 

Av f y d v

As f y 



cot  



1,83512   2,007  0.51,663.5 cot36.55o    0.971.83  0.9 

As 60  340.62  1,886.18  2,226.8 kips As ( req)  37.11 in.2  As (bot)  37.44 in.2

•

OK

Check case of maximum negative moment and associated shear Mu(max) = -4,813 kip-ft Vu(assoc) = -1,405 kips Compute x and  for this particular location under this loading: As(top)

= 28.08 in.2

dv

= 73.2 in.

nu = 0.0607 ; x = 0.00139;  f c¢

= 36.4o

1.866073.2 cot 36.4 o s 6 Vu 1,405  1,846.7 kips    1,561 kips  0.9

Vs 

Av f y d v

cot  





Provide 22 - # 11 (As = 34.32 in.2) as top reinforcement •

Check case of maximum positive moment and associated shear Mu(max) = 1,835 kip-ft Vu(assoc) = -2,007 kips Moment and shear in this case is similar to maximum shear and associated moment case.

Note: Using the same process, one can determine the size and spacing of the stirrups at other locations.

Chapter 12 – Concrete Bent Caps

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12.5.1.11

Determine Additional Details Additional details are discussed as follows.

12.5.1.11.1

Construction Reinforcement As shown in Figure 12.5-18, the concrete in the superstructure is poured in two stages during the construction process. It is a Caltrans practice to assume that 10 ft of the soffit on each side of the bent cap contributes to this dead load: 

Stage 1 (first pour) includes the soffit slab and the girder stems. The deck slab is not included.



Stage 2 (second pour) includes the deck slab.

Height of the Stage 1 pour = (6.75)(12) – 9 – 3 = 69 in. = 5.75 ft Width of the bent cap 

= 8 ft

Dead load due to exterior girder stem DL = [(5.75 - 8.25/12) / cos 26.6 (1)] (10 +10) (0.15) = 17 kips



Dead load due to cap and soffit Width of soffit slab causing negative moment = 10 + 10 = 20 ft DL = {(5.75)(8) + (20)(8.25/12)}(0.15) = 9 kip/ft Effective column diameter = 5.32 ft Negative moment at the face of the column Mu = 1.25 {[17 (7.12 – 5.32/2) + 9 (7.12 – 5.32/2)2 / 2 } = 207 kip-ft

Note: Assume a reduced value for fc as the concrete has not reached its specified compressive strength at the Stage 1 pour. Hence, use fc = 2.5 ksi. Caltrans standard practice is to provide four #10 bars as minimum construction reinforcement in the bent cap. Since there is no prestressing steel or compression reinforcement, for simplicity of calculations, the overhangs will be neglected and nominal moment is calculated by:

æ a hƒ ö æ aö M n = As ƒ s ç ds - ÷ + 0.85ƒ¢c b - bw hƒ ç - ÷ 2ø è è2 2ø

(

b

)

(12.5-4)

= 96 in.

ds = 69 – 3 = 66 in. As = 1.27(4) = 5.08 in.2

Chapter 12 – Concrete Bent Caps

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 c

= (5.08)(60) / [(0.85)( 2.5)(0.85)(96)] = 1.76 in.

a

= c  = 1.494

Mn = 5.08 (60)(66-1.494/2) = 19,889 kip-in. = 1,657 kip-ft Mr = Mn = 1,491 kip-ft > 1.33Mu = 275 kip-ft Provide 4 # 10 (As = 5.08 in.2) as construction reinforcement.

Second Pour First Pour

Figure 12.5-18 Concrete Pour Stages

Chapter 12 – Concrete Bent Caps

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12.5.1.11.2

Side Face / Skin Reinforcement The Caltrans standard practice is to provide side face reinforcement, which is 10 percent of the maximum longitudinal reinforcement. Maximum longitudinal reinforcement, As(bot) = 37.44 in.2 Side face reinforcement = 0.1(37.44) = 3.74 in.2 Provide 10 #6 bars. Since the construction reinforcement is also provided, two #10 bars would also count as side-face reinforcement. So, provide four #6 bars on each side of the bent cap. Spacing = {(5.75)(12) – (5.7+ 1.63) – 3} / 5 = 11.73 in. < 12 in. OK

12.5.1.11.3

End Reinforcement End reinforcements as shown in Figure 12.5-19 should be provided as a crack control measure, as discussed in Section 12.4.3 in accordance with BDD 7 (Caltrans, 1986).

U-bars in the Bent Cap

Figure 12.5-19 End Reinforcements The U-bars are designed for shear friction. Depending on whether the bent cap width is less than or more than 7 ft, designer to use one or two loops of the U-bars, as shown in Figure 12.5-20. Length of U-bars shall extend a development length beyond the inside face of the exterior girder.

Chapter 12 – Concrete Bent Caps

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Figure 12.5-20 Length of End Reinforcements 12.5.1.11.4

Bent Cap Reinforcement Since the bent cap skew = 20, bent cap reinforcement shall be detailed as shown in Figure 12.5-21. A dropped deck section may be required if main cap bars are bundled vertically. Distribution bars and bottom transverse bars may have to be terminated farther from the bent cap than three in. (standard) to allow vertical clearance for main bent bars . 

Slab reinforcement parallel to skew



Bent cap reinforcement as high as possible

Figure 12.5-21 Bent Cap Reinforcement (Skew ≤ 20)

Chapter 12 – Concrete Bent Caps

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2.5.2

Drop Bent Cap A three-span bridge with reinforced concrete drop bent cap is shown in Figures 12.5-22 through 12.5-25.

Figure 12.5-22 Elevation

Figure 12.5-23 Plan

Chapter 12 – Concrete Bent Caps

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Figure 12.5-24 Typical Section

Figure 12.5-25 Side View

Chapter 12 – Concrete Bent Caps

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12.5.2.1

Member Proportioning The bent cap depth should be deep enough to develop the column longitudinal reinforcement without hooks (SDC 7.3.4 and SDC 8.2.1). The minimum bent cap width required for adequate joint shear transfer shall be column sectional width in the direction of interest, plus two feet (SDC 7.4.2.1). dcap = 6 ft wcap = 5.5 + 2 = 7.5 ft

12.5.2.2

Classification of Bent Cap (AASHTO 5.8.1.1) Bent caps may fall under the classification of flexural beam or deep beam. The classification dictates the type of analytical theory that would most accurately estimate the internal forces of the bent cap. If either of these two cases is satisfied, then the bent cap may be considered a deep beam: Lv_zero < 2de If any girder produces more than half the bent cap shear at the support and is located less than 2de from the face of support, follow this: Lv_zero = 8.25 ft de = dcap – 2.5 = 69.5 in., assume 2 in. clear and an additional 0.5 in. to the centroid of flexural bar reinforcing 2de = 11.6 ft Both deep beam criteria are met for this example drop cap, so it should be evaluated using the strut-and-tie method per the AASHTO LRFD design specifications. Bent caps have typically been designed by using the sectional method which has been proved to be acceptable as historical data does not suggest design inadequacies in Caltrans. This design example will follow the sectional method for its conservativeness and ease of application. Caltrans will continue to use the sectional method until the strut-and-tie method is adopted agency-wide.

12.5.2.3

Material Properties Material properties are as follows: fc

=

4 ksi

fy

=

60 ksi

Es

=

29,000 ksi

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12.5.2.4

Loads CTBridge, LEAP Bridge, or other computer analysis programs can be used to determine the dead and live loads along the length of the bridge. For the bent cap design, the analysis is performed with only a single lane and/or truck so that wheel line loads may be generated and subsequently implemented in the frame analysis of the bent cap. Generally, dead loads such as DCsuperstructure and DW are distributed by tributary area (or width) for PC/PS I girder, steel girder, and bulb T girder bridges. DCbent_cap, however, is distributed along the length of the bent cap as a tributary load. In very stiff superstructures, such as cast-in-place prestressed concrete box girders, DCsuperstructure and DW may be distributed equally despite varying girder spacing. For this drop cap example, the deck, girders, bent cap, and columns will be modeled as individual elements. Based on the longitudinal analysis, the following dead loads are applied to bent cap analytical model as shown in Figure 12.5-26. PDC_barrier = 46.5 kips wDC_deck

= 12.8 kip/ft

wDW

= 4.4 kip/ft

PDC_girder

= 178 kips

wDC_bent_cap = 7.2 kip/ft Force effects, on the bent cap, from live loads are determined similarly to the methods used for the longitudinal analysis. The live loads are discretized into wheel line loads with fixed and variable spacing to represent spacing between wheel lines, as well as spacing between trucks and lanes. Influence lines are generated to determine the governing force effects on the bent cap element. We will begin this process by generating wheel line loads from CTBridge program results.

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Legend: 1 = DCbarrier

2 = DW

3 = DCdeck

4 = DCgirders

5 = DCbent cap Figure 12.5-26 Dead Loads in Elevation View 12.5.2.4.1 Unfactored Bent Reactions from Longitudinal Analysis For each design vehicular live load, the unfactored bent reactions at the bent can be obtained from the output of the longitudinal analysis. Results shown in Figures 12.5-27 to 12.5-29 are due to a single truck or lane load. The location designating "Col Bots" and "Col Tops" are the force effects to the bottoms and tops of all columns from the single truck or lane load. Results from three live loads, LRFD design vehicle (also known as HL-93), LRFD permit vehicle, and the LRFD fatigue

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vehicle are shown in Figures 12.5-27 to 12.5-29, respectively. The LRFD design vehicle consists of a truck and a lane. We are only interested in the maximum axial load from the truck or lane, and those values are boxed accordingly and shown as follows: LLHL-93_truck

= design truck (1 lane only)

(AASHTO 3.6.1.2.2)

LLHL-93_lane

= lane load (1 lane only)

(AASHTO 3.6.1.2.4)

LLpermit

= permit vehicle (1 lane only)

LLfatigue

= fatigue vehicle (1 lane only)

(CA 3.6.1.8) (AASHTO 3.6.1.4)

Figure 12.5-27 LRFD Design Vehicle

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Figure 12.5-28 LRFD Permit Vehicle

Figure 12.5-29 LRFD Fatigue Vehicle

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The lane load is considered uniformly distributed over a 10-ft width. However, we will simplify the analysis by combining the HL-93 lane load with the HL-93 truck wheel line loads. Per AASHTO 3.6.1.2.4, the IM shall be applied only to the truck load. The dynamic load allowance is applied during the combination of the truck and lane load are IMHL-93_truck = 1.33; and IMPermit = 1.25. LLHL-93_single_truck = 117.24 kips LLHL-93_single_lane = 115.19 kips LLpermit_single_truck = 368.22 kips LLfatigue_single_truck = 70.63 kips

LLHL93_ wheel _ line = =

(

)

LLHL93_single_truck IM HL93_ truck + LLHL93_single_lane

(

2

)

117.24 1.33 + 115.19

LLPermit _ wheel _ line = LLFatigue _ wheel _ line =

2

= 135.6 kips

(

LLPermit_single_truck IM permit 2 LLFatigue_single_truck 2

) = 368.22(1.25) = 230.1 kips 2

=

70.63 = 35.3 kips 2

12.5.2.4.2 Determine Number of Live Load Lanes Maximum number of whole live load lanes is obtained as: Clear bridge width between curbs and/or barriers = 39 ft. N = Integer part of (w/12) = Integer part of (39/12) = 3 12.5.2.4.3 Moving Live Load Transverse Analysis for Traffic Lanes For this drop cap example, the design vehicle variations are shown below. This is based on the maximum number of live load lanes—a total of three—that can possibly fit in the clear roadway width. Note that the multiple presence factor, m, for one lane of permit vehicle is 1.0 (AASHTO 3.6.1.8.2), and the factor does not apply to the fatigue vehicle (AASHTO 3.6.1.1.2). Figure 12.5-30 shows traffic lanes with a multiple presence factor.

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Figure 12.5-30 Types and Number of Live Loads to Apply to Bent Cap Both the design truck and the 10 ft loaded width in each lane are positioned along the clear bridge width to produce extreme force effects. The design load is positioned transversely such that the center of any wheel load is not closer than 2 ft from the edge of the design lane (AASHTO 3.6.1.3.1). For the moving load transverse analysis, the wheel lines may move anywhere within the 12 ft lane as long as AASHTO 3.6.1.3.1 is satisfied. Figures 12.5-31 and 32 show possible wheel line placement within the same 12-ft lane configuration. The designer must determine the placement of wheel lines that produces maximum force effects in the bent cap.

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Figure 12.5-31 Example Placement of Live Load

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Figure 12.5-32 Another Example of Placement of Live Load Additionally, the 12-ft lanes may move within the confines of the clear bridge width as long as no 12-ft lane overlaps another 12-ft lane. The designer must place the 12-ft lane, as well as the wheel lines, to garner the maximum force effects on the bent cap. The designer must consider analyzing the transverse model with one, two, and three truck configurations since it is not entirely evident that three trucks will always result in the maximum force effects in the bent cap. For the cap overhang, a single vehicle placed as close to the edge of the design lane as possible may result in the maximum negative moment demand. Note that by placing only one vehicle, the

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multiple presence factor, m, is maximized and will result in a higher negative moment demand than placing two trucks with a lower multiple presence factor. For a bent cap supported by multiple columns, it is advisable to use a structural analysis program, such as CSiBridge, capable of generating combinations of lane configurations and influence lines from moving live loads. CSiBridge is used for this example. The geometry (Figure 12.5-36) and model (Figure 12.5-37) as shown in Figures 12.5-33 and 12.5-34 consider the deck and girders atop the bent cap. Some designers may choose to construct a hybrid frame in which the deck, girders, and bent cap are represented by an integrated horizontal frame member. For the purpose of maintaining a simplified representation, we are opting to keep the members separate.

Figure 12.5-33 Geometric Model of Bent Cap

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Figure 12.5-34 CSiBridge Model Showing Member Designations For each vehicular load (HL-93, Permit, and Fatigue), the output will be organized in the following manner: Max M3, associated V2 Min M3, associated V2 Max V2, associated M3 Min V2, associated M3 Breaking down the demands in such a manner allows us to combine them with dead loads and apply the appropriate load factors. The associated force effects are necessary for checking moment-shear interaction (AASHTO 5.8.3.5). The bent cap locations of interest will be positive moments at midspan, negative moments at faces of column, and shear at faces of column.

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12.5.2.4.4 Frame Element 33 (Max Positive Moment at Midspan) Figures 12.5-35 to 12.5-38 are excerpts from the CSiBridge output to illustrate how moments are extracted from the analysis.

Figure 12.5-35 Positive Moment from Dead Loads (DC and DW) at Midspan of Cap

Figure 12.5-36 Maximum Positive Moment from HL-93 at Midspan of Cap

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Figure 12.5-37 Max Positive Moment from Permit Truck at Midspan of Cap

Figure 12.5-38 Max Positive Moment from Fatigue Truck at Midspan of Cap Maximum positive moments for Frame Element 33 obtained from CSiBridge output are summarized in Tables 12.5-6. Table 12.5-6 Maximum Positive Moments for Frame Element 33 Dead Loads

Live Load (M3)

Live Load (Associated V2)

MDC_33 = 71.2 kip-ft

MHL-93_33 = 1181.9 kip-ft

VHL-93_33 = 2 kips

VDC_33 = 4.2 kips

MPermit_33 = 2005.6 kip-ft

VPermit_33 = 3.4 kips

VDW_33 = 73.8 kip-ft

MDC_33 = 71.2 kip-ft

MDC_33 = 71.2 kip-ft

MDC_33 = 71.2 kip-ft

This example calculation will evaluate the bent cap at the right face of the left column, also referred to as the left end of Frame Element 34. Maximum negative

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moments for Frame Element 34 obtained from CSiBridge are summarized in Table 12.5-7. Table 12.5-7 Maximum Negative Moment for Frame Element 34 Dead Loads

Live Load (M3)

MDC_34 = -1127.5 kip-ft

MHL-93_34 = -904.9 kip-ft

VDC_34 = 344.5 kips

MPermit_34 = -1530.7 kip-ft

MDW_34 = -82.6 kip-ft

MFatigue_34 = -196.3 kip-ft

VDW_34 = 71.2 kip-ft

12.5.2.5

Design for Flexure The software WinConc are used to design for flexure in this example. Note that in WinConc, there are optional inputs for other loads discussed thoroughly in AASHTO Article 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, and 3.14. Of those loads, CR, SH, wind load on live load (WL), wind load on structure (WS), temperature loads (TU), and differential settlement (SE) can impose force effects in the bent cap. By virtue of experience, flexural and shear design is almost always governed by the Strength I and Strength II limit states. Since WL and WS are not considered in either of these load combinations (AASHTO Table 3.4.1-1), force effects from wind are not computed. However, uniform temperature is considered in both Strength I and Strength II limit states, but because of the relatively short distance between the two columns, it is anticipated that the force effects generated by TU is insignificant. Differential shrinkage pertains to strains generated between material of different age or composition. This example drop cap will be built monolithically; therefore, SH is not considered. Creep is a force effect that is generated by prestressed concrete elements. This example drop cap will be conventionally reinforced with mild reinforcement; therefore CR is also not considered. For this example, differential settlement will not be considered. Generally, the geotechnical engineer dictates the consideration of differential settlement. SE may impose force effects in the bent cap if the soil profiles between the two columns differ, thereby causing one column to settle more than the other column. Two WinConc models were assembled: one for the design of the bottom flexure reinforcing (positive moment) and the other for the design of the top flexure reinforcing (negative moment). WinConc allows the user to enter the bent cap dimensions and loads. Then, it tabulates the rebar size and quantity that satisfies the AASHTO LRFD design specifications. Figures 12.5-39 and 12.5-40 are excerpts from the output files to show how the results can be used

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Figure 12.5-39 Partial WinConc Output for Positive Flexure Design

Figure 12.5-40 Partial WinConc Output for Negative Flexure Design

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When WinConc is run in design mode, the “final results” show the various bar size and spacing configurations that satisfy the strength, service, fatigue, and extreme limit states, as well as crack control (AASHTO 5.7.3.4) and minimum reinforcement (AASHTO 5.7.3.3.2) requirements. It is worth noting that any bar configuration flagged with “space code” is undesirable because of substandard bar spacing. The following “load controls” are checked, and the governing “load control” is summarized in the final results: S-As = Service I, bar spacing for crack control

(AASHTO 5.7.3.4)

F-As = Fatigue I, fatigue stress in mild steel

(AASHTO 5.5.3.2)

Str-I

= HL-93 loads

Str-II = Permit loads Str-III = No HL-93 loads, wind > 55 mph Str-IV = No HL-93 loads, governs when DL to LL ratio is high Str-V = HL-93 loads, wind = 55 mph Ext-I = Earthquake Ext-II = Ice, train, vehicle, or vessel collision Arb-I = User defined load Min1 = Minimum reinforcement requirement, Mcr Min2 = Waiver of minimum reinforcement requirement, 1.33 Mu For flexural design in positive bending, Min2 governed for all bar sizes except the #18 bar. Of the available bars, #7, #8, #9, #10, and #11 provide acceptable capacity while satisfying bar spacing requirements. A total of 16 #8 bars are used. For flexural design in negative bending, Min2 also governed for all bar sizes except the #18 bar. Of the available bars, #8, #9, #10, #11, and #14 provide acceptable capacity while satisfying bar spacing requirements. We'll specify a total of 16 #10 bars, primarily so that they lineup with the 16 #8 main bottom bars. Although 16 #9 bars would suffice, some engineers believe it is good practice to specify bars sizes that are not too similar in size when bars can potentially be mixed up during construction. In summary, the bar reinforcement areas corresponding to positive and negative moments are as such: As positive = 16 (0.79 in.2) = 12.6 in.2 As negative = 16 (1.27 in.2) = 20.3 in.2 12.5.2.6

Design for Shear The AASHTO LRFD shear design method is based on the modified compression field theory. Contrary to the traditional shear design methodology, it assumes a

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variable angle truss model instead of the 45 truss analogy. The LRFD method accounts for interaction between shear, torsion, flexure, and axial load. The LRFD method notes that shear design will be considered to the distance, dv, from the face of support (AASHTO 5.8.3.2). However, Caltrans' practice is to evaluate shear to the face of support. 12.5.2.6.1 Calculate Factored Shear Unfactored shears and associated moments for the left face of the left column, Frame Element 34, extracted from CSiBridge output are summarized in Table 12.5-8. Table 12.5-8 Maximum Shears and Associated Moments for Frame Element 34 ( Interior Face of Column) Shear Demand (V2)

Associated Moments (M3)

VDC_34 = 344.5 kip

MDC_34 = -1127.5 kip-ft

VDW_34 = 47.3 kip

MDW_34 = -82.6 kip-ft

VHL-93_34 = 300.6 kip

MHL-93_34 = -196.3 kip-ft

VPermit_34 = 510.1 kip

MPermit_34 = 114.8 kip-ft

Two load combinations, Strength I and II typically govern for shear design of bent caps. Factored shear demand and associated factored moments for frame element 34 are calculated as: Vu_34_Strength_1 = 1.25 (VDC_34) + 1.5 (VDW_34) + 1.75 (VHL-93_34) = 1,027.6 kips Mu_34_Strength_1 = 1.25 (MDC_34) + 1.5 (MDW_34) + 1.75 (MHL-93_34) = -1,876.8 kip-ft Vu_34_Strength_I1 = 1.25 (VDC_34) + 1.5 (VDW_34) + 1.35 (VPermit_34) = 1,190.2 kips Mu_34_Strength_I1 = 1.25 (MDC_34) + 1.5 (MDW_34) + 1.35 (MPermit_34) = -1,378.3 kip-ft Vu_34 =Vu_34_Strength_I1 = 1,190.2 kips

 Strength II governs

Mu_34 =Mu_34_Strength_I1 = -1378.3 kip-ft 12.5.2.6.2 Determine  and  Cross section of the drop cap is shown in Figure 12.5-41. bv is width of web taken as 96 in; de is effective depth taken as 69.5 in; dv is effective shear depth taken as the max (dv1, 0.9 de, 0.72dcap); dcap is depth of cap taken as 72 in.

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Figure 12.5-41 Cross Section of Drop Cap Assuming fs = fy, we have

a

A

s negaitve f y

 As

positive

0.85 f cbv

dv1 = de -

fy

  20.360  12.660  1.42 in. 0.85496

a = 68.8 in. 2

dv = max (dv1, 0.9 (de), 0.72 (dcap) = 68.8 in. vu 

Vu 34 1,190.2   0.2 ksi  v bv d v 0.99668.8

(AASHTO 5.8.2.9-1)

Shear stress factor: vu 

vu 0.2   0.05 f c 4

Assume (0.5cot  = 1) and use absolute values for Mu and Vu for strain calculation, we have:

M u 34 x 

dv

 0.5Vu 34 cot 

2 E s As negative

 1,378.312   1,190.2   68 . 6   0.00121   229,00020.3

From AASHTO Table B5.2-1, β = 2.23 and θ = 36.4 are obtained. Use θ = 36.4 and recalculate εx as follows:

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M u 34 x 

dv

 0.5Vu 34 cot 

2 E s As negative

 1,378.312   0.51,190.2 cot 36.4 o   68.6   0.00089   229,00020.3





Form AASHTO Table B5.2-1, β = 2.23 and θ = 36.4 are obtained again. And convergence is reached. 12.5.2.6.3

Determine Shear Reinforcement Concrete contribution to shear resistance: Vc  0.0316 f c bv d v  0.03162.23 4 9668.6  928.1 kips

(AASHTO 5.8.3.3) V 1,190.2 Vs  u 34  Vc   928.1  394.3 kips v 0.9

Required shear stirrups:

Av Vs 1,190.2    0.212 in.2 /in. o s f y d v cot  6068.6 cot 36.4





Check stirrup ratio with minimum allowed transverse reinforcement ratio per AASHTO 5.8.2.5:

b 96  Av   0.0316 f c v  0.0316 4  0.101 in.2 /in.   fy 60  s  min Use stirrup ratio = 0.212 in.2/in. Compute stirrup spacing: Try four legs of #6 bar reinforcing (Ab = 0.44 in.2): Av = 4 (0.44 in.2) = 1.8 in.2 Required spacing, s = 1.8/0.212 = 8.49 in. Check maximum spacing requirement (AASHTO 5.8.2.7): vu f c'

 0.05

v  smax  if  u'  0.125, min(18 in., 0.8d v ), min(12 in., 0.4d v )   18 in. f   c 

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For design, provide stirrup spacing of 8 in at the face of column. Repeat the shear design steps as demonstrated above, for points along the drop cap length and produce a shear design chart. The design chart (Figure 12.5-42) shows the spacing requirements needed to satisfy the factored shear demand, Vu, as well as the minimum allowed transverse reinforcing ratios along the length of the bent cap.

Figure 12.5-42 Shear Design Chart 12.5.2.7

Check Longitudinal Reinforcement (Shear-Flexural Interaction) As discussed in Section 12.3.3, direct loading/support cases are typical for the drop bent cap beams. Since longitudinal reinforcements are continuous in Caltrans practice, there is no need to check shear-flexural interaction.

12.5.2.8

Detailing of Drop Cap Additional details are discussed as follows.

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12.5.2.8.1 Spacing of Longitudinal Reinforcement The clear distance between parallel bars shall not be less than lager of 1.5(nominal diameter of bars), 1.5(maximum size of coarse aggregate) and 1.5 inches (AASHTO 5.10.3.1.1). slongit max = 1.5 (1.27) = 1.9 in. The following calculation determines clear spacing between the positive flexural reinforcement, accounting for clear cover and approximate outside diameters of the #6 transverse reinforcement and #10 longitudinal bars: slongi provided 

7.5 12  2  2  0.88  0.88  241.44  2.2 in. 23

12.5.2.8.2 Side Face Reinforcement AASHTO 5.7.3.4 specifies that for sections that exceed three inches deep, longitudinal skin reinforcement shall be uniformly distributed along both side faces of the component for a distance de / 2 nearest the flexural tension reinforcing. The area of skin reinforcing, Ask (in.2/ft), of height on each side face shall satisfy the following equation, AASHTO required Ask min  0.012d e  30  0.47 in.2 /ft

(AASHTO 5.7.3.4-2) The total area of longitudinal skin reinforcing (per face) need not exceed Ask max. Aps = 0 in.2 Our drop cap example does not contain prestressing.

Ask max =

max éë( As positive )( As negative ) ùû + Aps 4

= 7.62 in.2

The maximum spacing of the skin reinforcing shall not exceed de / 6 or 12 in.

de  11.6 in. 6 Specify six #6 bars at each face. Spacing between bars will be approximately 9 inches. SDC (SDC Equation 7.4.4.3-3) requires that the total longitudinal side face reinforcement in the bent cap shall be at least equal to 0.1As_positive or 0.1As_negative and shall be placed near the side faces of the bent cap with a maximum spacing of 12 inches.

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





SDC Required Ask min  max 0.1( As positive ) 0.1( As negative )  3.05 in.2 /ft

Bent cap reinforcements are shown in Figures 12.5-43 to 12.5-45:

Figure 12.5-43 Elevation of Drop Cap Showing Stirrup Spacing

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Figure 12.5-44 Section A-A

Figure 12.5-45 Section B-B

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NOTATIONS a

=

distance from the center of gravity (CG) of the superstructure to the bottom of column = height of the column + depth to CG of the bent cap (12.3.1.1.7)

a

=

c1; depth of the equivalent stress block (in.) (12.3.1.1.7)

Aps

=

area of prestressing steel (in.2) (12.3.1.1.7)

As

=

area of mild steel tension reinforcement (in.2) (12.3.1.1.7)

As

=

area of compression reinforcement (in.2) (12.3.1.1.7)

Av

=

area of shear reinforcement (in.2) within a distance s (12.3.2)

b

=

top width of superstructure (12.3.1.1.7)

b

=

width of the compression face of the member (in.) (12.3.1.1.7)

bf

= flange width (12.1.1.3)

bledge

=

ledge width

bstem

=

stem width (12.1.1.3)

bv

=

effective web width (in.) taken as the minimum web width within the depth dv (12.3.2)

bw

=

web width or diameter of a circular section (in.) (12.3.1.1.7)

c

=

distance from the extreme compression fiber to the neutral axis (12.3.1.1.5)

c

=

distance from the centerline (CL) of the columns to the edge of deck (e + f) (12.3.1.1.7)

d

=

distance between the CL of columns (12.3.1.1.7)

d

=

distance from the compression face to the centroid of tension reinforcement (in.) (12.1.2)

dc

=

thickness of concrete cover measured from extreme tension fiber to center of the closest flexural reinforcement (in.) (12.3.1.2)

de

=

effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) (12.3.2)

dledge =

(12.1.1.3)

ledge depth (12.1.1.3)

dp

=

distance from extreme compression fiber to the centroid of prestressing tendons (in.) (12.3.1.1.7)

ds

=

distance from extreme compression fiber to the centroid of non-prestressed tensile reinforcement (in.) (12.3.1.1.7)

dstem

=

stem depth (12.1.1.3)

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dt

=

distance from the extreme compression fiber to the centroid of extreme tension steel (12.3.1.1.5)

ds

=

distance from extreme compression fiber to the centroid of compression reinforcement (in.) (12.3.1.1.7)

dv

=

effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compression force due to flexure, it need not be taken to be less than the greater of 0.9de or 0.72h (in.) (12.3.2.1)

e

=

distance from the CL of exterior girder to the edge of deck (EOD) (12.3.1.1.7)

Es

=

modulus of elasticity of reinforcing bars (ksi) (12.3.2.1)

Ep

=

modulus of elasticity of prestressing tendons (ksi) (12.3.2.1)

f

=

distance from the CL of the column to the CL of the exterior girder (12.3.1.1.7)

fcpe =

compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) not including the effects of secondary moment, at extreme fiber of section where tensile stress is caused by externally applied loads (ksi) (12.3.1.1.2)

fmin

=

minimum live load stress resulting from the Fatigue I load combination, combined with more serve stress from either the permanent loads or the permanent loads, shrinkage, and creep-induced external loads; . positive if tension, negative if compression (ksi) (12.3.1.3)

fpo

=

a parameter taken as modulus of elasticity of prestressing tendons multiplied by the locked-in difference in strain between the prestressing tendons and surrounding concrete (ksi) (12.3.2.1)

fps

=

average stress in prestressing steel at nominal bending resistance (ksi) (12.3.1.1.7)

fr

=

modulus of rupture of the concrete (ksi) (12.3.1.1.2)

fs

=

stress in the mild steel tension reinforcement at nominal flexural resistance (ksi) (12.3.1.1.7)

fss

=

tensile stress in steel reinforcement at the service limit state (ksi) (12.3.1.2)

fy

=

yield strength of transverse reinforcement (ksi) (12.3.2)

fc

=

specified compressive strength of concrete (ksi) at 28 days (ksi) 12.3.1.1.7)

fs

=

stress in the mild steel compression reinforcement at nominal flexural resistance (ksi) (12.3.1.1.7)

hcap

=

bent cap depth (12.1.1.3)

h

=

overall thickness or depth of the component (in.) (12.3.1.2)

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hf

=

compression flange depth of an I or T member (in.) (12.3.1.1.7)

Ig

=

moment of inertial of the gross concrete section about the centroidal axis, neglecting the reinforcement (in.4) (12.3.1.1.2)

k

=

constant that depends on the type of tendon (12.3.1.1.7)

Lv,zero

= distance from the point of zero shear to the face of the support (in.) (12.1.2)

Mcr

=

cracking moment (kip-in.) (12.3.1.1.2)

Mdnc

=

total unfactored dead load moment acting on the monolithic or noncomposite section (kip-in.) (12.3.1.1.2)

MDC

=

unfactored moment demand at the section from dead load of structural components and nonstructural attachments (12.3.1.1.1)

MDW

=

unfactored moment demand at the section from dead load of wearing surfaces and utilities (12.3.1.1.1)

Mu(HL93) =

factored moment demand at the section from HL93 Vehicle (12.3.1.1.1)

Mu(P-15) =

factored moment demand at the section from the Permit Vehicle (12.3.1.1.1)

Mu

= absolute value of the factored moment, not to be taken less than Vu  V p d v (kip-in.) (12.3.2.1)

Nu

=

factored axial force, taken as positive if tensile and negative if compressive (kip) (12.3.2.1)

s

=

spacing of stirrups (in.) (12.3.2)

Sc

=

section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3) (12.3.1.1.2)

Snc

=

section modulus for the extreme fiber of the monolithic or noncomposite section where tensile stress is caused by externally applied loads (in.3) (12.3.1.1.2)

Vn

=

nominal shear resistance (kip) (12.3.2)

Vp

=

component in the direction of the applied shear of the effective prestressing force (kip); positive if resisting the applied shear; typically zero for conventionally reinforced bent caps (12.3.2)

Vr

=

factored shear resistance (kip) (12.3.2)

Vu

=

factored shear (kip) (12.3.2.1)

Yt

=

distance from the neutral axis to the extreme tension fiber (in.) (12.3.1.1.2)



=

angle of inclination (°) of transverse reinforcement to longitudinal axis; typically 90° (12.3.2)

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

=

angle of inclination (°) of diagonal compressive stresses (12.3.2)



=

factor indicating ability of diagonally cracked concrete to transmit tension and shear (12.3.2)

1

=

stress block factor (12.3.1.1.7)

x

=

longitudinal strain at the mid-depth of member (12.3.2.1)

εcu

=

failure strain of concrete in compression (12.3.1.1.5)

εt

=

net tensile strain (12.3.1.1.5)



=

load factor for Fatigue I (12.3.1.3)

1

=

flexural cracking variability factor (12.3.1.1.2)

2

=

prestress variability factor (12.3.1.1.2)

3

=

ratio of specified minimum yield strength to ultimate tensile strength of the reinforcement (12.3.1.1.2)

γe

=

exposure factor (12.3.1.2)



=

resistance factor (12.3.1.1.4)

f

=

resistance factor for moment (12.3.3)

c

=

resistance factor for axial load (12.3.3)

v

=

resistance factor for shear (12.3.3)

f

=

live load stress range (ksi) (12.3.1.3)

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, American Association of State Highway and Transportation Officials, Washington, DC. 2. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications— 6th Edition, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2013). Caltrans Seismic Design Criteria—Version 1.7, California Department of Transportation, Sacramento, CA. 4. Caltrans, (2000). Bridge Design Specifications—LFD Version, California Department of Transportation, Sacramento, CA, 5. Caltrans, (1988). Memo to Designers 15-17 Future Wearing Surface, California Department of Transportation, Sacramento, CA. 6. Caltrans, (1986). Bridge Design Details—Section 7, California Department of Transportation, Sacramento, CA.

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CHAPTER 13 CONCRETE COLUMNS TABLE OF CONTENTS 13.1

INTRODUCTION ......................................................................................................... 13-1

13.2

TYPES OF COLUMNS ....................................................................................... 13-1

13.3

DESIGN LOADS............................................................................................... 13-1

13.4

DESIGN CRITERIA ......................................................................................... 13-2 13.4.1 Limit States ...................................................................................................... 13-2 13.4.2 Forces ............................................................................................................... 13-2

13.5

APPROXIMATE EVALUATION OF SLENDERNESS EFFECTS ............... 13-2 13.5.1 Moment Magnification Method ....................................................................... 13-3

13.6

COMBINED AXIAL AND FLEXURAL STRENGTH ................................... 13-5 13.6.1 Interaction Diagrams ........................................................................................ 13-5 13.6.2 Pure Compression ............................................................................................ 13-6 13.6.3 Biaxial Flexure ................................................................................................. 13-7

13.7

COLUMN FLEXURAL DESIGN PROCEDURE ............................................ 13-8 13.7.1 Longitudinal Analysis (CTBridge) ................................................................... 13-8 13.7.2 Transverse Analysis (CSiBridge) ..................................................................... 13-8 13.7.3 Column Live Load Input Procedue .................................................................. 13-8 13.7.4 Wind Loads (WS, WL) .................................................................................. 13-14 13.7.5 Braking Force (BR) ........................................................................................ 13-14 13.7.6 Prestress Shortening Effects (CR, SH) ........................................................... 13-14 13.7.7 Prestressing Secondary Effect Forces (PS) .................................................... 13-14 13.7.8 Input Loads into WinYIELD .......................................................................... 13-14 13.7.9 Column Design/Check ................................................................................... 13-14

13.8

COLUMN SHEAR DESIGN PROCEDURE.................................................. 13-15 13.8.1 Longitudinal Analysis .................................................................................... 13-15 13.8.2 Transverse Analysis ....................................................................................... 13-15

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13.8.3 Column Live Load Input Procedure ............................................................... 13-16

13.9

COLUMN SEISMIC DESIGN PROCEDURE ............................................... 13-18

13.10 DESIGN EXAMPLE ....................................................................................... 13-18 13.10.1 Design Column One at Bent Two .................................................................. 13-19 13.10.2 Flexural Check of Main Column Reinforcemen (As) ..................................... 13-21 13.10.3 Shear Design for Transverse Reinforcement (Av) .......................................... 13-44

NOTATION ................................................................................................................. 13-55 REFERENCES ............................................................................................................ 13-58

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CHAPTER 13 CONCRETE COLUMNS 13.1

INTRODUCTION Columns are structural elements that support the superstructure, transfer vertical loads from superstructure to foundation, and resist the lateral loads acting on the bridge due to seismic and various service loads.

13.2

TYPES OF COLUMNS Columns are categorized along two parameters (Chen, 2014 and MacGregor, 1988): shape and height: 

Columns sections are usually round, rectangular, solid, hollow, octagonal, or hexagonal.



Columns may be short or tall. The column is called either short or tall according to its effective slenderness ratio (Klu/r). where: K = effective length factor lu = unsupported length of a compression member r = radius of gyration

13.3

DESIGN LOADS The considered design loads as specified in AASHTO 3.3.2 are:



Dead loads (DC)



Added dead loads (DW)



Design vehicular live loads: 1. Design vehicle HL-93 shall consists of a combination of (Truck + Lane) or (design tandem + Lane) including dynamic load allowance (IM). 2. Permit vehicle (P15) including the dynamic load allowance (IM).



Wind loads (WS, WL)

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13.4



Braking force (BR)



Thermal effects (TU)



Prestress shortening effects (CR, SH)



Prestressing secondary effects (PS)

DESIGN CRITERIA Columns are designed for Service, Strength, and Extreme Event limit states (AASHTO, 2012 and Caltrans, 2014). The Extreme Event I limit state must be in accordance with the current the Caltrans Seismic Design Criteria (SDC) version 1.7 (Caltrans, 2013). Columns should be designed as ductile members to deform inelastically for several cycles without significant degradation of strength or stiffness under the design earthquake demand (see SDC seismic design criteria chapters 3 and 4 for more details). Columns supporting a superstructure that is built using balanced cantilevered construction, or other unusual construction loads, are not addressed herein.

13.4.1

Limit States As stated above, columns are designed for three limit states:  

 13.4.2

Strength Limit State Service Limit State Extreme Event Limit State

Forces Bridge columns are subjected to axial loads, bending moments, and shears in both the longitudinal and transverse directions of the bridge.

13.5

APPROXIMATE EVALUATION OF SLENDERNESS EFFECTS The slenderness of the compression member is based on the ratio of Klu/r (AASHTO 5.7.4.3), while the effective length factor, K (AASHTO 4.6.2.5), is to compensate for rotational and transitional boundary conditions other than pinned ends. Theoretical and design values of K for individual members are given in AASHTO Table C4.6.2.5.-1. Slenderness s effect is ignored if: Klu/r < 22

Chapter 13 – Concrete Columns

(members not braced against sidesway)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Klu/r < 34 – 12 (M1 / M2)

(members braced against sidesway)

where: M1 = smaller end moment, should be positive for single curvature flexure M2 = larger end moment, should be positive for single curvature flexure lu

= unsupported length of a compression member

r

= radius of gyration = 0.25 times the column diameter for circular columns = 0.3 times the column dimension in the direction of buckling for rectangular columns

If slenderness ratio exceeds the above-mentioned limits, the moment magnification procedure (AASHTO 4.5.3.2.2b) can approximate the analysis. Note: If Klu/r exceeds 100, columns may experience appreciable lateral deflections resulting from vertical loads or the combination of vertical loads and lateral loads. For this case, a more detailed second-order non-linear analysis should be considered, including the significant change in column geometry and stiffness.

13.5.1

Moment Magnification Method The factored moments may be increased to reflect effects of deformation as follows: Mc = b M2b + s M2s

(AASHTO 4.5.3.2.2b-1)

where: Mc = magnified factored moment M2b = moment on compression member due to factored gravity loads that result in no sideway, always positive M2s = moment on compression member due to factored lateral or gravity loads that result in sideway, , greater than lu/1500, always positive b = moment magnification factor for compression member braced against sidesway s = moment magnification factor for compression member not braced against sidesway The moment magnification factors (b and s) are defined as follows:

b 

Cm 1 Pu 1  k Pe

Chapter 13 – Concrete Columns

(AASHTO 4.5.3.2.2b-3)

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s 

1  Pu 1  k  Pe

(AASHTO 4.5.3.2.2b-4)

For members braced against sideway s is taken as one unless analysis indicates a lower value.For members not braced against sideway b is to be determined as for a braced member and s for an unbraced member. Pu = factored axial load Pe = Euler buckling load, which is determined as follows: Pe 

 2 Ec I

( Klu ) 2

Ec = the elastic modulus of concrete I = moment of inertia about axis under consideration

k = stiffness reduction factor; 0.75 for concrete members and 1 for steel members Cm = a factor, which relates the actual moment diagram to an equivalent uniform moment diagram, is typically taken as one However, in the case where the member is braced against sidesway and without transverse loads between supports, Cm may be based on the following expression:

C m  0.6  0.4

M 1b M 2b

(AASHTO 4.5.3.2.2b-6)

To compute the flexural rigidity EI for concrete column in determining Pe, AASHTO 5.7.4.3 (AASHTO, 2012) recommends that the larger of the following be used: Ec I g

 Es I s 5 1  d

EI 

(AASHTO 5.7.4.3-1)

Ec I g EI  2.5 1 d

(AASHTO 5.7.4.3-2)

where: Ig = the gross moment of inertia (in.4) Es = elastic modulus of reinforcement (ksi) Is = moment of inertia of longitudinal steel about neutral axis (ksi) d = ratio of maximum factored permanent load moment to the maximum factored total load moment, always positive

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13.6

COMBINED AXIAL AND FLEXURAL STRENGTH

13.6.1

Interaction Diagrams Flexural resistance of a concrete member is dependent upon the axial force acting on the member. Interaction diagrams for a reinforced concrete section are created assuming a series of strain distributions and computing the corresponding moments and axial forces. The results are plotted to produce an interaction diagram as shown in Figure 13.6-1. c = 0.003 Po

P c = 0.003

s ≤ y = 0.002

Compression Controlled

s = y = 0.002

Pb

Balanced Strain Condition Tension Controlled

Mo

Mb

c = 0.003

Mn

s ≥ 0.005

Figure 13.6-1 Typical Strength Interaction Diagram for Reinforced Concrete Section with Grade 60 Reinforcement When combined axial compression and bending moment act on a member having a low slenderness ratio and where column buckling is not a possible mode of failure, the strength of the member is governed by the material strength of the cross section. For this so–called short column, the strength is achieved when the extreme concrete compression fiber reaches the strain of 0.003. In general, one of three modes of failure will occur: tension controlled, compression controlled, or balanced strain condition (AASHTO 5.7.2.1). These modes of failure are detailed below:

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13.6.2



Tension controlled: Sections are tension controlled when the net tensile strain in the extreme tension steel is equal to or greater than 0.005 just as the concrete in compression reaches its assumed strain limit of 0.003.



Compression controlled: Sections are compression controlled when the net tensile strain in the extreme tension steel is equal to or less than the net tensile strain in the reinforcement (y = 0.002) at balanced strain condition at the time the concrete in compression reaches its assumed strain limit of 0.003.



Balanced strain condition: Where compression strain of the concrete (c = 0.003) and yield strain of the steel (for Grade 60 reinforcement y = 0.002) are reached simultaneously, the strain is in a balanced condition.

Pure Compression For members with spiral transverse reinforcement, the axial resistance is based on: Pr =Pn = Po =  fc AgAstAst fy(AASHTO  For members with tie transverse reinforcement, the axial resistance is based on: Pr =Pn = Po =  fc AgAstAst fy AASHTO where: Pr = factored axial resistance Pn = nominal axial resistance, with or without flexure  = resistance factor specified in AASHTO 5.5.4.2 Po = nominal axial resistance of a section at zero eccentricity fc = specified strength of concrete at 28 days, unless another age is specified Ag = gross area of section Ast = total area of main column reinforcement fy = specified yield strength of reinforcement

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.6.3

Biaxial Flexure AASHTO 5.7.4.5 specifies the design of non-circular members subjected to biaxial flexure and compression based on the stress and strain compatibility using one of the following approximate expressions: For the factored axial load, Pu ≥ 0.1fcAg

1 1 1 1    Prxy Prx Pry Po

(AASHTO 5.7.4.5-1)

where: Po = 0.85fc (Ag – Ast) + Astfy

(AASHTO 5.7.4.5-2)

For the factored axial load, Pu ≤ 0.1fcAg

M ux M uy  1 M rx M ry

(AASHTO 5.7.4.5-3)

where: Prxy = factored axial resistance in biaxial flexure Prx = factored axial resistance determined on the basis that only eccentricity ey is present Pry = factored axial resistance determined on the basis that only eccentricity ex is present Pu = factored applied axial force Mux = factored applied moment about x axis Muy = factored applied moment about y axis Mrx = uniaxial factored flexural resistance of a section about x axis corresponding to the eccentricity produced by the applied factored axial load and moment Mry = uniaxial factored flexural resistance of a section about y axis corresponding to the eccentricity produced by the applied factored axial load and moment

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.7

COLUMN FLEXURAL DESIGN PROCEDURE Column flexure design steps for permanent and transient loads are presented in the following sub-sections.

13.7.1

Longitudinal Analysis (CTBridge) Perform a longitudinal analysis of the bridge under consideration using Caltrans CTBridge software. Results will determine:

13.7.2



Axial load (Ax) and longitudinal moment (Mz) at top of the column for DC and DW



Maximum unfactored axial load (Ax) and associated longitudinal moment (Mz) of design vehicular live loads for one lane per bent



Maximum unfactored longitudinal moment (Mz) and associated axial load (Ax) of design vehicular live loads for the one lane per bent

Transverse Analysis (CSiBridge) Perform a transverse analysis of bent cap (BDP Chapter 12, Bent-Cap) using commercial software CSiBridge. Results of the analysis is used to determine: 

Column axial load (P) and transverse moment (M3) for DC and DW



Maximum axial load (P) and associated transverse moment (M3) for design vehicular live loads



Maximum transverse moment (M3) and associated axial load (P) for design vehicular live loads

Note: WinYIELD (Caltrans, 2008) uses the x-axis for longitudinal direction and yaxis for the transverse direction. The CTBridge output renames Mz as Mx and Ax as P. The CSiBridge output renames the transverse moment, M3, as My.

13.7.3

Column Live Load Input Procedure

13.7.3.1

Output from Longitudinal 2D Analysis (CTBridge) Column unfactored live load forces and moments for one lane from longitudinal analysis (CTBridge) are summarized in Table 13.7-1 below:

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.7-1 Unfactored Bent Reactions for One Lane, Dynamic Load Allowance Factors Not Included Design Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft)

A  A 

M  

T max CT

Truck

T z assoc CT

M  

Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft)

A  A 

T assoc CT

Truck

P max CT

M  

P z assoc CT

M   M  

T z max CT

Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft)

A



P assoc CT

M  

P z max CT

L z max CT

L assoc CT

Lane

A 

L z assoc CT

L max CT

Lane

Permit Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft)

where:

A 

T max CT

= maximum axial force for truck load

M  

A M A M

T z assoc CT

     

L max CT

= longitudinal moment associated with maximum axial force for truck load = maximum axial force for lane load

L z assoc CT

P max CT

= maximum axial force for permit vehicle load

P z assoc CT

M   A  M  A  M   A 

T z max CT

T assoc CT L z max

L assoc CT

P z max CT

P assoc CT

= longitudinal moment associated with maximum axial force for lane load

= longitudinal moment associated with maximum axial force for permit vehicle load

= maximum longitudinal moment for truck load = axial force associated with maximum longitudinal moment for truck load = maximum longitudinal moment for lane load = axial force associated with maximum longitudinal moment for lane load = maximum longitudinal moment for permit vehicle load = axial force associated with maximum longitudinal moment for permit vehicle load

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.7.3.2

Output from 2D Transverse Analysis (CSiBridge) Axial forces presented in Table 13.7-1 are converted to two pseudo wheel loads including dynamic allowance factor to be used in transverse analysis (see BDP Chapter 12) to be used in transverse analysis. 

Include dynamic load allowance factor for Table 13.7-1.



Column reaction = 1.33(reaction/2) for truck = 1(reaction/2) for lane = 1.25(reaction/2) for P-15

The transverse analysis column forces for pseudo truck and permit wheel loadings are presented in Table 13.7-2. Table 13.7-2 Unfactored Column Reaction, Including Dynamic Load Allowance Factors Design Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft)

P 

T max CSi

Truck

M  

T 3 assoc CSi

Maximum transverse moment and associated axial load P (kip) M3 (kip-ft)

P 

T assoc CSi

Truck

M  

T 3 max CSi

Permit Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft)

P 

P max CSi

M  

P 3 assoc CSi

Maximum transverse moment and associated axial load P (kip) M3 (kip-ft)

P 

P assoc CSi

M  

P 3 max CSi

where:

P  M   T max CSi

T 3 assoc CSi

P  M   P max Csi

= maximum axial force due to pseudo truck wheel loads =

wheel loads. =

P 3 assoc CSi =

M   P 

T 3 max CSi

T assoc CSi

transverse moment associated with maximum axial force due to pseudo truck

maximum axial force due to pseudo permit wheel loads transverse moment associated with maximum axial force due to pseudo permit wheel loads

= maximum transverse moment due to pseudo truck wheel loads = axial force associated with maximum transverse moment due to pseudo truck wheel loads

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

M   P 

P 3 max CSi

P assoc CSi

= maximum transverse moment due to pseudo permit wheel loads

=

axial force associated with maximum transverse moment due to pseudo permit wheel loads

13.7.3.3

CTBridge output including Dynamic Load Allowance Factors Multiply dynamic allowance factor for values in Table 13.7-1 divided by number of bent columns to get reactions per column (Table 13.7-3).

Table 13.7-3 Unfactored Column Reactions for One Lane, Including Dynamic Load Allowance Factors Design Vehicle Maximum axial load and associated longitudinal moment P (kip) Mx (kip-ft) Truck Lane

P  P 

T max CT

L max CT

M   M  

Permit Vehicle Maximum axial load and associated longitudinal moment P (kip) Mx (kip-ft)

T x assoc CT

 

L x assoc CT

 

Maximum longitudinal moment and associated axial load P (kip) Mx (kip-ft) Truck Lane

13.7.3.4

P  P 

T assoc CT

L assoc CT

M   M  

T x max CT L x max CT

P 

P max CT

M  

P x assoc CT

 

 

Maximum longitudinal moment and associated axial load P (kip) Mx (kip-ft)

M  

P 

P x max CT

P assoc CT

Truck and Lane Loads for Transverse Analysis (CSiBridge) Split truck reactions results of transverse analysis (Table 13.7-3) into truck and lane loads as follows:

    P =   P 

     P 

T  Pmax CT Ratio of truck load per design vehicle =  T L P   max CT Pmax

Ratio of lane load per design vehicle

CT

   R1 

L max CT T L max CT max CT

   R2 

Unfactored column reactions (Table 13.7-4) including dynamic load allowance (CSiBridge): R1 = truck load ratio of design vehicle (values of Table 13.7-2) R2 = lane load ratio of design vehicle (values of Table 13.7-2)

Chapter 13 – Concrete Columns

13-11

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.7-4 Unfactored Column Reactions, Including Dynamic Load Allowance Factors Design Vehicle Maximum axial load and associated transverse moment P (kip) My (kip-ft) Truck Lane

P  P 

T max CSi L max CSi

M   M  

Lane

13.7.3.5

P  P 

T assoc CSi L assoc CSi

M  

P 

T y assoc CSi

P y assoc CSi

P max CSi

L y assoc CSi

Maximum transverse moment and associated axial load P (kip) My (kip-ft) Truck

Permit Vehicle Maximum axial load and associated transverse moment P (kip) My (kip-ft)

M   M  

Maximum transverse moment and associated axial load P (kip) My (kip-ft)

M  

P 

T y max CSi

P y max CSi

P assoc CSi

L y max CSi

Combination of Longitudinal and Transverse Output

Combine forces and moments of Tables 13.7-3 and 13.7-4. 

Case 1: Maximum My (Table 13.7-5)



Case 2: Maximum Mx (Table 13.7-6)



Case 3: Maximum P (Table 13.7-7)

Table 13.7-5 Case 1: Maximum Transverse Moment (My) My (kip-ft) Mx (kip-ft) P (kip)

P-truck

H-truck

M yP max CSi

M Ty max CSi

      P  Passoc     M    P CSi M xP assoc. CT  P  Pmax CT   P   T assoc CSi T max CT

P 

P assoc CSi

Chapter 13 – Concrete Columns

P 

T x assoc CT

T assoc CSi

Lane

M    P  M     P   L y max CSi

L assoc CSi L max CT

L x assoc CT

P 

L assoc CSi

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.7-6 Case 2: Maximum Longitudinal Moment (Mx)

 

 M   

P  Pmax  P  Pmax

 

  M    

 

  P   

My (kip-ft)

  

Mx (kip-ft)

P (kip)

P-truck

P Passoc . CT P Pmax CT



P y. assoc . CSi

P x . max CT

CSi CT

P  Passoc .  P CT  Pmax CT

P max . CSi

H-truck

 

 M   

T  Pmax  T  Pmax

 

  M    

 

  P   

  

T Passoc . CT T Pmax CT



T y. assoc . CSi

T x . max CT

CSi CT

T  Passoc .  T CT  Pmax CT

  

Lane

 

 M   

L Passoc . CT L Pmax CT



L y. assoc . CSi

   M      L  Passoc  L .  L CT  Pmax . CSi  Pmax CT 

L  Pmax  L  Pmax

T max . CSi

L x. max CT

CSi CT

Table 13.7-7 Case 3: Maximum Axial Load (P) P-truck

My (kip-ft) Mx (kip-ft) P (kip)

13.7.3.6

    P     M   P  

H-truck

M yP assoc. CSi

P max CSi P max CT

    P     M   P  

Lane

  

M Ty assoc. CSi



P x assoc . CT

T max CSi T max CT

P 

M yL assoc. CSi



T x assoc . CT

 

L  Pmax  L  Pmax

P 

P max CSi

T max CSi

  M    L Pmax CSi CSi CT



L x assoc . CT

WinYIELD Live Load Input

Transfer Tables 13.7-5, 13.7-6, and 13.7-7 data into Table 13.7-8, which will be used as load input for the WinYIELD program. Table 13.7-8 Input for Column Live Load Analysis of WinYIELD Program. Case 1: Max Transverse (My) Lane P-truck H-truck Load My Trans Mx Long P Axial

TABLE 13.7-5 Data

Chapter 13 – Concrete Columns

Case 2: Max Longitudinal (Mx) Lane P-truck H-truck Load TABLE 13.7-6 Data

Case 3: Max Axial (P) Ptruck

H-truck

Lane Load

TABLE 13.7-7 Data

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.7.4

Wind Loads (WS, WL) Calculate wind moments and axial loads for column (see BDP Chapter 3).

13.7.5

Braking Force (BR) Calculate braking force moments and axial load for column (see BDP Chapter 3).

13.7.6

Prestress Shortening Effects (CR, SH) Calculate prestress shortening moments as shown in design example (13.10).

13.7.7

Prestressing Secondary Effect Forces (PS) Calculate secondary prestress moments and axial loads (from CTBridge output).

13.7.8

Input Loads into WinYIELD Transfer all loads into WinYIELD’s load table.

13.7.9

Column Design/Check Run WinYIELD to design/check the main vertical column reinforcement.

Chapter 13 – Concrete Columns

13-14

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.8

COLUMN SHEAR DESIGN PROCEDURE Column shear demand values are calculated from longitudinal and transverse analyses.

13.8.1

Longitudinal Analysis Perform a longitudinal analysis (CTBridge) to determine:  

Longitudinal shear (Vy) and moment (Mz) for DC and DW at top and bottom of the column. Maximum longitudinal shear (Vy) and associated moment (Mz) for design vehicular live loads at top and bottom of the bent unfactored reactions for one lane as shown in Table 13.8-1.

Table 13.8-1 Longitudinal Unfactored Bent Reactions for One Lane, Dynamic Load Allowance Factors Not Included. Design Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column Vy (kip) Mz (kip-ft) Truck ((VyT)max)CT ((MzT)assoc)CT Lane ((VyL)max)CT ((MzL)assoc)CT Maximum longitudinal shear and associated longitudinal moment at bottom of the column Vy (kip) Mz (kip-ft) Truck ((VyT)max)CT ((MzT)assoc)CT Lane ((VyL)max)CT ((MzL)assoc)CT

Permit Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column Vy (kip) Mz (kip-ft) ((VyP)max)CT ((MzP)assoc)CT Maximum longitudinal shear and associated longitudinal moment at bottom of the column Vy (kip) Mz (kip-ft) ((VyP)max)CT ((MzP)assoc)CT

where:

((VyT)max)CT = maximum longitudinal shear at top and bottom of column for truck load ((MzT)assoc)CT = longitudinal moment at top and bottom of column associated with maximum shear for truck load ((VyL)max)CT = maximum longitudinal shear at top and bottom of column for lane load ((MzL)assoc)CT = longitudinal moment at top and bottom of column associated with ,aximum shear for lane load ((VyP)max)CT = maximum longitudinal shear at top and bottom of column for permit load ((MzP)assoc)CT = longitudinal moment at top and bottom of column associated with maximum shear for permit load

Chapter 13 – Concrete Columns

13-15

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.8.2

Transverse Analysis Perform a transverse analysis (CSiBridge) to determine: 



Column transverse shears (V2) and associated moment (M3) for DC and DW Maximum transverse shear (V2) and associated moment (M3) for design vehicular live loads at top and bottom of the column with dynamic load allowance factors included, as shown in Table 13.8-2

Table 13.8-2 Transverse Unfactored Column Reactions Including Dynamic Load Allowance Factors Design Vehicle Maximum transverse shear and associated transverse moment at top of the column V2 (kip) M3 (kip-ft) Truck ((V2T)max)CSi ((M3T)assoc)CSi Maximum transverse shear and associated transverse moment at bottom of the column V2 (kip) M3 (kip-ft) Truck ((V2T)max)CSi ((M3T)assoc)CSi

Permit Vehicle Maximum transverse shear and associated transverse moment at top of the column V2 (kip) M3 (kip-ft) ((V2P)max)CSi ((M3P)assoc)CSi Maximum transverse shear and associated transverse moment at bottom of the column V2 (kip) M3 (kip-ft) ((V2P)max)CSi ((M3P)assoc)CSi

where: ((V2T)max)CSi

= maximum longitudinal shear at top and bottom of column for truck load

((M3T)assoc)CSi = transverse moment at top and bottom of column associated with maximum shear for truck load ((V2P)max)CSi

= maximum transverse shear at top and bottom of column for permit load

((M3P)assoc)CSi = transverse moment at top and bottom of column associated with maximum shear for permit load

13.8.3

Column Live Load Input Procedure

13.8.3.1

Output from Longitudinal 2D Analysis (CTBridge)

Include dynamic load allowance factors per column for CTBridge output (Table 13.8-1) and summarize the results in Table 13.8-3.

Chapter 13 – Concrete Columns

13-16

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.8-3 Unfactored Column Longitudinal Shear and Associated Longitudinal Moment for One Lane, Including Dynamic Load Allowance Factors (CTBridge) Design Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column Vy (kip) Mz (kip-ft) Truck ((VyT)max)CT ((MzT)assoc)CT Lane ((VyL)max)CT ((MzL)assoc)CT Maximum longitudinal shear and associated longitudinal moment at bottom of the column Vy (kip) Mz (kip-ft) Truck ((VyT)max)CT ((MzT)assoc)CT Lane ((VyL)max)CT ((MzL)assoc)CT

13.8.3.2

Permit Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column Vy (kip) Mz (kip-ft) ((VyP)max)CT ((MzP)assoc)CT Maximum longitudinal shear and associated longitudinal moment at bottom of the column Vy (kip) Mz (kip-ft) ((VyP)max)CT ((MzP)assoc)CT

Output from 2D Transverse Analysis (CSiBridge)

Reform Table 13.8-2 to split truck reactions of CSiBridge analysis (Table 13.8-2) into truck and lane loads (13.7.3.4) as shown in Table 13.8-4. Table 13.8-4 Unfactored Column Reactions, Including Dynamic Load Allowance Factors (CSiBridge) Design Vehicle Maximum transverse shear and associated longitudinal moment at top of the column V2 (kip) M3 (kip-ft) Truck ((V2T)max)CSi ((M3T)assoc)CSi Lane ((V2L)max)CSi ((M3L)assoc)CSi Maximum transverse shear and associated longitudinal moment at bottom of the column V2 (kip) M3 (kip-ft) Truck ((V2T)max)CSi ((M3T)assoc)CSi Lane ((V2L)max)CSi ((M3L)assoc)CSi

Chapter 13 – Concrete Columns

Permit Vehicle Maximum transverse shear and associated longitudinal moment at top of the column V2 (kip) M3 (kip-ft) ((V2P)max)CSi ((M3P)assoc)CSi Maximum transverse shear and associated longitudinal moment at bottom of the column V2 (kip) M3 (kip-ft) ((V2P)max)CSi ((M3P)assoc)SAP

13-17

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Since the longitudinal shears and associated longitudinal moments are per one lane from CTBridge, the total longitudinal shears and associated longitudinal moments should be calculated as shown in Table 13.8-5. Table 13.8-5 Total Longitudinal Shear (Vy) and Associated Longitudinal Moment (Mz) (Vy)max (kip) (Mz)assoc. (kip-ft)

13.9

  

 

P-truck

    M    P    P   V    

P Pmax CSi P Pmax CT

 

P  Pmax  P  Pmax

CSi

P y max CT

  

P z assoc CT

CT

H-truck

  V       Mz    

T Pmax CSi T Pmax CT T max CSi T max CT

T y max CT

T z assoc CT

Lane

   V       P    P   M     



L Pmax CSi L Pmax CT

L y max CT

L max CSi L max CT

L z assoc CT





Determine factored shear and associated factored moment for Strength I and Strength II Limit States.



Design for shear for controlling case as per AASHTO 5.8.3.



The following example in Section 13.10 will demonstrate the shear design in details.

COLUMN SEISMIC DESIGN PROCEDURE Column seismic design and details shall follow the Caltrans Seismic Design Criteria 1.7.

13.10

DESIGN EXAMPLE The bridge shown in Figures 13.10-1 and 13.10-2 are a three-span PS/CIP box girder bridge with 20º skew and two column bents. The superstructure depth is 6.75 ft. Columns’ heights from top of footing to superstructure soffit are 44 ft at bent two and 47 ft at bent three. The columns are round with a diameter of 6 ft. The centerline distance between columns is 34 ft.

Chapter 13 – Concrete Columns

13-18

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.10.1

Design Column One at Bent Two

Figure 13.10-1 Elevation View of Example Bridge.

Chapter 13 – Concrete Columns

13-19

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Figure 13.10-2 Typical Section of Example Bridge.

Chapter 13 – Concrete Columns

13-20

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.10.2

Flexural Check of Main Column Reinforcement (As)

13.10.2.1

Longitudinal Analysis

From CTBridge output, determine Mz for Dead Load (DC) and Added Dead Load (DW). Table 13.10-1 Dead Load Unfactored Column Forces

Table 13.10-2 Additional Dead Load Unfactored Column Forces.

Controlling moments, Mz, are as follows: DC Mz = -925.2 kip-ft DW Mz = -110.1 kip-ft

Chapter 13 – Concrete Columns

13-21

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 13.10.2.2

Design Vehicular Live Loads

From CTBridge output, determine bent two unfactored reactions for one lane (no dynamic load allowance factors) for the design vehicle as: 

Maximum Ax and associated Mz at top of the column



Maximum Mz and associated Ax at top of the column

Table 13.10-3 Live Load, Controlling Unfactored Bent Reactions

From the CTBridge output, determine unfactored bent two reactions for one lane (no dynamic load allowance factors) of permit vehicle load as follows: 

Maximum Ax and associated Mz at top of the column



Maximum Mz and associated Ax at top of the column

Chapter 13 – Concrete Columns

13-22

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.10-4 Bent 2 Reactions, LRFD Permit Vehicle

13.10.2.3

Transverse Analysis

From CSiBridge output, determine the axial loads and transverse moments for DC and DW. Table 13.10-5 Axial loads and Transverse Moment for Dead Load and Added Dead Load

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.10.2.4

Live Loads

From CSiBridge output, determine the unfactored column reactions for design vehicle including the dynamic load allowance factors which are: 

Maximum P and associated M3



Maximum M3 and associated P

Table 13.10-6 Maximum Axial Load (P) for Design Vehicle

Table 13.10-7 Maximum Longitudinal Moment (M3) for Design Vehicle

From CSiBridge output, determine the unfactored column reactions for permit vehicle including the dynamic load allowance factors which are: 

Maximum P and associated M3



Maximum M3 and associated P

Chapter 13 – Concrete Columns

13-24

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 13.10-8 Maximum Axial Load (P) for Permit Vehicle.

Table 13.10-9 Maximum Longitudinal Moment (M3) for Permit Vehicle.

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.10.2.5

Output from Longitudinal 2D Analysis (CTBridge)

Column unfactored live load forces and moments for one lane from longitudinal analysis (CTBridge) are presented in Table 13.10-10. Table 13.10-10 Unfactored Bent Reactions for One Lane, Dynamic Load Allowance Factors Not Included Design Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft) Truck -115 -65 Lane -99 -167 Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft) Truck -44 332 Lane -42 239

13.10.2.6

Permit Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft) -360 -201 Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft) -231 -1486

Output from Transverse 2D Analysis (CSiBridge)

Two pseudo wheel loads including dynamic allowance factor to be used in transverse analysis (see Section 13.7.3.2). The transverse analysis column forces for pseudo truck and permit wheel loadings are presented in Table 13.10-11. Table 13.10-11 Unfactored Column Reaction, Including Dynamic Load Allowance Factors. Design Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft) Truck -569 98 Maximum transverse moment and associated axial load P (kip) M3 (kip-ft) Truck -261 401

Chapter 13 – Concrete Columns

Permit Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft) -961 -193 Maximum transverse moment and associated axial load P (kip) M3 (kip-ft) -469 718

13-26

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

13.10.2.7

Unfactored Column Reactions for One Lane, Including Impact (CTBridge)

Multiply dynamic allowance factor for values in Table 13.10-10 and calculate reaction per column (Table 13.10-12). Table 13.10-12 Unfactored Column Reactions for One Lane, Including Dynamic Load Allowance Factors (CTBridge) Design Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft) Truck -76 -43 Lane -50 -84 Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft) Truck -29 221 Lane -21 119

13.10.2.8

Permit Vehicle Maximum axial load and associated longitudinal moment Ax (kip) Mz (kip-ft) -225 -126 Maximum longitudinal moment and associated axial load Ax (kip) Mz (kip-ft) -145 -929

Unfactored Column Reactions, Including Dynamic Load Allowance Factors (CSiBridge)

Split the truck reactions results of transverse analysis (Section 13.7.3.4) into truck and lane loads as follows: Ratio of truck load per design vehicle = (76.2) / (76.2 + 49.605) = 0.606 Ratio of lane load per design vehicle = (49.6) / (76.2 + 49.605) = 0.394 Truck load of design vehicle

= 0.606 (values of Table 13.10-11)

Lane load of design vehicle

= 0.394 (values of Table 13.10-11)

Table 13.10-13 summarizes the truck and lane loads for both design and permit vehicles of transverse analysis. Table 13.10-13 Unfactored Column Reactions, Including Dynamic Load Allowance Factors (CSiBridge) Design Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft) Truck -345 59 Lane -224 39 Maximum transverse moment and associated axial load P (kip) M3 (kip-ft) Truck -158 243 Lane -103 158

Chapter 13 – Concrete Columns

Permit Vehicle Maximum axial load and associated transverse moment P (kip) M3 (kip-ft) -961 -193 Maximum transverse moment and associated axial load P (kip) M3 (kip-ft) -469 718

13-27

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Combine load results as shown in Tables 13.7-5, 13.7-6, 13.7-7, and 13.7-8 to get WinYEILD input loads as shown in Table 13.10-14. Table 13.10-14 WinYIELD Column Live Load Input Case 1 Max TransverseMy PHLane Truck Truck Load

My-Trans (kip-ft) Mx-Long (kip-ft) P-Axial (kip)

13.10.2.9

Case 2 Max LongitudinalMx HLane P-Truck Truck Load

Case 3 Max Axial-P PTruck

HTruck

Lane Load

718

243

158

-124

23

16

-193

60

39

-262

-90

-173

-3965

1003

533

-537

-195

-377

-469

-158

-103

-617

-132

-95

-961

-345

-224

Wind Load (WS, WL)



Wind on structure (WS): Average bridge height = 50.25 ft Assume bridge is in “Open Country,” from AASHTO Table 3.8.1.1-1 Vo

= 8.2 mph

Zo

= 0.23 ft

V   Z  V DZ  2.5 Vo  30  ln    VB   Z o 

(AASHTO 3.8.1.1-1)

100   50.25  V DZ  2.5 8.2    110 .4 mph (design wind velocity)  ln  100   0.23  2

V  PD  PB  DZ  for wind skew direction = 0˚  VB 

(AASHTO 3.8.1.2.1-1)

From AASHTO Table 3.8.1.2.1-1 PB = 0.05 for superstructure

(skew angle of wind = 0˚)

PB = 0.04 for columns

(skew angle of wind = 0˚)

2

110.4  PD  0.05    0.061 ksf  100 

(Superstructure)

2

110.4  PD  0.04   0.049 ksf  100 

(Columns)

The base wind pressure, PB, for various angles of wind directions may be taken as specified in AASHTO Table 3.8.1.2.2-1 (AASHTO, 2012).

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

where: PB = base wind pressure, corresponding to VB =100 mph PD = wind pressure on structures, LRFD equation 3.8.1.2.1-1 VDZ = design wind velocity (mph) at design elevations VB = base wind velocity of 100 mph at 30 ft height Vo = friction velocity (mph), LRFD Table 3.8.1.1-1 Z

= height of structure (ft) at which wind loads are being calculated as measured from low ground, or from water level, > 30 ft

Zo = friction length (ft) upstream fetch, LRFD Table 3.8.1.1-1 The wind pressure, PD, is calculated at various angels using the base wind pressure, PB, as per AASHTO Table 3.8.1.2.2-1. Table 13.10-15 lists the wind pressure, PD, at various angles of wind. Table 13.10-15 Wind Pressure at Various Skew Angles of Wind Superstructure Skew angle of wind (degrees) 0 15 30 45 60

Columns

(PD)Trans (ksf)

(PD)Long (ksf)

(PD)Trans (ksf)

(PD)Long (ksf)

0.061 0.054 0.050 0.040 0.021

0 0.007 0.015 0.020 0.023

0.049 0.043 0.040 0.032 0.017

0 0.006 0.012 0.016 0.019

Load on span

= (6.75 + 2.67)PD

Load on columns = (6)PD Loads on both superstructure and columns at various winds skew directions are shown in Table 13.10-16: Table 13.10-16 Wind Loads at Various Skew Angles of Wind Superstructure Skew angle of wind (degrees) 0 15 30 45 60

Columns

(PD)Trans (kip/ft)

(PD)Long (kip/ft)

(PD)Trans (kip/ft)

(PD)Long (kip/ft)

0.575 0.509 0.471 0.377 0.198

0 0.066 0.141 0.188 0.217

0.294 0.258 0.24 0.192 0.102

0 0.036 0.072 0.096 0.114

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Model wind as a user-defined load in CTBridge as shown below:

Figure 13.10-3 User Defined Loads for Wind Loads

From CTBridge output: o

Case of maximum transverse wind takes place at wind direction with skew = 0˚

o

Case of maximum longitudinal wind takes place at wind direction with skew = 60˚

Chapter 13 – Concrete Columns

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 13.10-17 User Loads, Unfactored Column Forces, WS Trans Skew 0˚

Table 13.10-18 User Loads, Unfactored Column Forces, WS Trans Skew 60˚.

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

Wind on live load (WL): Apply 0.1k/ft acting at various angles (AASHTO Table 3.8.1.3-1) as shown in Table 13.10-19:

Table 13.10-19 Wind on Live Load (WL) at Various Angles Skew angle of wind (degrees) 0 15 30 45 60

Normal component (k-ft) 0.1 0.088 0.082 0.066 0.034

Parallel component (k-ft) 0 0.012 0.024 0.032 0.038

Using CTBridge for wind on live load, the results are: o

Case of maximum transverse wind takes place at skew angle of wind = 0˚

o

Case of maximum longitudinal wind takes place at wind direction with skew = 60˚

Table 13.10-20 User Loads, Unfactored Column Forces, WL Trans Skew 0˚

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Table 13.10-21 User Loads, Unfactored Column Forces, WL Trans Skew 60˚

Table 13.10-22 Summary of Wind Loads Reactions for Column 1 at Bent 2

My (kip-ft) Mx (kip-ft) P (kip)

Wind on Structure Max. Trans. Max. Long. 474 -468 205 1102 34 -7

Wind on Live Load Max. Trans. Max. Long. 80 -74 34 173 6 -1

13.10.2.10 Braking Force (BR)

The braking force (AASHTO 3.6.4) shall be taken as the greater of: = 0.25(72) = 18 kips  25% design truck 

25% design tandem

= 0.25(50) = 12.5 kips



5% design truck + lane

= 0.05[72 + 0.64(412)] = 16.8 kips



5% design tandem + lane = 0.05[50 + 0.64(412)] = 15.7 kips

Controlling force

= 18 kips

Number of lanes

= [58.83-2(1.42)]/12 = 4.66

Use four lanes, MPF = 0.65 Total breaking force = 18(4) (0.65) = 46.8 kips Apply the braking force longitudinally then design for the moment and shear force effects. The braking force can be modeled in CTBridge as a user defined load in the direction of local X direction as shown below:

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Figure 13.10-4 User Defined Loads for Braking Force

Braking forces output from CTBridge are shown in Table 13.10-23. Table 13.10-23 User Loads, Unfactored Column Forces, Braking Force

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13.10.2.11 Thermal Effects (TU)

For a three-span bridge, the point of no movement is shown in Figure 13.10-5:

Point of No Movement Figure 13.10-5 Point of No Movement

Design temperature ranges from 10 to 80˚F

(AASHTO Table 3.12.2.1-1)

For normal weight concrete ˚F

(AASHTO 5.4.2.2)

Load factor for moment in column due to thermal movement TU = 0.5 (AASHTO 3.4.1) Thermal movement = 100 ft)(12) = 0.504 in. /100 ft

E  33,000K1w1c.5 f c'

(AASHTO 5.4.2.4-1)

For f′c = 3.6 ksi, E  33 ,000 (1)( 0 .15 ) 1.5 3 .6  3637 ksi Ig 

r 4 for circular column 4

For 6 ft diameter column, I g 

(3)4  63.6 ft 4 4

Point of no movement calculation: k

3EI , L3

P = k∆ then,

P

3EI  L3

I (two columns per bent) = 2(63.6) =127.2 ft4

PBent2  PBent3 

3(3637)(127.2)(12) 4 (1) (44(12))3 3(3637)(127.2)(12) 4 (1)

Chapter 13 – Concrete Columns

(47(12))3

 195.51 kips  160.4 kips

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where: 

coefficient of thermal expansion

k

= column stiffness



= lateral displacement

L

= column height

PBent2 = lateral force due to lateral displacement () of 1 in at bent-2 PBent3 = lateral force due to lateral displacement () of 1 in at bent-3 Table 13.10-24 Point Of No Movement Units are kips and ft P at 1inch. (kip) Distance (D) (ft) PD (kip-ft)

Abut1

Bent2

Bent3

Abut4

SUM

0 0 0

195.5 126 24,633

160.4 294 47,157.6

0 412 0

355.9 832 71,790.6

Distance from CL of support at Abut (X) = (71790.6 / 355.9) = 201.72 ft Distance from point of no movement from Bent 2 = 201.72 – 126 = 75.72 ft Note: The point of no movement can be read directly from the CTBridge output. For this example, the point of no movement is 75.72 ft from bent two, as shown in Figure 13.10-6.

Figure 13.10-6 Point of No Movement

Thermal displacement () = (0.504 / 100) (75.72) = 0.38 in.

M TH  =

3EI g TH L2

 TU

3(3637)(63.6)(12) 4 (0.38)

Chapter 13 – Concrete Columns

(44(12)) 2

0.5 = 9807 kip-in. = 817 kip-ft

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(MTH)x = M cos cos(20) = 767.6 kip-ft (MTH)y = M sin sin(20) = 279.4 kip-ft where: MTH = column moment due to thermal expansion  = skew angle TU = load factor for uniform temperature 13.10.2.12 Prestress Shortening Effects (Creep and Shrinkage)

The anticipated shortening due to prestressing effects occurs at a rate of 0.63 in. per 100 ft (MTD 7-10). Displacement = 0.63 (75.72 / 100) = 0.48 in.

Mcsh 

3EIg  L2

3(3637)(63.6)(12)4 (0.48) 0.5=12387 kip-in.=1032 kip-ft p = (44x12)2

(Mcsh)x = M cos cos(20) = 970 kip-ft (Mcsh)y = M sin sin(20) = 353 kip-ft where: Mcsh = column moment due to prestress shortening (creep and shrinkage) p

= load factor for permanent load due to creep and shrinkage

13.10.2.13 Prestress Secondary Effects (PS)

The secondary effect of prestressing after long term losses is shown in Table 13.10-25. Table 13.10-25 Prestressing Secondary Effects

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13.10.2.14 WinYIELD Input for Column 1 at Bent 2

Design of column reinforcement is performed by running WinYIELD starting by general form as shown in Figure 13.10-7.

Figure 13.10-7 WinYIELD General Form

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Column form for circular column with diameter of 72 inches is shown in Figure 13.10-8.

Figure 13.10-8 WinYIELD Column Form

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Material form (Figure 13.10-9) shows concrete specified compressive strength, fʹc = 3.6 ksi and steel rebar specified minimum yield strength, fy = 60 ksi.

Figure 13.10-9 WinYIELD Material Form

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Figure 13.10-10 shows the rebar form with: Out to out distance = 72  2(2) = 68 in. (for cover = 2 in.) Assume #14 bundle total 36 and #8 hoops Loop radius = [72  2(2)  2(1.13)  2(1.88/2)]/2 = 31.9 in.

Figure 13.10-10 WinYIELD Rebar Form

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Use AASHTO Chapter 4 to determine Kx and Ky, considering AASHTO C4.6.2.5-1 to be used in load-1 form (Figure 13.10-11).

Figure 13.10-11 WinYIELD Load-1 Form

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Load-2 (Figure 13.10-12) input data is taken from Table 13.10-14.

Figure 13.10-12 WinYIELD Load-2 Form

13.10.2.15 WinYIELD Output

Winyield output sheet (Figure 13.10-13) shows the steel reinforcement required for the column.

Figure 13.10-13 WinYIELD Output Results

The final design could be summarized as: Provided number of bars = 18 bundle > required number of bars = 10.6 (OK) Min. clearance and spacing for #14 bundle horizontally = 7.5 in. Distance between bundles = 2(31.93) / 18 = 11.1 in. > 7.5 in. (OK) Chapter 13 – Concrete Columns

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13.10.3

Shear Design for Transverse Reinforcement (Av)

The procedure of determining column transverse reinforcement is presented in consequent sections. 13.10.3.1

Longitudinal Analysis

From CTBridge output (Tables 13.10-26 and 13.10-27), determine longitudinal shear (Vy) and moment (Mz) at top and bottom of columns for DC and DW. Combine output in Table 3.10-28. Table 13.10-26 Dead Load, Unfactored Column Forces

Table 13.10-27 Additional Dead Load, Unfactored Column Forces

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Table 13.10-28 Longitudinal Shear (Vy) and Longitudinal Moment (Mz) for DC and DW Top of Column

Vy (kip) Mz (kip-ft)

DC 21 -925.2

DW 2.5 -110.1

Bottom of Column DC DW 21 2.5 0 0

Determine maximum longitudinal shear (Vy) and associated moment (Mz) for design vehicular live loads at top and bottom of the bent unfactored reactions for one lane as shown in Table 13.10-29. Table 13.10-29 Unfactored Bent Reactions For Design Vehicle

Determine maximum longitudinal shear (Vy) and associated moment (Mz) for permit vehicular live loads at top and bottom of the bent unfactored reactions for one lane as shown in Table 13.10-30.

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Table 13.10-30 Unfactored Bent Reactions For Permit Vehicle

Re-arrange the longitudinal shear and moment output from CTBridge are for two columns (Table 13.10-31).

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Table 13.10-31 Unfactored Bent Reactions for One Lane, Dynamic Load Allowance Factors Not Included Design Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (Mz)assoc (kip) (kip-ft) Truck 10.3 -455 Lane 7.6 -336 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Mz)assoc (Vy)max (kip) (kip-ft) Truck 10.3 0 Lane 7.6 0

Permit Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (Mz)assoc (kip) (kip-ft) -12.28 540.25 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Vy)max (Mz)assoc (kip) (kip-ft) 33.78 0

Apply dynamic allowance factor to Table 13.10-31 for one column as shown in Table 13.10-32. Table 13.10-32 Unfactored Column Longitudinal Shear and Associated Longitudinal Moment for One Lane, Including Dynamic Load Allowance Factors. Design Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (kip) (Mz)assoc (kip-ft) Truck 6.8 -303 Lane 3.8 -168 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Vy)max (kip) (Mz)assoc (kip-ft) Truck 6.8 0 Lane 3.8 0

Chapter 13 – Concrete Columns

Permit Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (kip) (Mz)assoc (kip-ft) -7.7 338 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Vy)max (kip) (Mz)assoc (kip-ft) 21 0

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13.10.3.2

Transverse Analysis

CSiBridge output for load cases of dead load (DC) and added dead load (ADL) is shown in Table 13.10-33. Table 13.10-33 Transverse Shear (V2) and Moment (M3) at Top and Bottom of Columns due to Dead Load (DC) and Added Dead Load (DW)

Combine output in Table 3.10-34. Table 13.10-34 Transverse Shear (V2) and Moment (M3) for DC and DW

V2 (kip) M3 (kip-ft)

DC -10.5 462

Top of column DW -0.5 23

Bottom of column DC DW -10.5 -0.5 0 0

CSiBridge output for maximum shear (V2) and associated and moment (M3) for design vehicle including dynamic load allowance as shown in Table 13.10-35.

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Table 13.10-35 Maximum Shear (V2) and Associated Moment (M3) for Design Vehicle

CSiBridge output for maximum shear (V2) and associated and moment (M3) for permit vehicle including dynamic load allowance as shown in Table 13.10-36. Table 13.10-36 Maximum Shear (V2) and Associated Moment (M3) for Permit Vehicle

Re-arrange the transverse shear and moment output from CSiBridge in Table 13.10-37.

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Table 13.10-37 Unfactored Column Reaction, Including Dynamic Load Allowance Factors Design Vehicle Maximum transverse shear and associated transverse moment at top of the column (M3)assoc (V2)max (kip-ft) (kip) Truck -9.1 400 Maximum transverse shear and associated transverse moment at bottom of the column (V2)max (M3)assoc (kip) (kip-ft) Truck -9.1 0

Permit Vehicle Maximum transverse shear and associated transverse moment at top of the column (V2)max (M3)assoc (kip-ft) (kip) -16.3 718 Maximum transverse shear and associated transverse moment at bottom of the column (V2)max (M3)assoc (kip) (kip-ft) -16.3 0

Use the procedure shown in 13.7.4 and arrange output in Table 13.10-38. Table 13.10-38 Unfactored Column Reactions, Including Dynamic Load Allowance Factor Design Vehicle Maximum transverse shear and associated longitudinal moment at top of the column Truck -5.5 243 Lane -3.6 157 Maximum transverse shear and associated longitudinal moment at bottom of the column (M3)assoc (V2)max (kip) (kip-ft) Truck -5.5 0 Lane -3.6 0

13.10.3

Permit Vehicle Maximum transverse shear and associated longitudinal moment at top of the column -16.3 718 Maximum transverse shear and associated longitudinal moment at bottom of the column (V2)max (M3)assoc (kip) (kip-ft) -16.3 0

Total Longitudinal Shear and Associated Moments Total column longitudinal total shear and associated moment as per 13.8.3 is presented in Table 13.10-39.

Table 13.10-39 Unfactored Column Total Longitudinal Shear and Associated Longitudinal Moment, Including Dynamic Load Allowance Factors Design Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (kip) (Mz)assoc (kip-ft) Truck 31 -1367 Lane 17 -759 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Vy)max (kip) (Mz)assoc (kip-ft) Truck 31 0 Lane 17 0

Chapter 13 – Concrete Columns

Permit Vehicle Maximum longitudinal shear and associated longitudinal moment at top of the column (Vy)max (kip) (Mz)assoc (kip-ft) -12 519 Maximum longitudinal shear and associated longitudinal moment at bottom of the column (Vy)max (kip) (Mz)assoc (kip-ft) 32 0

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13.10.3.9

Summary of Column Shear Loads

Column shear loads are summarized in Table 13.10-40. Table 13.10-40 Longitudinal Shear and Associated Longitudinal Moment Top of Column (Vy)max. (kip) (Mz)assoc (kip-ft) 21 -925 2.5 -110 31 -1367 17 -759 -12 519

Load Case DC DW H-Truck Lane P-Truck

Bottom of Column (Vy)max. (kip) (Mz)assoc (kip-ft) 21 0 2.5 0 31 0 17 0 32 0

Table 13.10-41 Transverse Shear and Associated Transverse Moment. Top of Column (V2)max (kip) (M3)assoc (kip-ft) -10.5 462 -0.5 23 -5.8 258 -3.3 143 -16.3 718

Load Case DC DW H-Truck Lane P-Truck

Bottom of Column (V2)max (kip) (M3)assoc (kip-ft) -10.5 0 -0.5 0 -5.8 0 -3.3 0 -16.3 0

Since this example uses circular columns, the design shears and moments should be taken as the square root of the sum of the squares: Table 13.10-42 Square Root of the Sum of the Squares Top of Column V (kip) (M)assoc (kip-ft) 23 1034 3 112 32 1392 17 772 20 886

Load Case DC DW H-Truck Lane P-Truck

Bottom of Column V (kip) (M)assoc (kip-ft) 23 0 3 0 32 0 17 0 36 0

13.10.3.10 Strength Shear Limit States

Determine strength I and strength II limit states for shear and associated moments. 

Strength I: Vu = 1.25 (23) + 1.5 (3) + 1.75 (32 + 17) = 119 kips

(controls)

Mu = 1.25 (1034) + 1.5 (112) + 1.75 (1392 + 772) = 5248 kips 

Strength II: Vu = 1.25 (23) + 1.5 (3) + 1.35 (20) = 60 kips Mu = 1.25 (1034) + 1.5 (112) + 1.35 (886) = 2,657 kip-ft

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Vn = Vc + Vs Vs 

vu 

Av f y d v s

(AASHTO 5.8.3.3-1) cot 

(AASHTO 5.8.3.3-4)

Vu bv d v

(AASHTO 5.8.2.9-1)

Column loop radius = 31.93 in. (from WinYIELD input) Using simplified procedure for nonprestressed sections (AASHTO 5.8.3.4.1)

  Vc  0.0316 f c bv d v  0.03162 3.6 7250.16  433 kips  119 kips  where: Av = area of shear reinforcement within a distance s (in.2) bv = effective web width dv = effective shear depth s

= spacing of transverse reinforcement measured in a direction parallel to the longitudinal reinforcement (in.)

Vc = concrete shear capacity Vn = nominal shear capacity Vs = transverse shear reinforcement capacity Vu = factored shear force Mu = factored moment



= factor indication ability of diagonally cracked concrete to transmit tension and shear as specified in article 5.8.3.4

Use minimum shear reinforcement (AASHTO 5.8.2.5-1). f c 3 .6  Av   0.0316 bv  0.0316 x72  0.072 in. 2 /in.   60 s f  min  y

Av = 0.79 in.2 for #8 hoops, so s min 

0 . 79  11 in. 0 .072

(Use s = 6 in.)

Check maximum spacing:

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For

vu  0.125 f c

Smax = 0.8 dv ≤ 18 in.

(CA 5.8.2.7-1)

vu Smax = 0.4 dv ≤ 12 in. (AASHTO 5.8.2.7-2)  0.125 f c v 0.0483 Since u   0.0134  0.125 , then Smax = 0.8 (50.16) = 40.1 in. > 18 in. f c 3.6

Smax = 18 in. > 11 in.

(OK)

Note: Use #8 hoops @ 6 in. Seismic shear demands should be checked per the current SDC. Column confinement/shear steel, in most normal cases, will be governed by the plastic hinge shear. Check shear-flexure interaction: As f y 

M u  VU    0.5Vs  cot  d v   

2(18)(2.25)(60)³

(AASHTO 5.8.3.5.3-1)

5248(12) 117  + -0 cot45 0.9(50.16)  0.9 

4860 kips ≥ 1525 kips (OK), then #14 tot. 18 bundle as shown in Figure 13.10-14 are OK

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Y-Axis (Long)

#8 hoop at 6 in.

#14 tot. 18 bundles Y-Axis (Trans)

Figure 13.10-14 Column Details—Reinforcement of Column

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NOTATION Ag

= gross area of section (in.2) (13.6.2)

As

=

main column reinforcement (13.10.2)

Ast

=

total area of main column reinforcement (in.2) (13.6.2)

Av

=

area of shear reinforcement within a distance s (in2) (13.10.3.10)

Ax

=

axial load (13.7.1)

bv

=

effective web width (13.10.3.10)

Cm

=

a factor, which relates the actual moment diagram to an equivalent uniform moment diagram, is typically taken as 1 (13.5.1)

dv

=

effective shear depth ( 13.10.3.10)

Ec

=

the elastic modulus of concrete (ksi) (13.5.1)

Es

=

elastic modulus of reinforcement (ksi) (13.5.1)

fc

= specified strength of concrete at 28 days, unless another age is specified (ksi) (13.6.2)

fy

=

specified yield strength of reinforcement (ksi) (13.6.2)

I

=

moment of inertia about axis under consideration (in.4) (13.5.1)

Ig

=

the gross moment of inertia (in.4) (13.5.1)

Is

=

moment of inertia of longitudinal steel about neutral axis (ksi) (13.5.1)

K

=

the effective length factor (13.2)

k

= column stiffness (k/in)(13.10.2.11)

L

= column height (13.10.2.11)

lu

= the unsupported length of a compression member (in.) (13.2)

MTH

= column moment due to thermal expansion (13.10.2.11)

Mcsh

=

M1

= the smaller end moment, should be positive for single curvature flexure (13.5)

M2

=

the larger end moment, should be positive for single curvature flexure (13.5)

M2b

=

moment on compression member due to factored gravity loads that result no sidesway, always positive (kip-ft) (13.5.1)

M2s

=

moment on compression member due to factored lateral or gravity loads that result in sidesway, , greater than lu/1500, always positive (kip-ft) (13.5.1)

M3

=

transverse moment (13.7.2)

Mb

=

balanced moment resistance at balanced strain condition (13.6.1)

column moment due to prestress shortening (creep and shrinkage) (13.10.2.11)

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Mc

= magnified factored moment (13.5.1)

Mo

=

nominal flexural resistance of a section at zero eccentricity (13.6.1)

Mn

=

nominal flexural resistance (13.6.1)

Mrx

=

uniaxial factored flexural resistance of a section about x-axis corresponding to the eccentricity produced by the applied factored axial load and moment (13.6.2)

Mry

=

uniaxial factored flexural resistance of a section about y-axis corresponding to the eccentricity produced by the applied factored axial load and moment (13.6.2)

Mu

=

factored moment (13.10.3.10)

Mux

=

factored applied moment about x-axis (kip-in.) (13.6.2)

Muy

=

factored applied moment about y-axis (kip-in.) (13.6.2)

My

=

transverse moment (13.7.2)

Mz

= longitudinal moment (13.7.1)

P

=

PB

= base wind pressure, corresponding to VB =100 mph (13.10.2.9)

Pb

=

balanced axial resistance at balanced strain condition (13.6.1)

PBent2

=

lateral force due to lateral displacement () of 1 in at bent-2 (13.10.2.11)

PBent3

= lateral force due to lateral displacement () of 1 in at bent-3 (13.10.2.11)

PD

= wind pressure on structures

Pe

=

Euler buckling load (13.5.1)

Pn

=

nominal axial resistance, with or without flexure (13.6.2)

Po

=

nominal axial resistance of a section at 0 eccentricity (kip) (13.6.1)

Pr

=

factored axial resistance (13.6.2)

Prx

=

factored axial resistance determined on the basis that only eccentricity ey is present (kip) (13.6.2)

Prxy

=

factored axial resistance in biaxial flexure (kip) (13.6.2)

Pry

=

factored axial resistance determined on the basis that only eccentricity ex is present (kip) (13.6.2)

Pu

=

factored axial load (kip) (13.5.1)

r

=

radius of gyration (in.) (13.2)

R1

=

truck load of design vehicle (13.7.3)

R2

=

lane load of design vehicle (13.7.3)

column axial load (13.7.2)

Chapter 13 – Concrete Columns

(13.10.2.9)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

S

=

spacing of transverse reinforcement measured in a direction parallel to the longitudinal reinforcement (in) (13.10.3.10)

V2

=

transverse analysis (13.8.2)

VB

=

base wind velocity of 100 mph at 30 ft height (13.10.2.9)

Vc

=

concrete shear capacity (13.10.3.10)

VDZ

=

design wind velocity (mph) at design elevations (13.10.2.9)

Vn

=

nominal shear capacity (13.10.3.10)

Vo

=

friction velocity (mph) (13.10.2.9)

Vs

=

transverse shear reinforcement capacity (13.10.3.10)

Vu

=

factored shear force (13.10.3.10)

Vy

=

longitudinal shear (13.8.1)

Z

=

height of structure (ft) at which wind loads are being calculated as measured from low ground, or from water level, > 30 ft (13.10.2.9)

Zo

=

friction length (ft) upstream fetch (13.10.2.9)



 coefficient of thermal expansion(13.10.2.11)



=

factor indication ability of diagonally cracked concrete to transmit tension and shear (13.10.3.10)

d

=

ratio of maximum factored permanent moment to the maximum factored total load moment, always positive (13.5.1)

p

=

load factor for permanent load due to creep and shrinkage (13.10.2.12)

TU

=

load factor for uniform temperature (13.10.2.11)



=

lateral displacement (13.5.1)

c

= compression strain of the concrete (13.6.1)

y

=

b

= moment magnification factor for compression member braced against sidesway

yield strain of the steel (13.6.1) (13.5.1)

s

= moment magnification factor for compression member not braced against sidesway (13.5.1)



=

skew angle (13.10.2.11)



=

resistance factor specified in AASHTO 5.5.4.2 (13.6.2)

k

=

stiffness reduction factor; 0.75 for concrete members and 1 for steel members (13.5.1)

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, American Association of State Highway and Transportation Officials, 6th Edition, Washington, DC. 2. Caltrans, (2014). California Amendments to AASHTO LRFD Bridge Design Specifications— Sixth Edition, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2013). Caltrans Seismic Design Criteria—Version 1.7, California Department of Transportation, Sacramento, CA. 4. Caltrans, (2008). WinYIELD (2008): Column Live Load Input Procedure, California Department of Transportation, Sacramento, CA. 5. Chen, W.F. and Duan, L. Ed. (2014). Bridge Engineering Handbook—2nd Edition, CRC press, Boca Raton, FL. 6. CSI, (2015). CSiBridge 2015, Version 17.0.0, Computers and Structures, Inc. Walnut

Creek, CA. 7. MacGregor, J.G. (1988). Reinforced Concrete Mechanics and Design, Prentice-Hall, Englewood Cliffs, NJ.

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CHAPTER 15 SHALLOW FOUNDATIONS TABLE OF CONTENTS 15.1

INTRODUCTION .......................................................................................................... 15-1

15.2

COMMON TYPES OF SPREAD FOOTINGS FOR BRIDGES................................... 15-1

15.3

PROPORTIONING AND EMBEDMENT OF FOOTINGS ......................................... 15-1 15.3.1 Sizing of Spread Footings ................................................................................ 15-2 15.3.2 Embedment and Depth of Footings .................................................................. 15-2

15.4

DESIGN LOADS ........................................................................................................... 15-3

15.5

BEARING STRESS DISTRIBUTION .......................................................................... 15-3

15.6

GENERAL DESIGN REQUIREMENTS ...................................................................... 15-5 15.6.1 Settlement Check.............................................................................................. 15-5 15.6.2 Bearing Check .................................................................................................. 15-5 15.6.3 Eccentricity Limits ........................................................................................... 15-6 15.6.4 Sliding Check ................................................................................................... 15-6

15.7

STRUCTURAL DESIGN OF FOOTINGS ................................................................... 15-7

15.8

DESIGN EXAMPLE...................................................................................................... 15-7 15.8.1 Bridge Footing Data ......................................................................................... 15-7 15.8.2 Design Requirements ....................................................................................... 15-9 15.8.3 Footing Thickness Determination .................................................................. 15-10 15.8.4 Calculation of Factored Loads ....................................................................... 15-11 15.8.5 Footing Size Determination............................................................................ 15-14 15.8.6 Flexural Design .............................................................................................. 15-21 15.8.7 Shear Design .................................................................................................. 15-24

APPENDIX .............................................................................................................................. 15-27 NOTATION ............................................................................................................................. 15-29 REFERENCES ......................................................................................................................... 15-31

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CHAPTER 15 SHALLOW FOUNDATIONS 15.1

INTRODUCTION Shallow foundations (spread footings) are advantageous to pile foundations considering lower cost, easier construction, and fewer environmental constraints. However, weak soil and seismic considerations may limit use of spread footings and impact the foundation type selection. In general, size of the spread footing is determined based on bearing resistance of the supporting soil or rock and also permissible level of settlement. Design of spread footings requires constant communication between the Structural Designer (SD) and the Geotechnical Designer (GD) throughout the design process. Factored loads are provided by the SD and factored resistance for the supporting soil and rock, that is permissible net contact stress qpn and factored gross nominal bearing resistance qR are calculated and reported by the GD. The structural design is performed by the SD. Consistency between the SD and the GD in the use of required gross or net stresses is important. Caltrans Bridge Memo to Designers (MTD) 4-1 (Caltrans, 2014b) provides general guidance on design process and also the minimum level of required communications between the SD and the GD. The analysis and design of a spread footing based on the 6th Edition of the AASHTO LRFD Bridge Design Specifications (AASHTO, 2012) and the California Amendments (Caltrans, 2014a), and Seismic Design Criteria (SDC) Version 1.7 (Caltrans, 2013), will be illustrated through an example.

15.2

COMMON TYPES OF SPREAD FOOTINGS FOR BRIDGES Spread footings can be used as isolated footings to support single columns or as combined footings to support multi-columns when columns are closely spaced. Elongated spread footings under abutments and pier walls act as strip footings where moments act only in the short direction. Strip footings under abutments or piers can be analyzed and designed similar to column footings, with moments acting in one direction only.

15.3

PROPORTIONING AND EMBEDMENT OF FOOTINGS The designer should consider several parameters such as axial force and biaxial moment acting on the footing, right of way, existing structures, and also depth of footing when selecting size and location of the footing. Although square footings are more common for footings supporting pinned columns, rectangular shapes may be more efficient when column is fixed at the base, since moments acting on the footing in two directions may be very different. Considering various load combinations specified in AASHTO (2012) and Caltrans (2014a), and variation of geotechnical

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resistances with eccentricities of loads acting on the footing any type of optimization can be rigorous.

15.3.1

Sizing of Spread Footings The trial minimum size of the spread footing can be selected based on footings of similar conditions and past experience. Size of a spread footing is usually governed by the column size, magnitude of loads acting on the footing, and resistances of the substrate. The effective length to effective width (L′/B′) ratio is commonly 1.0 ~ 2.0. The GD should be consulted for selection of the ratios. The allowable settlement will be assumed as 1 in. or 2 in. according to MTD 4-1 (Caltrans, 2014b) and based on continuity of the superstructure. Larger limits can be used if structural analysis shows that the superstructure can tolerate such settlement without adverse serviceability impacts (Caltrans, 2014a). The footing size is usually proportioned based on “permissible net contact stress” at the service limit state and checked for “factored gross nominal bearing resistance” at strength and extreme event limit states. These stresses are functions of the effective width as well as the effective length to effective width ratio, therefore they are presented by a family of curves and also a table as shown in the design example. The SD needs to use double interpolation to extract the information required for design under different load combinations using corresponding effective dimensions. If necessary, the GD may be contacted to revise the information and provide a new set of curves and tables to avoid extrapolation.

15.3.2

Embedment and Depth of Footings The footing embedment shall be carefully determined for degradation and contraction scour for the base (100 year) flood, as well as short term scour depth. The embedment depth of the footing should be adequate to ensure the top of the footing is not exposed when total scour has occurred, as shown in Figure 15.3-1. If the footing is not in water and freezing is not of concern, a minimum cover of 2 to 3 ft is recommended.

Degradation, contraction, and local pier scour depth

Figure 15.3-1 Minimum Embedment for Scour Protection

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The depth (thickness) of the footing is preliminary selected based on the required development length of the column reinforcement and then designed for flexural and shear strength.

15.4

DESIGN LOADS The factored shear forces (Vx and Vy), column axial force (P) and bending moments (Mx and My) resulting from structural analysis are usually reported at the base of the column and must be transferred to the bottom of the footing in order to calculate contact bearing stresses. Therefore, the resultant moment at the base of the columns must be modified to include the additional moment caused by shear force transfer. The modified moment in a generic format can be written as M + (V × dfooting), where dfooting is the actual footing depth.

15.5

BEARING STRESS DISTRIBUTION The sign convention shown in MTD 4-1 (Caltrans, 2014b) is to avoid mistakes in communications between the SD and the GD. The footing local X axis is defined along the longer dimension of the footing (L), and the Y axis along the short dimensions (B) as shown in Figure 15.5-1. Forces and moments resulting from superstructure analysis acting at the column base are resolved in the directions of local axes if local axes do not coincide longitudinal and transverse directions of the bridge. Bearing stress distribution depends on relative stiffness of the footing and supporting soil and rock. For determination of the footing size based on the bearing resistance and settlement requirements, the bearing stress is assumed to be uniformly distributed for footings on soil and linearly distributed for footings on rock. For structural design of the footing, bearing stress is assumed to be linearly distributed. For eccentrically loaded footings on soil, the effective footing dimensions (B′ and L′) specified in AASHTO Article 10.6.1.3 (AASHTO, 2012) shall be used for design of settlement and bearing resistance. Bearing stress distribution over effective footing area is assumed to be uniform. The effective dimensions for a rectangular footing are shown in Figure 15.5.1 and shall be taken as follows:

′ = B

B − 2 ey

′ L − 2 ex L=

(AASHTO 10.6.1.3-1)

where: B, L ey , ex A′ q

= = = =

actual dimensions of the footing (ft) eccentricities parallel to dimensions B and L, respectively (ft) reduced effective area of the footing = B′× L′ (ft2) uniform bearing stress = P/A′ (ksf)

Chapter 15 – Shallow Foundations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Y L L/2

B'

ey

Reduced effective area

Point of load application

B

B/2

ex L'

Figure 15.5-1 Effective Footing Area For footings on rock and for structural design of footings, the bearing stress is assumed to be linearly distributed. If the eccentricity is less than B/6 (or L/6) the maximum bearing stress is calculated as: qmax =

M P My ± + x A Sy Sx

(15.5-1)

where: P M x, M y Sx, Sy A

= vertical force acting at the center of gravity of the bottom of the footing area (kip) = moments acting at the bottom of the footing about X and Y directions, respectively (kip-ft) = section modulus of the footing area about X and Y directions, respectively (ft3) = actual footing area = B × L (ft2)

Equation (15.5-1) is valid only if stresses calculated at corners of the footing are all positive (compression), otherwise the reduced contact area of footing must be determined and rocking must be considered in analysis. Bearing stresses can be calculated as “net” or ‘gross”. The weight of the footing and all overburden soil from top of the footing to finished grade must be included when calculating “gross bearing stress”. The weight of overburden soil between bottom of footing and original grade at excavation time is subtracted from gross bearing stress to calculate “net bearing stress.” Net bearing stress under AASHTO Service I Load Combination is used to evaluate footing settlement.

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15.6

GENERAL DESIGN REQUIREMENTS The bearing stresses calculated under various AASHTO LRFD limit states must be checked against acceptable stresses provided by the GD. After receiving foundation information and scour data (if applicable), the GD will provide “permissible net contact stress” used for Service Limit States checks, and “factored gross nominal bearing resistance” used for Strength and Extreme Event Limit States checks, respectively. The stresses are functions of the effective width as well as effective length to effective width ratio, therefore information will be provided as a family of data points for different values of L/B ratios for a given “B”. The SD needs to use double interpolation to extract the information required for design under different load combinations using corresponding effective dimensions. If necessary, the GD may be contacted to revise the information and provide a new set of curves and table to avoid extrapolation.

15.6.1

Settlement Check For Service Limit State, the following requirement must be met:

qn, u

≤ q pn

qn, amx ≤ q pn

for footing on soil

(15.6-1)

for footing on rock

(15.6-2)

where: qpn qn,u qn,max

15.6.2

=

permissible net contact stress provided by the GD and calculated based on a specified allowable settlement (ksf) = net uniform bearing stress calculated using Service-I Limit State loads assuming uniform stress distribution for footings on soil (ksf) = net maximum bearing stress calculated using Service-I Limit State loads assuming linear stress distribution for footings on rock (ksf)

Bearing Check For Strength and Extreme Event Limit States, the design requirement is written as:

q g ,u

≤ qR

for footing on soil

(15.6-3)

qg ,max

≤ qR

for footing on rock

(15.6-4)

where: qg,u

=

qg,max

=

qR

=

gross bearing stress calculated based on uniform stress distribution for footings on soil (ksf) gross maximum bearing stress calculated based on linear stress distribution for footings on rock (ksf) factored gross nominal bearing resistance provided by the GD = φbqn (ksf)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

qn φb

15.6.3

= =

gross nominal bearing resistance (ksf) resistance factor

Eccentricity Limits The eccentricity limits for Service and Extreme Event Limit States specified in AASHTO (2012) and Caltrans (2014a) are summarized as: Table 15.6-1 AASHTO (2012) Eccentricity Limits AASHTO Article Number 10.5.2.2

Limit State

Footing on Soil

Footing on Rock

Service

B/6 or L/6

B/4 or L/4

Extreme Event (Seismic ) γEQ = 0

B/3 or L/3

B/3 or L/3

10.6.4.2 and 11.6.5.1

Extreme Event (Seismic ) γEQ = 1.0

2B/5 or 2L/5

2B/5 or 2L/5

10.6.4.2 and 11.6.5.1

Note: Seismic forces should be applied in all directions per SDC (Caltrans 2013). It is not necessary to include live load (design or permit truck) in Extreme Event Limit State load combinations therefore γEQ = 0.

15.6.4

Sliding Check Shear force acting at the interface of footing and substrate should be calculated and compared to the factored nominal sliding resistance specified as:

RR = ϕRn = ϕτ Rτ + ϕep Rep

(AASHTO 10.6.3.4-1)

The contribution of soil passive pressure (second term) is generally negligible and equation is summarized to RR = φRn = φτ Rτ . For cohesionless soil Rτ is written as:

Rτ V tan ( δ ) =

(AASHTO 10.6.3.4-2)

where: Rτ = nominal sliding resistance between soil and concrete (kip) V = total force acting perpendicular to the interface (kip) δ = friction angle at interface of footing and soil = φf, internal friction angle of the drained soil for concrete cast against soil (degree) φτ = resistance factor against sliding = 0.8 for cast-in-place concrete against sand (AASHTO Table 10.5.5.2.2-1).

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

15.7

STRUCTURAL DESIGN OF FOOTINGS Structural design of the footing includes the following steps: •

Select footing thickness based on required development length of the column reinforcement

•

Design flexural reinforcement in both directions considering minimum reinforcement required for shrinkage and temperature

•

Check thickness of the footing for one-way and two-way shears and design shear reinforcement if required

•

Check seismic details per Caltrans SDC (Caltrans, 2013a) and other practice manuals

Table 15.7-1 provides highlights of requirements for structural design of the footings specified in AASHTO (2012) and Caltrans (2014a). Application of these requirements will be illustrated in the design example. Table 15.7-1 AASHTO (2012) and Caltrans (2014a) Requirements for Structural Design of Footings Topic

AASHTO Articles

Strut & tie Applicability Flexural design

5.7.3.2

Direct shear design

5.8.3.3

Shear friction Reinforcement spacing

5.6.3

5.8.4 5.7.3.3, 5.7.3.4, 5.10.3, 5.10.8

Application Requirement check Reinforcement design Footing depth and reinforcement design Shear key design Design and detailing

Reinforcement development

5.11.2

Structural design of footings

Concrete cover

5.12.3

Footing depth and detailing

Footings

5.13.3

Footing depth

15.8

DESIGN EXAMPLE

15.8.1

Bridge Footing Data Design process for a bridge bent spread footing is illustrated through the following example. A circular column of 6 ft diameter with 26#14 main rebars, and #8 hoops spaced at 5 in. is used for a two-span post-tensioned box girder bridge. Footing as shown in Figure 15.8-1 rests in cohesionless soil with internal friction

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

angle of 38°. Original ground (OG) elevation is 48 ft, finished grade (FG) elevation is 48 ft, and bottom footing elevation (BOF) is 39 ft. • •

Concrete material f c′ = 3,600 psi Reinforcement fy = 60,000 psi (A706 steel).

•

Governing unfactored live load forces at the base of the column are listed in Table 15.8-1.

•

Unfactored dead load and seismic forces at the base of the column are listed in Table 15.8-2.

•

Plastic moment and shear applied at the column base are: Mp = 15,573 kip-ft; Vp = 716 kips Overturning column axial force in transverse push is 992 kips.

Note: To facilitate communications of the SD and the GD, local coordinate of foundation have been defined as X and Y. As shown in Figure 15.8-2a. Local X axis is parallel to long dimension plan of footing (L) and the local Y axis is perpendicular to X. The global coordinates L (Longitudinal) and T (Transverse) are commonly used for bridge analysis. The structural designer needs to transfer forces and moments acting on the footing to shear forces and moments acting in local coordinates. All communications between the SD and the GD shall be based on forces/moments calculated in local coordinates of the footing. In this example local and global coordinates coincide, that is X = T and Y = L. Therefore, local and global coordinates may have been used interchangeable, as shown in Figure 15.8-2b.

Bottom of footing elevation = 39 ft

Figure 15.8-1 Elevation of the Spread Footing Table 15.8-1 Unfactored Live Load Forces at Column Base Load Case MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL (kip)

I -206 250 217 -12 12

Chapter 15 – Shallow Foundations

HL-93 Truck II -40 1,456 238 -1 81

III -80 552 479 -2 26

I -348 171 367 -16 8

Permit Truck II III 19 34 2,562 354 439 760 4 7 144 17

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Table 15.8-2 Unfactored Dead Load and Seismic Forces Applied at Column Base Load Case

DC

DW

PS

MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL (kip)

62 833 1,503 4 44

9 139 227 1 7

0 -14 -21 0 -16

L

Seismic-I (Mp applied) 15,574 0 992 716 0

Seismic-II (Mp applied) 0 15,574 0 0 716

Y T

B

X

L>B (a) General Case Y=L ML

MT X=T

B = LT

L = LT (b) Example Problem

Figure 15.8-2 Local Footing Coordinates vs. Global Structure Coordinates Upon calculation of effective dimensions under any load combination, the larger effective dimension is designated as “L” and smaller as “B” to calculate qpn and qR from information provided by the GD.

15.8.2

Design Requirements Perform the following design portion for the footing in accordance with the AASHTO LRFD Bridge Design Specifications, 6th Edition (AASHTO, 2012) with the California Amendments (Caltrans, 2014a), and design peak ground acceleration (PGA) = 0.6g .

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

15.8.3

•

Determine the minimum footing thickness required to develop the column reinforcement. (Assume #9 bars for footing bottom reinforcement)

•

Calculate LRFD factored forces for Service, Strength, and Extreme Event limit states applicable to footing design

•

Determine the minimum size of the square footing adequate for applicable LRFD limit states

•

Calculate required rebar spacing if #5 and #9 bars are used for top and bottom mats, respectively

•

Check footing thickness for one-way and two-way shears

Footing Thickness Determination Minimum footing thickness is equal to the minimum clearance from the bottom of footing to the bottom mat of footing reinforcement, plus the deformed diameters of the bars used for the bottom of footing reinforcement, plus the required development length of the main column reinforcement. dmin.

=

clr. + 2(db) + ld′

(15.8-1)

where: dmin. clr. db

ld′

= minimum footing thickness (ft) = minimum clearance from the bottom of footing to the bottom mat of footing reinforcement (in.) = nominal diameter of bar used for the bottom of footing reinforcement (in.) = required development length of the main column reinforcement (in.)

From AASHTO Table 5.12.3-1, clr. = 3 in., and for #9 bars, db = 1.25 in. The development length is calculated in accordance with AASHTO Articles 5.11.2.2, and 5.11.2.4. Development of Deformed Bars in Compression: ldb ≥ 0.63 (1.693)(60) / (3.6)0.5 = 33.7 in.

(AASHTO 5.11.2.2.1-1)

ldb ≥ 0.3(1.693)(60) = 30.5 in.

(AASHTO 5.11.2.2.1-2)

AASHTO Article 5.11.2.2.2 states that the basic development length may be multiplied by applicable modification factors, and requires that reinforcement is enclosed within a spiral of not less than 0.25 in. in diameter and spaced at not more than a 4 in. pitch, in order to use modification factor of 0.75. This reduction does not apply because we have the main column hoops spaced at 5 inches. Hooks shall not be considered effective in developing bars in compression, therefore, development length required for compression is equal to 33.7 inches.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Development of Standard Hooks in Tension lhb = 38.0 (1.693) / (3.6)0.5 = 33.9 in.

(AASHTO 5.11.2.4.1-1)

Basic development length shall be multiplied by applicable modification factors (AASHTO 5.11.2.4.2). Concrete Cover – For #11 bar and smaller, side cover (normal to plane of hook) not less than 2.5 in., and for a 90 degree hook cover on bar extension beyond the hook not less than 2 in., then modification factor = 0.70. Note - For determining modification factors, the specifications refer to the portion of the bar from the critical section to the bend as the “hook”, and the portion of the bar from the bend to the end of the bar as the “extension beyond the hook”. Ties or Stirrups – Hooks for #11 bar and smaller, enclosed vertically or horizontally within ties or stirrup-ties spaced along the full development, ldh, at a spacing not greater than 3db, where db is diameter of hooked bar, then modification factor = 0.80. None of the modification factors are applied, since #14 bars have been used for columns, therefore, development length of standard hooks in tension = 33.9 in. say 34 in. (Also greater than 8 × 1.693 in. and 6 in.). Development length for tension (34 in.), controls over the development length for compression (33.7 in.). The required footing thickness is calculated as: dmin. = clr. + 2(db) + l′d = 3 + 2(1.25) + 34 = 39.5 in. = 3.29 ft Try footing thickness dfooting = 4.0 ft

15.8.4

Calculation of Factored Loads Considering live load movements in the longitudinal and transverse directions, the following three cases of live load forces have been considered in this example: Case I) Maximum Transverse Moment (MT) and associated effects Case II) Maximum Longitudinal Moment (ML) and associated effects Case III) Maximum Axial Force (P) and associated effects Moments and shears at the column base must be transferred to the bottom of the footing for the footing design. The following unfactored forces are obtained to include the additional moment (V × dfooting) caused by shear force transfer. For example, HL-93 Truck – Load Case I, Forces applied at the column base are: MT = − 206 kip-ft VT = – 12 kip

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

For the footing thickness dfooting = 4 ft, forces applied at the bottom of footing are obtained as follows: MT = - 206 + (-12)(4) = -254 kip-ft VT =

-12 kip

The unfactored live load forces (without impact) at the bottom of the footing are calculated in Table 15.8-3. Table 15.8-3 Unfactored Live Load Forces at Bottom of Footing Load Case MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL (kip)

HL-93 Truck I - 254 298 217 -12 12

II - 44 1,780 238 -1 81

Permit Truck III - 88 656 479 -2 26

I - 412 203 367 - 16 8

II 35 3,138 439 4 144

III 62 422 760 7 17

The design for live loads for Case-III (both HL-93 and Permit Trucks) is only illustrated in this example, however all three cases need to be considered in practice. Forces and moments resulting from seismic analysis in transverse and longitudinal directions are also shown as Seismic-I and Seismic-II, respectively. As PGA > 0.5g shallow foundation will be designed for column plastic hinging, (rocking is not allowed). For the footing thickness dfooting = 4 ft, overstrength moment and shear applied at the bottom of the footing are calculated as: Mo = 1.2 [15,574 + (716)(4)] = 22,126 kip-ft VTo = 1.2(716) = 859 kip The unfactored dead load forces and seismic forces at the bottom of the footing are shown in Table 15.8-4. Table 15.8-4 Unfactored Forces Applied at Bottom of Footing Load Case

DC

DW

PS

MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL (kip)

78 1,009 1,503 4 44

13 167 227 1 7

0 -78 -21 0 -16

Seismic-I (Mo applied) 22,126 0 992 859 0

Seismic-II (Mo applied) 0 21,126 0 0 859

The LRFD load combinations (AASHTO, 2012) used in foundation design and corresponding load factors (AASHTO Tables 3.4.1-1 and 3.4.1-2) are summarized in Table 15.8-5. The upper and lower limits of permanent load factors (γp) are shown as “U” and “L”, respectively.

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Table 15.8-5 Load Factors for Footing Design Load Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

DC 1.25 0.90 1.25 0.90 1.25 0.90 1.25 0.9 1.00 1.00

DW 1.5 0.65 1.50 0.65 1.50 0.65 1.5 0.65 1.00 1.00

PS 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

EV 1.35 0.90 1.35 0.90 1.35 0.90 1.35 0.90 1.00 1.00

HL-93 1.75 1.75 1.35 1.35 1.00 -

P-15 1.35 1.35 -

EQ 1.00

The LRFD load factors are applied to axial force, shear forces, and moments in longitudinal and transverse directions to calculate factored loads for Strength, Service and Extreme Event limit states at the base of the column, as summarized in Table 15.8-6. Only the governing seismic case that is Seismic-I is used in Extreme Event-I load combination. Table 15.8-6 Factored Forces at Column Base for Footing Design

Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L

MT (kip-ft) -37 -75 201 162 117 79 -2 -40

ML (kip-ft) 2,582 2,087 2,003 1,508 1,434 939 2,319 1,824

P (kip) 3,037 2,318 3,224 2,505 2,198 1,479 2,845 2,126

VT (kip) 3 1 16 14 7 4 4 2

VL (kip) 95 74 72 51 50 28 85 63

Service I

3

1,754

2,188

3

61

VTotal (kip) 95 74 74 53 50 28 85 63 61

Extreme Event I

22,126

0

2,701

864

35

865

Factored Loads

Example: Calculation of gross axial force at bottom of footing for Strength-II-U limit state: Pg = 1.25(1,503)+1.5(227)+1(–21)+1.35(760) = 3,224 kips

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

15.8.5

Footing Size Determination In order to design a spread footing all live load combinations (Cases I, II and III) should be considered for both design and permit trucks. It is recommended to consider maximum axial case (Case III) for initial sizing of the footing and check footing size and stresses for the other two cases (I and II), however this example only considers Case-III. Based on preliminary analysis of the footing, reasonable estimates for “width of the footing” as well as “length to width ratios” are provided to the GD to be used in design (Appendix A). Refer to MTD 4-1 (Caltrans, 2014b) for other information to be submitted to the GD. The GD will provide graphs and also a table of “permissible net contact stress” (used for Service-I limit state check), and “factored gross nominal bearing resistance” (used for strength and extreme event limit states) for numerous “B' ” and “L'/B' ” ratios, as shown in Figures 15.8-3 to 15.8-5, and Table 15.8-7 for given ranges of footing widths and also effective length to effective width ratios.

Permissible Net Contact Stress

qpn (ksf)

10.0 L'/B' = 1.00

9.0

L'/B' = 1.25

For Permissible Settlement = 1''

L'/B' = 1.50

8.0

L'/B' = 1.75

7.0

L'/B' = 2.00

6.0 5.0 4.0 3.0 2.0 0.0

5.0

10.0

15.0

20.0

25.0

30.0

Footing Effective Width (ft)

Figure 15.8-3 Variations of Permissible Net Contact Stress

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Factored Gross Nominal Bearing Resistance qR (ksf)

45.0 L'/B' = 1.00 L'/B' = 1.25 40.0

L'/B' = 1.50 L'/B' = 1.75 L'/B' = 2.00

35.0

30.0

Resistance Factor φb = 0.45 25.0 0.0

5.0

10.0

15.0

20.0

25.0

30.0

Footing Effective Width (ft)

Figure 15.8-4 Variations of Factored Gross Nominal Bearing Resistance (Strength Limit State) Factored Gross Nominal Bearing Resistance qR (ksf)

100.0 L'/B' = 1.00 L'/B' = 1.25

90.0

L'/B' = 1.50 L'/B' = 1.75 80.0

L'/B' = 2.00

70.0

60.0

Resistance Factor φb = 1.00

50.0 0.0

5.0

10.0

15.0

20.0

25.0

30.0

Footing Effective Width (ft)

Figure 15.8-5 Variations of Factored Gross Nominal Bearing Resistance

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(Extreme Event Limit State) Table 15.8-7 Variations of Bearing Resistance for Different Limit States Effective Footing Size

Effective Footing Size Ratio

Factored Gross Nominal Bearing Resistance (Extreme Event Limit)

Factored Gross Nominal Bearing Resistance (Strength Limit)

Permissible Net Contact Stress (Service Limit)

No.

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

B′ (ft)

L′ (ft)

L′/B′

qR (ksf)

qR (ksf)

qpn (ksf)

10.00 13.75 17.50 21.25 25.00 10.00 13.75 17.50 21.25 25.00 10.00 13.75 17.50 21.25 25.00 10.00 13.75 17.50 21.25 25.00 10.00 13.75 17.50 21.25 25.00

10.00 13.75 17.50 21.25 25.00 12.50 17.19 21.88 26.56 31.25 15.00 20.63 26.25 31.88 37.50 17.50 24.06 30.63 37.19 43.75 20.00 27.50 35.00 42.50 50.00

1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25 1.25 1.50 1.50 1.50 1.50 1.50 1.75 1.75 1.75 1.75 1.75 2.00 2.00 2.00 2.00 2.00

69.8 74.5 79.4 84.5 89.6 66.8 72.2 77.9 83.7 89.5 64.8 70.7 76.9 83.1 89.5 63.3 69.7 76.2 82.8 89.4 62.3 68.8 75.6 82.5 89.4

31.4 33.5 35.8 38.0 40.3 30.1 32.5 35.1 37.7 40.3 29.2 31.8 34.6 37.4 40.3 28.5 31.3 34.3 37.2 40.2 28.0 31.0 34.0 37.1 40.2

9.7 7.0 5.5 4.6 3.9 8.7 6.3 5.0 4.1 3.5 8.0 5.8 4.6 3.8 3.2 7.4 5.4 4.2 3.5 3.0 7.0 5.1 4.0 3.3 2.8

As the first trial, a square footing of 20×20 ft is selected and contact stresses under service, strength, and extreme event factored loads are calculated as summarized in the following tables. Stresses are compared to “permissible net contact stress” (Service-I), and “factored gross nominal bearing resistance” (Strength and Extreme Event), as explained in MTD 4-1. Since the footing rests on soil, contact stress distribution is assumed uniform over the effective area of the footing. The bearing stresses should be calculated as net for Service-I limit state and gross for all strength and extreme event limit states as shown in Figure 15.8-6, therefore, weight of overburden soil and footing with corresponding load factors have been considered in the axial forces shown in Table 15.8-8.

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FG

OG

Poverburde

Pgross

Pnet=Pgross - Poverburden

Figure 15.8-6 Definition of Gross and Net Bearing Stresses For example: Strength I-U Pgross = Pgross at column base + factored weight at soil on footing + factored weight of footing Pgross = 3,037 + (20×20–28.26)(48–39–4) (120/1,000)(1.35) + (20×20×4)(150/1,000)(1.25) = 3,638 kips Service-I Pnet = Pnet at column base + weight of soil on footing + weight of footing – excavated soil (over burden) Pnet = 2,188 + (20×20−28.26)(48–39-4)(120/1,000) + (20×20×4)(150/1,000)−(48-39)(20×20)(120/1,000) = 2,220 kips Detailed calculations for Strength I-U limit state can be summarized as: MT = –37 kip-ft;

P = 3,638 kips

eT = 37/3,638 = 0.01 ft;

LT′ = 20–2(0.01) = 19.98 ft

ML = 2,582 kip-ft,

eL = 2,582/3,638 = 0.71ft

LL′ = 20–2(0.71) = 18.58 ft Ae = 19.98(18.58) = 371 ft2;

qg,u = 3,638/371 = 9.80 ksf

L'/B' = 19.98/18.58 = 1.08, therefore qR = 36.23 (From Figure 15.8-3) Since qR is greater than qg,u, bearing resistance is adequate. Similar calculation is required for every load combination as shown in Table 15.8-8.

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Table 15.8-8 Detailed Check for Footing Size (First Trial) Load Combination Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

MT (kip-ft) -37 -75 201 162 117 79 -2 -40 -3 22,126

ML (kip-ft) 2,582 2,087 2,003 1,508 1,434 939 2,319 1,824 1,754 0

P (kip) 3,638 2,735 3,826 2,923 2,800 1,896 3,447 2,543 2,220 3,164

eT (ft) 0.01 0.03 0.05 0.06 0.04 0.04 0.00 0.02 0.00 6.99

eL (ft) 0.71 0.76 0.52 0.52 0.51 0.50 0.67 0.72 0.79 0.00

L'T (ft) 19.98 19.94 19.90 19.89 19.92 19.92 20.19 19.97 20.00 6.01

L'L (ft) 18.58 18.47 18.95 18.97 18.97 19.01 18.65 18.56 18.42 20.00

Table 15.8-8 Detailed Check for Footing Size (First Trial) (Continued) Load Combination

Ae (ft2)

L'/B'

qo (ksf)

qpn or qR (ksf)

qo/qpn or qo/qR Ratio

Check

Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

371.2 368.5 377.1 377.2 377.9 378.6 373.1 370.7 368.3 119.1

1.08 1.08 1.05 1.05 1.05 1.05 1.07 1.08 1.09 3.32

9.80 7.42 10.15 7.75 7.41 5.01 9.24 6.86 6.02 26.30

36.23 36.15 36.52 36.53 36.53 36.56 36.28 36.22 5.11 40.43*

0.27 0.21 0.28 0.21 0.20 0.14 0.25 0.15 1.18 0.65

OK OK OK OK OK OK OK OK NG OK

* L'/B′ is out of range, therefore, factored nominal bearing resistance was calculated by extrapolation. For design purposes, the SD needs to ask the GD to provide adequate data to cover all applicable cases, without any need to extrapolation.

In Table 15.8-8: L'L, L'T = eL , eT Ae B' L' qo qpn qR

effective dimensions of the footing in the directions of L and T, respectively (ft). L'T = LT – 2eT and L'L = LL – 2eL = eccentricities calculated from ML and MT, respectively (ft) = effective area of the footing (ft2) = shorter effective dimension (ft) = longer effective dimension (ft) = uniform bearing stress calculated as net for service (qn,u) and gross for Strength and Extreme Event limits (qg,u) (ksf) = permissible net contact stress (ksf) = factored gross nominal bearing resistance (ksf)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

The permissible eccentricity under Service-I Load is calculated as: B/6 = L/6 = 20/6 = 3.33 ft. Therefore the eccentricity calculated under Service-I loads (0.79 ft) is acceptable. Under Extreme Event, the calculated eccentricity of 6.99 ft is larger than the permissible eccentricity of B/3 = L/3 = 20/6 = 6.66 ft and is not acceptable. Examination of stresses shows that contact stress calculated under Service-I limit state is higher than permissible net stress calculated from information (chart or table) provided by the GD. Therefore, size of the footing is increased to 24 ft ×24 ft and stresses are recalculated as shown in Table 15.8-9. Table 15.8-9 Detailed Check for Footing Size (Second Trial) Load Combination

MT (kip-ft)

ML (kip-ft)

P (kip)

eT (ft)

eL (ft)

L′T (ft)

L′L (ft)

Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

-37 -75 201 162 117 79 -2 -40 3 22,126

2,582 2,087 2,003 1,508 1,434 939 2,319 1,824 1,754 0

3,913 2,925 4,101 3,113 3,074 2,086 3,721 2,733 2,241 3,376

0.01 0.03 0.05 0.05 0.04 0.04 0.00 0.01 0.00 6.55

0.66 0.71 0.49 0.48 0.47 0.45 0.62 0.67 0.78 0.00

23.98 23.95 23.90 23.90 23.92 23.92 24.00 23.97 24.00 10.89

22.68 22.57 23.02 23.03 23.07 23.10 22.75 22.66 22.43 24.00

Table 15.8-9 Detailed Check for Footing Size (Second Trial) (Continued) Load Combination

Ae (ft2)

L'/B'

qo (ksf)

qpn or qR (ksf)

qo/qpn or qo/qR Ratio

Check

Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

543.9 540.6 550.3 550.3 551.9 552.7 546.1 543.3 538.4 261.3

1.06 1.06 1.04 1.04 1.04 1.04 1.05 1.06 1.07 2.20

7.20 5.41 7.45 5.66 5.57 3.77 6.82 5.03 4.16 12.91

38.85 38.77 39.08 39.09 39.11 39.13 38.90 38.84 4.22 63.1*

0.15 0.14 0.15 0.14 0.14 0.10 0.18 0.13 0.99 0.23

OK OK OK OK OK OK OK OK OK OK

* L'/B′ is out of range, therefore, factored nominal bearing resistance was calculated by extrapolation. For design purposes, the SD needs to ask the GD to provide adequate data to cover all applicable cases, without any need to extrapolation.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

As shown in above tables, a 24 ft ×24 ft footing size satisfies stress requirements. Furthermore, the calculated eccentricities under service and extreme event limit states (0.78 ft and 6.55 ft, respectively) are smaller than the limits (4 ft and 8 ft, respectively). The ratio of length of footing from column face to face of footing, to thickness of footing, Lftg/Dftg = (0.5)(24-6)/4 = 2.25 that is slightly over the limit of 2.2 required by SDC 7.7.1.3 (Caltrans 2013) for rigidity of the footing. However, the SDC limitation is mostly applicable to pile cap and it is less critical for spread footings. The factored nominal sliding resistance between footing and soil is calculated as:

RR = ϕRn = ϕτ Rτ + ϕep Rep

(AASHTO 10.6.3.4-1)

Assuming that soil passive pressure is negligible, RR= φτRτ , and for cohesionless soil: Rτ = V tan(δ)

(AASHTO 10.6.3.4-2)

where: Rτ = V =

nominal sliding resistance between soil and foundation (kip) total vertical force (kip)

For concrete cast against soil: δ = φf = internal friction angle of drained soil The factored resistance against sliding failure for cast-in-place concrete on sand is calculated using φτ = 0.8 for strength limit states and φτ = 1.0 for extreme event limit state (AASHTO Table 10.5.5.2.2-1). Similar to axial force, LRFD load factors are applied to unfactored resultant shear forces (VRes.) to calculate factored shear forces for each load combination. Table 15.8-10 shows that requirements of AASHTO Article 10.6.3.4 for sliding failure is met, therefore footing size of 24 ft ×24 ft is acceptable and will be used throughout this example. Table 15.8-10 Sliding Check for Footing Load Combination Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Extreme Event I

Chapter 15 – Shallow Foundations

Factored Resultant Shear (kip) 95 74 74 53 50 28 83 63 865

Factored Vertical Load (kip) 3,913 2,925 4,101 3,113 3,074 2,086 3,721 2,733 3,376

RR (kip)

Check

2,444 1,827 2,561 1,544 1,520 1,303 2,324 1,707 2,635

OK OK OK OK OK OK OK OK OK

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Upon finalizing the footing size, “Foundation Design Data Sheets” shown in Appendix are completed and forwarded to the GS to be used for preparation of “Foundation Design Recommendations”.

15.8.6

Flexural Design For structural design of the footing, distribution of contact stresses is assumed linear (trapezoidal or triangular) irrespective of the substrate stiffness (resting on soil or rock). If the eccentricity (e = M/P) is less than L/6 then the soil under the entire area of the footing is in compression and contact stresses can be determined based on trapezoidal distribution. Maximum forces acting at the bottom of column for case-III can be summarized as: Service:

P = 2,188 kips;

ML = 1,754 kip-ft;

MT = 3 kip-ft

Strength I-U:

P = 3,638 kips;

ML = 2,582 kip-ft;

MT = –37 k-ft

The area and section modulus of the footing contact surface are: 576 ft2 and 2,304 ft3, respectively. Maximum and minimum contact stresses acting along the edges of the footing (q1 and q2) are calculated using the generic equation of (P/A) ± (M/S): •

•

Strength Limit State: L Direction:

q1= 7.44 ksf;

q2= 5.20 ksf

T Direction:

q1= 6.33 ksf;

q2= 6.30 ksf

Service Limit State: L Direction:

q1 = 4.56 ksf; q2 = 3.04 ksf

T Direction:

q1 = 3.80 ksf; q2 = 3.80 ksf

Since the column has a circular cross section, it is transformed into an effective square section for footing analysis with equivalent column width of: (28.26)0.5 = 5.32 ft. Assuming #5 (db=0.69 in.) and #9 (db=1.25 in.) bars are used for top and bottom mat reinforcement, the minimum effective depths (de) of the footing for the top and bottom mats are calculated as 43.96 in. and 43.1 in., respectively. Critical sections for moment and shear calculations:

•

Bending moment at the face of the column (AASHTO 5.13.3.4)

•

One-way shear at distance “dv” from the face of the column (AASHTO 5.8.3.2)

•

Two-way (punching) shear on the perimeter of a surface located at distance “dv,avg” from the face of the column (AASHTO 5.13.3.6)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

where: dv

=

effective shear depth of the section (ft)

dv,avg

=

average of effective shear depths for both directions (ft)

Using critical contact stresses (q1 and q2), maximum moments at the face of the column for unit foot width of the footing are calculated as: •

Strength Limit State: ML = 311.8 kip-ft;

MT = 276.1 kip-ft

•

Service Limit State: ML =190.4 kip-ft;

MT = 165.8 kip-ft

Assuming 3 in. concrete cover, and using 42#9 bars for bottom mat, the spacing of rebars is calculated as: s = [24(12) –2(3) –1.25]/(42–1) = 6.85 in. The calculated spacing is less than maximum spacing of 12 in. specified in AASHTO Article 5.10.8, and it is acceptable. The area of steel contributing to unit width of the footing is: (1.0)(12)/6.85 =1.75 in.2, therefore the depth of the concrete stress block and resisting moment are calculated as: a=

(1.75)(60) = 2.86 in. (0.85)(3.6)(12)

Corresponding depth of the neutral axis will be c = (2.86)/(0.85) = 3.36 in. and the net tensile strain in extreme tension steel reinforcement is calculated as: εs = (0.003)(43.1–3.36)/(3.36) = 0.035 Since calculated strain is larger than 0.005, the section is considered as tensioncontrolled and resistance factor φ is taken as 0.9 (AASHTO 5.5.4.2). The factored moment is calculated as: Mr = φMn = (0.9)(1.75)(60)(43.1–0.5×2.86)(1/12) = 328.1 kip-ft > ML = 311.8 kip-ft

OK

Therefore, selected number of bars is adequate for strength in both directions. However AASHTO Article 5.7.3.3.2 requires minimum amount of reinforcement to be provided for crack control. The factored flexural resistance Mr is required at least equal to as the smaller of Mcr and 1.33 Mu as follows (gross section properties are used instead of transformed sections): Modulus of rupture:

fr = (0.24)(3.6)0.5 = 0.455 ksi

Gross section modulus: Sc = Snc = (12)(48)2 /6 = 4,608 in.3 Mcr = γ3 γ1 fr Sc = 1.6(0.75)(0.455)(4,608) = 2,516 kip-in. = 209.7 kip-ft (AASHTO 5.7.3.3.2-1) 1.33Mu = 1.33(311.8) = 414.7 kip-ft

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

φM n = Mr = 328.1 kip-ft >

 M cr = 209.7 kip-ft  smaller of  209.7 kip-ft  = 1.33M u = 414.7 kip-ft  OK (AASHTO 5.7.3.3.2)

AASHTO Article 5.7.3.4 requires maximum limits of rebar spacing for crack control. s ≤

700 γ e β s f ss

− 2d c

(AASHTO 5.7.3.4-1)

Assuming exposure factor γe = 1 (class-I exposure) and dc = 3 + (1.25/2)=3.625 in. βs = 1 +

3.625 0.7(48 − 3.625)

= 1.117

Cracked concrete section is used to calculate tensile stress in steel reinforcement under service loads: Ec = 57(3,600)0.5 = 3,420 ksi n = 29,000/3,420 = 8.48 Per AASHTO Article 5.7.1, n is rounded to the nearest integer number, therefore n = 8 will be used. The distance of the neutral axis to the top of the footing is calculated as: yb = 8.93 in. The moment of inertia for unit width (12 in.) of the transformed section (based on concrete) is calculated as: Itrans = (12)(8.933)/3+(1.75)(8)(43.1– 8.93)2 = 19,194 in.4 Tensile stress in steel reinforcement at the service limit state is calculated as: M ( d e − yb ) (190.4) (12) (43.1 − 8.93) = f ss n= (8) = 32.54 ksi I 19,194

The maximum spacing is checked as: s 6.85 in. < =

700 γ e 700 (1) dc − 2= − 2(3.625) = 12 in. (1.117)(32.54) βs f ss

OK

Therefore, 42#9 bars are acceptable for the bottom mat. The shrinkage and temperature reinforcement for the top mat per unit foot width shall satisfy (AASHTO Article 5.10.8): As >

1.3bh 1.3(12 × 24)(48) = = 0.446 in.2 2(b + h) f y 2(12 × 24 + 48)(60)

(AASHTO 5.10.8.1)

0.11 ≤ As ≤ 0.6

(AASHTO 5.10.8-2)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Since thickness of the footing is greater than 18 in., spacing of the rebar shall not exceed 12 in. If 42#5 bars are considered: s = [24(12) –2(3) –0.69]/(42–1) = 6.85 in. < 2 in.

OK

As = (0.31)(12)/(6.85) = 0.543 in.2 > 0.446 in.2/ft OK (AASHTO 5.10.8-1) 0.11 < As = 0.543 < 0.6

OK (AASHTO 5.10.8-2)

Therefore, 42#5 bars in each direction will be used for the top mat. Note: For square footings the reinforcement shall be distributed uniformly across the entire width of the footing. (AASHTO Article 5.13.3.5)

15.8.7

Shear Design According to AASHTO Article 5.13.3.6.1, both one-way and two-way shears shall be considered in footing design: •

The critical section for one-way action extends in a plane across the entire width and located at a distance as specified in AASHTO 5.8.3.2 (that is mostly at distance dv from the face of the column).

•

The critical section for two-way action is perpendicular to the plane of the footing and located so that its perimeter b0, is a minimum but not closer than 0.5dv to the perimeter of the concentrated load or reaction area.

where: dv = d – 0.5a = 43.1–0.5(2.86) = 41.67 in. ≈ 3.5 ft., however, dv should be greater than both 0.9de = 0.9(43.1) = 38.79 in. and 0.72h = 0.72(48) = 34.56 inch. Therefore, dv = 3.5 ft will be used in calculations. 15.8.7.1

Direct (One-Way) Shear The extreme contact stresses for most critical strength limit state case (L direction) are 5.20 ksf and 7.44 ksf. As shown in Figure 15.8-7, assuming a linear stress distribution the contact stress at distance dv from face of the column is calculated: q3 = 7.44 – (7.44 – 5.20)(12 –2.66 –3.5)/(24) = 6.89 ksf Shear force at critical section for unit width: Vu = (7.44+ 6.89)(12 –2.66 –3.5)/2 = 41.84 kips The maximum shear resistance of the section (considering shear reinforcement contribution) is limited to 0.25 f c′bv d v (AASHTO 5.8.3.3-3): Vn, max = 0.25(3.6)(12)(41.67) = 450 kips This maximum shear resistance is much higher than factored shear force of 41.84 kips, and is not governing.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Shear resistance of concrete (Vc) is 0.0316 β f c′bv d v , where β = 2.0.

= Vc 0.0316(2) = 3.6(12)(41.67) 59.96 kips Assuming that no shear reinforcement will be used, Vs= 0 and φVn = 0.9(60) = 54 kips > 39.15 kips

OK

5.32'

Vu 5.20 ksf

7.44 ksf 3.5′

6.89 ksf 24'

Figure 15.8-7 Direct Shear Force Calculation 15.8.7.2

Punching (Two-Way) Shear The critical section is located at the distance of 0.5 dv,avg. from face of the column as shown in Figure 15.8-8. Critical section for two way shear

24'

24'

6'

9.5'

Figure 15.8-8 Critical Section for Two-Way Shear

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

Using conservative assumption of dv,avg=3.5 ft results in b0 = π (6+3.5) = 29.83 ft = 358 in. For square footing βc = 1, and nominal shear resistance is as:



Vn =  0.063 +



0.126  βc

0.126   5, 350 kips  0.063 +  3.6 ( 358 )( 41.67 ) =  f c′bo d v = 1.0   

 Vn = 5,350 kips > 0.126 f c′ bo d v = 0.126 3.6 (358)(41.67 ) = 3,566 kips ∴ Use Vn = 3,566 kips

φVn = 0.9(3,566) = 3,210 kips The punching shear force acting on the critical surface is calculated by subtracting the force resulting from soil contact stress acting on the critical surface from the axial force of the column: P2= 3,638 − − way

3,638 2 π (4.75) = = Vn 3,210 kips 3,190 kips < φ (24) (24)

OK

Shear reinforcement is not required and the footing depth d = 4.0 ft is acceptable. Note: Although seismic loads were considered in sizing of the footing, structural design of the footing was only based on service and strength I-U (Case-III) limit states. Refer to Caltrans SDC (Caltrans, 2013) for other design and detailing requirements.

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APPENDIX COMMUNICATIONS OF STRUCTURAL AND GEOTECHNICAL DESIGNERS (REFER TO MTD 4-1 AND MTD 1-35) Preliminary Foundation Design Data Sheet (Trial Footing Size) Table 15.A-1 Preliminary Foundation Data

Support No.

Finished Grade Elevation (ft)

Abut 1 Bent 2 Abut 3

48

BOF Elevation (ft)

39

Estimated Footing Dimensions (ft) B

L

10

10

Permissible Settlement under Service-I Load (in.) 1 or 2 1 1 or 2

Approximate Ratio of

 Permanent     Total  Service –I Load 0.75 -

Table 15.A-2 Scour Data Long-term (Degradation and Contraction) Scour Elevation (ft) N/A 48 N/A

Support No. Abut 1 Bent 2 Abut 3

Short-term (Local) Scour Depth (ft) N/A 0.00 N/A

Foundation Design Data Sheet (Final Footing Size) Table 15.A-3 Foundation Data Support No.

Finished Grade Elevation (ft)

Abut 1 Bent 2 Abut 3

Chapter 15 – Shallow Foundations

48

BOF Elevation (ft) 39

Footing Dimensions (ft) B 24

L 24

Permissible Settlement under Service Load (in.) 1 or 2 1 1 or 2

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Table 15.A-4 LRFD Service-I Limit State Loads for Controlling Load Combination Total Load Support No. Abut 1 Bent 2

PTotal (kip) Net 2,241

Permanent Load

Mx (kip-ft)

My (kip-ft)

Vx (kip)

1,754

N/A 3

N/A N/A

N/A

N/A

Abut 3

Vy (kip)

N/A

PPerm (kip) Gross 1,762

Mx (kip-ft)

My (kip-ft)

Vx (kip)

Vy (kip)

1,098

N/A 85

N/A N/A

N/A

N/A

N/A

Table 15.A-5 LRFD Strength/Construction and Extreme Event Limit States Load Data Strength/Construction Limit State (Controlling Group)

Extreme Event Limit State (Controlling Group)

Support No.

PTotal (kips) Gross

Mx (kip-ft)

My (kip-ft)

Vx (kip)

Abut 1 Bent 2

N/A 4101

N/A 2003

N/A 201

N/A N/A

N/A 3,376

Abut 3

N/A

N/A

N/A

N/A

N/A

Vy (kip)

PTotal Mx (kip) (kip-ft) Gross

My (kip-ft)

Vx (kip)

Vy (kip)

N/A 0

N/A 22,126

N/A 864

N/A 35

N/A

N/A

N/A

N/A

Note: Load tables may be modified to submit multiple lines of critical load combinations for each limit state, if necessary.

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NOTATION A

=

actual footing area (ft2)

A′

=

reduced effective area of the footing (ft2)

B, L

=

actual dimensions of the footing (ft)

B′, L′ =

effective dimensions of the footing (ft)

clr.

=

minimum clearance from the bottom of footing to the bottom mat of footing reinforcement (in.)

db

=

diameter of bar (in.)

dfooting = footing depth (ft) dmin.

=

minimum footing depth (ft)

dv

=

effective shear depth of the section (ft)

dv,avg

=

average of effective shear depths for both directions (ft)

ey , ex

=

eccentricities parallel to dimensions B and L, respectively (ft)

=

28-day compressive strength of concrete (psi)

fy

=

specified minimum yield strength of steel (ksi)

ldb

= development length for deformed bars (in.)

lhb

= development length for deformed bars (in.)

ld′

=

f c′

ML, MT = Mp

=

required development length of the main column reinforcement (in.) moments acting about L and T directions, respectively (kip-ft) plastic moment at column base (kip-ft)

M x, M y =

moments acting X and Y directions, respectively (kip-ft)

P

=

vertical force acting at the center of gravity of the bottom of the footing area (kip)

q

= uniform bearing stress (ksf)

qg,u

=

gross uniform bearing stress (ksf)

qg,max

=

gross maximum bearing stress (ksf)

qn

=

gross nominal bearing resistance (ksf)

qn,max

=

net maximum bearing stress calculated using Service-I Limit State loads assuming linear stress distribution for footings on rock (ksf)

qn,u

= net uniform bearing stress calculated using Service-I Limit State loads assuming uniform stress distribution for footings on soil (ksf)

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qpn

=

permissible net contact stress provided by the GD and calculated based on a specified allowable settlement (ksf)

qR

=

factored gross nominal bearing resistance provided by the GD = ϕbqn (ksf)

Rτ

=

nominal sliding resistance between soil and concrete (kip)

Sx, Sy

=

section modulus of the footing area about X and Y directions, respectively (ft3)

V

=

total force acting perpendicular to the interface (kip)

VL, VT = Vp

=

shears acting along L and T directions, respectively (kip) plastic shear at column base (kip)

Vx, Vy =

shears acting along X and Y directions, respectively (kip)

δ

=

friction angle at interface of concrete and soil (degree)

ϕf

= internal friction angle of drained soil (degree)

ϕb

=

resistance factor for bearing

ϕτ

=

resistance factor against sliding

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REFERENCES 1. AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 2012 (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 2. Caltrans, (2013). Caltrans Seismic Design Criteria, Version 1.7, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2014a). California Amendment to AASHTO LRFD Bridge Design Specifications - 6th Edition, California Department of Transportation, Sacramento, CA. 4. Caltrans, (2014b). Bridge Memo to Designers 4-1: Shallow Foundations, California Department of Transportation, Sacramento, CA. 5. Caltrans, (2008). Bridge Memo to Designers 1-35: Foundation Recommendation and Reports, California Department of Transportation, Sacramento, CA.

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CHAPTER 16 DEEP FOUNDATIONS TABLE OF CONTENTS 16.1

16.2

16.3

INTRODUCTION .......................................................................................................... 16-1 16.1.1

Types of Piles and Shafts .............................................................................. 16-2

16.1.2

Constructability Issues .................................................................................. 16-2

16.1.3

General Design Considerations – Pile/Shaft Group ...................................... 16-3

ANALYSIS/DESIGN OF PILE/SHAFT GROUPS IN COMPETENT SOIL (DESIGN EXAMPLE) .................................................................................................................... 16-8 16.2.1

Determine Pile Cap Layout and Depth ....................................................... 16-11

16.2.2 States

Determine Factored Loads for Service, Strength, and Extreme Event Limit 16-13

16.2.3

Check Pile/Shaft Capacity........................................................................... 16-15

16.2.4

Design Pile Cap for Flexure ........................................................................ 16-19

16.2.5

Design Pile Cap for Shear ........................................................................... 16-24

16.2.6

Design Pile Cap for Joint Shear .................................................................. 16-28

16.2.7

Communication to Geotechnical Services .................................................. 16-32

ANALYSIS/DESIGN OF SHAFT GROUPS IN SOFT/LIQUEFIABLE SOIL UNDER EXTREME EVENT I LIMIT STATE........................................................................... 16-34 16.3.1

Introduction ................................................................................................. 16-34

16.3.2

Caltrans Design Practice ............................................................................. 16-34

16.3.3

Practice Bridge Geometry ........................................................................... 16-40

16.3.4

Site Conditions and Foundation Recommendations ................................... 16-40

16.3.5

Material Properties ...................................................................................... 16-49

16.3.6

Minimum Pile-Cap Depth ........................................................................... 16-51

16.3.7

Shaft-Group Layout .................................................................................... 16-53

16.3.8

Seismic Forces on Shaft-Group Foundations .............................................. 16-53

16.3.9

Structural Modeling of Shaft-Group Foundations ...................................... 16-56

16.3.10

Inelastic Static Analysis of Shaft-Group Foundations ................................ 16-57

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16.4

ANALYSIS AND DESIGN OF LARGE DIAMETER COLUMN-SHAFTS ..................................................................................................... 16-63 16.4.1

Introduction ................................................................................................. 16-63

16.4.2

Design Practice ........................................................................................... 16-63

16.4.3

Lateral Stability Check of Type I and II Shafts .......................................... 16-66

16.4.4

Reinforcement Spacing Requirements of Column-Shafts .......................... 16-67

16.4.5

Design Process ............................................................................................ 16-70

16.4.6

Design Example .......................................................................................... 16-72

NOTATION

................................................................................................................... 16-106

REFERENCES

................................................................................................................... 16-112

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CHAPTER 16 DEEP FOUNDATIONS 16.1

INTRODUCTION This chapter discusses the design practice of deep foundations, which comprises pile and shaft foundations. A pile is defined as a slender deep foundation unit, entirely or partially embedded in the ground and installed by driving, vibration, or other method. A drilled shaft is defined as a foundation unit, entirely or partially embedded in the ground, constructed by placing concrete in a drilled hole with or without steel reinforcement. Within Caltrans terminology, “pile” is often used as a general term referring to both driven piles and drilled shafts. However, the term “piles” is referred as “Driven Piles” in the AASHTO LRFD Bridge Design Specification (AASHTO, 2012). Both piles and drilled shafts develop their geotechnical capacities from the surrounding soil. Pile/shaft groups in competent soil are addressed in Sections 16.1.3 and 16.2, shaft groups in soft/liquefiable soil are addressed in Section 16.3, and column shafts (Type I and Type II) are addressed in Section 16.4. Pile/shaft foundations can be an economical/necessary alternative to spread footings, particularly when: (i)

competent soil strata are far from original ground;

(ii)

liquefaction and/or lateral-spreading potential exist;

(iii)

scour depth is large;

(iv)

removal of existing soil is undesirable, e.g., soil contaminated by hazardous material; or

(v)

space limitations prohibit the use of spread footings.

The structural system of a pile/shaft group is an array of piles or shafts that are connected to a relatively thick reinforced concrete or composite cap and that work interactively together to support the bridge bents/piers. The forces and moments acting at the base of the bent/pier are directly transferred to the pile cap, and resulting displacements and rotations of the cap generate axial force, shear force, and bending moment in the piles/shafts. Design provisions for driven piles and drilled shafts are specified in AASHTO Articles 10.7 and 10.8, respectively, with corresponding CA Amendments (Caltrans, 2014a). Furthermore, Caltrans Memo to Designers 3-1 (Caltrans, 2014b) provides general guidance for selection and design of the piles or shafts and detailed communication procedures between the Structural Designer (SD) and the Geotechnical Designer (GD).

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16.1.1

Types of Piles and Shafts Application of different types of piles and shafts are discussed in Memo to Designers 3-1 (Caltrans, 2014b). Standard Plan Piles (Class Piles) are structurally predesigned piles or shafts mostly used in pile groups to support columns or at abutments and piers. Upper limits of structural resistance of Standard Plan (Class) Piles in compression and tension, as well as structural details, are given in the Standard Plans. The most common types of driven piles are steel H-Pile (HP) or pipe piles, precast pre-stressed concrete piles, and Cast-in-Steel Shell (CISS) piles. In selection of driven piles, environmental constraints such as acceptable limits of noise and vibration, construction constraints such as required overhead, and geotechnical condition of the soil are of importance. Drilled shafts also known as Cast-in-Drilled Hole (CIDH) concrete piles are often recommended when: (i)

pile driving is not viable, e.g., when there is interference of pile driving with overhead power or telephone lines or nearby underground utilities;

(ii)

large vertical or lateral resistance is required; and

(iii)

noise and vibration mitigation plans are either not feasible or too expensive.

However, disposal of hazardous drill spoils may be costly. Drilled shafts may be used in a group similar to driven piles or as large diameter isolated shafts, that is, pile extensions and Types I/II shafts. Memo to Designers 3-1 (Caltrans, 2014b) includes provisions that improve constructability of the shaft, such as the use of temporary/permanent casing and also construction joint in Type-II shafts. For more information on isolated large diameter shafts (Type I and II shafts), refer to Section 16.4.

16.1.2

Constructability Issues If ground water is anticipated during construction, drilled shafts must be at least 24 in. in diameter, and PVC inspection pipes should be installed to allow GammaGamma Logging (GGL) or Cross-Hole Sonic Logging (CSL) test of the shafts for quality assurance mostly performed by the Foundation Testing Branch of Geotechnical Services. Memo to Designers (MTD) 3-1 (Caltrans, 2014b) illustrates requirements for proper placement of the inspection pipes. Inspection pipes are laid out by the (SD) and must be shown in the structure plans where applicable. Drilled shafts need to allow for additional concrete cover for placement of the rebar cage. Minimum cover requirements for various drilled shaft sizes are shown in Table 16.11. The minimum cover is not related to protection of the reinforcing steel (refer to CA Amendment (Caltrans, 2014a) Table 5.12.3-1) but rather as an aid for construction. The minimum cover allows for rebar cage deformations that occur during placement as well as for some tolerance for the final shaft and column location. For non-Standard Plan Piles, irrespective of the actual cover, only 3 in. of cover is assumed effective and used in the structural capacity calculations.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 16.1-1 Minimum Cover Requirement for Drilled Shafts Diameter of Drilled Shaft, D

Concrete Cover

16 in. and 24 in. Standard Piles

See Standard Plan B2-3

24 in. ≤ D ≤ 36 in.

3 in.

42 in. ≤ D ≤ 54 in.

4 in.

60 in. ≤ D < 96 in.

5 in.

96 in. and larger

6 in.

For Type-II shafts use of a construction joint below the column cage will facilitate construction. The plans should show the location of the construction joint and also any permanent casing used to allow workers to prepare the joint. If the joint is more than 20 ft deep, the District should be contacted to obtain classification of the site as gassy/non-gassy from Cal-OSHA Mining and Tunneling Unit as explained in topic 110 of the Highway Design Manual and MTD 3-1 (Caltrans, 2014b). The most common types of driven piles are steel pipe, steel HP shapes, Cast-inSteel Shell (CISS), and precast pre-stressed concrete piles. The Structural Designer should check availability with the cost estimating branch if HP sections are to be used. Timber piles are not commonly used in Caltrans’ projects unless for temporary construction. Vibration and noise generated by pile driving should be considered from early stages of the project and when developing the Advanced Planning Study (APS). District should be consulted regarding acceptable levels of noise and vibration based on environmental, geotechnical, and structural constraints. The Project Engineer may present mitigation methods to avoid elimination of driven piles which are usually cheaper than other alternatives. Redundancy of the steel piles, shells, and casings can affect quality assurance of welding and, therefore, impact the cost and schedule of the project. Definition of Redundant (R) and Non-redundant (N) piles is covered in the Caltrans Standard Special Provisions [49-2.02B(1)(a), 2011] and may differ from the commonly used definition of structural redundancy. If ground water is anticipated during construction, steel casings may be used to facilitate construction of drilled shafts and to avoid caving problems. Unlike driven shells (used in CISS piles), casings can be installed by vibration or oscillations, and usually, contribution of cased portion of the shaft to geotechnical capacity is negligible. Contribution of casing to confinement or flexural strength and stiffness of the shaft may be considered in design calculations.

16.1.3

General Design Considerations – Pile/Shaft Group Columns and piers can be supported by a foundation system consisting of a concrete cap attached to a group of piles or shafts. The structural redundancy of pile/shaft group is advantageous. However, excavation and backfill required for construction may impact its selection.

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16.1.3.1

Pile/Shaft Spacing Table 16.1-2 summarizes the current recommendations of AASHTO (AASHTO, 2012) and CA Amendments (Caltrans, 2014a) for pile/shaft spacing in a pile/shaft group, where D is the diameter of the pile/shaft. Table 16.1-2 Pile/Shaft Group Spacing Type of Piles or Shafts

Minimum center-to-center spacing of piles/shafts

Driven Piles

36 in. or 2D (whichever greater) 2.5D

Drilled Shafts

Minimum spacing between face of the pile/shaft to face of the cap (for edge/corner piles/shafts) 9 in. or 0.5D (whichever greater) 12 in.

The limit of 2.5D for drilled shafts shall not be violated for better constructability. Furthermore, use of larger spacing is recommended to avoid interference of adjacent piles/shafts and to economize geotechnical design. 16.1.3.2

Scour Protection To avoid loss of geotechnical capacity and structural problems caused by washout of the surrounding soil, the pile cap should be deep enough to prevent pile/shaft exposure during service life of the bridge. All components of scour that is degradation, contraction, and local pier scour must be considered in design. The minimum required depth of the cap to eliminate scour problem is shown in Figure 16.1-1.

Figure 16.1-1 Required Embedment Depth for Scour Protection

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When evaluating the geotechnical/structural capacity of the pile/shaft group, the combination of different components of scour should follow Table 16.1-3 (see Section 3.7.5 of CA Amendments (Caltrans, 2014a). Table 16.1-3 Percentage of Scour Used in Design for Different Limit States Limit State

Service Strength Extreme Event

16.1.3.3

Maximum Aggradation/Degradation and Contraction Scour to be considered for footing design, shown as a % of total 100 100 100

Maximum Local Scour to be considered for footing design, shown as a % of total

100 50 0

Standard (Class) Piles Based on structural capacity, piles and shafts are classified as standard and nonstandard. Standard piles, including drilled shafts and driven piles, have a precalculated structural capacity. Caltrans Standard Plans, Sheets B2-3, B2-5, and B2-8 provide pile details for class 90, 140, and 200 kip standard piles. Class of a pile or shaft refers to Design Compression Strength of the pile/shaft, generally used for Working Stress Design (WSD). Design Tensile Strength (WSD) for the above piles is 0.4 times design compressive strength as shown in the standard plans. The LRFD Nominal Resistance in Compression of Pile Class 90, 140 and 200 is twice the class of the pile, i.e., 180, 280, and 400 kips, respectively. The LRFD Nominal Resistance in Tension is half of the compression. Due to lack of solid information on joint performance, the pile-to-cap connection of standard piles is assumed as a pin connection. Corrosion mitigation provisions are covered by construction specifications. Therefore, standard plans are valid for both corrosive and non-corrosive sites, except for pipe piles “Alternative W” shown on B2-5 and B2-8. Designers may use this alternative if applicable corrosion allowance is considered, and structural resistance of the reduced cross section of the pile is recalculated and checked based on design life of the structure (commonly 75 years). Standard piles must be checked and redesigned if used for seismic critical applications.

16.1.3.4

General Design Assumptions A pile/shaft group is an indeterminate structure and is generally subjected to axial force and biaxial moment and shear. The following assumptions are commonly used in analysis of pile/shaft groups: 

Rigid Pile Cap: The pile cap can be assumed as rigid when the length-tothickness ratio of the cantilever measured from face of the column/pier to the edge of the cap is less than or equal to 2.2 according to Seismic Design Criteria (SDC) 7.7.1.3 (Caltrans, 2013).

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Pile/Shaft-to-Cap Pin Connection: When surrounded by competent soil, the lateral movement of the piles/shafts under lateral loads such as earthquake is very small. Therefore, moments in the pile/shaft can be ignored and a pin connection can be assumed between piles/shafts and the pile cap.

These assumptions will result in a linear distribution of pile/shaft forces and facilitate analysis under lateral forces as explained in this chapter. 16.1.3.5

Analysis for Service and Strength Limit State Loads The maximum compression (Cmax) and tension (Tmax) axial forces applied to a pile/shaft in a symmetrical group are calculated as: Cmax 

M y Cx P M xC y   N Ix Iy

(16.1.3.5-1)

Tmax 

M y Cx P M xC y   N Ix Iy

(16.1.3.5-2)

where P, Mx and My are axial force, bending moment about x axis, and bending moment about y axis, respectively, acting at the top of pile (bottom of pile cap). N is the total number of piles/shafts, and Ix and Iy are equivalent moments of inertia of pile/shaft groups in the x and y directions calculated as:

  I y   N y  C x2  I x   N x  C y2

(16.1.3.5-3) (16.1.3.5-4)

In the above equations Nx and Ny are number of piles/shafts in a row parallel to x or y directions, and Cy and Cx are perpendicular distances of the row under consideration from center of gravity of the pile/shaft group, respectively. In the above equations compression is assumed positive. 16.1.3.6

Analysis for Extreme Event (Seismic) Loads For Extreme Event-I Limit State (seismic) the pile/shaft group is analyzed under column overstrength moment (Mo) and associate shear force (Vo) acting at the base of the column and applied at all different directions. The plastic moment (Mp) at the base of the column should be calculated using fiber method analysis (for example, xSECTION analysis) and considering the seismic induced overturning effect on the column axial force for multi-column bents. The overstrength moment (Mo) is equal to 1.2Mp. The overstrength moment and shear should be transferred to the bottom of the pile cap for pile/shaft group analysis, and therefore, the moment to be used in pile/shaft analysis will be Mo + Vo Df, where Df is the depth of the pile cap. Analysis of pile/shaft group for seismic forces depends on the type of the soil. Caltrans SDC 6.2.2 (Caltrans, 2013) classifies soil as competent, marginal, and poor. This

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 classification is based on physical and mechanical properties of the soil, as well as possibility of seismic-associated effects such as liquefaction and lateral spreading. Pile/shaft group analysis for seismic forces in competent soil will be similar to analysis for Service or Strength limit state load combinations. The analysis of pile/shaft group in marginal or poor (soft/liquefiable) soil under Extreme Event I limit state is addressed in Section 16.3. 16.1.3.7

Design Process MTD 3-1 (Caltrans, 2014b) lays out the design process for deep foundations. The SD provides factored loads acting on the pile/shaft for different load combinations, and the GD provides tip elevations for compression, tension, and settlement. The settlement tip is calculated based on service-I limit state loads, while compression and tension tips are calculated based on strength and extreme event limit state loads. The factored weight of the footing (pile cap) and overburden soil should be added to the factored axial force calculated at the base of the column to provide the “gross” factored axial force. The factored weight of the soil from Original Ground (OG) to bottom of the pile cap is subtracted from factored gross axial force to obtain factored “net” axial force. Pile/shaft load calculations are based on net axial force for Service limit state and gross axial force for Strength and Extreme Event limit states. The lateral tip elevation is provided by SD. The seismic moment and shear are applied at the cut-off point of the pile/shaft, and deflection at the cut-off point is recorded. Then, the length of the pile/shaft is changed, the deflection is recalculated, and the variation of the deflection vs. length of the pile/shaft is drawn. “Critical Depth” of the pile/shaft is the shallowest depth at which any increase in the length of the pile/shaft does not change the cut-off deflection. The critical length is used to specify “lateral tip” on the plans. A determination of the lateral tip elevation is not necessary for pile/shaft groups in competent soil. For pile/shaft groups in marginal or soft/liquifiable soil it is not necessary to use a factor of safety for determination of the lateral tip elevation. MTD 3-1 explains the design process and information to be communicated between the SD and the GD, as well as information to be shown in the Pile Data Table. Examples of different types of piles/shafts are provided in the Attachments of MTD 3-1.

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16.2

ANALYSIS/DESIGN OF PILE/SHAFT GROUPS IN COMPETENT SOIL (DESIGN EXAMPLE) The design process for a column-to-pile cap pile foundation is illustrated through the following example. Note: A fixed column-pile cap connection was assumed for this design illustration. However, a more efficient pile cap design may utilize a pinned column-pile cap connection. Given: A circular column of 6 ft diameter with 26 #14 main bars and #8 hoops at 5 in. is used for a two-span post-tensioned box girder bridge. OG elevation is 48 ft, Finished Grade (FG) elevation is 48 ft, and bottom of the cap elevation is 38.75 ft. Furthermore: 

Concrete material fc = 3,600 psi.



Reinforcement fy = 60,000 psi.



Un-factored live loads at the base of the column are listed in Table 16.2-1.



Plastic moment and shear at the base of the column are calculated as Mp = 15,455 kip-ft and Vp = 716 kips.



Un-factored dead load and seismic forces at the base of the column are listed in Table 16.2-2.



Geotechnical information classifies soil as competent (SDC 7.7.1.1). Density of soil is 120 lb./ft3



Pile Cap rests on 30 in. CIDH piles (drilled shafts). Estimate of the shaft factored nominal geotechnical resistance is 600 kips compression and 300 kips tension. Shaft moments are assumed negligible (SDC 7.7.1.1). Shaft layout is assumed as shown in Figure 16.2-1.

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Figure 16.2-1 Shaft Group Layout Note: When the distance between the center of applied load (column) and the supporting reactions (piles/shafts) is less than twice the depth (per AASHTO 5.6.3.1), a strut and tie model may be used. Caltrans practice allows use of simplified analysis and design of a pile cap in lieu of a strut and tie model. 

The typical section of the shaft is shown in Figure 16.2-2.

2ID

Figure 16.2-2 Shaft Details

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Table 16.2-1 Un-factored Live Load Forces at Column Base Case MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL(kip)

I -203.8 248.3 217.3 0 0

Design Truck II -39.6 1442.2 237.6 0 0

III -79.8 547.1 479.2 0 0

I -344.1 169.1 366.5 0 0

Permit Truck II 18.7 2537.6 438.7 0 0

III 32.2 351.4 760.4 0 0

Note: Case I: Maximum Transverse Moment (MT) and associated effects Case II: Maximum Longitudinal Moment (ML) and associated effects Case III: Maximum Axial Force (P) and associated effects

Table 16.2-2 Un-factored Dead Load and Seismic Forces Applied at the Column Base. Un-factored, without impact Loads DC DW EV PS Seismic-I+

MT (kip-ft) 61.0 11.4 0.0 0.0 18545.8

ML (kip-ft) 826.1 167.8 0.0 -141.7 0.0

P (kip) 1164.9 227.4 312.3 -20.9 992.0

VT (kip)

VL (kip)

859.0

0.0

Seismic-ISeismic-II Seismic-III+ Seismic-III-

18545.8 0.0 13115.8 13115.8

0.0 18545.8 13115.8 13115.8

-992.0 0.0 496.0 -496.0

859.0 0.0 607.0 607.0

0.0 859.0 607.0 607.0

Note: Seismic forces and moments caused by overturning effects are calculated using an assumed overstrength moment and shear force of : Mo=1.2Mp= 18,545.8 kip-ft, Vo=1.2Vp=859 kips. However, in practice, the magnitude of overstrength moment and shear depends on the applied axial force.

VT, VL shown in Tables 16.21-1 and 16.21-2 are forces applied at the top of the footing. Requirements: 1. Determine pile cap layout and depth. 2. Determine LRFD factored loads for service, strength, and extreme event limit states. 3. Check pile/shaft capacity. 4. Design pile cap for flexure. 5. Design pile cap for shear. 6. Design pile cap for joint shear.

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16.2.1

Determine Pile Cap Layout and Depth A pile/shaft layout of 16 shafts (4 rows of 4 shafts) is assumed as shown in Figure 16.2-1. Per CA Amendments 10.8.1.2, the minimum spacing for CIDH piles is 2.5D. The minimum edge distance for CIDH piles is 12 in. clear (AASHTO 10.8.1.2). The shaft spacing is 2.5 × 30 in. = 75 in = 6.25 ft, and minimum overall pile cap width is 75 in. × 3 + (12 in. + 30 in./2) × 2 = 279 in. Pile/shaft layout and pile cap size meet minimum size and spacing criteria. Note: If Geotechnical Services indicate group effects control, it may be warranted to increase pile/shaft spacing to raise pile/shaft tip elevations. To ensure the full moment capacity of a column can be developed, the minimum pile cap depth is equal to the minimum clearance from the bottom of pile cap to the bottom mat of cap reinforcement, plus the bar diameters used for the bottom of pile cap reinforcement, plus the required development length of the main column reinforcement. Dftg,min = clr. + 2(dbd) + ld Dftg,min = minimum pile cap depth. clr.

= minimum clearance from the bottom of pile cap to the bottom mat of pile cap reinforcement = 6 in. (BDD 6.71)

dbd

= diameters of the bars used for the bottom of pile cap reinforcement.

l d

= required development length of the main column reinforcement.

Assuming #11 bars for the pile cap bottom reinforcement: dbd = 1.63 in. (BDD 13-10) Calculate the development length according to the specifications given by AASHTO 5.11.2.2 and AASHTO 5.11.2.4, which are shown below in section 16.2.1.1.

16.2.1.1

Development of Deformed Bars in Compression AASHTO states: ldb  0.63 (1.693)(60) / (3.6)0.5 = 33.7 in.

(AASHTO 5.11.2.2.1-1)

ldb  0.3(1.693)(60) = 30.5 in.

(AASHTO 5.11.2.2.1-2)

AASTHO 5.11.2.2.2 states that the basic development length may be multiplied by applicable modification factors. AASHTO 5.11.2.2.2: Reinforcement is enclosed within a spiral of not less than 0.25 in. in diameter and not more than a 4 in. pitch, modification factor = 0.75. (This reduction does not apply because we have the main column hoops at 5 in.). Hooks shall not be considered effective in developing bars in compression. Therefore, development length required for compression is equal to 33.7 in.

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16.2.1.2

Development of Standard Hooks in Tension lhb = 38.0 (1.693) / (3.6)0.5 = 33.9 in.

(AASHTO Eq. 5.11.2.4.1-1)

Basic development length shall be multiplied by applicable modification factors: 

Concrete Cover: For #11 bar and smaller, side cover (normal to plane of hook) not less than 2.5 in., and for 90 degree hook, cover on bar extension beyond hook not less than 2 in., modification factor = 0.70.

Note: For determining modification factors the specifications refer to the portion of the bar from the critical section to the bend as the “hook,” and the portion of the bar from the bend to the end of the bar as the “extension beyond the hook.”



Ties or Stirrups: For #11 bar and smaller, hook enclosed vertically or horizontally within ties or stirrup-ties spaced along the full development ldh not greater than 3 db, where db is diameter of hooked bar, modification factor = 0.80.

None of the modification factors are applied, since #14 bars have been used for columns. Therefore: Development of standard hooks in tension = 33.9 in., say 34 in. (Also greater than 8 × 1.88 in. and 6 in.) Development length for tension (34 in.) controls over the development length for compression (33.7 in.). The required pile cap thickness is calculated as: Dftg,min = clr. + 2(dbd) + l ft.d = 6 in. + 2(1.63 in.) + 34 in. = 43.3 in. Note: If the development length of the pile/shaft reinforcement is a concern, the pile cap depth should be similarly checked for this reinforcement. Recommendation for balanced footing/pile cap depth is: 0.7 x Dc = 0.7 x 6.0 ft = 4.2 ft; Use Dftg = 50 in. = 4.17 ft

(SDC 7.6.1-2)

Check minimum pile cap depth for rigid footing assumption. (SDC 7.7.1.3) Lftg/Dftg ≤ 2.2 where: Lftg = cantilever length of pile cap from face of column = (23.25-6.0)/2 = 8.63 ft Dftg = 50 in. = 4.17 ft Lftg/Dftg = 2.06; rigid footing assumption

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16.2.2

Determine Factored Loads for Service, Strength, and Extreme Event Limit States The following three cases of live load forces should be considered in design: Case I: Maximum Transverse Moment (MT) and associated effects Case II: Maximum Longitudinal Moment (ML) and associated effects Case III: Maximum Axial Force (P) and associated effects Analysis results for other applicable loads acting on the pile cap are given in Table 16.2-2. Forces and moments resulting from seismic analysis in transverse, longitudinal, and 45 degree combination thereof are shown as Seismic I(+/-), Seismic II, and Seismic III(+/-). For Seismic I and Seismic III, the + represents the compression column while the – represents the tension column due to overturning forces. A combination of seismic forces should be taken at 15 degree intervals, however, for this example, one location at 45 degrees was used. EV was calculated in Table 16.2-2 as: (23.25 ft  23.25 ft – ((6 ft)2/4 ))  (48.0 ft – 38.75 ft – 4.17 ft)  (0.12 k/ft3) = 312.3 kips The LRFD load combinations used in foundation design and corresponding load factors (AASHTO Table 3.4.1-1) are summarized in the following table. The upper and lower limits of permanent load factors (p) are shown as U and L respectively. Table 16.2-3 LRFD Load Factors Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

DC 1.25 0.9 1.25 0.9 1.25 0.9 1.25 0.9 1 1

DW 1.5 0.65 1.5 0.65 1.5 0.65 1.5 0.65 1 1

PS 1 1 1 1 1 1 1 1 1 1

EV 1.35 0.9 1.35 0.9 1.35 0.9 1.35 0.9 1 1

HL-93 1.75 1.75 0 0 0 0 1.35 1.35 1 0

P-15 0 0 1.35 1.35 0 0 0 0 0 0

Seismic 0 0 0 0 0 0 0 0 0 1

The PS load factor of 1.0, as shown in Table 16.2-3, is recommended when the column’s cracked moment of inertia is used in analysis. However, for load cases other than Extreme Event-I a load factor of 0.5 may be used; see AASHTO Table 3.4.1-3 (AASHTO, 2012). In order to determine loads at the bottom of the pile cap, the cap size and depth will be needed. For this example a pile cap depth of 50 in. with a length and width of 23 ft - 3 in. is used.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The overall un-factored pile cap weight (DL) = 23.25 ft × 23.25 ft × 4.17 ft × 0.15 kip/ft3 = 338 kips The LRFD load factors are applied to axial force and moments in longitudinal and transverse directions to calculate factored loads for Strength, Service, and Extreme Event limit states, as summarized in Table 16.2-4 below. Loading shown in the table below is for live load case III only. Table 16.2-4 Case III Maximum Axial Force Factored Loads Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I, Seismic I+ Extreme Event I, Seismic IExtreme Event I, Seismic II Extreme Event I, Seismic III+

MT (kip-ft) -46 -77 137 106 93 62 -14 -45 -7 22128 22128 0 15645

ML (kip-ft) 2100 1668 1617 1185 1143 711 1881 1450 1399 0 0 22128 15645

P (kip) 3459 2599 3647 2787 2621 1761 3267 2407 2501 3014 1030 2022 2518

Extreme Event I, Seismic III-

15645

15645

1526

Shown below are sample calculations for the factored loads shown in Table 16.2-4: 

Example 1: Calculation of axial force at the bottom of pile cap for Strength-II limit state: P = 1.25(1164.9) + 1.25(338) + 1.5(227.4) + 1(-20.9) + 1.35(760.4) + 1.35(312.3) = 3647 kips



Example 2: Calculation of transverse moment at the bottom of pile cap for Seismic I+: MT = 1(18545.8 + 4.17 × 859.0) = 22,128 kip-ft



Example 3: Calculation of gross axial force at the bottom of pile cap for Service I limit state: P = 1(1164.9) + 1(338) + 1(227.4) + 1(-20.9) + 1(479.2) + 1 (312.3) = 2501 kips

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However, the net Service I loads should be reported to Geotechnical Services as shown in the footnotes of Table 16.2-9. The net axial force is calculated by subtracting the weight of the overburden soil from gross axial force. Soil Weight = 23.25 ft × 23.25ft × (48 ft - 38.75 ft) × 0.12k/ft 3 = 600 kips Pnet = 2501 – 600 = 1,901 kips Note:

represents loads used in Table 16.2-9, Foundation Design Data Sheet.

Similarly, the net permanent loads are to be calculated and reported to Geotechnical Services. P = 1(1164.9) + 1(338) + 1(227.4) + 1 (-20.9) + 1(312.3) = 2,022 kips Pnet = 2,022 – 600 = 1,422 kips

16.2.3

Check Pile/Shaft Capacity The general equation for moment in a rigid pile cap under seismic demand (SDC 7.7.1-2) is written as: Mocol + Vo Col × Dftg + M(i)pile - Rs × (Dftg – DRs) – (C(i)pile × C(i)) – (T(i)pile × C(i)) = 0 Since the pile cap is surrounded by competent soil, the simplified foundation model may be used; Eq.16.1.3.5-1 and Eq.16.1.3.5-2 shall apply. The moment of inertia of the pile/shaft group in both directions is calculated as: Table 16.2-5 – Transverse Pile/Shaft Layout Row 1 Row 2 Row 3 Row 4

Cy (ft) -9.38 -3.13 3.13 9.38

# piles/shafts 4 4 4 4

N × Cy2 352 ft2 39 ft2 39 ft2 352 ft2

Ix =  (N ×Cy2) =

782 ft2

Similarly, Iy = Ix = 782 ft2 Strength and Service loads shown below in Table 16.2-6 are for Case II, which is the controlling load case for shaft loading for this example. The last two columns list maximum force (Pmax) and minimum force (Pmin) in the piles under the various loading conditions. Negative forces show tension in the shafts.

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Table 16.2-6 – Case II Maximum Longitudinal Moment Factored Loads: kips, ft

MT (kip-ft)

ML (kip-ft)

P (kip)

Pc/N (kip)

My×cx/Iy (kip) 44.0

Pmax (kip)

Pmin (kip)

190

Mx×cy/Ix (kip) 0.3

Strength I-U

24

3666

3036

Strength I-L

-7

3235

2176

234.0

145.5

136

-0.1

38.8

174.9

97.1

Strength II-U

118

4568

Strength II-L

87

4136

3213

201

1.4

54.8

257.0

144.5

2353

147

1.0

49.6

197.7

Strength III-U

93

96.4

1143

2620

164

1.1

13.7

178.6

148.9

Strength III-L Strength V-U

62

711

1761

110

0.7

8.5

119.3

100.8

40

3089

2941

184

0.5

37.1

221.4

146.3

Strength V-L

9

2658

2081

130

0.1

31.9

162.1

98.1

Service I

33

2294

2259

141

0.4

27.5

169.1

113.3

Seismic-I(+)

22128

0

3014

188

265.5

0.0

453.9

-77.2

Seismic-I(-)

22128

0

1030

64

265.5

0.0

329.9

-201.2

Seismic II

0

22128

2022

126

0.0

265.5

391.9

-139.2

Seismic III+

15645

15645

2518

157

187.7

187.7

532.8

-218.1

Seismic-III-

15645

15645

1526

95

187.7

187.7

470.8

-280.1

Extreme Event I

16.2.3.1

Check Geotechnical Requirements The CA Amendments 10.5.5.2.4 and 10.5.5.3.3 specify strength reduction factors (ϕ) for strength and extreme event limit states as 0.7 and 1, respectively. Compare factored loads on piles/shafts to factored resistance for strength limit state: For compression 257 kips For tension 0 kips

< (0.7) × 600 = 420 kips < (0.7) × 300 = 210 kips

OK OK

Compare factored loads on piles/shafts to factored resistance for extreme event limit state: For compression 533 kips For tension 280 kips

< (1.0) × 600 = 600 kips < (1.0) × 300 = 300 kips

OK OK

Therefore, shafts meet LRFD geotechnical requirements. 16.2.3.2

Check Pile/Shaft Structural Requirements Strength Limit State: 

Pile/Shaft Tension Capacity =  Pn =  (Ast × fy) (AASHTO 5.7.6.1) 2 = 0.9 (9 bars × 1in. /bar) 60 ksi = 486 kips > 0 kips OK

Note: No tension in shafts for Strength limit state.

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

Pile/Shaft Compression Capacity  Pn = 0.75{0.85[0.85 × fc (Ag-Ast) + fyAst]} (AASHTO 5.7.4.4) = 0.75{0.85[0.85 × 3.6(706.9-9) + 60(9)]} = 1706 kips > 257 kips 1706 kips > 257 kips

OK

Extreme Event Limit State (Seismic): for shear  

16.2.3.3

(SDC 3.2.1)

Pile/Shaft Tension Capacity Pn = (9 bars × 1in.2/bar) 60 ksi = 540 kips > 280 kips

OK

Pile/Shaft Compression Capacity Pn = 1{0.85[0.85 × 3.6(706.9-9) + 60(9)]} = 2274 kips > 533k

OK

Piles/Shafts Shear Capacity Caltrans’ practice is not to check the shear capacity for pile/shaft groups in competent soil. However, formal guidance material for this policy is currently not available for LRFD design. The following is an example calculation showing how the pile shear check calculation may be performed. Ignoring the pile cap passive pressure on the front face as well as the friction on the two side faces of the pile cap, the pile/shaft shear may be conservatively approximated as the total shear divided by the number of piles/shafts. Assuming the maximum shear will occur at the top of the pile/shaft, the maximum factored shear force is as follows: Seismic III-: Vu = 1 (6072 + 6072) 0.5/16 = 53.7 kips The structural shear capacity of a reinforced concrete pile/shaft can be calculated as follows: Vn = Vc + Vs + Vp ≤ 0.25 fc bv dv + Vp

(AASHTO 5.8.3.3)

Vc = 0.0316(fc)0.5(bv)(dv) Vs = [Av (fy)(dv) (cot  + cot ) sin ] / s Vp = 0 (no pre-stressing in pile/shaft) bv = D = 30 in. dv = 0.9 de = 0.9 (21.8 in.) = 19.6 in. Dr = D – 2 (clr + hoop dbd + long dbd/2) = 30 – 2(3 + 0.69 + 1.25/2) = 21.4 in. de = D/2 + Dr/ in.

 (AASHTO C5.8.2.9-2)

 = 90° Av = 0.31in.2 × 2 = 0.62 in.2, s = 6 in. (#5 hoops at 6 in. spacing)

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Check for minimum transverse reinforcement: Av ≥ 0.0316 ( fc)0.5(bv)s/fy = 0.0316(3.6)0.5(30)(6)/60 = 0.18 in.2

OK

Nu = 280.1 kips (shear will be controlled by maximum tensile member) Mu = 0 (minimal moment demand assumed at top of pile/shaft) Aps = 0 (no pre-stressing steel in pile/shaft)

s = [(|Mu|/dv) + 0.5 Nu + |Vu – Vp| – Aps fpo]/[(Es As + Ep Aps)] (AASHTO 5.8.3.4.2-4)

s = [0.5 (280.1) + |53.7|]/[(29,000)(4.5)] = 0.00148





4.8 1  750 s

(AASHTO 5.8.3.4.2-1)

4.8  2.27 1  7500.00148

= 29 +3500s = 29 +3500(0.00148) = 34.2

(AASHTO 5.8.3.4.2-3) o

As = 9(1 in.2)/2 = 4.5 in.2 Vc = 0.0316(2.27)(3.6)0.5(30)(19.6) = 80.02 kips Vs = 0.62(60)(19.6)[cot(34.2°)+cot(90°)]/6 = 178.8 kips Vn = 80.02 + 178.8 = 258.8 kips < 0.25(3.6)(30)(19.6) = 529.2 kips

OK

Vr =  Vn =  (258.8) = 232.9 kips > 53.7 kips

OK

It is worth noting that a more refined analysis that accounts for the passive pressure on the front face as well as the friction on the two side faces may be warranted if the pile shear demand exceeds its structural shear capacity.

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16.2.3.3

Pile/Shaft Moment Capacity For the pile group in competent soil, due to small pile deflections, the bending moment demand has been assumed not to control, and therefore, does not require further analysis.

16.2.4

Design Pile Cap for Flexure The critical section of the pile cap for flexure is at the face of the column as shown in Figure 16.2-3. Since the column has a circular cross-section, it is transformed into an effective square section for pile cap analysis with equivalent column width of: (28.26)0.5 = 5.32 ft

P1

P2

P3

P4

Figure 16.2-3 Pile Cap Loading Mcap (transverse) = (P1transv × X1transv)+(P2transv × X2transv) –Wft × Xfttransv – Ws × Xstransv Due to symmetry: X1transv = X1long, X2trans = X2long, Xfttransv = Xstransv = Xftlong = Xslong

where: X1

= distance from face of column to row 1 of shafts = (6.25/2 + 6.25) – 5.32/2 = 6.715 ft

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 X2

= distance from face of column to row 2 of shafts = 6.25/2 – 5.32/2 = 0.465 ft

Pi

= (No. shafts per row x) × [P/N + │Mx × cy/Ix│]

Xfttransv = 0.5 distance from face of column to edge of pile cap = 1/2((23.25-5.32)/2) = 4.48 ft Wft

= weight of pile cap portion (from face of column to edge of cap) = ((23.25-5.32)/2)(23.25)(50/12)(0.15) = 130.3 kips

Ws

= weight of soil portion (from face of column to edge of cap) = ((23.25-5.32)/2)(23.25)(48-38.75-4.17)(0.12) = 127.1 kips



= load factor, as specified in CA Amendments Tables 3.4.1-1 and 3.4.1-2

The maximum compression forces and the corresponding moment, Mcap, as shown in Table 16.2-6, are for Case II loading. Loading Case I and III should also be checked but are not shown here. Table 16.2-6 Pile/Shaft Compression Forces and Corresponding Mcap (Strength and Service)

Factored Loads: Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I

P1 (kip) 760 544 809 592 660 443 737 521 566

Transverse P2 (kip) 759 544 805 590 657 441 736 520 565

Mcap (kip-ft) 3958 2870 4306 3214 3235 2142 3793 2700 2912

P1 (kip) 935 699 1022 787 710 474 884 648 675

Longitudinal P2 Mcap (kip) (kip-ft) 818 5159 596 3935 876 5774 654 4549 673 3581 452 2356 785 4799 563 3574 602 3658

Example Calculation: For Strength II, max P1 force and Mcap in longitudinal direction: P1long = (#shafts per row 1) × [P/N + │Mx×cy/Ix│] = (4) × [3213/16 + (4568×9.38)/782] ≅ 1022 kips (dead load components) = 1.25 (earth vertical pressure) = 1.35 Mcap = (1022 × 6.715) + (876 × 0.465) – 1.25(130.3)(4.48) – 1.35(127.1)(4.48) ≅ 5,774 kip-ft

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In this example, tension forces were developed only under seismic loads. Maximum Compression and Tension Forces for Extreme Event I loading are shown in Table 16.2-7. The sign indicates tension forces. The corresponding Mcap values are also shown. Table 16.2-7 Shaft Compression Forces and Corresponding Mcap (Seismic)

Factored Loads:

P1 (kip)

Seismic-I(+) Seismic-I(-) Seismic II Seismic III(+) Seismic-III(-)

1815 1319 505 1380 1132

Seismic-I(+) Seismic-I(-) Seismic II Seismic III(+) Seismic-III(-)

-309 -805 505 -122 -370

Transverse P2 Mcap (kip) (kip-ft) Maximum Compression 1107 11551 611 7990 505 2475 880 8524 632 6744 Maximum Tension 399 -3040 -97 -6602 505 2475 379 -1794 131 -3575

P1 (kip)

Longitudinal P2 (kip)

Mcap (kip-ft)

753 257 1567 1380 1132

753 257 859 880 632

4256 694 9771 8524 6744

753 257 -557 -122 -370

753 257 151 379 131

4256 694 -4821 -1794 -3575

Example Calculation: For Seismic-I(+), Mcap in transverse direction:



 (seismic) = 1.0 Mcap = (1815 × 6.715) + (1107 × 0.465) – 1(130.3)(4.48) – (127.1)(4.48) ≅ 11,551 kip-ft Maximum moments acting on the pile cap at the face of the column for SeismicI(+), Seismic-I(-), and Service I and are shown in Tables 16.2-6 and 16.2-7. Dividing by the 23.25 ft pile cap width, the maximum design moments and associated values per unit width are calculated as: Strength Limit State: Strength II

MT = 185 kip-ft/ft; ML = 248 kip-ft/ft

Extreme Event Limit State: Seismic-I (+) MT = 497 kip-ft/ft; ML = 183 kip-ft/ft Extreme Event Limit State: Seismic-I (-) MT = -284 kip-ft/ft; ML = 29 kip-ft/ft Service Limit State:

MT = 125 kip-ft/ft; ML = 157 kip-ft/ft

Assuming that #9 (dbd = 1.25 in.) bars with 3 in. cover and #11 (dbd = 1.63 in.) bars with 6 in. cover are used for top and bottom mat reinforcement, the minimum effective depths (de) of the pile cap for the top and bottom mats are calculated as 45.13 in. and 41.55 in., respectively. Critical sections for moment and shear calculations are:

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Bending moment at the face of the column (AASHTO 5.13.3.4).



One-way shear at distance dv from the face of the column (AASHTO 5.8.3.2).



Two-way (punching) shear on the perimeter of a surface located at distance dv,avg from the face of the column (AASHTO 5.13.3.6).

where dv is effective shear depth of the section, and dv,avg is the average of effective shear depths for both directions. 16.2.4.1

Pile Cap Bending Moment Check: Bottom Bars Due to Maximum Pile/Shaft Compression Assuming 3 in. side concrete cover and using 46-#11 bars for bottom mat, the spacing of the rebar is calculated as: s = (23.25(12)-2(3)-1.63)/(46-1) = 6 in. The calculated spacing is less than maximum spacing of 12 in. specified in AASHTO 5.10.8, and it is acceptable. The area of steel contributing to unit width of the pile cap is: (1.56)(12)/6=3.12 in.2, therefore the depth of the concrete stress block is calculated as: ac

(3.12)(60)  5.1in. (0.85)(3.6)(12)

The bending moment capacity for non-seismic loading is computed as follows: Mr = Mn = (0.9)(3.12)(60)(41.55 – 0.5 × 5.1)(1/12) = 547.6 kip-ft/ft > 248 (AASHTO 5.7.3.2) OK where  = 0.9 is based upon tensile controlled reinforcement (AASHTO 5.5.4.2) εt = 0.003(de – c)/c ≅ 0.003(de – c)/c = 0.003(41.55-5.1)/5.1 = 0.0214 > 0.005 (AASHTO Fig.C5.7.2.1-1) Therefore, the section is tensile controlled,  = 0.9

OK

For flexural capacity under seismic loading, the moment capacity for capacity protected members is determined from expected material properties. (SDC 3.4) where expected fc = 5 ksi, fy = 68 ksi a

(3.12)(68)  4.16in. (0.85)(5.0)(12)

Mne = (1)(3.12)(68)(41.55 – 0.5 × 4.16)(1/12) = 697.8 kip-ft/ft > 497 OK

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16.2.4.2

Pile Cap Bending Moment Check: Top Bars Due to Maximum Pile/Shaft Tension: Assuming 3 in. side concrete cover and using 46 #9 bars for top mat, the spacing of the rebar is calculated as: s = (23.25(12) – 2(3) – 1.25)/(46 – 1) = 6 in. The calculated spacing is less than maximum spacing of 12 in. specified in AASHTO 5.10.8, and it is acceptable. The area of steel contributing to unit width of the pile cap is: (1)(12)/6 = 2 in.2, and therefore, the depth of the concrete stress block and resisting moment are calculated as: a

(2.0)(60)  3.27in. (0.85)(3.6)(12)

Flexural capacity check is needed for seismic loading only as top reinforcement is not in tension due to strength and service loading. The moment capacity is determined from expected material properties. (SDC 3.4) a

(2.0)(68)  2.7in. (0.85)(5.0)(12)

Mne = (1)(2)(68)(45.13 – 0.5 × 2.7)(1/12) = 496.2 kip-ft/ft > 284 OK Therefore, selected number of bars is adequate for strength, however AASHTO Eq. 5.7.3.3.2 requires minimum amount of reinforcement to be provided for crack control. Crack control is for service load case and is neglected for top bars, as top bars are not in tension for service loading condition. To check the crack control requirement for the bottom reinforcement (Strength II), the cracking moment (Mcracking) is calculated as smaller of Mcr and 1.33 Mu as follows: Modulus of rupture

= fr = (0.24)(3.6)0.5 = 0.455 ksi

Gross section modulus = (12)(50)2/6 = 5000 in.3/ft M cr   1 3 f r S x  1.6(0.7)(0.455)(5,000)  2,548

kip - in. kip - ft  212 ft ft

1.33Mu = 1.33(248) = 329.8 kip-ft/ft Therefore, Mcr = 212 kip-ft/ft governs, and Mr = Mn = 547.6 kip-ft/ft > 212 OK AASHTO 5.7.3.4 requires maximum limits of rebar spacing for crack control.

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s

700 e  2d c  s f ss

Assuming e = 1 (class-I exposure) and dc = 6+(1.63+1.63/2) = 8.45 in.:

s  1 

8.45

0.750  8.45

 1.29

Cracked section is used to calculate tensile stress in steel reinforcement under service loads: f ss 

(n)( M )(d  x) I cr

(Chen and Duan 2000)

Ec = 1820(3.6)0.5 = 3453 ksi

(AASHTO C5.4.2.4-1)

n = 29000/3453 = 8.40 M (maximum service moment) = ML = 157 kip-ft d = de = 41.55 in. x can be solved by a quadratic equation, where for a rectangular section: B = [n × As + (n – 1) × As′]/b C = 2[n × d × As + (n – 1) × d′ × As′]/b

x  B 2  C  B  10.9 Icr = (1/3) × b × x3 + n × A s× (d – x)2 + (n – 1) × As′ × (x – d′)2 = 30,400 in.4 Using the above information fss is calculated as:

f ss 

8.40157  1241.55  10.9  16.0 ksi 30,400

The maximum spacing is checked per AASHTO Eq. 5.7.3.4-1:

7001  28.45  17.0 ksi 1.2916

 6.0 in.

OK

Therefore, 46#11 bars are acceptable for the bottom mat. Note: For square pile caps the reinforcement will be distributed uniformly across the entire width of the cap. (AASHTO 5.13.3.5)

16.2.5

Design Pile Cap for Shear According to AASHTO 5.13.3.6.1, both one-way and two-way shear shall be considered in pile cap design:

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The critical section for one-way action extends in a plane across the entire width and is located at a distance as specified in 5.8.3.2 (that is mostly at distance dv from the face of the column).



The critical section for two-way action is perpendicular to the plane of the pile cap and is located so that its perimeter bo is a minimum but not closer than 0.5dv to the perimeter of the concentrated load or reaction area.

where dv = d – 0.5a = 41.55 – 0.5(5.1) = 39 in. = 3.25 ft dv should be greater than both 0.9d = 37.1 in. and 0.72h =36 in. (AASTHO 5.8.2.9) use dv = 39 in. = 3.25 ft The dv value calculated above is based upon the strength loading case and is conservatively used for both the strength and seismic shear capacity calculations in this example. 16.2.5.1

Direct (One-Way) Shear The applied one-way shear acting at distance dv away from the face of the column will engage one shaft row. The maximum shear force Vu is 687 kips for strength and 1558 kips for the extreme event. Strength:

Vu = 1022 – 1.35(127.1) – 1.25(130.3) = 687.4 kips

Seismic I+:

Vu = 1815 – 1.0(127.1) – 1.0(130.3) = 1558 kips

Note: For circular columns the distance dv can be taken from the face of an equivalent square column.

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dv

P1

P2

P3

P4

Figure 16.2-4 One-Way Shear Therefore, use Vu= 1558 / 23.25 = 67 kip/ft The maximum shear resistance of the section (considering shear reinforcement contribution) is limited to 0.25 fcbvdv + Vp (AASHTO 5.8.3.3-2) Vn, max = 0.25(3.6)(12)(39) + 0 = 421.2 kip/ft > 67 kip/ft Shear resistance of concrete (Vc) is 0.0316(fc) bvdv 0.5

Shear resistance of steel (Vs) is Av × fy × dv × cots where  = 2, = 45.0°

OK (AASHTO 5.8.3.3-2) (AASHTO C5.8.3.3-1) (AASTHO 5.8.3.4.1)

Vc  0.03162 3.6 1239  56.1 kip/ft  67 kip/ft, shear reinforcement is required. Assuming #5 bars at 12 in. both ways: Vs = (0.31in.2/ft)(60ksi)(39 in.)(cot(°) / (12 in.) = 60.5 kip/ft Factored nominal resistance of the steel and concrete: Vn =  (Vc + Vs) Vn = 0.9(56.1 + 60.5) = 104.94 kip/ft > 67 kip/ft

OK

Check Minimum Transverse Reinforcement: (AASHTO 5.8.2.5)

Av ≥ 0.0316 f c bv×s/fy

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 0.31 in.2/ft ≥ 0.0316 3.6 (12 in.)(12 in.)/60 = 0.144 in.2/ft

OK

Check Maximum Spacing of Transverse Reinforcement: If u < 0.125 fc, then smax = 0.8 dv or 24 in.

(AASHTO 5.8.2.7-1)

u = Vu/[ (bv) (dv)] u = 67 kip/ft /[0.9 (12 in.) (39 in.)] = 0.16 ksi < (0.125 × 3.6 ksi) smax = 0.8dv = 0.8 (39 in.) = 31.2 in., use smax = 24 in. > 12 in. 16.2.5.1

OK

Punching (Two-Way) Shear The critical section is located at the distance of 0.5 × dv,avg. from face of the column as shown in Figure 16.2-5.

Figure 16.2-5 Pile Cap Critical Section The actual punching shear force acting on the critical surface is calculated by subtracting the force resulting from the piles/shafts acting on critical surface from the axial force (Pu) of the column. Determine the controlling punching shear force (Pu). Strength II: Pu = [1.25(1164.9) + 1.5(227.4) + 1(-20.9) + 1.35(760.4)] = 2803 Seismic I+: Pu = [1(1164.9) + 1 (227.4) + 1(-20.9) + 1(992)] = 2363 Therefore, use Pu = 2803 kips,  = 0.9 As the 4 shafts shown above are not fully within the effective zone, only a portion of these shafts should be removed from the punching shear force. The number of shafts within the critical surface will be approximated as 2 of 16 shafts.

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P2 way  2803(1 

2 )  2453 16

Using conservative assumption of dv,avg = 39 in. = 3.25 ft, results in b0 =  (6 + 3.25) = 29.1 ft = 348.7 in. For 2-way action with transverse reinforcement, the nominal shear resistance shall be taken as: Vn  0.192 f c 348.739

Vn  Vc  Vs  0.192 fc (bo )(dv ) Vn  4,954 kips Vn max = 0.9(4,954) = 4,459 kips

Vc  0.03162 3.6 348.739 1,631 kips Vs 

Av  f y  dv s

; for = 45°

(AASHTO 5.13.3.6.3-2) (AASHTO 5.13.3.6.3-4)

Av = 0.31 × (348.7/12) = 9 in.2 Vs = 9(60)(39)/(12) = 1755 kips Nominal resistance of the steel and concrete: Vn = Vc + Vs = 0.9(1631+1755) = 3,047 kips  4,459 kips Vn  Vn max 3,047 kips  4,459 kips Vn > P2-way 3,047 kips > 2,453 kips

OK

Note: For large capacity piles, localized pile punching shear failure and the development of flexural reinforcement beyond the exterior piles should be investigated. (SDC 7.7.1.6)

16.2.6

Design Pile Cap for Joint Shear Footing joint shear is evaluated in accordance with SDC 7.7.1.4. Principal Compression, pc ≤ 0.25 × fc = 0.25 × (3.6) = 0.9 ksi Principal Tension,

t 12 f c  12 3600  720 psi  0.72 ksi 1/2

2  fv   fv  pt       v 2jv   2  2   

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  f 2  f pc  v    v   v 2jv   2  2   

fv 

v jv 

Pcol ftg A jh

T jv ftg Beff  D ftg

ftg A jh  ( Dc  D ftg )  ( Bc  D ftg ) ftg Beff  2  Dc

T jv  Tc  T(i ) pile T(i)pile = summation of the hold down force in the tension piles/shafts Note: The column tensile force (Tc) can be determined from the xSECTION output file or CSiBridge. After determining Tc associated with Mp, multiply by 1.2 to determine Tc associated with Mo. Check Maximum Compressive Column: Pc = Pu = [1(1164.9) + 1(227.4) + 1(-20.9) + 1(992)] = 2363 kips For Maximum Compressive Column, Tc (at M = Mp) = 2578 kips (from CSiBridge) Tc ( at M = Mo) = 1.2 × 2578 kips = 3094 kips Use T(i)pile = 0; conservatively ignore tensile piles. Note: If tensile piles are used, only the tensile piles within the joint shear area should be considered.

fv 

Pc A jhftg



2,363 2,363   0.159 72  5072  50 14,884

Tjv = 3,094 - 0 = 3,094 vjv = 3,094/(101.8  50) = 0.608 1/2

2  0.159   0.159  pc     0.6082    2   2  

 0.69ksi  0.9ksi

1/2

2  0.159   0.159  pt     0.6082    2   2  

Chapter 16 – Deep Foundations

 0.53 ksi  0.72ksi

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Check Maximum Tensile Column: Pc = Pu = [1.0(1164.9) + 1.0(227.4) + 1.0(-20.9)-1.0(992)] = 379 kips For Maximum Tensile Column, Tc (at M = Mp) = 3000 kips (from CSiBridge) Tc ( at M =Mo) = 1.2 × 3000 kips = 3600 kips T(i)pile = 0 kips 𝑃𝑐 379 𝑓𝑣 = 𝑓𝑡𝑔 = = 0.025 14,884 𝐴𝑗 ℎ Tjv = 3600 – 0 = 3600 vjv =3600 / (101.8  50) = 0.707 1/ 2

2  0.025   0.025  2 pc      0.707   2   2  

 0.720 ksi  0.9 ksi

1/ 2

2  0.025   0.025  2 pt      0.707   2   2  

  0.694 ksi  0.72 ksi

If   3.5 f c then T heads are required in stirrups: (SDC 7.7.1.7) 3.5 f c  3.5 3600  210 psi  0.21ksi

Use pt = 0.694 ksi > 0.21 ksi

NG

Therefore, T headed stirrups are required in pile cap to account for joint shear effects. The pile cap region within Dc/2 from the face of the column should have T heads at the bottom of stirrups. See the pile cap reinforcement detail, Figure 16.2-6, for T headed stirrup locations.

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23-3

Figure 16.2-6 Reinforcement Layout Note: Fully lapped stirrups with 180 degree hooks at opposite ends may be used in lieu of T-headed stirrups (SDC 7.7.1.7). For other seismic design and detailing requirements, refer to Caltrans’ Seismic Design Criteria (Caltrans, 2013).

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16.2.7

Communication to Geotechnical Services The following attachments provide examples of communication processes between the Structural Designer and Geotechnical Services Refer to MTD 3-1 (2014b) and MTD 1-35 (2008). Attachment-I: Example of Preliminary Information to be sent from Structural Designer to Geotechnical Services: Table 16.2-8 Preliminary Foundation Design Data Sheet Support No.

Foundation Type(s) Considered

Estimate of Maximum Factored Compression Loads (kips)

Abut 1 Bent 2

Spread Footing 30 in. CIDH Piles

3500

Abut 3

Spread Footing

Attachment-II: General Foundation and Load Information to be sent from Structural Designer to GD for LRFD Strength and Extreme Event Limit States Load Data: Table 16.2-9 Foundation Design Data Sheet Foundation Design Data Sheet Finished Cut-off Pile Cap Size Permissible Number of Support Design Grade Elevation (ft) Settlement Pile Type Piles No. Method Elevation under Service per Support (ft) (ft) B L Load (in.)* Abut 1

WSD

Bent 2

LRFD

Abut 3

WSD

1.0 or 2.0 30 in. CIDH

48

39

23.25 23.25

1.0

16

1.0 or 2.0

*Note: Based on Caltrans’ current practice, the total permissible settlement is 1 in. for multi-span structures with continuous spans or multi-column bents, 1 in. for single span structures with diaphragm abutments, and 2 in. for single span structures with seat abutments. Different permissible settlement under service loads may be allowed if a structural analysis verifies that required level of serviceability is met.

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Table 16.2-10 Foundation Design Loads Foundation Design Loads Strength Limit State Extreme Event Limit State Service-I Limit State (Controlling Group, kips) (Controlling Group, kips) (kips) Permanent Compression Tension Compression Tension Support Total Load Loads No. Max. Max. Max. Max. Max. Per Per Per Per Per Per Per Per Per Per Per Support Support Support Support Support Support Pile Pile Pile Pile Pile Abut 1 Bent 2

1901

N/A

Abut 3

1422

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

3647

257

0

0

3014

533

0

280

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

Note: For Geotechnical Services:   

Support loads shown are per column. Service I loads are reported as net loads. Strength and Extreme loads are reported as gross loads.

Loads from Table 16.2-10 are shown in bold and highlighted in Sections 16.2.2 and 16.2.3. Load tables may be modified to submit multiple lines of critical load combinations for each limit state, if necessary.

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16.3

ANALYSIS/DESIGN OF SHAFT GROUPS IN SOFT/LIQUEFIABLE SOIL UNDER EXTREME EVENT I LIMIT STATE

16.3.1

Introduction The behavior of a pile/shaft group depends on the characteristics of its surrounding soil. This problem pertains to a class of Soil-Structure Interaction (SSI) problems. The lateral behavior of a pile/shaft group is governed by the soil near the ground surface, whereas its axial behavior is governed by the soil at a deeper depth. These two behaviors are practically independent of each other (Parkers and Reese, 1971). The axial behavior of a pile/shaft group embedded in a soft/liquefiable soil is similar to that embedded in a competent soil; whereas its lateral behavior depends on the type of soil in which it is embedded. For a pile/shaft group embedded in a competent soil, the soil near the ground surface provides substantial soil resistance to the lateral displacement of the pile cap, which results in small displacement and moment demands in the piles/shafts. For a pile/shaft group embedded in a soft/liquefiable soil, the majority of this soil resistance is lost, which results in large displacement and moment demands in the piles/shafts. The main objective of this section is to illustrate the analysis and design of shaft groups in soft/liquefiable soil under Extreme Event I Limit State (seismic loading).

16.3.2

Caltrans Design Practice Foundation components of Ordinary Standard Bridges shall be designed to remain essentially elastic when resisting the column’s overstrength moment, the associated overstrength shear, and the axial force at the base of the column (Caltrans, 2013). Bridge features that lead to complex response during seismic events, e.g., irregular geometry, unusual framing, and/or unusual geologic conditions, are considered non-standard. The current Caltrans design practice for Ordinary NonStandard Bridges is that the formation of plastic hinges in piles/shafts is not desirable, and piles/shafts should remain elastic during the design seismic hazard. For a soft or liquefiable soil (unusual geologic conditions), this might be uneconomical due to the excessive curvature demand imposed on the piles/shafts. Project-specific design criteria may permit plastic hinging at the top of the piles/shafts with a maximum displacement ductility demand of 2.5 (estimated at the bottom of the pile cap); the formation of a second set of plastic hinges at some distance below the bottom of the pile cap shall not be permitted (Caltrans, 2013). Shaft groups with permitted plastic hinging at the top of the shafts should be designed to meet the performance and strength criteria described in the following subsections.

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16.3.2.1

Shaft-Group Foundation Performance Criteria

16.3.2.1.1 Demand Ductility Criteria (SDC 4.1.2) The displacement ductility demand of a shaft group, D, is defined as D=D /Y, where D is the global displacement demand of the shaft group and Y is the yield displacement of the shaft group from its initial position to the formation of the first plastic hinge in the shafts (SDC 2.2.3). Shaft groups with permitted plastic hinging at the top of the shafts shall have a maximum displacement ductility demand of 2.5. For ordinary standard bridges, the global displacement demand, D, is typically estimated using a linear elastic analysis (equivalent static or dynamic) of the bridge assuming effective section properties for ductile members (SDC 2.2, 5.2, and 5.6). For a shaft-group foundation, however, the forces transferred to the foundation are limited by the overstrength moment capacity of the column, Mocol. The global displacement demand of a shaft group, D, shall therefore be defined as the lateral displacement (measured at the bottom of the pile cap) resulting from the application of the column’s overstrength moment and associated overstrength shear at the base of the column: see Figure 16.3-1.

(a)

(b)

Figure 16.3-1 Schematic Views of a Shaft-Group Foundation with Permitted Plastic Hinging for Two Loading Cases In the above figure: (a) corresponds to the formation of the first plastic hinge at the top of a shaft, which occurs at a lateral force VY = Vocol and bending moment MY = Mocol, where  is a constant (for shafts with permitted plastic hinging ≤ 1 and for elastic shafts > 1); and (b) corresponds to the application of the columns’ overstrength moment and associated overstrength shear at the base of the column.

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16.3.2.1.2 Capacity Ductility Criteria (SDC 4.1.3) The local displacement ductility capacity of an isolated shaft within a shaft group c is defined as c = c /yshaft, where yshaft is the idealized yield displacement of the shaft at the formation of the first plastic hinge (SDC 3.1.4.1), and c is the local displacement capacity of the shaft at its collapse limit state. The value of c is calculated for an equivalent member that approximates a guided-guided column (SDC 3.1.3, 3.1.4, and 3.1.4.1). See Figure 16.3-2. c1 = y1shaft + p1 ;

c2 = y2shaft + p2 ;

(SDC 3.1.3-1)

y1shaft = Y1L12/3 ;

y2shaft = Y2L22/3 ;

(SDC 3.1.3-7)

p1 = p1(L1 – Lp1 /2) ;

p2 = p2(L2 – Lp2 /2) ;

(SDC 3.1.3-8)

p1 = p1Lp1 ;

p2 = p2Lp2 ;

(SDC 3.1.3-9)

c1 = c1/y1shaft ;

c2 = c2/y2shaft ;

(SDC 3.1.4-2)

where L1, L2 are the distances from the two points of maximum moments to the point of contra-flexure (assumed equal, i.e., L1  L2 = L/2); L is the distance between the two points of maximum moments; Y1, Y2 are the idealized yield curvatures of the top and lower plastic hinges, respectively (assumed equal, i.e., Y1  Y2); p1, p2 are the idealized plastic curvatures of the top and lower plastic hinges, respectively (assumed equal, i.e., p1  p2); p1, p2 are the idealized local plastic displacement capacities due to the rotations of the top and lower plastic hinges, respectively; p1, p2 are the plastic rotation capacities of the top and lower plastic hinges, respectively; Lp1, Lp2 are the equivalent analytical lengths of the top and lower plastic hinges, respectively. The top plastic hinge of a shaft within a shaft group is analogous to that of a column, whereas the lower plastic hinge of a shaft is analogous to that of a noncased type I pile shaft; therefore, Lp1, Lp2 are given by: Lp1 = 0.08 L1 + 0.15 fye dbl ≥ 0.3 fye dbl Lp2 = D + 0.08 L2 ,

(SDC 7.6.2.1-1 )

(Analogous to SDC 7.6.2.3-1 for non-cased type I shaft)

where dbl and fye are the nominal diameter and the expected yield stress of the longitudinal reinforcement of the shaft, respectively, and D is the shaft diameter. Individual shafts within a shaft group shall have a minimum local displacement ductility capacity of 3 (i.e., c1 ≥ 3; c2 ≥ 3) to ensure dependable rotational capacity in the plastic hinge regions regardless of the displacement demand imparted to them (SDC 3.1.4.1).

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(a)

(b)

(c)

Figure 16.3-2 Plastic Hinges of a Shaft-Group Foundation In the above figure: (a) corresponds to a schematic view of a shaft-group foundation subjected to lateral loading near collapse; (b) corresponds to a segment of a shaft between the two points of maximum moment idealized as a guided-guided column; and (c) corresponds to an idealized curvature diagram of the shaft segment shown in (b). 16.3.2.1.3 Global Displacement Criteria (SDC 4.1.1) The global displacement demand of a shaft group, D, shall be less than its global displacement capacity, C: see Figure 16.3-3. The global displacement capacity of a shaft group, C, is defined as the lateral displacement (measured at the bottom of the pile cap) corresponding to the first plastic hinge reaching its plastic rotation capacity (SDC 3.1.3).

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Figure 16.3-3 An Idealized Force-Deflection Curve of a Shaft Group in Soft/Liquefiable Soil Note: The models shown in the inset diagrams approximately depict the behavior of the shaft group, assuming that the liquefied-soil stiffness is negligibly small. Figure 16.3-3 shows a schematic plot of the idealized force-deflection curve (solid lines) of a shaft group in soft/liquefiable soil. The broken lines correspond to a hypothetical case, where the column is retrofitted such that its plastic moment capacity is sufficiently large to force the formation of the lower set of plastic hinges. The lateral behavior of a shaft group can be divided into three distinct regions. Region I, from point 0 to point 1, where the lateral behavior is approximated by a guided-guided shaft, with L* = L; Region II, from point 1 to point 2, where the lateral behavior is approximated by a guided-free shaft (assuming that the top set of plastic hinges form simultaneously). The additional lateral displacement (measured at the

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 bottom of the pile cap) beyond point 1 is ( L*/2), where L* = L-Lp1/2, and  (rad) is the rotation of the top of the shafts beyond point 1 (= 0 at point 1); Region III, from point 2 to point 3, where the lateral behavior is approximated by a pinned-free shaft (assuming that the lower set of plastic hinges form simultaneously). The additional lateral displacement (measured at the bottom of the pile cap) beyond point 2 is ( L*), where L* = L – Lp1/2 – Lp2/2, and  (rad) is the additional rotation of the top of the shafts beyond point 2. The points 0, 1, 2, and 3 shown in Figure 16.3-3 identify the boundaries of these three regions. The global displacement capacity of a shaft group is given by C = Y1 + P1, where Y1 is the global yield displacement of the shaft group from its initial position to the formation of the first plastic hinge, and P1 is the global plastic displacement of the shaft group corresponding to the first plastic hinge reaching its plastic rotation capacity. The value of Y1 is obtained from the inelastic static analysis of the shaft group (discussed in Section 16.3.10 in this Chapter). The value of P1 can be estimated based on the model describing the behavior of the shaft group in region II, i.e., P1 = p1 (L – Lp1/2)/2. It is worth noting that point 3 shown in Figure 16.3-3 can fall either before or after point 2 (corresponding to the formation of the lower set of plastic hinges). The expression P1 = p1 (L – Lp1/2)/2 is derived for the case where point 3 falls before point 2. If point 3 falls after point 2, both region II and region III contribute to the expression for P1. The expression P1 = p1 (L – Lp1/2)/2, however, can still be conservatively used to verify that (C /D) > 1. The reason here is that point 2 has to fall after point C since the formation of any of the lower plastic hinges is not permitted before the column reaches its overstrength capacity; and subsequently, point 3 has to fall after point C, i.e., (C /D) > 1. The use of the expression P1 = p1 (L – Lp1/2)/2 simplifies the analysis since it eliminates the need for tracking the formation of the lower set of plastic hinges. 16.3.2.2

Shaft-Group Foundation Strength Criteria

16.3.2.2.1 Minimum Lateral Strength (SDC 3.5) Shaft groups with permitted plastic hinging shall have a minimum lateral flexural capacity (based on the expected material properties) to resist a lateral force VY of not less than 10% of the dead load on the shaft group PP, i.e., VY / PP ≥ 0.1, where VY corresponds to the formation of the first plastic hinge in the shaft group: see Figure 16.3-1(a). 16.3.2.2.2 P- Effects (SDC 4.2) The Caltrans Seismic Design Criteria has established a conservative limit for lateral displacements induced by axial loads for columns meeting the ductility demand limits. This limit for columns shall be adopted for shafts within a shaft group since the lateral soil springs of a liquefied layer are most likely yielded before the

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 formation of plastic hinges at the top of the shafts. For a ductile shaft approximated as a guided-guided column, this limit is defined by: Pdl  r /2 ≤ 0.20  Mpshaft, where Pdl is the axial force in an individual shaft attributed to dead load (with no overturning effects); Mpshaft is the idealized plastic moment capacity of the shaft associated with Pdl; r is the relative lateral offset (of the displacement demand) between the top and the lower points of maximum moment along the shaft: see Figure 16.3-1(b). 16.3.2.2.3 Force Demands (SDC 6.2.3.1) Foundation elements shall be designed to resist the column’s overstrength moment, Mocol, the associated overstrength shear, Vocol, and the axial load, PP. The moment and the shear demands for the pile cap and the shafts shall be determined from a static analysis of the shaft-group foundation under the actions of Mocol, Vocol, and PP at the base of the column. The capacity of concrete components to resist all seismic force demands, except for shear, shall be based on the expected material properties (SDC 3.2.1) with a strength reduction factor of = 1. The seismic shear capacity of all concrete components (ductile and capacity-protected) shall be conservatively based on the nominal concrete and steel strengths (SDC 3.2.1) with a strength reduction factor of = 0.9. For shaft-group foundations with permitted plastic hinging, verifying the moment demand of the ductile shafts is not required, since the moment demand shall not exceed the plastic moment capacity of the shafts. The moment demand of the capacity-protected pile cap shall not exceed its expected nominal moment capacity, Mne (SDC 3.4).

16.3.3

Practice Bridge Geometry Figure 16.3-4 shows schematically the elevation and the typical section of a fourspan cast-in-place pre-stressed concrete box-girder bridge having a total length of 525 ft (Yashinsky et al., 2007). The bridge crosses a dry-bed canyon at zero skew. The bridge is supported on two seat-type abutments on pile foundations, and three single-column piers on shaft-group foundations. Each shaft-group foundation consists of a 25  25  4.5 ft pile cap on a 5  5 array of 24-in. diameter, 60-ft long drilled shafts spaced at 5.25 ft in both directions. The length of the drilled shafts is limited to 30 times the shaft diameter to ensure constructability and quality control (Caltrans, 2014b).

16.3.4

Site Conditions and Foundation Recommendations Figure 16.3-5 shows the idealized soil profile at Pier 3, which consists of a 24-ft thick layer of medium dense sand underlain by a 60-ft thick layer of dense sand. Groundwater was encountered at a depth of 9 ft. The estimated future long-term

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 scour (degradation and contraction) elevation is 91.5 ft. The potential local pier scour depth corresponding to the 100-year base flood shall not be considered in the Extreme Event I Limit State (seismic): see CA Amendments (Caltrans, 2014a) Table 3.7.5-1. The 15 ft thick layer of saturated medium dense sand below the pile cap is determined to be liquefiable under the design seismic hazard. Figure 16.3-6 shows the Acceleration Response Spectrum curve for this project site. It should be noted that the project site is underlain by cohesionless soil layers and shallow groundwater; therefore, the use of 16-in. diameter CIDH piles is not recommended due to potential construction difficulties (e.g., caving of sand into the drilled hole, and the difficulties associated with the cleaning and inspection of the bottom of the drilled hole). Thus, only 24-in. (or larger) diameter CIDH piles should be used (Caltrans, 2014b). A series of analyses is performed on the 24-in. diameter shafts selected for this practice bridge in order to provide the necessary geotechnical information for foundation design (Malek and Islam, 2010)

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Figure 16.3-4 Geometry of the Practice Bridge

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Figure 16.3-5 Idealized Soil Profile for the Project Site The following notation is used in the idealized soil profile: N is the standard blow count, t (lb./ft3) is the unit weight of sand,  (degrees) is the friction angle, and k (lb./in.3) is a soil modulus parameter for sand.

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SPECTRAL ACCELERATION (g)

1.6

1.2

0.8

0.4

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

PERIOD (sec)

Figure 16.3-6 Acceleration Response Spectrum Curve for the Project Site The computer program LPILE (Reese et al., 2007) is used to develop the soil lateral reaction versus the lateral deflection (p-y) curves (also known as lateral soil springs). The p-y curves are generally non-linear and vary along the length of shafts. The soil lateral reaction, p, is the lateral load per unit length of a given diameter shaft and is obtained by integrating the unit lateral stress around the shaft. The force p acts in the opposite direction of the lateral deflection y: see Figure 16.3-7. Several methods are used by Geotechnical Services to develop the p-y curves for the liquefied soil layer. A preliminary analysis has indicated that the p-reduction factor method is the appropriate method for this project. In this method, the p-y curve for a liquefied soil layer is obtained by reducing the p-values of the corresponding non-liquefied soil layer using a reduction factor, which depends on the density of the liquefied soil layer. For the idealized soil profile shown in Figure 16.3-5, a p-reduction factor of 0.1 is selected since the liquefied soil layer has a density that falls within the lower range of the medium density. The recommended p-y curves for an individual 24-in. diameter shaft installed in the soil profile shown in Figure 16.3-5 are presented in Figure 16.3-8. It is worth mentioning that the lateral displacement of a pile cap mobilizes the soil in front of it. This results in a build-up of passive pressure, which depends on the amount of lateral displacement of the pile cap. This passive pressure is typically provided by Geotechnical Services in the form of a passive resistance-displacement relationship. The designer, however, should evaluate the amount of displacement required to produce the ultimate passive resistance of the soil. For this practice bridge, the 6.5-ft thick soil layer overlain on the liquefiable layer is relatively thin and likely to crack during liquefaction; therefore, the pile cap passive pressure is negligible. The t-z method (a well-known method of soil-structures interaction) is used by Geotechnical Services to develop the load transfer curves for the axial side resistance

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (t-z curves) and the end bearing resistance (Q-z curve). The t-z curves are a set of non-linear curves, which vary along the length of a shaft and represent the axial load transfer per unit length of a given diameter shaft as a function of the vertical shaft displacement at the corresponding depth. All soil within and above the liquefiable zone shall be considered not to contribute to axial compressive resistance (AASHTO 10.7.4). The Q-z curve is a non-linear curve representing the axial soil reaction in end bearing as a function of the axial displacement at the shaft tip: see Figure 16.3-9. The recommended t-z and Q-z curves for an individual 24-in. diameter shaft installed in the soil profile shown in Figure 16.3-5 are presented in Figures 16.3-10 and 16.3-11, respectively. It should be noted that both the tip resistance (end bearing) and the side resistance (skin friction in cohesionless soil or adhesion in cohesive soil) develop in response to the vertical displacement of the shaft. However, the peak value for the side resistance typically occurs at a smaller vertical displacement than the peak value for the tip resistance. Geotechnical Services usually discards the tip resistance for service and strength limit states, particularly in wet conditions, where soft compressible drill spoils and questionable concrete quality are both possible at the tip of the shaft. For extreme event I limit state, however, Geotechnical Services may include some fraction of the tip resistance.

Elevation

Plan

Figure 16.3-7 A Schematic Diagram Showing the Distribution of Soil Stress Reaction p(z) at a Depth z due to a Lateral Deflection y of an Individual Shaft

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6.0 5.0

p (kip/ft)

4.0

Depth 0 to 5 ft

3.0

Depth 5 to 10 ft Depth 10 to 15 ft

2.0 1.0 0.0 0.0

0.5

1.0

1.5

2.0

y (ft)

450 375

Depth 15 to 25 ft

p (kip/ft)

300

Depth 25 to 35 ft

225

Depth 35 to 45 ft 150

Depth 45 to 60 ft

75 0 0.0

0.5

1.0

1.5

2.0

y (ft)

Figure 16.3-8 Idealized Soil p-y Curves for an Individual 24-in. Diameter Shaft at Various Depths Measured from the Bottom of the Pile Cap

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Figure 16.3-9 Soil Stress Reaction The above figure is a schematic diagram showing the distribution of soil stress reaction t along a unit length of the shaft at a depth d due to a vertical displacement z(d). Also shown is the soil reaction Q at the shaft tip due to a vertical displacement z(L).

12 10

Depth 15 to 19 ft t (kip/ft)

8

Depth 19 to 25 ft

6

Depth 25 to 35 ft Depth 35 to 45 ft

4

Depth 45 to 60 ft 2 0 0.00

0.05

0.10

0.15

0.20

z (ft)

Figure 16.3-10 Idealized Soil t-z Curves The above figure illustrates the idealized soil t-z curves for an individual 24-in. diameter shaft at various depths measured from the bottom of the pile cap.

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120 100

Q (kip)

80 60 40 20 0 0.0

0.2

0.4

0.6

0.8

z (ft)

Figure 16.3-11 Idealized Soil Q-z Curve for an Individual 24-in. Diameter Shaft The lateral capacity of a closely spaced shaft group is less than the sum of the capacities of the individual shafts within the group. This behavior can be attributed to overlapping of shear zones within the group. The p-y curves are typically developed for an individual shaft. In order to apply these p-y curves to a shaft group, a scale factor is applied to the load component, p, of the p-y curve. This scale factor is referred to as the P-multipliers for the lateral capacity of closely spaced shafts. For this practice bridge, the P-multipliers are based on the CA Amendments 10.7.2.4 (Caltrans, 2014a). Interpolating the results for a center-to-center spacing of 2.625 shaft diameters, and accounting for the side-by-side effects, the P-multipliers corresponding to row 1 (leading), row 2, and rows 3 and higher, are 0.572, 0.392, and 0.284, respectively: see Figure 16.3-12(a). The vertical capacity of a shaft group in sand is less than the sum of the capacities of the individual shafts due to overlapping zones of shear deformation in the soil surrounding the shafts and loosening of soil during construction (AASHTO 10.8.3.6.3). Figure 16.3-12(b) shows a schematic diagram of the overlapping zones of influence for individual shafts under vertical loads. Figure 16.3-12(c) shows a schematic diagram of the stress conditions below the tips of individual shafts and below the shaft group modeled as a block foundation. The t-z and Q-z curves are typically developed for an individual shaft. In order to apply these t-z and Q-z curves to shaft groups, a scale factor is applied to the load component t of the t-z curves, and to the load component Q of the Q-z curve. This scale factor is referred to as the Group Efficiency Factor (GEF). Based on AASHTO 10.8.3.6.3, the GEF for a shaft group in compression corresponding to a center-to-center spacing of 2.625 shaft diameters in cohesionless soil is 0.679.

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Figure 16.3-12 Overlapping Zones of Closely Spaced Shaft G Foundation The above figure is of schematic diagrams showing: (a) overlapping zones of influence for a closely spaced shaft group under a lateral load; (b) overlapping zones of influence for closely spaced frictional shafts under an axial compression load; and (c) vertical stress condition below tips of individual shafts (broken lines) and below the entire shaft group modeled as a block foundation (solid line) under axial compression load.

16.3.5

Material Properties

16.3.5.1

Reinforcing Steel ASTM A706 (Grade 60) Properties (SDC 3.2.3) Modulus of elasticity

Es = 29,000 ksi

Specified minimum yield strength

fy

Expected yield strength

fye = 68 ksi

Specified minimum tensile strength

fu = 80 ksi

Expected tensile strength

fue = 95 ksi

Nominal yield strain

y = 0.0021

Expected yield strain

ye = 0.0023

Ultimate tensile strain:

for #10 and smaller bars

su = 0.12

for #11 and larger bars

su = 0.09

Reduced ultimate tensile strain: for #10 and smaller bars

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= 60 ksi

Rsu = 0.09

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Onset of strain hardening:

16.3.5.2

for #11 and larger bars

Rsu = 0.06

for #8 bars

sh = 0.015

for #10 and 11 bars

sh = 0.0115

Normal-Weight Portland Cement Concrete Properties (SDC 3.2.6) Specified 28-day compressive strength of concrete

f'c = 3.6 ksi

Expected concrete compressive strength ( = 1.3 f'c ≥ 5 ksi

f'ce = 5 ksi

Unit weight of reinforced concrete

w = 0.150 kip/ft3

Unit weight of plain concrete

wc = 0.14396 kip/ft3

Modulus of elasticity ( = 33,000  (wc)1.5  √fce )

Ec = 4030.5 ksi

Ultimate unconfined compression (spalling) strain

sp = 0.005 co = 0.002

Unconfined concrete strain at maximum compressive strength

For confined concrete, the confined compressive strain, cc, and the ultimate compression strain, cu, are defined by Mander’s constitutive stress-strain model: see Figure 16.3-13.

(a)

(b) Figure 16.3-13 Constitutive Models

The above figure shows constitutive models for: (a) park’s stress-strain model for rebar; and (b) mander’s confined and unconfined stress-strain models for concrete.

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16.3.6

Minimum Pile-Cap Depth The depth of a pile cap shall be sufficient to:

16.3.6.1

(i)

ensure the development of the longitudinal column reinforcement (AASHTO 5.11.2.2 and 5.11.2.4);

(ii)

satisfy the column–pile cap joint shear requirements (SDC 7.7.1.4);

(iii)

resist the one-way action shear (AASHTO 5.13.3.6.2);

(iv)

resist the two-way action shear (AASHTO 5.13.3.6.3); and

(v)

satisfy the rigidity requirement if the infinitely rigid pile-cap assumption is made when calculating the axial force demand in the piles/shafts (SDC 7.7.1.3).

Pile Cap-Column Proportions The SDC recommends that the depth of the pile cap, Dftg, be greater than or equal to 0.7 times the column diameter, Dc (SDC 7.6.1). Maintaining this ratio should produce a reasonably well-proportioned structure and satisfy the joint-shear requirements. For this practice bridge: Dftg ≥ 0.7Dc ≥ 0.7  5 ≥ 3.5 ft

16.3.6.2

( SDC 7.6.1-1)

Development Length Requirements The minimum pile cap depth, Dftg,min, is defined as: Dftg,min = Concrete Cover + 2  dbd + ld where concrete cover on reinforcement is 6 in. for a pile cap on concrete shafts (Caltrans, 1990), dbd is the deformed diameter of the bottom-mat reinforcement (dbd = 1.44 in. for #10 bars (Caltrans, 1984)), and ld is the development length of the longitudinal column reinforcement, which is calculated in the following subsections.

16.3.6.2.1 Development of Deformed Bars in Compression (AASHTO 5.11.2.2) The development length, ld, for deformed bars in compression is equal to the product of the basic development length specified in AASHTO 5.11.2.2.1 and the applicable modification factors specified in AASHTO 5.11.2.2.2; but not less than 8 in. The basic development length, ldb, for the #11 column’s longitudinal rebar (having a nominal diameter db = 1.41 in. (Caltrans, 1984) in compression is the larger of:

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ldb  0.63 db fy /√ f'c  0.63  1.41  60/√3.6  28.1in. [Governs] (AASHTO 5.11.2.2.1-1) ldb  0.30 db fy

 0.30  1.41  60

 25.4 in.

(AASHTO 5.11.2.2.1-2)

Confinement requirement: Column reinforcement is enclosed within #8 hoops, which are not less than 0.25 in. in diameter and spaced at a 4.0 in. pitch; therefore, a modification factor of 0.75 shall apply (AASHTO 5.11.2.2.2) as follows: ld = 0.75  28.1 = 21.1 in. (which is greater than the minimum value of 8 in.) 16.3.6.2.2 Development of Standard Hooks in Tension (AASHTO 5.11.2.4) The development length, ldh, in inches, for deformed rebar in tension terminating in a 90o standard hook shall not be less than: (i)

the product of the basic development length, lhb, and the applicable modification factors;

(ii)

8 bar diameters, but not less than 6 in.

The basic development length, lhb, for the #11 column’s longitudinal hooked rebar of yield strength, fy, not exceeding 60 ksi, is given by: lhb = 38 db / √fc = 38  1.41/√3.6 = 28.2 in.

(AASHTO 5.11.2.4.1-1)

The basic development length shall be multiplied by the following modification factors: 

Cover requirement: For #11 rebar and smaller, side cover (normal to plane of hook) not less than 2.5 in., and for 90˚ hook, cover on bar extension beyond hook not less than 2 in., a modification factor of 0.7 shall apply (AASHTO 5.11.2.4.2).



Confinement requirement: For #11 rebar and smaller, hooks enclosed vertically or horizontally within ties or stirrup-ties spaced along the full development ldh at a spacing not exceeding 3db, where db is the nominal diameter of the hooked rebar, a modification factor of 0.8 shall apply (AASHTO 5.11.2.4.2).

The development length, ldh, for #11 bars in tension terminating in a standard hook is: ldh = 0.7  0.8  28.2 = 15.8 in. (which is more than 6 in. and 8db = 8  1.41 = 11.28 in.) The development length for compression, ld = 21.1 in., governs over the development length for tension, ldh = 15.8 in. Therefore, the minimum pile cap thickness is given by: Dftg,min = clr + 2(dbd) + ld = 6 in. + 2  1.44 in. + 21.1 in. = 30 in.

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16.3.6.3

Rigid Pile-Cap Requirement If the rigid pile cap assumption is made to estimate the axial forces in the shafts, the SDC requires that Lftg/Dftg ≤ 2.2 (SDC 7.7.1.3), where Lftg is the cantilever length of the pile cap measured from the column face to the edge of the pile cap. For this practice bridge: Lftg

= (25 – 5)/2 = 10 ft

Dftg,min = Lftg / 2.2 = 10/ 2.2 = 4.5 ft

16.3.7

(SDC 7.7.1.3-1)

Shaft-Group Layout According to the CA Amendments 10.8.1.2, the minimum center-to-center spacing for shafts is 2.5 times the shaft diameter; and the minimum “clear” edgedistance for shafts is 12 in. For this practice bridge, the shafts are spaced at 5.25 ft, which is greater than the minimum shaft spacing of 2.5  2 ft = 5 ft. The selected 12in. “clear” edge distance for shafts in this practice bridge meets the minimum “clear” edge-distance requirement.

16.3.8

Seismic Forces on Shaft-Group Foundations In order to determine the force demand on the pile cap (a capacity protected member), a 20% overstrength magnifier is applied to the plastic moment capacity at the base of the column, i.e., Mocol = 1.2Mpcol, to account for the material strength variations between the column and the pile cap and the possibility that the actual moment capacity of the column may exceed its estimated value (SDC 4.3.1). Note that for Extreme Event I Limit State, the load factor for permanent loads, p, is 1.0 (CA Amendments 3.4.1).

16.3.8.1

Axial Force Effects The axial force effects due to the self-weight of superstructure and the future wearing surfaces are both obtained from a static analysis of the bridge and are given by: Self-weight of box-girder, integral pier cap, and concrete barrier 2

Self-weight of future wearing surfaces (35 lb./ft ) Self-weight of column

= 0.15 lb./ft  (5 ft) /4  22 ft 3

2

= 1153.587 kip =

131.250 kip

=

64.795 kip

Axial load at column base = 1153.60 kip + 131.25 kip + 64.795 kip = 1349.632 kip Self-weight of pile cap = 0.15 lb./ft3  25 ft  25 ft  4.5 ft

=

421.875 kip

Overburden soil weight = 0.115 lb./ft (25 ft  25 ft – (5 ft) /4)(2 ft) =

139.234 kip

3

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Axial load on pile cap, PP = 1349.632 kip + 421.875 kip + 139.234 kip = 1910.741 kip Average axial load/shaft

16.3.8.2

= 1910.741 kip / 25 shafts

=

76.430 kip

Plastic Moment Capacities for Ductile Concrete Members The plastic moment capacity for the column’s and the shaft’s cross-sections are estimated by moment-curvature analyses using the computer program x-SECTION based on the expected material properties (SDC 3.3.1). The results of the analyses are summarized in Table 16.3-1. Figure 16.3-14 shows the cross-section details of the column and the shaft, as well as the pile cap (a capacity-protected member).

Table 16.3–1 Summary of the Output Results of the x-SECTION Program Axial load (kip) Cross-Sectional Area, A (ft2) Idealized Cracked Moment of Inertia, Icr (ft3) Idealized Plastic Moment Capacity, Mp (kip-ft)

Column 1349.63 19.63 14.51 9812.0

Shaft 76.43 3.142 0.209 313.2

Figure 16.3-14 Cross-Section Details of the Column, Shaft and Pile Cap The figure above shows the cross-section details of: (a) the column; (b) the shaft; and (c) the pile cap for estimating the plastic moment capacity.

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16.3.8.3

Shear Force Effects Figure 16.3-15 shows the response of Pier 3 under lateral loading in both the transverse and the longitudinal directions. Neglecting the effect of the weight of the column on the plastic moment capacity for the column’s cross-section, the overstrength moment at the top and the bottom of the column is given by: Mocol@top = Mocol@bot = 1.2  Mpcol = 1.2  9812 = 11774.4 kip-ft The column’s overstrength shear in the transverse and the longitudinal directions can be estimated as follows: Vocol(Trans) = Mocol@bot /H = 11774.4/(22 ft + 3 ft 8 in.) = 458.74 kips Vocol(Long) = (Mocol@top + Mocol@bot)/Hc = (11774.4 + 11774.4)/(22 ft) = 1070.4 kip Since Vocol(Long) > Vocol(Trans),  longitudinal direction controls.

(a)

(b)

Figure 16.3-15 Schematic Views of the Deformed Shape of Pier 3

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The above figure shows schematic views of the deformed shape of Pier 3 in (a) the transverse direction and (b) the longitudinal direction, due to a lateral pushover force Vocol.

16.3.9

Structural Modeling of Shaft-Group Foundations A shaft-group foundation is a three-dimensional problem, which can be approximately modeled as a two-dimensional problem by assuming the pile cap is infinitely rigid in the direction perpendicular to the longitudinal analysis plane, i.e., the transverse direction. For this practice bridge, the shaft-group foundation is divided into five strips of equal width parallel to the longitudinal axis. Each shaftgroup foundation strip consists of 25  5  4.5 ft pile-cap strip on a 5  1 array of 24-in. diameter shafts spaced at 5.25 ft: see Figure 16.3-16. Note that the depth of the pile-cap strip equals the depth of the pile cap ( = 4.5 ft), but its width is only one-fifth the width of the pile cap, i.e., (1/5)  25 = 5 ft. The column is modeled as a 5-ft wide frame member endowed with one-fifth the actual cross-sectional properties of the column, i.e., Acol/5, Icrcol/5, Mpcol/5.

Figure 16.3-16 Shaft Group Foundation Schematic Plan Views The above figure shows: (a) a schematic plan view of a shaft-group foundation; (b) schematic views of the plan and the elevation of a two-dimensional shaft-group foundation strip model of the shaft-group foundation.

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16.3.10

Inelastic Static Analysis of Shaft-Group Foundations For this practice bridge, the inelastic static analysis of the shaft-group foundation strip in the longitudinal direction is performed using three design tools, namely: w-FRAME (Seyed, 1995), CSiBridge (CSI, 2015), and LPILE (Reese et al., 2007). The use of each design tool is based on a number of assumptions, which differ from one design tool to another. A detailed study, however, has shown that results obtained by the three design tools are comparable for design purposes. The selection of the appropriate design tool for a specific project is based upon the discretion of the Structural Designer. The aim of the inelastic static analysis of the shaft-group foundation strip is to determine the parameters required to verify the performance and strength criteria of the shaft group, namely, Y, VY, P1Y, Y1, p1, D, r, P1, V1, V4, and L. Table 16.3-2 summarizes the value of the parameters obtained from wFRAME, CSiBridge, and LPILE. Table 16.3-2 Parameters Obtained from w-FRAME, CSiBridge, and LPILE Y (ft) VY/5 (kip) P1Y (kip) Y1@P1Y (rad/in.) p1@P1Y (rad/in.) D (ft) r (ft) P1 (kip) V1 (kip) V4 (kip) L (in.)

w-FRAME 0.15333 178.807 221.4 0.000218 0.001562 0.25443 0.23833 245.9 53.26 36.18 192

CSiBridge 0.15763 179.827 229.18 0.000225 0.00149 0.27931 0.26233 255 54.79 36.37 192

LPILE 0.14333 180.53  P1 = 262.496 0.000219 0.001476 0.30083 0.28074 262.496 55.123 39.429 201.6

The following subsections verify the shaft-group performance and strength criteria based on the parameters obtained from the w-FRAME program: see Table 16.3-2. 16.3.10.1

Shaft-Group Foundation Performance Criteria

16.3.10.1.1 Demand Ductility Criteria (SDC 4.1.2) The global displacement ductility demand, D, of the shaft group is given by: D =D /Y = 0.25443 / 0.15333 = 1.66 ≤ 2.5

OK

16.3.10.1.2 Capacity Ductility Criteria (SDC 4.1.3) In order to calculate the local displacement ductility capacity of an isolated shaft within a shaft group, the location of the potential lower plastic hinge in this shaft

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 needs to be determined. The location of the potential lower plastic hinge in a shaft can be assumed at the same location as the peak value of bending moment below the bottom of the pile cap, i.e., at a depth of L = 192 in. The point of contra-flexure shown in Figure 16.3-2 can be assumed to be the midpoint between the two points of maximum moment along the shaft. The lengths L1 and L2 defining the distances from the points of maximum moment to the point of contra-flexure (cf. Figure 16.3-2) are given by: L1  L2  L / 2 = 192 in. / 2 = 96 in. The equivalent analytical length of the top and the lower plastic hinges, Lp1 and Lp2, respectively (Figure 16.3-2), are given by: Lp1 = 0.08 L1 + 0.15 fye dbl ≥ 0.3 fye dbl Lp2 = D + 0.08 L2

(SDC 7.6.2.1-1 for columns)

(Analogous to SDC Eq. 7.6.2.3-1 for non-cased type I shaft)

where fye (= 68 ksi) and dbl (= 1 in. for #8) are the expected yield stress and the nominal diameter of the shaft’s longitudinal reinforcement, and D (= 24 in.) is the shaft diameter, Lp1 = 0.08  96 + 0.15  68  1 = 17.88 in. ≥ 0.3  68  1 = 20.40 in. Lp1 = 20.4 in. Lp2 = 24 + 0.08  96  Lp2 = 31.68 in. Neglecting the difference between the axial force at the top and the potential lower plastic hinges, both the yield and the plastic curvatures at the top and the potential lower plastic hinges can be assumed equal, i.e., Y1  Y2 (= 0.000218 rad/in.) and p1  p2 (= 0.001562 rad/in.): see Table 16.3-2. The plastic rotation capacities of the top and the potential lower plastic hinges, p1 and p2, respectively, are given by: p1 = p1  Lp1 = 0.001562  20.40  0.0319 rad

(SDC 3.1.3-4)

p2 = p2  Lp2 = 0.001562  31.68  0.0495 rad

(SDC 3.1.3-4)

The idealized plastic displacement capacities p1 and p2 due to the rotation of the of the top and the potential lower plastic hinges, respectively, are given by: p1 = p1  (L1 – Lp1 /2)  0.0319  (96 – 20.4/2)  2.74 in.

(SDC 3.1.3-3)

p2 = p2  (L2 – Lp2 /2) = 0.0495  (96 – 31.68/2)  3.97 in.

(SDC 3.1.3-3)

The idealized yield displacements y1shaft and y2shaft associated with the formation of the top and the potential lower plastic hinges, respectively, are given by: y1shaft = Y1  L12/3 = 0.000218  962/3 = 0.67 in.

(SDC 3.1.3-2)

y2shaft = Y2  L22/3 = 0.000218  962/3 = 0.67 in.

(SDC 3.1.3-2)

The local displacement capacities of the leading shaft c1 and c2 are given by: c1 = y1shaft + p2 = 0.67 + 2.74 = 3.41 in.

(SDC 3.1.3-6)

c2 = y2

(SDC 3.1.3-6)

shaft

+ p2 = 0.67 + 3.97 = 4.64 in.

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The local displacement ductility capacities of the leading shaft c1 and c1 are given by: c1 = c1/y1shaft = 3.41/0.67 = 5.09 ≥ 3

OK

(SDC 3.1.4-2)

c2 = c2/y2shaft = 4.64/0.67 = 6.93 ≥ 3

OK

(SDC 3.1.4-2)

The local displacement ductility capacities of the other shafts can be calculated similarly. However, these shafts are subjected to smaller axial compressive forces (or to tension), therefore, they have larger values of c1 and c2 compared to those of the leading shaft. 16.3.10.1.3 Global Displacement Criteria (SDC 4.1.1) The yield displacement associated with the formation of the first plastic hinge in the shaft-group strip (at the top of the leading shaft) is Y1 = 0.15333 ft (= 1.84 in.). The global displacement demand of the shaft-group strip resulting from the application of the column’s overstrength moment and its associated overstrength shear is D = 0.25443 ft (= 3.05 in.). The global plastic displacement of the shaft group due to the plastic rotation capacity of the first plastic hinge, P1, is given by: P1 = (1/2)  p1  (L – Lp1/2) = (1/2)  0.0319  (192 – 20.4/2) = 2.9 in. The global displacement capacity of the shaft group (measured at the bottom of the pile cap), C, is given by: C = Y1 + P1 = 1.84 in. + 2.9 in. = 4.74 in.  (C/D) = 4.74/3.05 = 1.55 > 1

OK

Figure 16.3-17 Force-Displacement Curve for the Shaft-Group Foundation Strip

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16.3.10.2

Shaft-Group Foundation Strength Criteria

16.3.10.2.1 Minimum Lateral Strength (SDC 3.5) The lateral pushover force associated with the formation of the first plastic hinge in the shaft-group strip is given by: (VY/5) / (PP/5) = 178.807 / 382.148 = 0.468 ≥ 0.1

OK

16.3.10.2.2 P- Effects (SDC 4.2) For a shaft approximated as a guided-guided column, the P- effects can typically be ignored if the following limit is satisfied (SDC 4.2). (Pdl  r /2)/(0.20 × Mpshaft) ≤ 1.0 For this practice bridge, Pdl (= 76.430 kip) is the axial force in the shaft attributed to tributary dead load with no overturning effect; Mpshaft (= 313.2 kip-ft) is the idealized plastic moment capacity of the shaft corresponding to Pdl = 76.430 kip; and r is the relative lateral offset (of the displacement demand) between the top and the potential lower points of maximum moment: see Figure 16.3-1(b), given by: (Pdl  r/2)/(0.2  Mpshaft) = (76.430  0.2383/2)/(0.2313.2) = 0.145 < 1.0

OK

Therefore, the P- effects can be ignored. 16.3.10.2.3 Structural Shear Capacity of Ductile Shafts (SDC 3.6) For the shaft-group foundation strip of this practice bridge, only two shafts need to be checked for shear; namely, the (leading) row 1 shaft, which has the highest compression force, and the row 4 shaft, which has the lowest tension force. The shear capacity of a shaft, Vn, is based on the nominal material strengths (fc = 3.6 ksi, fy = 60 ksi) and a strength reduction factor  = 0.9, as follows: Vn = Vc + Vs,

(SDC 3.6.1-2)

where Vc and Vs are the nominal shear strengths provided by the concrete and the shear-reinforcement, respectively. The concrete shear capacity of a member designed for ductility is defined by: Vc = vc  Ae,

(SDC 3.6.2-1)

where vc is the permissible shear stress carried by concrete, and Ae is the effective shear area of the member’s cross-section, which can be expressed in terms of the gross cross-sectional area, Ag (=  (24)2/4= 452.39 in.2), of the shaft as follows: Ae = 0.8  Ag = 0.8 452.39 in.2 = 361.91 in.2

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(SDC 3.6.2-2)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 For shafts whose net axial load is in tension, vc = 0 (SDC 3.6.2). For shafts whose axial load is in compression, the value of vc at a section depends on the section location: 

Inside the plastic hinge zone: vc  Factor1 Factor 2 



f c  4 f c (psi)

(SDC 3.6.2-3)

Outside the plastic hinge zone: vc  3  Factor 2 

f c  4 f c (psi)

(SDC 3.6.2-4)

Factor 1 = 0.3 ≤ s fyh (ksi) / 0.15 + 3.67 – d ≤ 3

(SDC 3.6.2-5)

Factor 2 = 1 + P1 (lb.)/{2000  Ag (in.2)} < 1.5

(SDC 3.6.2-6)

where fyh (= 60 ksi) is the nominal yield stress of the shaft’s spiral reinforcement; P1 is the axial force in the row 1 shaft (=245900 lb.); and s is the ratio of the volume of spiral reinforcement to the core volume confined by the spiral, which can be expressed in terms of the area of the spiral reinforcement, Ab(#4) = 0.2 in.2, the crosssectional dimension of the confined concrete core measured between the centerline of the peripheral spiral, D = 24 – 2  (3 + 0.25) = 17.5 in., and the pitch of the spiral reinforcement, s = 6 in.

s = {4  Ab}/{D  s} = {4  0.2}/{17.5  6} = 0.0076

(SDC 3.8.1-1)

Factor 1 = 0.3 ≤ { 60/0.15 + 3.67–1.66} = 5.05 > 3  Factor 1 = 3 Factor 2 = 1 + 245,900/{2000  452.39} = 1.272 < 1. 

 Factor 2 = 1.272

Inside the plastic hinge zone: vc = 3  1.272  √3600 ≤ {4√3600 = 240 psi}

 vc = 228.96 psi

 Outside the plastic hinge zone: vc = 3  1.272  √3600 ≤ {4√3600 = 240 psi}

 vc = 228.96 psi

In calculating “Factor 1,” the global displacement ductility demand, D, is used in lieu of the local displacement ductility demand, d, since a significant portion of the global displacement of the shaft is attributed to its local deformation (SDC 3.6.2). The nominal shear strength provided by the concrete, Vc, is defined by: Vc = vc  Ae = 228.96  361.91 = 82,863 lb. = 82.863 kip  Vc = 0.90  82.863 = 74.577 kip The shear reinforcement capacity of a confined circular section is defined by: Vs = (Av fyh D / s),

(SDC 3.6.3-1)

where Av = n  /2  Ab is the area of shear reinforcement, and n (= 1) is the number of individual interlocking spiral core-sections, and given by:

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Av = 1  /2  0.2 = 0.314 in.2 ≥ {Av(min) = 0.025D s/fyh = 0.044 in.2} (SDC 3.6.3-2 & 3.6.5.2-1) Vs = 0.314  60  17.5/6 = 54.95 kip ≤ {Vs(max) = 8  √fc  Ae = 173.717 kip} (SDC 3.6.5.1-1) Vs = 0.90  54.950 = 49.455 kip For the row 1 shaft (highest compression shaft), the shear demand is V1 = 53.62 kip Vn = Vc + Vs = 74.577 kip + 49.455 kip = 124.032 kip Vn /V1 = 124.032/53.62 = 2.31 ≥ 1

OK

For the row 4 shaft (lowest tension shaft), the shear demand is V4=36.18 kip Vn = Vc + Vs = 0 kip + 49.455 kip = 49.455 kip Vn /V4 = 49.455/36.18 = 1.37 ≥ 1

OK

Verifications of the performance and strength criteria of the shaft group based upon the parameters obtained from CSiBridge and LPILE are performed similar to the verifications performed using the parameters obtained from the w-FRAME program. A summary of the verifications performed using the three design tools is presented in Table 16.3-3. Table 16.3-3 Summary of the Performance and Strength Criteria Verifications Performed Using the w-FRAME, the CSiBridge, and the LPILE Programs

Performance Criteria

I. II.

w-FRAME

CSiBridge

LPILE

Allowable Limits

(D = D /Y )

1.66

1.77

2.10

≤ 2.5

(c1 = c1/y1shaft )

5.09

4.78

4.69

(c2 = c2/y2shaft )

6.93

6.48

6.42

1.55

1.39

1.27

≥ 1.0

(VY /5)/(PP /5)

0.468

0.471

0.472

≥ 0.1

(Pdl r /2)/(0.2 Mpshaft)

0.145

0.160

0.171

≤ 1.0

(Vn /V1)

2.31

2.27

2.27

(Vn /V4)

1.37

1.36

1.25

Strength Criteria

III. (C /D) I. II. III.

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≥ 3.0

≥ 1.0

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16.4

ANALYSIS AND DESIGN OF LARGE DIAMETER COLUMN-SHAFTS

16.4.1

Introduction The design process of large diameter column-shafts (shafts) includes several steps. Some of the steps highly depend on the characteristics of the surrounding soil. The lateral response of the shaft is governed by the soil close to the ground surface, and soil-foundation-structure interaction (SFSI) analysis is required. However, the axial resistance of the shaft is controlled by the quality and the depth of the deeper soil layers. Shaft embedded in competent soil is capable of resisting ground shaking forces while experiencing small deformations, but for shaft embedded in soft or liquefiable soil, the majority of the soil resistance is lost, which results in large deformations and moment demands in the shaft. The analysis and design of the large diameter shafts (Types I and II) based on the 6th Edition of AASHTO LRFD Bridge Design Specifications (BDS) together with California Amendments, and Caltrans’ Seismic Design Criteria (SDC) Version 1.7 will be illustrated through an example.

16.4.2

Design Practice Column-shafts with two different types of geometry (Types I and II) are commonly used in Caltrans projects. Bridge foundations are designed to remain essentially elastic when resisting the column’s overstrength moment, the associated overstrength shear, and the axial force at the base of the column (Caltrans, 2010). However, prismatic Type I column-shafts are designed to form the plastic hinge below the ground in the shaft and, therefore, are designed as ductile components. The concrete cover and area of transverse and longitudinal reinforcement in Type I column-shafts may change from column to the shaft. However, the cross section of the confined core is the same for both the column and the shaft. Type II column-shafts are designed with enlarged shaft so that the plastic hinge will form at or above the shaft/column interface, thereby containing the majority of inelastic action to the ductile column element. The diameter of the shaft is at least 24 in. larger than the diameter of the column, and therefore, two separate cages are used for the column and the shaft. The column cage is embedded in the shaft to allow the transfer of forces from column to the shaft. Refer to SDC for embedment length requirements. Use of a construction joint at the base of the column allows concrete of the lapped portion to be cast under dry conditions, therefore eliminating need of inspection in that area. However, a construction joint is not used for shafts smaller than 5 ft. Type II shafts are designed to remain elastic, D  1 and the global displacement ductility demand, D for the shaft must be less than or equal to the D for the column

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 supported by the shaft. See SDC Section 7.7.3.2 for design requirements for Type II column-shafts. Type I column-shafts are appropriate for short columns, while a Type II columnshaft is commonly used in conjunction with taller columns. The use of Type II shafts will increase the foundation costs, compared to Type I shafts. However, there is an advantage of improved post earthquake inspection and repair. For short columns, designers have used isolation casing in Type-II column-shafts to increase ductility capacity and to reduce shear demand in the shaft. CISS piles are pipe piles driven to the desired tip, followed by removal of the soil within the steel shell. The interior surface is cleaned to remove any soil or mud. Then, the rebar cage is placed, and concrete is cast inside of the steel shell. CISS piles provide excellent structural resistance against lateral loads and are a good option if any of the following conditions exist: poor soil, soft bay mud or loose sands, liquefaction, scour, and lateral spreading. To improve composite action, a shear transfer mechanism such as studs or shear rings can be welded to the steel shell interior face of large diameter piles. For a more in-depth discussion about the shear rings or welded studs, please refer to Caltrans’ Research Project Report (Gebman and Restrepo, 2006). Following is a summary of current Caltrans’ policy on the use of shear rings in CISS piles and drilled shafts with permanent casing: 

Shear rings are not required for pile group foundations.



Shear rings are not required on corrugated metal pipe casings.



Shear rings are not required for piles ≤ 3 ft in diameter.



For Type I and II shafts less than 5 ft but greater than 3 ft in diameter use one ring as shown in Figure 16.4-1a.



For Type I and II shafts 5 ft or larger in diameter, use two rings as shown in Figure 16.4-1b.

The spacing and size of the shear rings is shown in Table 16.4-1.

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Figure 16.4-1 Single and Double Shear Ring Details

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Table 16.4-1 Shear Ring Table Pile Diameter < 6 ft 6 ft to 10 ft > 10 ft

16.4.3

Shear Ring Options Reinforcing Bar Size Bar Stock Size #4 1/2 in x 1/2 in #6 5/8 in. x 5/8 in. #8 1 in. x 1 in.

Fillet Weld* 3/8 in. 1/2 in. 5/8 in.

Lateral Stability Check of Type I and II Shafts The column-shaft is considered stable when substantial decrease in pile shaft length does not result in appreciable increase in deflection. Lateral stability analysis may be performed by gradually lowering the tip elevation and calculating corresponding deflection as shown in Figure16.4-2. The length (of the pile) at which lowering the tip elevation does not change deflection appreciably (say less than 5%) is called critical length of the shaft. The following guidance on lateral tip elevation calculations is applicable to both CIDH and CISS piles: 

For column-shafts (types I and II) in multi-column bents, lateral stability analysis must be performed to determine the critical length of the shaft. A factor of safety of 1.0 must be applied to determine the lateral tip elevation. If applicable, the combined effects of liquefaction and scour are to be considered in analysis.



For column-shafts (types I and II) without rock sockets in single-column bents, lateral stability analysis is performed to determine the critical length of the shaft. A factor of safety of 1.2 must be applied to determine the lateral tip elevation. If applicable, the combined effects of liquefaction and scour are to be considered in analysis.



For column-shafts (types I and II) with rock sockets in single-column bents, lateral stability analysis is performed to determine the critical length of the shaft. A factor of safety of 1.2 must be applied to the rock socket portion of the shaft in determining the lateral tip elevation. If applicable, the combined effects of liquefaction and scour are to be considered in analysis.

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Critical Column Shaft Length (LP)

Push-over analysis

Dtop (in.)

LC

LP

Critical Pile Length = ≈ 80 ft.

LP (ft)

Figure 16.4-2 Lateral Stability Analysis of Column-Shafts

16.4.4

Reinforcement Spacing Requirements of Column-Shafts

16.4.4.1

Minimum Reinforcement Spacing in Column-Shafts Construction of column-shafts does not require compaction or vibration of concrete except for the top 15 ft of a dry pour because the concrete slump is at least 7 in. Dry pour is defined when the drilled hole is dry or dewatered without the use of temporary casing to control water. Therefore, flow of concrete significantly affects integrity of the shaft and its resistance to applied loads. A clear window of 5 in. x 5 in. has been required for shaft reinforcement to minimize possibility of any anomalies. When the drilled hole cannot be dewatered and the concrete is poured in the wet using the slurry-displacement method of construction, inspection of the shaft is required. Clear spacing at inspection pipe locations may be reduced from 5 in. to 3 in. between the inspection pipe and adjacent longitudinal rebar or 8.5 in. between longitudinal rebars adjacent to inspection pipes as shown in Memo to Designers 3-1. For more details, see Table 16.4-10. Furthermore, if the spacing between the shaft and column reinforcing cages does not meet the minimum spacing requirement, the engineer must either increase the size of the shaft or require a mandatory construction joint (for wet pour).

16.4.4.2

Maximum Reinforcement Spacing in Column-Shafts The main concern in detailing a shaft is how to accommodate the minimum 8.5 in. longitudinal reinforcing steel clear spacing at inspection pipe locations as required

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 by MTD 3-1. The 8 in. maximum center-to-center spacing of longitudinal reinforcing steel has been revised by California Amendments (10.8.1.3) to 12 in. for shafts with diameters equal to and larger than 5 ft, and 10 in. for shafts with diameters smaller than 5 ft. The rationale for the required spacings is based on a recent study at University of California-San Diego. Preliminary findings show that use of the proposed spacing given above will not affect the confinement effectiveness of the transverse steel. Furthermore: 

Number 14 bars are generally the largest bars used for longitudinal steel in column-shafts of 5 ft diameter or larger. The 12 in. spacing was derived by assuming #14s bundled with 8.5 in. clear space at the inspection pipe locations as shown in Figure 16.4-3.

Partial Plan View

Figure 16.4-3 Typical Rebar Spacing in Shafts (Equal and Larger than 5 ft ϕ) 

Number 11 bars are generally the largest bars used for longitudinal steel in column-shafts less than 5 ft diameter. The 10 in. spacing was derived by assuming single #11s with 8.5 in. clear space at the inspection pipe as shown in Figure 16.4-4.

Partial Plan View

Figure 16.4-4 Typical Rebar Spacing in Shafts (Smaller than 5 ft ϕ)

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16.4.4.3

Cage Offset Requirements for Type-II Shaft Construction Specifications require a permanent steel casing to be placed in the area where the column and shaft cages of a Type-II column-shaft overlap and when a construction joint is required. This means the pour in that segment is in the dry condition and can be vibrated. The spacing of the column reinforcement in this area can follow the standard requirements for column steel reinforcement. In a Type-II column-shaft, the allowable offset between centerlines of the column and shaft reinforcement cages is limited by the required horizontal clearance between the two cages. In this case, the clear distance between the two cages must be at least 3.5 in. if a construction joint is used and 5 in. without a construction joint. Furthermore, the offset between centerlines of the shaft cage and the drilled hole (horizontal tolerance at cut-off point) must be limited to provide minimum concrete cover of 3 in. to the shaft outermost reinforcement at all locations. Figure 16.4-5 shows cage offset requirements for Type II Shafts:

Figure 16.4-5 Cage Offset Requirements for Type II Shaft

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16.4.4.4

Rock Socket Design Criteria When CIDH piles tipped in rock are analyzed for lateral loads, the p-y method reports shear demand forces that exceed the seismic overstrength shear, Vo calculated demand in the column. The abrupt change to high-stiffness p-y springs may amplify shear force to more than 5Vo within the rock socket. In current Caltrans practice, the designer must enlarge or reinforce the pile to resist the amplified shear force. However, there is ongoing debate over whether the design force is "real" and whether the discretization of distributed soil reaction to nodal springs is appropriate at the rock interface. Designing for the amplified shear force would increase rebar congestion for an uncertain benefit. The Caltrans policy imposes an upper limit on the design shear force, recognizing the general problems of the discretization of distributed soil reaction to nodal springs application in rock. The design shear force demand in CIDH shafts and rock sockets need not be taken more than two and half times the seismic overstrength shear force of the column: Vu ≤ 2.5Vo The in-ground amplification of shear forces in rock sockets deserves special consideration. Shear demand values from analysis can be misleading, as the discrete spring is not capable of handling a sudden transition to hard rock.

16.4.4.5

Minimum Lateral Reinforcement in Column-Shafts 

If Vu < ϕVc the minimum lateral reinforcement of the shaft must be # 5 hoops at 12 in. center-to-center spacing.



If Vu  ϕVc the minimum lateral reinforcement of the shaft must be the larger of Av ≥ 0.0316 (f'c)0.5(bv)s/fy (AASHTO 5.8.2.5), and # 5 hoops at 12 in. center-to-center spacing.

Radial bundling of longitudinal reinforcement is not allowed due to fabrication challenges.

16.4.5

Design Process The design process of large diameter column-shafts is presented by the flow chart on the following page.

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CSiBridge

CSiBridge

Figure 16.4-6 Typical Design Process for Type II Column-Shaft

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16.4.6

Design Example Design process for a Type II shaft foundation is illustrated through the following example. A fixed column-to-shaft connection has been assumed in this example. However, a pin column-to-shaft connection will reduce forces/moments transferred to the foundations. Given: The following example is a two span post-tensioned concrete box girder bridge with a 2-column bent. The superstructure is a post-tensioned concrete box girder and is supported by 2 Type-II shafts as shown in Figure16.4-7. The soil profile of the bent consists of loose to very loose sand with gravel for the top 34 ft, underlain by very dense sand (friable sandstone) to about 55 ft below ground. Bedrock consisting of very hard sandstone/siltstone was encountered below that depth and changed to fresh and hard, with Rock Quality Designation (RQD) of 60% up to 100% at lower depths. Ground water has been encountered near the surface of the streambed. There is a moderate to high potential for liquefaction. Scour potential is high with local scour at the supports estimated from top of streambed or pile cutoff with 15 ft of degradation and contraction.

Figure 16.4-7 Transverse Elevation of Example Bridge

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3.00

Spectral Acceleration (g)

2.50

2.00

1.50

1.00

0.50

0.00 0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

Period (seconds)

Figure 16.4-8 Recommended Seismic Design Criteria Curve Equivalent to Vs30 = 270 m/s with PBA 0.6g As shown in Figure 16.4-10, section A-A, the columns are 5.5 ft in diameter. The shafts are 97 inches in diameter with 60 #14 bundled reinforcing bars and #8 confining hoops at 7.5 in. and double #8 hoops at 7.0 in. spacing along the shaft and the rock socket, respectively. The concrete cover to shaft reinforcement outside and inside the rock socket region is 9 in. and 5 in., respectively (Figure 16.4-10, sections B-B, C-C, and D-D).

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Figure 16.4-9 Column-Shaft Reinforcement Details (not to scale)

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Figure 16.4-10 Typical Shaft and Column Section Details Note: CIDH Concrete Piling (Rock Socket Section D-D) shown in this example is between the standard diameters of 7ft and 7.5 ft as stated in MTD 3-1 (Caltrans 2013). Contractor can use a special auger to handle a case like this. It is highly recommended to have the rock socket as a standard diameter per MTD 3-1.

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Material properties are as follows: 

Concrete:



Reinforcement: fy = 60 ksi; fye= 68 ksi



Unfactored live loads at the base of the Column are listed in Table 16.4-2



Plastic moment and shear at the base of the column are calculated as Mp=12,309 k-ft, and Vp = 821 kips



Un-factored dead load and seismic forces at the base of the column are listed in Table 16.4-3



The nominal geotechnical resistance of the shaft in compression is 5,310 kips



The nominal geotechnical resistance of the shaft in tension is 2,655 kips

f c = 4 ksi; f ce = 5 ksi

Note: Concrete Strength of f c = 4 ksi was chosen in this example for a stronger concrete shear capacity and it is not far off from the default value of f c = 3.6 ksi. Table 16.4-2 Un-factored Live Load Forces at Column Base Case MT (kip-ft) ML (kip-ft) P (kip) VT (kip) VL (kip)

I -189.2 59.25 295.13 -13.76 1.39

Design Truck II III -70.77 -109.7 752.94 123.57 397.07 615.51 4.74 -5.2 17.18 2.9

I -296.02 98.91 458.23 -21.53 2.31

Permit Truck II -28.21 1,239.31 705.35 -2.57 128.26

III -36.89 199.08 922.37 2.4 4.65

Note: Case I: Maximum Transverse Moment (MT) and associated effects Case II: Maximum Longitudinal Moment (ML) and associated effects Case III: Maximum Axial Force (P) and associated effects

Table 16.4-3 Un-factored Dead Load and Seismic Forces Applied at the Column Base Un-factored, without impact Loads

MT (k-ft)

ML (k-ft)

P (kips)

VT (kips)

VL (kips)

DC DW PS Seismic-I Transverse Seismic-II Longitudinal

86.88 10.32 -193.6 14,771

305.65 35.5 -195.6

1,793.75 163.65 37.1 1,926

6.31 0.75 4.45 985

-19.8 -2.30 12.30

14,771

1,926

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Note: The maximum axial force of the column (DC + DW) without considering the overturning effect of seismic forces is 1,926 kips (SAP 2000 transverse analysis), and the plastic moment of the column under such load is 12,309 kip-ft. The corresponding overstrength moment and associated shear force are calculated as Mocol = 1.2Mpcol = 14,771 k-ft, and Vocol = 985 kips assuming that Lc = 30 ft. However, in practice, the magnitudes of overstrength moment and associated shear depend on the applied axial force. Requirements:  Calculate LRFD factored loads for service, strength, and extreme event-I (seismic) limit states applicable to the shaft design.

16.4.6.1



Check geotechnical capacity, assuming nominal geotechnical resistance of 5,310 kips in compression and 2,655 kips in tension.



Check development length for column reinforcement extended into (Type-II) shaft (SDC 8.2.4).



Check shaft structural resistance for non-seismic loads.



Design shaft flexural reinforcement (non-seismic effects).



Design shaft for shear reinforcement (non-seismic effects).



Check shaft for seismic effects.

LRFD Factored Loads for Service, Strength, and Extreme Event Limit States Considering effects of live load movements in the longitudinal and transverse directions, the following three live load cases are commonly considered in design: Case I: Maximum Transverse Moment (MT) and associated effects Case II: Maximum Longitudinal Moment (ML) and associated effects Case III: Maximum Axial Force (P) and associated effects Analysis results for other applicable un-factored loads acting on the shaft are given in Table 16.4-3, together with forces and moments resulting from seismic analysis in transverse and longitudinal directions. The LRFD load combinations used in foundation design and corresponding load factors (AASHTO Table 3.4.1-1) are summarized in Table 16.4-4. The upper and lower limits of permanent load factors (p) are shown as U and L, respectively.

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Table 16.4-4 LRFD Load Factors Strength I-U Strength I-L Strength II-U

DC 1.25 0.9 1.25

DW 1.5 0.65 1.5

PS 1 1 1

Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I

0.9 1.25 0.9 1.25 0.9 1 1

0.65 1.5 0.65 1.5 0.65 1 1

1 1 1 1 1 1 1

EV 1.35 0.9 1.35 0.9 1.35 0.9 1.35 0.9 1 1

HL-93 1.75 1.75 0

P-15 0 0 1.35

Seismic 0 0 0

0 0 0 1.35 1.35 1 0

1.35 0 0 0 0 0 0

0 0 0 0 0 0 1

The PS load factor of 1, as shown in Table 16.4-4, is recommended when a column’s cracked moment of inertia is used in analysis. However, for load cases other than extreme event-I, a load factor of 0.5 may be used. See AASHTO LRFD Table 3.4.1-3. In order to determine loads at the bottom of the shaft, the column diameter and length are needed. For this example, the column diameter is 5.5 ft with a length of 30 ft. The overall un-factored DC for column weight = 30 ft × π × (5.5)2/4 × 0.15 kip/ft3 = 106.92 kips. The LRFD load factors are applied to axial force and moments in longitudinal and transverse directions to calculate factored loads for strength, service, and extreme event limit states, as summarized in Tables 16.4-5, 16.4-6, and 16.4-7. Loading shown in those tables are for live load cases I, II, and III. Table 16.4-5 Case III: Maximum Axial Force P and Associated Effects Factored Loads Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I, Seismic I Extreme Event I, Seismic II

Chapter 16 – Deep Foundations

MT (k-ft) -261 -301 -119 -159 -70 -109 -218 -257 -206 14,771

ML (k-ft) 456 319 508 371 240 103 407 269 269 14,771

P (kip) 3,602 2,835 3,770 3,003 2,525 1,758 3,356 2,589 2,610 3,921 3,921

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The following are a few example calculations for the factored loads shown in Table 16.4-5: Axial force for Strength II-U limit state: P = 1.25(1793.75) + 1.5(163.65) + 1(37.1) + 1.35(922.37) = 3,770 kips Longitudinal moment for Strength II-L limit state: ML = 0.9(305.65) + 0.65(35.5) + 1(-195.6) + 1.35(199.08) = 371 k-ft Transverse moment for Strength II-L limit state: MT = 0.9(86.88) + 0.65(10.32) + 1(-193.6) + 1.35(-36.89) = -159 k-ft Gross axial force for Service I limit state: P = 1(1793.75) + 1(163.65) + 1(37.10) + 1(615.51) = 2,610 kips However, the net Service I loads need to be reported to Geotechnical Services for settlement calculations. For a large diameter shaft without any pile cap, the difference between Pnet and Pgross is small. Therefore: Pnet = Pgross = 2,610 kips Similarly, the net permanent loads are to be calculated and reported to Geotechnical Services. P = 1(1793.75) + 1(163.65) + 1(37.10) = 1,995 kips Table 16.4-6 Case I: Maximum Transverse Moment and Associated Effects

Factored loads Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I, Seismic I Extreme Event I, Seismic II

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MT

ML

P

(k-ft) -401 -440 -469 -508 -70 -109 -325 -364 -286 14,771

(k-ft) 343 206 373 236 240 103 320 183 205

(kip) 3,041 2,274 3,143 2,270 2,525 1,651 2,923 2,156 2,290 3,921 3,921

14,771

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Table 16.4-7 Case II: Maximum Longitudinal Moment and Associated Effects Factored Loads Strength I-U Strength I-L Strength II-U Strength II-L Strength III-U Strength III-L Strength V-U Strength V-L Service I Extreme Event I, Seismic (Case I) Extreme Event I, Seismic (Case II)

16.4.6.2

MT (k-ft) -193 -233 -108 -147 -70 -109 -165 -204 -167 14,771

ML (k-ft) 1,557 1,420 1,913 1,776 240 103 1,256 1,119 898 14,771

P (kips) 3,220 2,453 3,477 2,710 2,525 1,758 3,061 2,294 2,392 3,921 3,921

Shaft Geotechnical Capacity Check Strength and Service loads for Case II shown in Table 16.4.7 are the controlling load cases for shaft design in this example. The CA Amendment articles 10.5.5.2.4 and 10.5.5.3.3 specify resistance reduction factors (ϕ) for strength and extreme event limit states as 0.7 and 1.0, respectively. Compare factored loads on piles/shafts to factored resistance for Strength II-U limit state: Compression: 3,477 kips compression < (0.7) × 5,310 (Nominal) = 3,717 kips OK The tension demand in this example is zero, and the tension factored resistance is (0.7) x 2,655 (Nominal) = 1,859 kips Therefore, it is acceptable. Compare factored loads on piles/shafts to factored resistance for Extreme Event limit State: Compression: 3,921 kips (compression) < (1) × 5,310 (Nominal) = 5,310 kips OK The tension demand in this example is zero, and the tension factored resistance is (1) x 2,655 (Nominal) = 2,655 kips Therefore, it is acceptable.

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16.4.6.3

Check Development Length for Column Reinforcement into Type-II Shaft AASHTO Eq. 5.11.2.2: Development of Deformed Shaft Bars in Compression: Eq. 5.11.2.2.1-1: ldb  0.63(1.88)(60)/(4)0.5 = 35.53 x 1.2 = 42.64 in. (AASHTO 5.11.2.3) Eq. 5.11.2.2.1-2: ldb  0.3(1.88)(60) = 33.84 x 1.2 = 40.61 in. (AASHTO 5.11.2.3) AASHTO 5.11.2.2.2 states that the basic development length may be multiplied by applicable modification factors. AASHTO 5.11.2.2.2: Reinforcement is enclosed within a spiral of not less than 0.25 in. in diameter and not more than a 4 in. pitch, modification factor = 0.75. (This reduction does not apply because we have the pile shaft hoops at 7.5 in.). Development length of the bars in compression is 42.64 in. Column longitudinal reinforcement shall be extended into Type II (enlarged) shafts in a staggered manner with the minimum recommended embedment lengths of (Dc,max + Ld) and (Dc,max + 2 × Ld), where Dc,max is the larger cross-section dimension of the column, and Ld is the development length in tension of the column longitudinal bars. This practice ensures adequate anchorage in case the plastic hinge damage penetrates into the shaft (SDC 8.2.4). AASHTO 5.11.2.1: ldb = 1.25(1.56)(68)/(5)0.5 = 59.3 in. not less than ldb = 0.4(1.63)(68) = 44.33 in. Development length = ldb= 0.6 × 59.3 = 35.58 in. Development length = (Dc,max + Ld) = 66 + 35.58 = 101.58 in. Development length = (Dc,max + 2 x Ld) = 66 + 2 × 35.58 = 137.16 in. Development provided is 20 ft = 240 in. > 137.16 in.

16.4.6.4

OK

Check Shaft Structural Resistance: Check axial force for strength limit state loads: Shaft Tension Capacity = ϕPn = ϕ(Ast × fy); = 0.9 (60 × 2.25) (60) = 7,290 kips

(AASHTO 5.7.6.1) OK

Note: Tension in shaft for Strength limit state is negligible. Therefore, shaft is acceptable in tension. Shaft Compression Capacity = ϕPn = 0.75{0.85[0.85 × f′c (Ag – Ast) + fyAst ]}; (AASHTO 5.7.4.4)

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ϕPn = 0.75{0.85[0.85 × 4(7238.3 – 135) + 60(135)]} = 20,560 kips > 3,770 kips OK Check axial force for Seismic Extreme Event Loading: ϕ = 1 for seismic resistance factors (CA Amendment article 5.5.5). ϕ = 0.9 shear resistance factor (SDC 1.7 and 3.6.7) for shear during seismic and other extreme events. Shaft Tension Capacity: ϕPn = 1 (60 × 2.25) (60) = 8,100 kips > 0 kips

OK

Shaft Compression Capacity: ϕPn = 1{0.85[0.85 × 4(7,238.3 – 135) + 60(135)]} = 27,414 kips > 3,921 kips OK Note: In this example, the factored axial resistance is significantly higher than factored loads. Therefore, the interaction of axial force and bending moment is not checked. However, in general such interaction needs to be considered in design. Shafts are also to be checked for maximum shear for service and strength limit states. Assuming, the maximum shear will occur at the top of the shaft, the maximum factored strength II limit state shear force is as follows: Strength II: Vu = 1.35(128.262 + 2.572)0.5 = 173.2 kips The shear capacity of a reinforced concrete pile/shaft can be calculated as follows: Vn = Vc + Vs + Vp  0.25 f′c bv dv + Vp

(AASHTO 5.8.3.3)

Vc = 0.0316β(f′c) (bv)(dv) 0.5

Vs = [Av (fy)(dv) (cot θ + cot α) sin α]/ s Vp = 0 (no pre-stressing in pile/shaft) bv = D = 96 in.

(AASHTO 5.8.3.1)

dv = 0.9 de = 0.9 (70.91) = 63.82 in.

(AASHTO 5.8.2.9)

Dr = D – 2 (clr + hoop ddb + long ddb/2) = 96 – 2(9 + 1.13 + 1.88) = 71.98 in. de = D/2 + Dr/π = 96/2 + 71.98/π = 70.91 in.

(AASHTO C5.8.2.9-2)

α = 90 Av = 0.79 in.2, S = 7.5 in. down to the bottom of casing then Av = 1.58 in.2 S = 7 in., at rock socket section Check for minimum transverse reinforcement: Av ≥ 0.0316 (f′c)0.5(bv)s/fy = 0.0316(4)0.5(96)(7.5)/60 = 0.76 in.2

OK (AASHTO 5.8.2.5)

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Determine shear capacity for Strength II: β = 2, θ = 45°, α = 90° (For Strength II piles in compression, the simplified method, AASHTO 5.8.3.4.1, may be used.) Vc = 0.0316(2)(4)0.5 (96)(63.82) = 774.42 kips Vs = 0.79(60)(63.82)[(cot(45°)+cot(90°)) sin(90°)]/7.5 = 403.34 kips Vn = 774.42 + 403.34 = 1,177.76 kips < 0.25(4)(96)(63.82) = 6,126.72 kips

OK

Vr = ϕVn = 0.9(1,177.76) = 1,060 kips > 550 kips (See Figure 16.4-11)

OK

Note: AASHTO Eq. 5.8.3.4.2-4 or B5.2 may have been used instead of the simplified method to calculate a higher shear resistance, as follows: εx = [(|Mu|/dv) + 0.5Nu + 0.5|Vu – Vp|cot(θ) – Aps fpo]/[2(Es As + Ep Aps)] (AASHTO B5.2-1) Nu = 0 kips (no tensile load) Mu = 1916 kip-ft (moment demand assumed at top of pile/shaft) Vu = 173 kips (shear demand assumed at top of pile/shaft) Aps = 0 (no pre-stressing steel in pile/shaft) εx = [(|Mu|/dv) + 0.5Nu + 0.5|Vu – Vp|cot(θ) – Apsfpo]/[2(Es As + Ep Aps)] (AASHTO B5.2-1) εx = [(1916 × 12/63.82) + 0.5|173|cot (45)]/[2(29,000)(60 × 2.25)] = 0.000057 vu = |Vu – ϕVp |/(ϕbvdv) = |173|/[(0.9)(96)(63.82)] = 0.0313

(AASHTO 5.8.2.9)

vu/f′c = 0.0313/4 = 0.0078 β = 3.24, θ = 24.3,

(AASHTO Table B5.2-1)

Using θ = 24.3, resolving for εx = 0.000057; β = 3.24 Vc = 0.0316 (3.24)(4)0.5(96)(63.82) = 1,254.55 kips Vs = 0.79(60)(63.82)[(cot(24.3°) + cot(90°))sin(90°)]/7.5 = 893.30 kips Vn = 1,254.55 + 893.30 = 2,147.85 kips < 0.25(4)(96)(63.82) = 6,126.72 kips OK Vr = ϕVn = 0.9 (2,147.85 kips) = 1,933.07 kips > 550 kips

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OK

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Figure 16.4-11 Shear Diagram at Strength State for Single Shaft

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Determine the shear capacity with 0.5 in. casing using the following (AASHTO 5.8.3.4.1) simplified method: Determine shear capacity for Strength II: β = 2, θ = 45°, α = 90° (for Strength II piles in compression, the simplified method, AASHTO 5.8.3.4.1, may be used.) Vc = 0.0316(2)(4)0.5(96)(63.82) = 774.42 kips Vs = 0.79(60)(63.82)[(cot(45°) + cot(90°)) sin(90°)]/7.5 = 403.34 kips Vnp = 0.5Fcr Ag with Fcr = 0.78Es/(D/t)3/2 ≤ 0.58 Fy (AASHTO 6.12.1.2.3c) Fcr = 0.78Es/(D/t)3/2 = 0.78 × 29000/(97/0.5)3/2 = 8.37 ksi < 0.58 × 45 = 26.1 ksi Vnp = 0.5Fcr Ag Nominal shear resistance (pipe or casing) Vnp = 0.5 × 8.37 × π × 97 × 0.5 = 637.65 kips Vn = 774.42 + 637.65 + 403.34 = 1,815.41 kips < 6,126.72 kips

OK

Vr = ϕVn = 0.9 (1,815.41 kips) = 1,633.87 kips > 550 kips (See figure 16.4-11)

OK

If the pile shear demand exceeds the capacity, a more refined analysis may be performed accounting for the passive resistance of the soil against the pile. Maximum moments acting on the shaft at the bottom of the column for extreme event I, strength and service are shown in Tables 16.4-5, 16.4-6 and 16.4-7 as:

LRFD Limit State Strength Limit: Strength II Case II Strength Limit: Strength II Case I Extreme Event I Limit State: Seismic I/II Service Limit State Case II Service Limit State Case I

MT (kip-ft)

ML (kip-ft)

-108 -508 14,771 -167

1,913 236 14,771 898

M T2  M L2 (kip-ft) 1,916 560 20,889 913

-286

205

352

The maximum shaft bending moment demand for non-seismic loading (6,500 kip-ft) is extracted from Figure 16.4-12. The maximum shaft bending moment capacity for non-seismic loading is determined using the Xsection program (Caltrans, 2006): Mr = ϕ[email protected] = (0.9)26,928 = 24,235.2 k-ft > 6,500 k-ft

Chapter 16 – Deep Foundations

OK (AASHTO 5.5.4.2)

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Figure 16.4-12 Moment Diagram at Strength State for Single Shaft

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B

Communications of Structural Designer with Geotechnical Services (Refer to MTD 3-1 and MTD 1-35) The following table will be sent from Structure Design to Geotechnical Services: Table 16.4-8 Foundation Design Loads

Service-I Limit State (kips) Permanent Total Load Loads

Foundation Design Loads Strength Limit State (Controlling Group, kips)

Extreme Event Limit State (Controlling Group, kips)

Compression Tension Compression Tension Max. Max. Max. Support Per Max. Per Per Per Max. Per Per Per Per No. Support Per Pile Per Support Support Pile Support Per Pile Support Pile Support Pile Bent 2

2,610

2,610

1,995

3,770

3,770

0

0

3,921

3,921

0

Notes:  Support loads shown are per column  Service I loads are reported as net loads  Strength and Extreme Event loads are reported as gross loads  Load tables may be modified to submit multiple lines of critical load combinations for each limit state, if necessary

16.4.6.5

Shaft Seismic Analysis and Design Procedures

16.4.6.5.1 Design Approach The design of CIDH with permanent steel casing would be analyzed and designed similarly to a CISS. 16.4.6.5.2 Preliminary Substructure/Foundation Design Based on the geotechnical and hydraulics engineer requirements, the typical bent was designed as follows. The clear column height is 30 ft supported by an 8 ft diameter CIDH with 0.5 in. thick permanent casing for the top 34 ft with a 66 ft rock socket at diameter of 7 ft - 4 in. The selected pile system serves well in a 2-column bent arrangement to overcome the potential channel bed scour and soil liquefaction. 16.4.6.5.3 Preliminary Demand Assessment It is a common practice to design bridges for service and strength limits states and then refines the bridge system for seismic design requirements. Furthermore, the seismic design is a non-linear process and the bridge design may have to be iterated several times to reach a desirable solution. Finally, the designer may have to take the

Chapter 16 – Deep Foundations

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 seismic changes to the bridge back to the service load design and go through the entire design procedure until service, strength, and seismic requirements are all met. The engineer’s experience and judgment is a key factor in the amount of time required to perform these tasks. One possible method in the seismic design is the creation of a linear elastic model of the structure and performing of a modal dynamic analysis to obtain an estimate of the displacement demands. An alternative method, as used in this chapter, is to use the initial slope of the force-deflection curve from the push analyses of bents and frames and estimate the displacement demands from such analyses (SDC 5.2.1). Details of the displacement demand computation will be shown later. 16.4.6.5.4 Material Properties The material properties used in the seismic analysis are as follows: 

Concrete strength, f ce = 5000 psi



Concrete specific weight for calculation of modulus of elasticity = 145 lb./ft3



Other concrete properties are based on SDC sections 3.2.5 and 3.2.6.



Reinforcement (A706 steel) properties are based on SDC sections 3.2.2 and 3.2.3.

16.4.6.5.5 Soil Springs for CIDH Piles with Permanent Steel Casing The shaft is embedded in two soil layers. The topsoil layer with depth of 34 ft is sand with gravel and the bottom soil layer is sandstone rock material. The p-y and t-z curves reflect these characteristics. The p-y curves are used in the lateral modeling of soil as they interact with the large diameter shafts. The geotechnical engineer generally produces these curves and the values are converted to proper soil springs within the push analysis. The spacing of the nodes selected on the pile members would naturally change the values of spring stiffness. However, a minimum of 10 elements per pile is advised (recommended optimum is 20 elements or 2 ft to 5 ft pile segments). The t-z curves are used in the modeling of skin friction along the length of piles. Vertical springs are attached to the nodes to support the dead load of the bridge system and to resist overturning effects caused by lateral bridge movement. The bearing resistance at the tip of the pile is usually modeled as a q-z spring. This spring may be idealized as a bilinear spring placed in the boundary condition section of the push analysis input file.

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Figures 16-4.13 and 16-4.14 show Idealized Soil Springs for the right (RC) and left (LC) columns, respectively:

Idealized Soil P-Y Curve RC P (kip/ft.)

200 Depth 0 to 6 ft. Depth 6 to 10 ft. Depth 10 to 15 ft. Depth 15 to 22 ft. Depth 22 to 28 ft.

150 100 50 0 0

0.2

0.4

0.6

y (ft.)

Idealized Soil P-Y Curve RC

1000

Depth 28 to 34 ft. Depth 34 to 40 ft. Depth 40 to 46 ft. Depth 46 to 53 ft. Depth 53 to 60 ft.

P (kip/ft.)

800 600 400 200 0 0

0.05

0.1

0.15

y (ft.)

Idealized Soil P-Y Curve RC P (kip/ft.)

2500 Depth 60 to 67 ft. Depth 67 to 74 ft. Depth 74 to 81 ft. Depth 81 to 88 ft. Depth 88 to 95 ft.

2000 1500 1000 500 0 0

0.05

0.1

0.15

y (ft.)

Figure 16.4-13 Idealized Soil Springs for Right Column

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Idealized Soil P-Y Curve LC

P (kip/ft.)

200 150

Depth 0 to 4 ft. Depth 4 to 10 ft. Depth 10 tp 16 ft. Depth 16 to 23 ft.

100 50 0 0

0.1

0.2

0.3

y (ft.)

Idealized Soil P-Y Curve LC P (kip/ft.)

300 200

Depth 33 to 38 ft. Depth 38 to 44 ft. Depth 44 to 50 ft. Depth 23 to 29 ft.

100 0 0

0.5

1 y (ft.)

1.5

2

P (kip/ft.)

Idealized Soil P-Y Curve LC 1000 500

Depth 56 to 65 ft. Depth 50 to 56 ft. Depth 29 to 33 ft.

0 0

0.2

0.4

0.6

y (ft.)

P (kip/ft.)

Idealized Soil P-Y Curve LC 2000 1000 0 0

0.05

0.1

0.15

Depth 65 to 71 ft. Depth 71 to 77 ft. Depth 77 to 83 ft. Depth 83 to 91 ft.

y (ft.)

Figure 16.4-14 Idealized Soil Springs for Left Column

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16.4.6.6

Ductility and Transverse Push-Over Analysis and Design

16.4.6.6.1 Displacement capacity versus demand (C>D) CIDH Piles with permanent steel casing, column-shaft design: The concept of placing a reinforced concrete flexural element within a pipe/casing element is used in this example. The pipe/casing is installed to the desired tip elevation and cleaned out, leaving a soil plug or a rock socket in place at the tip of the pile. The column-shaft rebar cage is then placed within the pipe, followed by the concrete pour. The pipe/casing and the reinforced concrete element are considered two parallel elements where strength or stiffness of elements can be added. Since composite action is not guaranteed, it is conservative to use the non-composite properties of the system. The pipe/casing piles are ASTM A252, Grade 3 as required by Caltrans’ Standard Specifications section 49-2.02B(3) 2010 with a minimum yield strength of 45 ksi. The yield moment of the pipe/casing is estimated based on the yield stress of 45 ksi and then is increased by 25% to represent the plastic moment of the pipe/casing. The compression portion of the pipe/casing is continuously supported against buckling. The elongation required for ASTM A252, for Grade 3 pipe/casing material is estimated at 12% and a factor of safety of 2 is used, based on engineering judgment, to represent the minimum strain at peak stress of the pipe/casing steel at 6%. Properties for 97 in. diameter by 0.5 in. thick pipe:

Is  S



 Do4  Di4 64



 97  4  96  4       12   12  

 

64

 8.5 ft 4

Is 8.5   2.1 ft3 R 4.04

where R = outer radius of the pipe Do= outer diameter of the pipe Di= inner diameter of the pipe My = Sσy = 2.1  123  45 = 163,296 kip-in. = 13,608 kip-ft where σy= (45 ksi) yield stress of the pipe Mp = 1.25My = 17,010 kip-ft (EsIs)pipe = (29,000  144)8.5 = 3.55  107 kip-ft2

Design Steel Strain 

Chapter 16 – Deep Foundations

UltimateSteel Strain 0.12   cu   0.06 rad Factor of Safety 2

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Design Curvature  u 

0.06 0.06   0.00123 rad/in. Pipe Radius 48.5 in.

The concrete core and the rebar cage combine to produce a well-confined concrete element that is modeled within the xSECTION program (Caltrans, 2006) to generate section properties. The design requirements are met since the plastic hinges form in the column. Several cross sections along the column/shaft are analyzed. The cross section properties for various locations are tabulated in Table 16.4-9. Since the steel casing of the shaft is fully extended from the top of rock to the cut-off, the confinement of shaft cross sections could be modeled with the steel casing as the lateral confining element. The hoop reinforcement, #8 at 7.5 in., is included in the confinement computation. The axial forces of 1,926 and 2,004 kips are used for the column and shaft cross-sections respectively. 60˗#14 bars, A706 steel, #8˗hoops at 7.5 in. spacing Axial load = 2,004 kips Mp = 27, 220 kip-ft ϕp = 0.000465 rad/in. Icr = 82.42 ft4 Ec = 4,032 ksi ≈ 580,000 ksf (EcIcr)cage = 580,000 × 82.42 = 4.78 × 107 kip-ft2 For the cross section at the mid-height of the shaft casing, the following data is obtained by using xSECTION software. (See Typical xSECTION Figures 16.4-15 and 16.4-16). The combined effects of the 2 elements, steel pipe/casing, and concrete core are as follows: Mp = 17,010 + 27,220 = 44,230 kip-ft

I combined 

Es I s  pipe  Ec I cr cage Econcrete



3.55107  4.78107  143.64 ft 4 580,000

Ec = 4,032 ksi = 580,000 ksf ϕp = 0.000465 rad/in. based on minimum of the two elements The Icombined is computed for converting the combined properties of steel pipe/casing and concrete core into an equivalent concrete property. These parameters are to be used in the pushover analysis later. The smaller of the curvature of steel pipe and concrete core is selected as the ductility capacity limit.

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Figure 16.4-15 Shaft Cross Section with Moment Curvature with 1.87% Longitudinal Steel

Figure 16.4-16 Shaft Cross Section with Moment Curvature in Rock Region with 2.22% Longitudinal Steel

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B

Table 16.4-9 Cross-Section Properties for the Column and Shaft Location Column Shaft Shaft+Col Shaft @rock Cap beam + Cap beam -

Section Composition Core Core Shaft+Col Core only Unconfined conc Unconfined conc

Mp kip-ft 12,309 27,220 34,157 28,508 27,111

Concrete Core p rad/in. 0.000898 0.000465 0.000317 0.000864 0.000587

Icr ft4 19.78 82.42 95.07 71.97 86.93

25,602

0.000422

83.1

Mp kip-ft

Pipe/Casing p rad/in.

Icr ft4

17,010 17,010

0.00123 0.00123

8.5 8.5

Mp kip-ft 12,309 44,230 51,167 28,508 27,111

Combination Icr p ft4 rad/in. 0.000898 19.78 0.000465 143.62 0.000317 143.62 0.000864 71.97 0.000587 86.93

Ec ksf 580,000 580,000 580,000 580,000 580,000

25,602

0.000422

580,000

83.1

A typical bent is then modeled within the wFRAME program (Caltrans, 1995) using the above properties and foundation springs to perform a non-linear pushover analysis. The force-deflection curve shown in section 16.4.6.6.4 indicates that 2 plastic hinges form at the top of columns followed by 2 more plastic hinges developing at the bottom of columns as designed for a Type II shaft The period of this bent is around 1.17 seconds and the initial stiffness and the displacement demand (D) are 263 kips/in. and 9.42 in., respectively. Detailed computations are shown below: ki = Initial Slope of Force – Deflection Curve = Lateral Force/Yield Displacement ki = (0.39  W)/ Δyi = (0.39  3,536 kips)/5.25 in =263kip/in. where W = total of dead load plus added dead load (kips) T  period  0.32

W 3,536  0.32  1.17 seconds ki 263

Displacement demand: ΔD = (ARS x W)/Ki = (0.7 x 3536)/263 = 9.42 in. The plastic hinge length at top of column is calculated from SDC Section 7.6.2. The standard plastic hinge length (SDC 7.6.2.1) is used with the moment diagram. The rotational capacity (p) of the plastic hinge is similar to the equations shown in SDC section 3.1.3. The plastic component of the displacement capacity, however, is based on an effective length of column defined from the center of the lower plastic hinge to the center of the upper plastic hinge as follows (Figure 16.4-17): For the plastic hinge at top of column: Lp = length of plastic hinge = 0.08L1 + 0.15fyedbl  0.3 fyedbl

(SDC 7.6.2.1-1)

L1 = portion of column from plastic hinge to contraflexure (zero moment) L1 = 16 ft dbl = diameter of main bar = 1.63 in. for #11 bar Lp = 0.08 × 16 × 12 + 0.15 × 68 × 1.63 = 32 in.  0.3 × 68 × 1.63 = 33.25 in.

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θp = Lpp

(SDC 3.1.3-4)

Where p = plastic curvature capacity at top of column from cross section analysis θp = 33.25 × 0.000898 = 0.0298 rad ∆p = θp × L = 0.0298 × 30 ft × (12 in./ft) = 10.73 in. L = portion of column from center of top plastic hinge to center of bottom plastic hinge ∆c = ∆p + ∆y for each and every plastic hinge ∆c = 10.73 + 5.25 = 15.98 in. > ∆D = 9.42 in.

OK

The overturning effects increased the axial load from 1,926 kips to 2,695 kips in the compression column. However, the increase in plastic moment was less than 20%. There is additional reserve capacity in the cap beam to easily meet the higher column moment demand. Comparison of displacement capacity (C) and demand (D) shows adequate ductility in the transverse direction for this particular bent (C >D ).

Figure 16.4-17 Plastic Hinges and Deformation 16.4.6.6.2

P-Moment Check: The P- moment check is required per SDC section 4.2. (P ˗ ∆) Check: M P ˗ ∆ = P (∆D/2) = 1,926 kips (9.42 in./12)/2 = 756 kip-ft

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M P  756   0.061  0.2 OK MP 12,309

16.4.6.6.3

(SDC 4.2-1)

Minimum Ductility Requirements The minimum ductility requirement (SDC section 3.1.4.1) is based on the following values: The distances of L1 and L2 are 16 ft and 14 ft, respectively, from the moment diagram of the push over analysis per definition in SDC Figure 3.1.3-2. The yield curvature at the top and bottom of column is 0.000088 rad/in. The plastic curvature at the top and bottom is 0.000898 rad/in. (from column xSection analysis). From SDC Eq. 7.6.2.1-1: LP1 = 0.08L1 + 0.15fyedbl  0.3 fyedbl = 0.08 × 16 × 12 + 0.15 × 68 × 1.63 = 32 in. < 0.3 × 68 × 1.63 = 33.25 for the top plastic hinge (SDC 7.6.2.1-1) Lp2 = 0.08L2 + 0.15fyedbl  0.3 fyedbl = 0.08 × 14 × 12 + 0.15 × 68 × 1.63 = 30 in. < 0.3 × 68 × 1.63 = 33.25 for the bottom plastic hinge (SDC 7.6.2.1-1) p1 = Lp1p1 = 33.25 × 0.000898 = 0.0298 rad.

(SDC 3.1.3-9)

p2 = Lp2p2 = 33.25 × 0.000898 = 0.0298 rad.

(SDC 3.1.3-9)

 Lp 1    0.0298  16  12  33.25/2  5.22 in.  p 1   p 1   L1   2     Lp 2    0.0298  14  12  33.25/2  4.51 in.  p 2   p 2   L2   2   

(SDC3.1.3-8) (SDC 3.1.3-8)

col y1 

16  122 0.000088  1.08 in. L12  y1  3 3

(SDC 3.1.3-7)

col y2 

14  122 0.000088  0.83 in. L22 y2  3 3

(SDC 3.1.3-7)

 c1  col y1   p1  1.08  5.22  6.30 in.

(SDC 3.1.3-6)

 c 2  col y 2   p 2  0.83  4.51  5.34 in.

(SDC 3.1.3-6)

c1 

c 2 

 c1 col y1 c2 col y2



6.30  5.83  3 1.08

(SDC 3.1.4-2)



5.34  6.43  3 0.83

(SDC 3.1.4-2)

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16.4.6.6.4

Minimum Lateral Strength The lateral strength of the bent is 0.39g, as calculated from the force-deflection curve of the push over analysis as shown in Figure 16.4-18. This meets the requirement of SDC section 3.5, which requires minimum lateral strength of 0.1 g.

Figure 16.4-18 Lateral Load – Deflection Curve of Bent

16.4.6.7

Shear and Flexural Capacity To meet the over-strength requirements of SDC section 4.3.1, the plastic moment of the column is increased by 20% in the push analysis to determine the larger flexural demand in the cap beam. The push analysis including over-strength of the column in addition to overturning indicated a maximum negative moment demand of 18,146 kip-ft in the cap beam, while the capacity of the cap beam in negative moment was 25,602 kip-ft

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16.4.6.7.1 Shaft Shear Capacity with 0.5 in. Casing Excluding Scour and Liquefaction The plastic shear demand in the column is limited to: Vp 

M p TOP  M p BOT

L Plastic Himge to Plastic Hinge



212,309  820.6 30

kips

The following cases affecting a shear increase should be considered: Increase the shear demand by an additional 20% due to overturning, VO = 985 kips. The nominal shear capacity of the casing/pipe alone is calculated per AASHTO 6.12.1.2.3c and is less than the maximum shear demand as shown in Figure 16.4-19. Vnp = 0.5Fcr Ag with Fcr = 0.78Es/(D/t)3/2 ≤ 0.58 Fy ( AASHTO 6.12.1.2.3c) Fcr = 0.78Es/(D/t) = 0.78  29000/(97/0.5)3/2 = 8.37 ksi < 0.58  45 = 26.1 ksi 3/2

Vnp = 0.5Fcr Ag Nominal shear resistance (pipe or casing) Vnp = 0.5  8.37  π  97  0.5 = 637.65 kips The shear capacity of the shaft reinforced concrete outside the plastic hinge zone must be added. For this case the traditional reinforced concrete column shear capacity equations of SDC 3.6.2 are used. In this particular case SDC Eqs. 3.6.3-1 and 3.6.3-2 yield the following for the pipe and an axial load at the top of shaft of 2,031 kips: Vs 

 Av f yhD 2s

0.3  Factor1 

 0.796077 in.



27.5 in.

 s f yh

0.15ksi

 764 kips

 3.67   d  3

(SDC 3.6.2-5)

Factor 1 = 3 outside the plastic hinge zone

Factor 2  1  Factor 2  1 

Pc lb.



2,000 Ag in.2

(SDC 3.6.3-1)

(SDC 3.6.2-5)

  1.5

(SDC 3.6.2-6)

2,0311,000  1.14 2,00051.32144

(SDC 3.6.2-6)

vc psi  3Factor 2 f c psi  4 f c psi

(SDC 3.6.2-4)

vc psi  31.14 4,000  216.3 psi  4 4,000  252.98 psi (SDC 3.6.2-4) Vc = vc  Ac where Ac = 0.8Ag Vc  216.3 0.8 Ag  216.30.851.32  144 / 1.000  1,279 kips





(SDC 3.6.2-1) and (SDC 3.6.2-2)

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Vn   Vc  Vs  Vnp   0.91,279 764  637.65  2,413 kips  2,265 kips (SDC 3.6.1-1 and 3.6.1-2) The shear capacity of the shaft casing and the reinforced concrete section are all added. The total shear capacity of 2,413 kips is larger than the shear demand without scour and liquefaction. It is slightly less than the maximum shear demand of 2.5Vo (2,463 kips) required for the case with scour and liquefaction. To increase capacity, the designer may reduce the confinement spacing or increase the steel shell thickness of the shaft. The capacity calculated is acceptable for this example.

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Figure 16.4-19 Shear Diagram at Potential Collapse State for Single Shaft

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16.4.6.7.2 Shaft Shear Capacity at Rock Socket: The shear capacity of the shaft without casing at the rock interface outside the plastic hinge zone must also be checked. For this case, the traditional reinforced concrete column shear capacity equations of SDC 3.6.2 are used. In this particular case SDC Eqn. 3.6.3-1 yields the following for the pipe with an increase in confinement spacing from a single #8 confinement at 7.5 in. to a doubled #8 confinement at 7 in. and an axial load at the top of rock socket section of 2,260 kips. For the case without scour or liquefaction, see L-Pile or CSiBridge Demand see Figure 16.4-19. Vs 

 Av f yh D 2s



 20.796077

27

 s f yh

 1,638 kips  2,265 kips (SDC 3.6.3-1)

 3.67   d  3

(SDC 3.6.2-5)

Factor 1 = 3 outside the plastic hinge zone

(SDC 3.6.2-5)

0.3  Factor1 

Factor 2  1  Factor 2  1 

0.15ksi

Pc lb.



2,000 Ag in.2

  1.5

(SDC 3.6.2-6)

2,2601,000  1.19 2,00042.24144

(SDC 3.6.2-6)

vc psi  3Factor 2 f c psi  4 f c psi

(SDC 3.6.2-4)

vc psi  31.19 4,000  226 psi  4 4,000  252.98 psi

(SDC 3.6.2-4)

Vc = vc  Ac where Ac = 0.8Ag Vc  226 0.8 Ag  2260.842.24  144 / 1.000  1,100 kips





(SDC 3.6.2-1) and (SDC 3.6.2-2)

Vn   Vc  Vs   0.91,100 1,638  2,464 kips  2,265 kips (SDC 3.6.1-1 and 3.6.1-2) The shear demand for the case with scour and liquefaction effect as shown in Figure 16.4-19 is about 50% more than the shear capacity of the rock socket section as calculated above. Therefore, we limit the demand to be no more than 2.5Vo (2,463 kips), which happened to be almost equal to the calculated capacity of 2,464 kips in this example.

vc psi  31.19 4,000  226 psi  4 4,000  252.98 psi

Chapter 16 – Deep Foundations

(SDC 3.6.2-4)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Vc = vc  Ac where Ac = 0.8Ag Vc  226 0.8 Ag  2260.842.24  144 / 1.000  1,100 kips





(SDC 3.6.2-1) and (SDC 3.6.2-2)

Vn   Vc  Vs   0.91,100 1,638  2,464 kips  2,463 kips

OK

(2.5Vo where Vo = 985 kips) (See L-Pile or CSiBridge Demand, see Figure 16.4-19) 16.4.6.7.3 Shaft Seismic Flexural Capacity: The type II shaft flexural capacity under seismic loading outside the plastic hinge zone and away from the column inserted inside the type II shaft is determined from expected material properties. (Capacity protected component, per SDC 3.4) where expected f′ce = 5.0 ksi, fye = 68 ksi Using (Xsection) The type II shaft flexural demands under seismic loading outside plastic hinge zone and away from the column inserted inside type II shaft are determined from (LPile) output as shown below in Figure 16.4-20 for both cases. Mr = [email protected] = (1)(26,928(shaft) + 13,608(casing)) = 40,536 kip-ft > 1.25 (29,167) kip-ft (L-Pile, CSiBridge, or Wframe Demand) without liquefaction and scour demand (SDC 7.7.3.2) OK Mr = [email protected] = (1) (26,928(shaft) + 13,608(casing) = 40,536 kip-ft < 1.25 (36,667) kip-ft (L-Pile, CSiBridge, or Wframe Demand) with liquefaction and scour demand (SDC 7.7.3.2 ) NG The moment capacity is less than the moment demand of (1.25Md) per SDC by approximately 13% is acceptable for this example. Designers can add more reinforcement to increase the moment capacity by 13%. Designers need to be aware that the moment demand used in the example was extracted from L-Pile software and is based on a single pile, disregarding the framing action of a two-column system that can reduce the moment demand by at least by 15%.

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Figure 16.4-20 Moment Diagram at Potential Collapse State for Single Shaft

Chapter 16 – Deep Foundations

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Location

Type-I Column-Shaft with Diameter < 5ft Type-I Column-Shaft with Diameter  5ft Type-II ColumnShaft with Shaft Diameter < 5ft Type-II ColumnShaft with Shaft Diameter  5ft

Column and Shaft Column and Shaft

Clear Spacing Requirements for Longitudinal Bars Max2 (in.) Min (in.)

Clear Spacing Requirements for Transverse Bars Max3 (in.) Min5 (in.)

10-ndb

Note 4

Smax-mdb

5

12-ndb

Note 4

Smax-mdb

5

Shaft

10-ndb

Note 4

12

5

Shaft

12-ndb

Note 4

12

5

Note: 1. For Type II Shaft ≥ 5ft ϕ construction joint and permanent steel casing are mandatory, unless the drilled hole is determined by Geotechnical Services and Structure Representative to be dry. 2. n=1 is for single bars and n =2 for circumferential bundled bars. Refer to California Amendments to AASHTO LRFD Sixth Edition 10.8.1.3 and 5.13.4.5. 3. m is number of hoops in a bundle, otherwise m =1. Smax is maximum center-tocenter (C-C) spacing of single hoops per SDC 8.2.5 Version 1.7. 4. Per MTD3-1, minimum longitudinal clear spacing of rebars is 5 in. except at locations of inspection pipes where it is 8.5 in. to accommodate the pipes

5. Minimum Clear Distance between parallel longitudinal and transverse bars is 5 in. per California Amendments to AASHTO LRFD Sixth Edition 5.13.4.5.2.

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NOTATION A

= area of cross-section of a member (in.2) (16.3.8)

Ab

=

area of individual reinforcing steel bar (in.2) (16.3.10)

Ae

=

effective shear area of a cross-section (in.2) (16.3.10)

Ag

=

gross cross-sectional area (in.2) (16.2.3)

Ajh ftg

=

effective horizontal area for a moment-resisting pile cap joint (in.2) (16.2.6)

Aps

=

area of prestressing steel (in.2) (16.2.3)

ARS

=

5% damped elastic Acceleration Response Spectrum, expressed in terms of g (SDC 2.1) (16.4.6.6.1)

As

=

total area of non prestressed tension reinforcement (in.2) (16.2.3)

As′

=

total area of compression reinforcement (in.2) (16.2.4)

Ast

=

total area of longitudinal steel (in.2) (16.2.3)

Av

=

area of shear reinforcement normal to flexural tension reinforcement (in.2) (16.2.3)

a

=

depth of equivalent rectangular stress block (in.) (16.2.4)

B

=

coefficient used to solve a quadratic equation (16.2.4)

Beff ftg

=

effective width of the pile cap for calculating average normal stress in the horizontal direction within a pile cap moment-resisting joint (in.) (16.2.6)

bv

=

effective width of a member for shear stress calculations (in.) (16.2.3)

C

=

coefficient used to solve a quadratic equation (16.2.4)

C(i)pile

=

axial compression demand on a pile/shaft (kip) (16.2.3)

clr

=

minimum clearance for bottom-mat reinforcement (16.3.6.2)

D

=

shaft diameter (in.) (16.1.2)

Dc

=

column diameter (in.) (16.2.1)

Dftg

=

depth of pile cap (in.) (16.2.1)

Dftg,min =

minimum depth of pile cap (in.) (16.2.1)

DRs

=

depth of resultant soil-resistance measured from the top of pile cap (ft) (16.2.3)

Dr

=

diameter of the circle passing through the longitudinal reinforcement (in.) (16.2.3)

D´

=

cross-sectional dimension of a confined concrete core measured between the centerline of the peripheral hoop or spiral (in.) (16.3.10.2.3)

d

=

distance from the compression flange to the centroid of the compression steel (in.); depth below original ground (in.) (16.2.4)

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db

=

nominal bar diameter (in.) (16.2.1)

dbd

=

deformed bar diameter (in.) (16.2.1)

dbl

=

nominal bar diameter of the longitudinal reinforcement of a shaft (in.) (16.3.2)

dc

=

thickness of concrete cover measured from extreme tension fiber to center of closest bar (in.) (16.2.1)

de

=

effective depth from extreme compression fiber to the centroid of the tensile force in the tensile reinforcement (in.) (16.2.4)

dv

=

effective shear depth (in.) (16.2.3)

Ec

=

modulus of elasticity of concrete (ksi) (16.2.4)

Es

=

modulus of elasticity of reinforcing steel (ksi) (16.3.5)

f′c

=

specified 28-day compressive strength of unconfined concrete (ksi) (16.3.5)

f′ce

=

expected compressive strength of unconfined concrete (ksi) (16.3.5)

fpo

=

average stress in prestressing steel (ksi)

fr

=

modulus of rupture of concrete (ksi) (16.2.4)

fss

=

tensile stress in mild steel at the service limit state (ksi) (16.2.4)

fue

=

expected minimum tensile strength for A706 reinforcing steel (ksi) (16.3.5)

fy

=

nominal yield stress for A706 reinforcing steel (ksi) (16.2)

fye

=

expected yield stress for A706 reinforcing steel (ksi) (16.3.2)

fyh

=

nominal yield stress of transverse (spiral) shaft reinforcement (ksi) (16.3.10)

fu

=

specified minimum tensile strength for A706 reinforcing steel (ksi) (16.3.5)

H

=

height of column measured from the top of the pile cap to the center of gravity of the superstructure (in.) (16.3.8)

Hc

=

clear height of a column measured from the top of the pile cap to the soffit of the superstructure (in.) (16.3.8)

Icr

=

moment of inertia of the cracked cross-section of a member about its centroidal axis (in.4) (16.2.4)

Iy

=

moment of inertia of a pile/shaft group about the y-axis (in.4) (16.1.3)

k

=

a soil modulus parameter for sand (lb./in.3) (16.3.4)

L

=

distance between the top and the lower points of maximum moment along a shaft: see Figure 16.3-2 (in.) (16.3.4)

L1, L2

=

distances between the top and the lower points of maximum moment, respectively, and the point of contraflexure defined in Figure 16.3-2 (in.) (16.3.2)

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L*

=

effective length of the shaft segment between the top and the lower points of maximum moment along a shaft defined in Figure 16.3-3 (in.) (16.3.1)

Lftg

=

cantilever length of the pile cap measured from the face of the column to the edge of the pile cap along the principal axis of the pile cap (in.) (16.2.1)

Lp

=

equivalent analytical plastic hinge length (in.) (16.3.2)

l′d

=

development length of longitudinal column reinforcement (in.) (16.3.6)

ld

=

development length of deformed bars in compression (in.) (16.3.6)

ldb

=

basic development length of deformed bars in compression (in.) (16.2.1)

ldh

=

development length of deformed bars in tension (in.) (16.2.1)

lhb

=

basic development length of hooked bars in tension (in.) (16.2.1)

Mcap

=

moment demand in the pile cap (kip-ft) (16.2.4)

Mcr

=

cracking moment of a member’s cross-section (kip-ft) (16.2.4)

Mi

=

moment demand at the top of a row i shaft (kip-ft): see Figure 16.3-1 (16.3.2)

Miy

=

moment at the top of a row i shaft at the formation of the first plastic hinge (kip-ft): see Figure 16.3-1 (16.3.2)

ML

=

maximum longitudinal moment (kip-ft) (16.2)

Mn

=

nominal flexural resistance of a member’s cross-section (kip-ft) (16.2.4)

Mne

=

nominal moment capacity of a member’s cross-section based on the expected material properties and concrete strain c = 0.003 (kip-ft) (16.2.4)

Mocol

=

overstrength moment capacity of column’s cross-section (kip-ft) (16.2.3)

Mp

=

idealized plastic moment capacity of a member’s cross-section (kip-ft) (16.1.3)

Mpcol

=

idealized plastic moment capacity of a column’s cross-section (kip-ft) (16.3.8)

Mpshaft =

idealized plastic moment capacity of a shaft’s cross-section (kip-ft) (16.3.2)

Mr

=

factored flexural resistance of a section in bending (kip-ft) (16.2.4)

MT

=

maximum transverse moment (kip-ft) (16.2)

Mu

=

factored moment at a section (kip-ft) (16.2.3)

MY

=

moment at the column’s base (=Mocol) associated with the formation of the first plastic hinge in the shaft group (kip-ft): see Figure 16.3-1 (16.1.3)

N

=

total number of piles/shafts in a pile/shaft group; standard blow count per foot for the California Standard Penetration Test (16.1.3)

Nu

=

applied factored axial force taken as positive if tensile (kip) (16.2.3)

n

= modular ratio, Es/Ec or Ep/Ec; number of individual interlocking spiral coresections (16.3.10)

P

=

maximum axial force (kip) (16.1.3)

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Pc

=

the column axial load including the effects of overturning (kip) (16.2.6)

Pdl

=

axial load attributed to dead load (kip) (16.3.2)

Pi

=

axial force demand on a row i shaft (kip): see Figure 16.3-1

Piy

=

axial force at the top of a row i shaft at the formation of the first plastic hinge (kip): see Figure 16.3-1

Pn

=

nominal axial resistance of the pile/shaft (kip) (16.2.2)

Pnet

=

net effective load acting on the bottom of the pile cap (kip) (16.2.2)

PP

=

total axial load on a shaft-group foundation (kip) (16.3.2)

pc

=

principal compression stress (ksi) (16.2.3)

pt

=

principal tension stress (ksi) (16.2.6)

Rs

=

total resultant soil-resistance along the end and sides of pile cap (kip) (16.2.3)

Q

=

axial soil compressive resistance at the tip of a given diameter shaft (kip) (16.3.4)

s

=

pitch of spiral reinforcement measured along the length of the shaft (in.); spacing of reinforcing bars (in.) (16.2.4)

Tc

=

total tension force in the longitudinal reinforcement of a column associated with Mo (kip) (16.2.3)

Tjv

=

net tension force in a moment-resisting pile cap joint (kip) (16.2.6)

t

=

axial load transfer per unit length of a given diameter shaft (kip/ft) (16.3.4)

Vc

=

nominal shear strength provided by concrete (kip) (16.2.3)

Vi

=

shear demand at the top of a row i shaft (kip): see Figure 16.3-1

Viy

=

shear force at the top of a row i shaft at the formation of the first plastic hinge (kip): see Figure 16.3-1

VL

=

maximum longitudinal shear (kip) (16.2)

Vn

=

nominal shear resistance of a section (kip) (16.2.3)

Vnp

=

nominal shear resistance of a pipe or casing (kip) (16.4.6)

Vp

=

component of the pre-stressing force in the direction of applied shear (kip) (16.2)

Vocol

=

overstrength shear capacity of the column (kip) (16.3.2)

Vr

=

nominal shear resistance of cross-section (kip) (16.2.3)

Vs

=

nominal shear strength provided by shear reinforcement (kip) (16.2.3)

VT

=

maximum transverse shear (kip) (16.2)

Vu

=

factored shear force at a section (kip) (16.2.3)

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VY

=

shear force at the column’s base (=Vocol) associated with the formation of the first plastic hinge in the shaft group (kip-ft): see Figure 16.3-1 (16.3.2)

x

=

distance from the compression flange to the neutral axis (in.) (16.1.3)

y

=

lateral deflection of a shaft at a specific depth (in.) (16.1.3)

z

=

vertical deflection of a shaft at a specific depth (in.) (16.2.3)



=

angle of inclination of transverse reinforcement to longitudinal axis (16.2.3)



=

factor relating effect of longitudinal strain on the shear capacity of concrete; ratio of long side to short side of footing (16.2.3)

s

=

ratio of flexural strain at the extreme tension face to the strain at the centroid of the reinforcement layer nearest the tension face (16.2.4.2)

vc

=

permissible shear stress carried by concrete (psi) (16.3.10)

vjv

=

nominal shear stress in a moment-resisting joint (ksi)

vu

=

average factored shear stress on concrete (ksi) (16.2.2)

cc

=

confined concrete compressive strain at maximum compressive stress (16.3.5)

co

=

unconfined concrete compressive strain at maximum compressive stress (16.3.5)

cu

=

confined concrete ultimate compressive strain (16.3.5)

sh

=

tensile strain at the onset of strain hardening for A706 reinforcement (16.3.5)

sp

=

unconfined concrete ultimate compressive strain (spalling strain) (16.3.5)

su

=

ultimate tensile strain of A706 reinforcement (16.3.5)

Rsu

=

reduced ultimate tensile strain of A706 reinforcement (16.3.5)

ye

=

expected yield tensile strain of A706 reinforcement (16.3.5)

c

=

local member displacement capacity (in.) (16.3.2)

C

=

global displacement capacity of a shaft group (in.) (16.3.2)

D

=

global displacement demand of a shaft group (in.) (16.3.2)

pi

=

idealized local plastic displacement capacity of a shaft due to the rotation of the ith plastic hinge (in.): see Figure 16.3-2 (16.3.2.1.3)

Pi

=

global plastic displacement capacity of a shaft group due to the plastic rotation capacity of the ith plastic hinge (in.): see Figure 16.3-3

r

=

relative lateral offset between the top of the shaft and the lower point of maximum moment in a shaft (in.) (16.3.2)

Yi

=

yield displacement of a shaft group at the formation of the ith plastic hinge in the shafts (in.): see Figure 16.3-2 (16.3.2.1.3)

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  yishaft =

idealized local yield displacement of the ith shaft (in.) (16.3.10)

y2shaft

=

(16.3.10.1.2)



=

angle of internal friction (16.2.3)



=

strength reduction factor

p

=

idealized plastic curvature of a cross-section (rad/in.) (16.2.3)

Y

=

idealized yield curvature of a cross-section (rad/in.) (16.3.2)

e

=

crack control exposure condition factor (16.2.4)

p

=

load factor for permanent loads (16.2.2)

t

=

unit weight of soil (lb./ft3) (16.3.4)



=

non-dimensional constant defined in Figure 16.3-1(16.3.2)

c

=

local displacement ductility capacity (16.3.2)

d

=

local displacement ductility demand (16.3.10)

D

=

global displacement ductility demand (16.3.2)



=

angle of inclination of diagonal compressive stresses (rad) (16.2.3)

=

angle defined in Figure 16.3-3 (rad)

p

=

plastic rotation capacity (rad) (16.3.2)

s

=

ratio of volume of spiral reinforcement to the core volume confined by the spiral reinforcement (measured out-to-out) for circular cross-sections (16.3.10)



=

angle defined in Figure 16.3-3 (rad) (16.3.2)

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REFERENCES 1.

AASHTO, (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units (6th Edition), American Association of State Highway and Transportation Officials, Washington, DC.

2.

AISC, (2010). Specification for Structural Steel Buildings Manual, American Institute of Steel Construction Inc., Chicago, IL

3.

Caltrans, (1984). Bridge Design Details 13-10: Reinforcing Bar Data – Grade 60 (ACI 31883), California Department of Transportation, Sacramento, CA, July 1984.

4.

Caltrans, (1990). Bridge Design Details 6-71: Pile Footings, California Department of Transportation, Sacramento, CA, June 1990.

5.

Caltrans, (1995). wFRAME, Push Analysis of Bridge Bents and Bridge Frames – Version 1.12, California Department of Transportation, Sacramento, CA.

6.

Caltrans, (2006). xSECTION, Moment-Curvature Curves for Reinforced Concrete Structural Elements – Version 4.00,, California Department of Transportation, Sacramento, CA.

7.

Caltrans, (2014 a). California Amendments to AASHTO LRFD Bridge Design Specifications – Sixth Edition, California Department of Transportation, Sacramento, CA.

8.

Caltrans, (2014b). Bridge Memo to Designers 3-1: Deep Foundations, California Department of Transportation, Sacramento, CA, June 2014.

9.

Caltrans, (2008). Bridge Memo to Designers 1-35: Foundation Recommendation and Reports, California Department of Transportation, Sacramento, CA, June 2008.

10. Caltrans, (2010). Bridge Memo to Designers 20-1: Seismic Design Methodology, California Department of Transportation, Sacramento, CA, July 2010. 11. Caltrans, (2013). Caltrans Seismic Design Criteria, Version 1.7, California Department of Transportation, Sacramento, CA, April 2013. 12. Chen, W. F. and Duan, L. (2000). Bridge Engineering Handbook, Chapter 9, 15-16, CRC Press, Boca Raton, FL. 13. CSI, (2015). CSiBridge 2015, v. 17.0.0, Computers and Structures, Inc., Walnut Creek, CA. 14. Gebman, M. J. and Restrepo, A. J. (2006). “Axial Force Transfer Mechanisms within CastIn Steel Shell Piles.” Report: SSRP-06/16. University of California, San Diego, La Jolla, CA. 15. Parker, F., Jr. and Reese, L. C. (1971). “Lateral Pile-Soil Interaction Curves for Sand.” Proceedings of the International Symposium on Engineering Properties of Sea-Floor Soils and their Geophysical Identification, UNESCO-University of Washington, Seattle, WA, July 1971, 212-223. 16. Reese, L. C. Wang, S. T., Isenhower, W., and Arrellega, J. A. (2007). “Computer Program LPILE Plus Version 5.0.38.” A Program for the Analysis of Piles and Drilled Shafts under Lateral Loads, Technical Manual, Ensoft, Inc., Austin, TX.

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17. Seyed, M. (1995). Computer Programs w-FPREP and w-FRAME Version 1.12 — A Program for Two Dimensional Push Analysis of Bridge Bents and Frames, Sacramento, CA.

18. Yashinsky, M. et al. (2007). Pile Group Foundation Study, California Department of Transportation, Sacramento, CA, November 2007. 19.

Malek, A. M., and Islam M. S. (2010), “Seismic Analysis and Design of Pile Group Bridge Foundations in Soft and Liquefied Soil.” Journal of the Transportation Research Board, No. 2202, 183-191.

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CHAPTER 21 SEISMIC DESIGN OF CONCRETE BRIDGES TABLE OF CONTENTS 21.1

INTRODUCTION .......................................................................................................... 21-1

21.2

DESIGN CONSIDERATIONS ...................................................................................... 21-1

21.3

21.2.1

Preliminary Member and Reinforcement Sizes ........................................... 21-1

21.2.2

Minimum Local Displacement Ductility Capacity ...................................... 21-7

21.2.3

Displacement Ductility Demand Requirements .......................................... 21-9

21.2.4

Displacement Capacity Evaluation............................................................ 21-12

21.2.5

P- Effects ............................................................................................... 21-15

21.2.6

Minimum Lateral Strength ........................................................................ 21-15

21.2.7

Column Shear Design ................................................................................ 21-16

21.2.8

Bent Cap Flexural and Shear Capacity ...................................................... 21-18

21.2.9

Seismic Strength of Concrete Bridge Superstructures .............................. 21-18

21.2.10

Joint Shear Design ..................................................................................... 21-26

21.2.11

Torsional Capacity .................................................................................... 21-34

21.2.12

Abutment Seat Width Requirements ......................................................... 21-34

21.2.13

Hinge Seat Width Requirements ............................................................... 21-35

21.2.14

Abutment Shear Key Design ..................................................................... 21-36

21.2.15

No-Splice Zone Requirements .................................................................. 21-38

21.2.16

Seismic Design Procedure Flowchart ........................................................ 21-39

DESIGN EXAMPLE - THREE-SPAN CONTINUOUS CAST-IN-PLACE CONCRETE BOX GIRDER BRIDGE ...................................................................... 21-43 21.3.1

Bridge Data................................................................................................ 21-43

21.3.2

Design Requirements................................................................................. 21-43

21.3.3

Step 1- Select Column Size, Column Reinforcement, and Bent Cap Width ......................................................................................... 21-46

21.3.4

Step 2 - Perform Cross-section Analysis ................................................... 21-47

21.3.5

Step 3 - Check Span Configuration/Balanced Stiffness ............................ 21-48

21.3.6

Step 4 - Check Frame Geometry ............................................................... 21-49

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21.3.7

Step 5 – Calculate Minimum Local Displacement Ductility Capacity and Demand .............................................................................................. 21-49

21.3.8

Step 6 – Perform Transverse Pushover Analysis....................................... 21-51

21.3.9

Step 7- Perform Longitudinal Pushover Analysis ..................................... 21-56

21.3.10

Step 8- Check P - Δ Effects ..................................................................... 21-60

21.3.11

Step 9- Check Bent Minimum Lateral Strength ........................................ 21-60

21.3.12

Step 10- Perform Column Shear Design ................................................... 21-61

21.3.13

Step 11- Design Column Shear Key .......................................................... 21-64

21.3.14

Step 12 - Check Bent Cap Flexural and Shear Capacity ........................... 21-65

21.3.15

Step 13- Calculate Column Seismic Load Moments ............................... 21-67

21.3.16

col @ soffit Step 14 - Distribute M eq into the Superstructure ............................ 21-74

21.3.17

Step 15- Calculate Superstructure Seismic Moment Demands at Location of Interest ................................................................................... 21-74

21.3.18

Step 16- Calculate Superstructure Seismic Shear Demands at Location of Interest ................................................................................... 21-78

21.3.19

Step 17- Perform Vertical Acceleration Analysis ..................................... 21-82

21.3.20

Step 18- Calculate Superstructure Flexural and Shear Capacity ............... 21-82

21.3.21

Step 19- Design Joint Shear Reinforcement .............................................. 21-86

21.3.22

Step 20- Determine Minimum Hinge Seat Width ..................................... 21-95

21.3.23

Step 21- Determine Minimum Abutment Seat Width .............................. 21-95

21.3.24

Step 22 - Design Abutment Shear Key Reinforcement ............................. 21-96

21.3.25

Step 23 - Check Requirements for No-splice Zone ................................... 21-97

APPENDICES .......................................................................................................................... 21-98 NOTATION ........................................................................................................................... 21-137 REFERENCES ....................................................................................................................... 21-144

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CHAPTER 21 SEISMIC DESIGN OF CONCRETE BRIDGES 21.1

INTRODUCTION This chapter is intended primarily to provide guidance on the seismic design of Ordinary Standard Concrete Bridges as defined in Caltrans Seismic Design Criteria (SDC), Version 1.7 (Caltrans 2013). Information presented herein is based on SDC (Caltrans 2013), AASHTO LRFD Bridge Design Specifications (AASHTO 2012) with California Amendments (Caltrans 2014), and other Caltrans Structure Design documents such as Bridge Memo to Designers (MTD). Criteria on the seismic design of nonstandard bridge features are developed on a project-by-project basis and are beyond the scope of this chapter. The first part of the chapter, i.e., Section 21.2, describes general seismic design considerations including pertinent formulae, interpretation of Caltrans SDC provisions, and a procedural flowchart for seismic design of concrete bridges. In the second part, i.e., Section 21.3, a seismic design example of a three-span continuous cast-in-place, prestressed (CIP/PS) concrete box girder bridge is presented to illustrate various design applications following the seismic design procedure flowchart. The example is intended to serve as a model seismic design of an ordinary standard bridge using the current SDC Version 1.7 provisions.

21.2

DESIGN CONSIDERATIONS

21.2.1

Preliminary Member and Reinforcement Sizes Bridge design is inherently an iterative process. It is common practice to design bridges for the Strength and Service Limit States and then, if necessary, to refine the design of various components to satisfy Extreme Events Limit States such as seismic performance requirements. In practice, however, engineers should keep certain seismic requirements in mind even during the Strength and Service Limit States design. This is especially true while selecting the span configuration, column size, column reinforcement requirements, and bent cap width.

21.2.1.1 Sizing the Column and Bent Cap (1) Column size SDC Section 7.6.1 specifies that the column size should satisfy the following equations: Chapter 21 – Seismic Design of Concrete Bridges

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Dc  1.00 Ds D ftg

0.70  0.7 

(SDC 7.6.1-1) (SDC 7.6.1-2)

Dc

where: Dc Ds Dftg

= = =

column cross sectional dimension in the direction of interest (in.) depth of superstructure at the bent cap (in.) depth of footing (in.)

If Dc > Ds, it may be difficult to meet the joint shear, superstructure capacity, and ductility requirements. (2) Bent Cap Width SDC Section 7.4.2.1 specifies the minimum cap width required for adequate joint shear transfer as follows:

Bcap  Dc  2 21.2.1.2

(ft)

(SDC 7.4.2.1-1)

Column Reinforcement Requirements (1) Longitudinal Reinforcement Maximum Longitudinal Reinforcement Area, Ast ,max  0.04  Ag

(SDC 3.7.1-1)

Minimum Longitudinal Reinforcement Area:

Ast ,min  0.01( Ag )

for columns

(SDC 3.7.2-1)

Ast ,min  0.005( Ag )

for Pier walls

(SDC 3.7.2-2)

where: Ag

=

the gross cross sectional area (in.2)

Normally, choosing column Ast  0.015( Ag ) is a good starting point. (2) Transverse Reinforcement Either spirals or hoops can be used as transverse reinforcement in the column. However, hoops are preferred (see MTD 20-9) because of their discrete nature in the case of local failure.

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

Inside the Plastic Hinge Region The amount of transverse reinforcement inside the analytical plastic hinge region (see SDC Section 7.6.2 for analytic plastic hinge length formulas), expressed as volumetric ratio,s, shall be sufficient to ensure that the column meets the performance requirements as specified in SDC Section 4.1. 4( A )  s  ' b for columns with circular or interlocking cores (SDC 3.8.1-1) D (s) For rectangular columns with ties and cross ties, the corresponding equation for  s , is:

s  where: Av = Dc'

=

s

=

Av Dc' s

(SDC 3.8.1-2)

sum of area of the ties and cross ties running in the direction perpendicular to the axis of bending (in.2) confined column cross-section dimension, measured out to out of ties, in the direction parallel to the axis of bending (in.) transverse reinforcement spacing (in.)

In addition, the transverse reinforcement should meet the column shear requirements as specified in SDC Section 3.6.3. 

Outside the Plastic Hinge Region As specified in SDC Section 3.8.3, the volume of lateral reinforcement outside the plastic hinge region shall not be less than 50 % of the minimum amount required inside the plastic hinge region and meet the shear requirements.

(3) Spacing Requirements The selected bar layout should satisfy the following spacing requirements for effectiveness and constructability: 

Longitudinal Reinforcement Maximum and minimum spacing requirements are given in AASHTO Article 5.10 (2012).



Transverse Reinforcement According to SDC Section 8.2.5, the maximum spacing in the plastic hinge region shall not exceed the smallest of:

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

1

of the least column cross-section dimension for columns and ½ of the least cross-section dimension for piers  6 times the nominal diameter of the longitudinal bars  8 in. 5

Outside this region, the hoop spacing can be and should be increased to economize the design. 21.2.1.3

Balanced Stiffness (1) Stiffness Requirements For an acceptable seismic response, a structure with well-balanced mass and stiffness across various frames is highly desirable. Such a structure is likely to respond to a seismic activity in a simple mode of vibration and any structural damage will be well distributed among all the columns. The best way to increase the likelihood that the structure responds in its fundamental mode of vibration is to balance its stiffness and mass distribution. To this end, the SDC recommends that the ratio of effective stiffness between any two bents within a frame or between any two columns within a bent satisfy the following:

kie k ej

 0.5

 ke / m i 2   ie k /m j  j

For constant width frame

   0.5  

For variable width frame

(SDC 7.1.1-1)

(SDC 7.1.1-2)

SDC further recommends that the ratio of effective stiffness between adjacent bents within a frame or between adjacent columns within a bent satisfies the following:

kie k ej

 0.75

For constant width frame

 ke / m i 1.33   ie k /m j  j

   0.75 For variable width frame  

(SDC 7.1.1-3)

(SDC 7.1.1-4)

where: kie mi

= =

smaller effective bent or column stiffness (kip/in.) tributary mass of column or bent i (kip-sec2/ft)

k ej

=

larger effective bent or column stiffness (kip/in.)

mj

=

tributary mass of column or bent j (kip-sec2/ft)

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Bent stiffness shall be based on effective material properties and also include the effects of foundation flexibility if it is determined to be significant by the Geotechnical Engineer. If these requirements of balanced effective stiffness are not met, some of the undesired consequences include:  The stiffer bent or column will attract more force and hence will be susceptible to increased damage  The inelastic response will be distributed non-uniformly across the structure  Increased column torsion demands will be generated by rigid body rotation of the superstructure (2) Material and Effective Column Section Properties To estimate member ductility, column effective section properties as well as the moment-curvature ( M   ) relationship are determined by using a computer program such as xSECTION (Mahan 2006) or similar tool. Effort should be made to keep the dead load axial forces in columns to about 10% of their ultimate compressive capacity (Pu = Ag f c' ) to ensure that the column does not experience

brittle compression failure and also to ensure that any potential P- effects remain within acceptable limits. When the column axial load ratio starts approaching 15%, increasing the column size or adding an extra column should be considered. Material Properties 

Concrete Concrete compressive stress f c = 4,000 psi is commonly used for superstructure, columns, piers, and pile shafts. For other components like abutments, wingwalls, and footings, f c =3,600 psi is typically specified. SDC Section 3.2 requires that expected material properties shall be used to calculate section capacities for all ductile members. To be consistent between the demand and capacity, expected material properties will also be used to calculate member stiffness. For concrete, the expected compressive strength, f ce' , is taken as:

f ce'

 1.3( f c' )     Greater of  and   5,000 psi   

(SDC 3.2.6-3)

Other concrete properties are listed in SDC Section 3.2.6.

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

Steel Grade A706/A706M is typically used for reinforcing steel bar. Material properties for Grade A706/A706M steel are given in SDC Section 3.2.3.

Effective Moment of Inertia It is well known that concrete cover spalls off at very low ductility levels. Therefore, the effective (cracked) moment of inertia values are used to assess the seismic response of all ductile members. This is obtained from a moment-curvature analysis of the member cross-section. 21.2.1.4

Balanced Frame Geometry SDC Section 7.1.2 requires that the ratio of fundamental periods of vibration for adjacent frames in the longitudinal and transverse directions satisfy: Ti (SDC 7.1.2-1)  0.7 Tj where: Ti Tj

= =

natural period of the less flexible frame (sec.) natural period of the more flexible frame (sec.)

The consequences of not meeting the fundamental period requirements of SDC Equation 7.1.2-1 include a greater likelihood of out-of-phase response between adjacent frames leading to large relative displacements that increase the probability of longitudinal unseating and collision between frames at the expansion joints. For bents/frames that do not meet the SDC requirements for fundamental period of vibration and/or balanced stiffness, one or more of the following techniques (see SDC Section 7.1.3) may be employed to adjust the dynamic characteristics:        

Use of oversized shafts Adjust the effective column length. This may be achieved by lowering footings, using isolation casings, etc. Modify end fixities Redistribute superstructure mass Vary column cross section and longitudinal reinforcement ratios Add or relocate columns Modify the hinge/expansion joint layout, if applicable Use isolation bearings or dampers

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If the column reinforcement exceeds the preferred maximum, the following additional revisions as outlined in MTD 6-1 (Caltrans 2009) may help: 

Pin columns in multi-column bents and selected single columns adjacent to abutments at their bases Use higher strength concrete Shorten spans and add bents Use pile shafts in lieu of footings Add more columns per bent

   

21.2.2 Minimum Local Displacement Ductility Capacity Before undertaking a comprehensive analysis to consider the effects of changes in column axial forces (for multi-column bents) due to seismic overturning moments and the effects of soil overburden on column footings, it is good practice to ensure that basic SDC ductility requirements are met. SDC Section 3.1 requires that each ductile member shall have a minimum local displacement ductility capacity c of 3 to ensure dependable rotational capacity in the plastic hinge regions regardless of the displacement demand imparted to the member.

Δc  ΔYcol  Δp L2 (Y ) 3 Lp     p   p  L  2   ΔYcol 

(SDC 3.1.3-1) (SDC 3.1.3-2) (SDC 3.1.3-3)

 p  Lp p

(SDC 3.1.3-4)

 p  u  Y

(SDC 3.1.3-5)

col  c1  col Y 1   p1 ;  c 2   Y 2   p 2

(SDC 3.1.3-6)

L12 L22 (Y 1 ); col  (Y 2 ) Y2 3 3 L p1  L    ;  p 2   p 2  L2  p 2   p1   p1  L1    2  2     p1  L p1 p1;  p 2  L p 2 p 2 ΔYcol 1 

 p1  u1  Y1;  p 2  u 2  Y 2

(SDC 3.1.3-7) (SDC 3.1.3-8) (SDC 3.1.3-9) (SDC 3.1.3-10)

where: L

= distance from the point of maximum moment to the point of contra-flexure (in.)

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LP

=

Y

=

p

=

equivalent analytical plastic hinge length as defined in SDC Section 7.6.2 (in.) idealized plastic displacement capacity due to rotation of the plastic hinge (in.) idealized yield displacement of the column at the formation of the plastic hinge (in.) idealized yield curvature defined by an elastic-perfectly-plastic representation of the cross section’s M- curve, see SDC Figures 3.3.1-1 and 3.3.1-2 (rad/in.) idealized plastic curvature capacity (assumed constant over Lp) (rad/in.)

p

=

p

= =

plastic rotation capacity (radian) curvature capacity at the Failure Limit State, defined as the concrete strain

Ycol =

u

reaching cu or the longitudinal reinforcing steel reaching the reduced ultimate strain suR (rad/in.) It is Caltrans’ practice to use an idealized bilinear M- curve to estimate the idealized yield displacement and deformation capacity of ductile members. c Ycol

C.L. Column

p

C.G.

L

Force

Idealized Yield Curvature Capacity EquivalentCurvature

Lp

p

Actual Curvature

P

u

p





Y Y

c Displacement

Figure SDC 3.1.3-1 Local Displacement Capacity – Cantilever Column with Fixed Base

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B

C.L. Column

Y1

u1

p1

P1 Lp1

P2

Idealized

L1

c2

Yield Curvature

P1

Actual Curvature

colY1

colY2 c1

Idealized Equivalent Curvature

L2 Lp2

P2

u2

p2

Y2

Figure SDC 3.1.3-2 Local Displacement Capacity – Framed Column, Assumed as Fixed-Fixed

21.2.3

Displacement Ductility Demand Requirements The displacement ductility demand is mathematically defined as  D  D Y (i )

(SDC 2.2.3-1)

where:

D

Y(i)

= =

the estimated global/frame displacement demand the yield displacement of the subsystem from its initial position to the formation of plastic hinge (i)

To reduce the required strength of ductile members and minimize the demand imparted to adjacent capacity protected components, SDC Section 2.2.4 specifies target upper limits of displacement ductility demand values,  D , for various bridge components.

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D  4 D  5 D  5 D  1

Single Column Bents supported on fixed foundation Multi-Column Bents supported on fixed or pinned footings Pier Walls (weak direction) supported on fixed or pinned footings Pier Walls (strong direction) supported on fixed or pinned footings

In addition, SDC Section 4.1 requires each bridge or frame to satisfy the following equation:

D  C

(SDC 4.1.1-1)

where: = C

the bridge or frame displacement capacity when the first ultimate capacity is reached by any plastic hinge (in.) = the displacement generated from the global analysis, stand-alone analysis, or the larger of the two if both types of analyses are necessary (in.)

D

The seismic demand can be estimated using Equivalent Static Analysis (ESA). As described in SDC Section 5.2.1, this method is most suitable for structures with well-balanced spans and uniformly distributed stiffness where the response can be captured by a simple predominantly translational mode of vibration. Effective properties shall be used to obtain realistic values for the structure’s period and demand. The displacement demand,  D , can be calculated from Equation 21.2-1.

D 

ma ke

(21.2-1)

where: m

a ke

= tributary superstructure mass on the bent/frame = demand spectral acceleration = effective frame stiffness

For ordinary bridges that do not meet the criteria for ESA or where ESA does not provide an adequate level of sophistication to estimate the dynamic behavior, Elastic Dynamic Analysis (EDA) may be used. Refer to SDC Section 5.2.2 for more details regarding EDA.

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Figure SDC 3.1.4.1-1 Local Ductility Assessment Chapter 21 – Seismic Design of Concrete Bridges

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21.2.4

Displacement Capacity Evaluation SDC Section 5.2.3 specifies the use of Inelastic Static Analysis (ISA), or “pushover” analysis, to determine reliable displacement capacities of a structure or frame. ISA captures non-linear bridge response such as yielding of ductile components and effects of surrounding soil as well as the effects of foundation flexibility and flexibility of capacity protected components such as bent caps. The effect of soil-structure interaction can be significant in the case where footings are buried deep in the ground. Pushover analysis shall be performed using expected material properties of modeled members to provide a more realistic estimate of design strength. As required by SDC Section 3.4, capacity protected concrete components such as bent caps, superstructures and footings shall be designed to remain essentially elastic when the column reaches its overstrength capacity. This is required in order to ensure that no plastic hinge forms in these components. Caltrans’ in-house computer program wFRAME (Mahan 1995) or similar tool may be used to perform pushover analysis. If wFRAME program is used, the following conventions are applicable to both the transverse and longitudinal analyses:   



The model is two-dimensional with beam elements along the c.g. of the superstructure/bent cap and columns. The dead load of superstructure/bent cap, and of columns, if desired, is applied as a uniformly distributed load along the length of the superstructure/bent cap. The element connecting the superstructure c.g. to the column end point at the soffit level is modeled as a super stiff element with stiffness much greater than the regular column section. The moment capacity for such element is also specified much higher than the plastic moment capacity of the column. This is done to ensure that for a column-to-superstructure fixed connection, the plastic hinge forms at the top of the column below the superstructure soffit. The soil effect can be included as p-y, t-z, and q-z springs.

Though “pushover” is mainly a capacity estimating procedure, it can also be used to estimate demand for structures having characteristics outlined previously in Section 21.2.3. 21.2.4.1

Foundation Soil Springs The p-y curves are used in the lateral modeling of soil as it interacts with the bent/column foundations. The Geotechnical Engineer generally produces these curves, the values of which are converted to proper soil springs within the push analysis. The spacing of the nodes selected on the pile members would naturally

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change the values of spring stiffness, however, a minimum of 10 elements per pile is advised (recommended optimum is 20 elements or 2 ft to 5 ft pile segments). The t-z curves are used in the modeling of skin friction along the length of piles. Vertical springs are attached to the nodes to support the dead load of the bridge system and to resist overturning effects caused by lateral bridge movement. The bearing reaction at tip of the pile is usually modeled as a q-z spring. This spring may be idealized as a bi-linear spring placed in the boundary condition section of the push analysis input file. 21.2.4.2

Transverse Pushover Analysis During the transverse movement of a multi-column frame, a strong cap beam provides a framing action. As a result of this framing action, the column axial force can vary significantly from the dead load state. If the seismic overturning forces are large, the top of the column might even go into tension. The effect of change in the axial force in a column section due to overturning moments can be summarized as follows:   

Mp changes The tension column(s) will become more ductile while the compression column(s) will become less ductile. The required flexural capacity of cap beam that is needed to make sure that the hinge forms at column top will obviously become larger.

With the changes in column axial loads, the section properties (Mp and Ie) should be updated and a second iteration of the wFRAME program performed if using wFRAME for the analysis. The effective bent cap width to be used for the pushover analysis is calculated as follows:

Beff  Bcap  (12t )

(SDC 7.3.1.1-1)

where: t Bcap 21.2.4.3

= thickness of the top or bottom slab (in.) = bent cap width (in.)

Longitudinal Pushover Analysis Although the process of calculating the section capacity and estimating the seismic demand is similar for the transverse and longitudinal directions, there are some significant differences. For longitudinal push analysis: 

If wFRAME program is used, columns are lumped together

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

For prestressed superstructures, gross moment of inertia is used for the superstructure Bent overturning is ignored The abutment is modeled as a linear spring whose stiffness is calculated as described in this Section.

 

If the column or pier cross-section is rectangular, section properties along the longitudinal direction of the bridge as shown in Figure 21.2-1 must be calculated and used. If using xSECTION , this can be achieved by specifying in the xSECTION input file, the angle between the column section coordinate system and the longitudinal direction of the bridge as shown in the sketch below. Both left and right longitudinal pushover analyses of the bridge should be performed.

Bridge longitudinal direction

Bent Line

Figure 21.2-1 Bridge Longitudinal Direction It is Caltrans’ practice to design the abutment backwall so that it breaks off in shear during a seismic event. SDC Section 7.8.1 requires that the linear elastic demand shall include an effective abutment stiffness that accounts for expansion gaps and incorporates a realistic value for the embankment fill response. The abutment embankment fill stiffness is non-linear and is highly dependent upon the properties of the backfill. The initial embankment fill stiffness, Ki, is estimated at 50 kip/in./ft for embankment fill material meeting the requirements of Caltrans Standard Specifications and 25 kip/in./ft, if otherwise. The initial stiffness, K i shall be adjusted proportional to the backwall/diaphragm height as follows:

 h  K abut  K i w   5.5 

(SDC 7.8.1-2)

where: w h

= projected width of the backwall or diaphragm for seat and diaphragm abutments, respectively (ft) = height of the backwall or diaphragm for seat and diaphragm abutments, respectively (ft)

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The passive pressure resisting movement at the abutment, Pw, is given as:

 h or hdia  Pw  Ae (5 ) bw  5.5  

kip  ft

(SDC 7.8.1-3)

where:

hbw wbw Ae   hdia wdia

For seat abutments For diaphragm abutments

(SDC 7.8.1-4)

The terms hbw, hdia, wbw, and wdia, are defined in SDC Figure 7.8.1-2. SDC Section 7.8.1 specifies that the effectiveness of the abutment shall be assessed by the coefficient:

RA   D eff

(SDC 7.8.1-5)

where: RA

D eff

= = =

abutment displacement coefficient the longitudinal displacement demand at the abutment from elastic analysis the effective longitudinal abutment displacement at idealized yield

Details on the interpretation and use of the coefficient RA value are given in SDC Section 7.8.1.

21.2.5

P- Effects In lieu of a rigorous analysis to determine P- effects, SDC recommends that such effects can be ignored if the following equation is satisfied:

Pdl  r  0.20M col p

21.2.6

(SDC 4.2-1)

where: M col p =

idealized plastic moment capacity of a column calculated from M-

Pdl r

analysis dead load axial force relative lateral offset between the base of the plastic hinge and the point of contra-flexure

= =

Minimum Lateral Strength SDC Section 3.5 specifies that each bent shall have a minimum lateral flexural capacity (based on expected material properties) to resist a lateral force of 0.1Pdl,

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where Pdl is the tributary dead load applied at the center of gravity of the superstructure.

21.2.7

Column Shear Design The seismic shear demand shall be based upon the overstrength shear Vo , associated with the column overstrength moment M 0col . Since shear failure tends to be brittle, shear capacity for ductile members shall be conservatively determined using nominal material properties, as follows:

Vn  V0

(SDC 3.6.1-1)

Vn = Vc + Vs

(SDC 3.6.1-2)

where:

  0.90 21.2.7.1 Shear Demand Vo Shear demand associated with overstrength moment may be calculated from: V0 

M 0col L

(21.2-2)

where:

M 0col  1.2M col p L

(SDC 4.3.1-1)

= clear length of column

Alternately, the maximum shear demand may be determined from wFRAME pushover analysis results. The maximum column shear demand obtained from wFRAME analysis is multiplied by a factor of 1.2 to obtain the shear demand associated with the overstrength moment.

21.2.7.2

Concrete Shear Capacity Vc   c Ae

(SDC 3.6.2-1)

Ae  (0.8) Ag

(SDC 3.6.2-2)

where:

 c  f1 f 2 f c'  4 fc' = 3 f 2 f c'  4 f c'

(Inside the plastic hinge region)

(SDC 3.6.2-3)

(Outside the plastic hinge region)

(SDC 3.6.2-4)

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0.3  f1 

 s f yh 0.150

 3.67  d  3

( f yh in ksi)

 s f yh  0.35 ksi f2  1  21.2.7.3

(SDC 3.6.2-5) (21.2-3)

Pc  1.5 2,000Ag

( Pc is in lb, Ag is in in.2)

(SDC 3.6.2-6)

Transverse Reinforcement Shear Capacity Vs

 Av f yh D'   Vs   s  

(SDC 3.6.3-1)

where:

  Av  n  Ab (SDC 3.6.3-2) 2 n = number of individual interlocking spiral or hoop core sections 21.2.7.4

Maximum Shear Reinforcement Strength, Vs,max

Vs, max  8 f c' Ae 21.2.7.5

(SDC 3.6.5.1-1)

Minimum Shear Reinforcement

Av, min  0.025 21.2.7.6

(psi)

D's f yh

in.  2

(SDC 3.6.5.2-1)

Column Shear Key The area of interface shear key reinforcement, Ask in hinged column bases shall be calculated as shown in the following equations: 1.2( Fsk  0.25P) if P is compressive (SDC 7.6.7-1) Ask  fy Ask 

1.2( Fsk  P) fy

if P is tensile

(SDC 7.6.7-2)

where: Ask  4 in.2

Fsk

=

(21.2-4)

shear force associated with the column overstrength moment, including overturning effects (kip)

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P

=

absolute value of the net axial force normal to the shear plane (kip)

=

lowest column axial load if net P is compressive considering overturning effects = largest column axial load if net P is tensile, considering overturning effects

The hinge shall be proportioned such that the area of concrete engaged in interface shear transfer, Acv satisfies the following equations:

4.0 Fsk f c' Acv  0.67Fsk

Acv 

(SDC 7.6.7-3) (SDC 7.6.7-4)

In addition, the area of concrete section used in the hinge must be enough to meet the axial resistance requirements as provided in AASHTO Article 5.7.4.4 (AASHTO 2012), based on the column with the largest axial load.

21.2.8

Bent Cap Flexural and Shear Capacity According to SDC Section 3.4, a bent cap is considered a capacity protected member and shall be designed flexurally to remain essentially elastic when the column reaches its overstrength capacity. The expected nominal moment capacity Mne for capacity protected members may be determined either by a traditional strength method or by a more complete M-ϕ analysis. The expected nominal moment capacity shall be based on expected concrete and steel strength values when either R concrete strain reaches 0.003 or the steel strain reaches  SU as derived from the applicable stress-strain relationship. The shear capacity of the bent cap is calculated according to AASHTO Article 5.8 (AASHTO 2012). The seismic flexural and shear demands in the bent cap are calculated corresponding to the column overstrength moment. These demands are obtained from a pushover analysis with column moment capacity as M0 and then compared with the available flexural and shear capacity of the bent cap. The effective bent cap width to be used is calculated as follows:

Beff  B cap 12t  t

21.2.9

(SDC 7.3.1.1-1)

= thickness of the top or bottom slab

Seismic Strength of Concrete Bridge Superstructures When moment-resisting superstructure-to-column details are used, seismic forces of significant magnitude are induced into the superstructure. If the superstructure does not have adequate capacity to resist such forces, unexpected and unintentional

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hinge formation may occur in the superstructure leading to potential failure of the superstructure. According to SDC Sections 3.4 and 4.3.2, a capacity design approach is adopted to ensure that the superstructure has an appropriate strength reserve above demands generated from probable column plastic hinging. MTD 20-6 (Caltrans 2001a) describes the philosophy, design criteria, and a procedure for determining the seismic demands in the superstructure, and also recommends a method for determining the flexural capacity of the superstructure at all critical locations. 21.2.9.1

General Assumptions As discussed in MTD 20-6, some of the assumptions made to simplify the process of calculating seismic demands in the superstructure include: 

The superstructure demands are based upon complete plastic hinge formation in all columns or piers within the frame. Effective section properties shall be used for modeling columns or piers while gross section properties may be used for superstructure elements. Additional column axial force due to overturning effects shall be considered when calculating effective section properties and the idealized plastic moment capacity of columns and piers. Superstructure dead load and secondary prestress demands are assumed to be uniformly distributed to each girder, except in the case of highly curved or highly skewed structures. While assessing the superstructure member demands and available section capacities, an effective width, Beff as defined in SDC Section 7.2.1.1 will be used.

   

 D  2 Ds Beff   c  Dc  Ds

Box girders and slab superstructures Open soffit superstructures

(SDC 7.2.1.1-1)

where: Dc Ds 21.2.9.2

= =

cross sectional dimension of the column (in.) depth of the superstructure (in.)

Superstructure Seismic Demand The force demand in the superstructure corresponds to its Collapse Limit State. The Collapse Limit State is defined as the condition when all the potential plastic hinges in the columns and/or piers have been formed. When a bridge reaches such a state during a seismic event, the following loads are present: Dead Loads, Secondary Forces from Post-tensioning (i.e., prestress secondary effects), and Seismic Loads. Since the prestress tendon is treated as an internal component of the superstructure and is included in the member strength calculation, only the secondary effects which are caused by the support constraints in a statically indeterminate prestressed frame are considered to contribute to the member demand.

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The procedure for determining extreme seismic demands in the superstructure considers each of these load cases separately and the final member demands are obtained by superposition of the individual load cases. Since different tools may be used to calculate these demands, it is very important to use a consistent sign convention while interpreting the results. The following sign convention (see Figure 21.2-2a) for positive moments, shears and axial forces, is recommended. The sign convention used in wFRAME program is shown in Figure 21.2-2b. It should be noted that although the wFRAME element level sign convention is different from the standard sign convention adopted here, the resulting member force conditions (for example, member in positive or negative bending, tension or compression, etc.) are the same as furnished by the standard convention. In particular, note that the inputs Mp and Mn for the beam element in the wFRAME program correspond to tension at the beam bottom (i.e., positive bending) and tension at the beam top (i.e., negative bending), respectively. The engineer should also ensure that results obtained while using the computer program CTBridge (Caltrans 2007) are consistent with the above sign conventions when comparing outputs or employing the results of one program as inputs into another program. (i)

Beam

(j)

(j)

(i)

(j) Beam

(i) Column

Column

(i)

(a) Standard

(j)

(b) wFRAME

Figure 21.2-2 Sign Convention for Positive Moment, Shear and Axial force (Element Level) Prior to the application of seismic loading, the columns are “pre-loaded” with moments and shears due to dead loads and secondary prestress effects. At the Collapse Limit State, the “earthquake moment” applied to the superstructure may be greater or less than the overstrength moment capacity of the column or pier depending on the direction of these “pre-load” moments and the direction of the seismic loading under consideration. Figure 21.2-3 shows schematically this approach of calculating columns seismic forces.

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As recommended in MTD 20-6, due to the uncertainty in the magnitude and distribution of secondary prestress moments and shears at the extreme seismic limit state, it is conservative to consider such effects only when their inclusion results in increased demands in the superstructure. Once the column moment, Meq, is known at each potential plastic hinge location below the joint regions, the seismic moment demand in the superstructure can be determined using currently available Caltrans’ analysis tools. One such method entails application of Meq at the column-superstructure joints and then using computer program SAP2000 (CSI 2007) to compute the moment demand in the superstructure members. Another method involves using the wFRAME program to perform a longitudinal pushover analysis by specifying the required seismic moments in the columns as the plastic hinge capacities of the column ends. The pushover is continued until all the plastic hinges have formed.

+

+



+



 

M 0   M dl    M ps   M eq

M eq

M ps

M dl



When earthquake forces add to dead load and secondary prestress forces. When earthquake forces add to dead load and secondary prestress forces.

= Collapse Limit Collapse Limit State

+

State

M ps

M dl

+

-





 

 M 0   M dl    M ps   M eq

M eq

=



When earthquake forces counteract dead load and secondary prestress When earthquake forces counteract dead forces. load and secondary prestress forces.

Figure 21.2-3 Column Forces Corresponding to Two Seismic Loading Cases Note that CTBridge is a three-dimensional analysis program where force results are oriented in the direction of each member’s local axis. If wFRAME (a twodimensional frame analysis program) is used to determine the distribution of seismic forces to the superstructure, it must be ensured that the dead load and secondary prestress moments lie in the same plane prior to using them in any calculations. This must be done especially when horizontal curves or skews are involved.

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(1) Dead Load Moments, Additional Dead Load Moments, and Prestress Secondary Moments These moments are readily available from CTBridge output and are assumed to be uniformly distributed along each girder. (2) Earthquake Moments in the Superstructure (Reference MTD 20-6, SDC 4.3.2) The aim here is to determine the amount of seismic loading needed to ensure that potential plastic hinges have formed in all the columns of the framing system. To form a plastic hinge in the column, the seismic load needs to produce a moment at the potential plastic hinge location of such a magnitude that, when combined with the “pre-loaded” dead load and prestress moments, the column will reach its overstrength plastic moment capacity, M 0col . @ soffit col @ soffit M 0col @ soffit  M dlcol @ soffit  M col  M eq ps

(21.2-5)

It should be kept in mind that dead load moments will have positive or negative values depending on the location along the span length. Also, the direction of seismic loading will determine the nature of the seismic moments. Two cases of longitudinal earthquake loading shall be considered, namely, (a) bridge movement to the right, and (b) bridge movement to the left. col The column seismic load moments, M eq , are calculated from Equation 21.2-5

based upon the principle of superposition as follows:



col @ soffit @ soffit M eq  M 0col @ soffit  M dlcol @ soffit  M col ps



In the above equation, the overstrength column moment M

M ocol  1.2M col p

(21.2-6) col o

is given as: (SDC 4.3.1-1)

(3) Earthquake Shear Forces in the Superstructure A procedure similar to that used for moments can be followed to calculate the seismic shear force demand in the superstructure. As in the case of moments, the shear forces in the superstructure member due to dead load, additional dead load, and secondary prestress are readily available from CTBridge output.

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The superstructure seismic shear forces due to seismic moments can be obtained directly from the wFRAME output or calculated by using the previously computed values of the superstructure seismic moments, M eqL and M eqR , for each span. (4) Moment and Shear Demand at Location of Interest The extreme seismic moment demand in the superstructure is calculated as the summation of all the moments obtained from the above sections, taking into account the proper direction of bending in each case as well as the effective section width. The superstructure demand moments at the adjacent left and right superstructure span are given by: L M DL  M dlL  M ps  M eqL

(21.2-7)

R M DR  M dlR  M ps  M eqR

(21.2-8)

Similarly, the extreme seismic shear force demand in the superstructure is calculated as the summation of shear forces due to dead load, secondary prestress effects and the seismic loading, taking into account the proper direction of bending in each case and the effective section width. The superstructure demand shear forces at the adjacent left and right superstructure spans are defined as:

VDL  VdlL  VpsL  VeqL

(21.2-9)

VDR  VdlR  VpsR  VeqR

(21.2-10)

As stated previously in this section, the secondary effect due to the prestress will be considered only when it results in an increased seismic demand. Dead load and secondary prestress moment and shear demands in the superstructure are proportioned on the basis of the number of girders falling within the effective section width. The earthquake moment and shear imparted by column is also assumed to act within the same effective section width. (5) Vertical Acceleration In addition to the superstructure demands discussed above, SDC Sections 2.1.3 and 7.2.2 require an equivalent static vertical load to be applied to the superstructure to estimate the effects of vertical acceleration in the case of sites with Peak Ground Acceleration (PGA) greater than or equal to 0.6g. For such sites, the effects of vertical acceleration may be accounted for by designing the superstructure to resist an additional uniformly applied vertical force equal to 25% of the dead load applied upward and downward.

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21.2.9.3

B

Superstructure Section Capacity (1) General To ensure that the superstructure has sufficient capacity to resist the extreme seismic demands determined in Section 21.2.9.2, SDC Section 4.3.2 requires the superstructure capacity in the longitudinal direction to be greater than the demand distributed to it (the superstructure) on each side of the column by the largest combination of dead load moment, secondary prestress moment, and column earthquake moment, i.e., sup(R ) M ne   M dlR  M pR/ s  M eqR

(SDC 4.3.2-1)

sup(L ) M ne   M dlL  M pL / s  M eqL

(SDC 4.3.2-2)

where: sup R , L = M ne

expected nominal moment capacity of the adjacent right (R) or left (L) superstructure span

(2) Superstructure Flexural Capacity MTD 20-6 (Caltrans 2001a) describes the philosophy behind the flexural section capacity calculations. Expected material properties are used to calculate the flexural capacity of the superstructure. The member strength and curvature capacities are assessed using a stress-strain compatibility analysis. Failure is reached when either the ultimate concrete, mild steel or prestressing ultimate strain is reached. The internal resistance force couple is shown in Figure 21.2-4. p/s se

se sa

sa Ap/s

As

s

Ts Tp/s MLn

ds

C ’s

Stress

Strain

c

Cc

Cs Cp/s d’s

s ’s c

c

MR n T’

’s A’s

Strain

Note: Axial forces not shown

V M

Stress

s

col o

col o

Figure 21.2-4 Superstructure Capacity Provided by Internal Couple

Chapter 21 – Seismic Design of Concrete Bridges

dc

dp/s

N/A

N/A

e

d’c

  cs

s

dp/s Cc

p/s

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Caltrans in-house computer program PSSECx or similar program, may be used to calculate the section flexural capacity. The material properties for 270 ksi and 250 ksi prestressing strands are given in SDC Section 3.2.4. According to MTD 20-6, at locations where additional longitudinal mild steel is not required by analysis, a minimum of #8 bars spaced 12 in. (maximum spacing) should be placed in the top and bottom slabs at the bent cap. The mild steel reinforcement should extend beyond the inflection points of the seismic moment demand envelope. As specified in SDC Section 3.4, the expected nominal moment capacity, Mne, for capacity protected concrete components shall be determined by either M- analysis or strength design. Also, SDC Section 3.4 specifies that expected material properties shall be used in determining flexural capacity. Expected nominal moment capacity for capacity-protected concrete members shall be based on the expected concrete and steel strengths when either the concrete strain reaches its ultimate value based on the stress-strain model or the reduced ultimate prestress steel strain,  suR = 0.03 is reached. In addition to these material properties, the following information is required for the capacity analysis:   

Eccentricity of prestressing steel - obtained from CTBridge output file. This value is referenced from the CG of the section. Prestressing force - obtained from CTBridge output file under the “P/S Response After Long Term Losses” Tables. Prestressing steel area, Aps - calculated for 270ksi steel as A ps 

 

P jack

(21.2-11)

(0.75)(270)

Reinforcement in top and bottom slab, per design including #8 @12. Location of top and bottom reinforcement, referenced from center of gravity of section, slab steel section depth and assumed cover, etc.

Both negative (tension at the top) and positive (tension at the bottom) capacities are calculated at various sections along the length of the bridge by the PSSECx computer program. The resistance factor for flexure, flexure = 1.0, as we are dealing with extreme conditions corresponding to column overstrength. (3) Superstructure Shear Capacity MTD 20-6 specifies that the superstructure shear capacity is calculated according to AASHTO Article 5.8. As shear failure is brittle, nominal rather than expected material properties are used to calculate the shear capacity of the superstructure.

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21.2.10

Joint Shear Design

21.2.10.1

General (1) Principal Stresses In a ductility-based design approach for concrete structures, connections are key elements that must have adequate strength to maintain structural integrity under seismic loading. In moment resisting connections, the force transfer across the joint typically results in sudden changes in the magnitude and nature of moments, resulting in significant shear forces in the joint. Such shear forces inside the joint can be many times greater than the shear forces in individual components meeting at the joint. SDC Section 7.4 requires that moment resisting connections between the superstructure and the column shall be designed to transfer the maximum forces produced when the column has reached its overstrength capacity, M 0col , including the effects of overstrength shear V0col . Accordingly, SDC Section 7.4.2 requires all superstructure/column moment-resisting joints to be proportioned so that the principal stresses satisfy the following equations: For principal compression, pc: pc  0.25 fc For principal tension, pt:

pt  12 fc

(psi) (psi)

(SDC 7.4.2-1) (SDC 7.4.2-2)

2

pt 

( f h  fv )  f  fv  2   h   v jv 2 2  

(SDC 7.4.4.1-1)

2

( f  fv )  f  fv  2 pc  h   h   v jv 2  2  T v jv  c A jv Ajv  lacBcap

fv 

Pc A jh

(SDC 7.4.4.1-2) (SDC 7.4.4.1-3) (SDC 7.4.4.1-4) (SDC 7.4.4.1-5)

Ajh  Dc  Ds Bcap

(SDC 7.4.4.1-6)

Pb Bcap Ds

(SDC 7.4.4.1-7)

fh  where: fh

= average normal stress in the horizontal direction (ksi)

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f Bcap Dc Ds lac Pc Pb

= = = = = = =

Tc

=

h

=

average normal stress in the vertical direction (ksi) bent cap width (in.) cross sectional dimension of column in the direction of bending (in.) depth of superstructure at the bent cap for integral joints (in.) length of column reinforcement embedded into the bent cap (in.) column axial force including the effects of overturning (kip) beam axial force at the center of the joint, including the effects of prestressing (kip) column tensile force (defined as M 0col h ) associated with the column overstrength plastic hinging moment, M 0col . Alternatively, Tc may be obtained from the moment-curvature analysis of the cross section (kip) distance from the center of gravity of the tensile force to the center of gravity of the compressive force of the column section (in.)

In the above equations, the value of f h may be taken as zero unless prestressing is specifically designed to provide horizontal joint compression. (2) Minimum Bent Cap Width – See Section 21.2.1.1 (3) Minimum Joint Shear Reinforcement SDC 7.4.4.2 specifies that, if the principal tensile stress, pt is less than or equal to 3.5 f c ' (psi) , no additional joint reinforcement is required. However, a minimum area of joint shear reinforcement in the form of column transverse steel continued into the bent cap shall be provided. The volumetric ratio of the transverse column reinforcement (  s , min ) continued into the cap shall not be less than:

 s, min 

3.5 f c f yh

(psi)

(SDC 7.4.4.2-1)

If pt is greater than 3.5 f c ' , joint shear reinforcement shall be provided. The amount and type of joint shear reinforcement depend on whether the joint is classified as a “T” joint or a Knee Joint. 21.2.10.2

Joint Description The following types of joints are considered as “T” joints for joint shear analysis (SDC Section 7.4.3):  

Integral interior joints of multi-column bents in the transverse direction All integral column-to-superstructure joints in the longitudinal direction

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

Exterior column joints for box girder superstructures if the cap beam extends beyond the joint (i.e. column face) far enough to develop the longitudinal cap reinforcement

Any exterior column joint that satisfies the following equation shall be designed as a Knee joint. For Knee joints, it is also required that the main bent cap top and bottom bars be fully developed from the inside face of the column and extend as closely as possible to the outside face of the cap (see SDC Figure 7.4.3-1). S  Dc where:

S

=

Dc

=

(SDC 7.4.3-1)

cap beam short stub length, defined as the distance from the exterior girder edge at soffit to the face of the column measured along the bent centerline (see Figure SDC 7.4.3-1), column dimension measured along the centerline of bent

Bent Cap Top and Bottom Reinforcement

S

Dc

Figure SDC 7.4.3-1 Knee Joint Parameters 21.2.10.3 T Joint Shear Reinforcement (1) Vertical Stirrups in Joint Region Vertical stirrups or ties shall be placed transversely within a distance Dc extending from either side of the column centerline. The required vertical stirrup area Asjv is given as

Asjv  0.2  Ast

(SDC 7.4.4.3-1)

where Ast = Total area of column main reinforcement anchored in the joint. Refer to SDC Section 7.4.4.3 for placement of the vertical stirrups.

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(2) Horizontal Stirrups Horizontal stirrups or ties, Asjh , shall be placed transversely around the vertical stirrups or ties in two or more intermediate layers spaced vertically at not more than 18 inches. Asjh  0.1  Ast (SDC 7.4.4.3-2) This horizontal reinforcement shall be placed within a distance Dc extending from either side of the column centerline. (3) Horizontal Side Reinforcement The total longitudinal side face reinforcement in the bent cap shall at least be equal to the greater of the area specified in SDC Equation 7.4.4.3-3. top 0.1 Acap  Assf  max  0.1 Abot cap 

(SDC 7.4.4.3-3)

where: Acap = area of bent cap top or bottom flexural steel (in.2). The side reinforcement shall be placed near the side faces of the bent cap with a maximum spacing of 12 inches. Any side reinforcement placed to meet other requirements shall count towards meeting this requirement. (4) “J” Dowels For bents skewed more than 20o, “J” bars (dowels) hooked around the longitudinal top deck steel extending alternately 24 in. and 30 in. into the bent cap are required. The J-dowel reinforcement shall be equal to or greater than the area specified as:

Asj bar  0.08Ast

(SDC 7.4.4.3-4)

This reinforcement helps to prevent any potential delamination of concrete around deck top reinforcement. The J-dowels shall be placed within a rectangular region defined by the width of the bent cap and the distance Dc on either side of the centerline of the column. (5) Transverse Reinforcement Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio specified as: 

  2  lac , provided  

 s  0.4

Ast

Chapter 21 – Seismic Design of Concrete Bridges

(SDC 7.4.4.3-5)

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where: area of longitudinal column reinforcement (in.2) actual length of column longitudinal reinforcement embedded into the bent cap (in.) For interlocking cores,  s shall be based on area of reinforcement Ast of each core. All vertical column bars shall be extended as close as possible to the top bent cap reinforcement. Ast lac

= =

(6) Anchorage for Main Column Reinforcement The main column reinforcement shall extend into the cap as deep as possible to fully develop the compression strut mechanism in the joint. If the minimum joint shear reinforcement prescribed in SDC Equation 7.4.4.2-1 is met, and the column longitudinal reinforcement extension into the cap beam is confined by transverse hoops or spirals with the same volumetric ratio as that required at the top of the column, the anchorage for longitudinal column bars developed into the cap beam for seismic loads shall not be less than:

lac, required  24dbl

(SDC 8.2.1-1)

With the exception of slab bridges where the provisions of MTD 20-7 shall govern, the development length specified above shall not be reduced by use of hooks or mechanical anchorage devices. 21.2.10.4

Knee Joint Shear Reinforcement Knee joints may fail in either “opening” or “closing” modes (see Figure SDC 7.4.5-1). Therefore, both loading conditions must be evaluated. Refer to SDC Section 7.4.5 for the description of Knee joint failure modes.

(a)

(b)

Figure SDC 7.4.5-1 Knee Joint Failure Modes Chapter 21 – Seismic Design of Concrete Bridges

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Two cases of Knee joints are identified as follows: Case 1:

S

Dc 2

Dc  S  Dc 2 The following reinforcement is required for Knee joints. Case 2:

(SDC 7.4.5.1-1) (SDC 7.4.5.1-2)

(1) Bent Cap Top and Bottom Flexural Reinforcement - Use for both Cases 1 and 2 The top and bottom reinforcement within the bent cap width used to meet this provision shall be in the form of continuous U-bars with minimum area: Asu barmin  0.33Ast

(SDC 7.4.5.1-3)

where: Ast

=

total area of column longitudinal reinforcement anchored in the joint (in.2)

The “U” bars may be combined with bent cap main top and bottom reinforcement using mechanical couplers. Splices in the “U” bars shall not be located within a distance, ld, from the interior face of the column. (2) Vertical Stirrups in Joint Region - Use for both Cases 1 and 2 Vertical stirrups or ties, Asjv as specified in SDC Equation 7.4.5.1-4, shall be placed transversely within each of regions 1, 2, and 3 of Figure SDC 7.4.5.1-1.

Asjv  0.2  Ast

(SDC 7.4.5.1-4)

The stirrups provided in the overlapping areas shown in Figure SDC 7.4.5.1-1 shall count towards meeting the requirements of both areas creating the overlap. These stirrups can be used to meet other requirements documented elsewhere including shear in the bent cap.

(3) Horizontal Stirrups - Use for both Cases 1 and 2 Horizontal stirrups or ties, Asjh , as specified in SDC Equation 7.4.5.1-5, shall be placed transversely around the vertical stirrups or ties in two or more intermediate layers spaced vertically at not more than 18 inches (see Figures SDC 7.4.4.3-2, 7.4.4.3-4, and 7.4.5.1-5 for rebar placement). Asjh  0.1 Ast

(SDC 7.4.5.1-5)

The horizontal reinforcement shall be placed within the limits shown in Figures SDC 7.4.5.1-2 and SDC 7.4.5.1-3.

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jv

in each of

As 1

2

Bent cap stirrups

3

2

CL Bent

3

Bcap Dc 1

CL Girder

S Dc/2

S < Dc

Figure SDC 7.4.5.1-1 Location of Knee Joint Vertical Shear Reinforcement (Plan View) (4) Horizontal Side Reinforcement- Use for both Cases 1 and 2 The total longitudinal side face reinforcement in the bent cap shall be at least equal to the greater of the area specified as:

Assf

top 0.1  Acap    or 0.1  Abot cap 

(SDC 7.4.5.1-6)

where: top Acap = Area of bent cap top flexural steel (in.2) bot Acap =

Area of bent cap bottom flexural steel (in.2)

This side reinforcement shall be in the form of “U” bars and shall be continuous over the exterior face of the Knee Joint. Splices in the U bars shall be located at least a distance ld from the interior face of the column. Any side reinforcement placed to

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meet other requirements shall count towards meeting this requirement. Refer to SDC Figures 7.4.5.1-4 and 7.4.5.1-5 for placement details. (5) Horizontal Cap End Ties for Case 1 Only The total area of horizontal ties placed at the end of the bent cap is specified as: Asjhcmin  0.33Asu bar

(SDC 7.4.5.1-7)

This reinforcement shall be placed around the intersection of the bent cap horizontal side reinforcement and the continuous bent cap U-bar reinforcement, and spaced at not more than 12 inches vertically and horizontally. The horizontal reinforcement shall extend through the column cage to the interior face of the column. (6) J-Dowels - Use for both Cases 1 and 2 Same as in Section 21.2.10.3 for T joints, except that placement limits shall be as shown in SDC Figure 7.4.5.1-3. (7) Transverse Reinforcement Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio as specified in SDC Equations 7.4.5.1-9 to 7.4.5.1-11.

s 

0.76 Ast Dclac, provided



(For Case 1 Knee joint)



Ast  2  lac, provided 

 s  0.4

(SDC 7.4.5.1-9)

(For Case 2 Knee joint, Integral bent cap) (SDC 7.4.5.1-10)





Ast  2  lac, provided 

 s  0.6

(For Case 2 Knee joint, Non-integral bent cap) (SDC 7.4.5.1-11)

where: lac,provided = Ast

=

Dc

=

actual length of column longitudinal reinforcement embedded into the bent cap (in.) total area of column longitudinal reinforcement anchored in the joint (in.2) diameter or depth of column in the direction of loading (in.)

The column transverse reinforcement extended into the bent cap may be used to satisfy this requirement. For interlocking cores, ρs shall be calculated on the basis of Chapter 21 – Seismic Design of Concrete Bridges

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Ast and Dc of each core (for Case 1 Knee joints) and on area of reinforcement, Ast of each core (for Case 2 Knee joints). All vertical column bars shall be extended as close as possible to the top bent cap reinforcement.

21.2.11

Torsional Capacity There is no history of damage to bent caps of Ordinary Standard Bridges from previous earthquakes attributable to torsional forces. Therefore, these bridges are not usually analyzed for torsional effects. However, non-standard bridge features (for example, superstructures supported on relatively long outrigger bents) may experience substantial torsional deformation and warping and should be designed to resist torsional forces.

21.2.12

Abutment Seat Width Requirements Sufficient seat width shall be provided to prevent the superstructure from unseating when the Design Seismic Hazards occur. Per SDC Section 7.8.3, the abutment seat width measured normal to the centerline of the bent, N A , as shown in Figure SDC 7.8.3-1 shall be calculated as follows: CL Brg. NA

p/s+cr+sh+temp

eq

4

Minimum Seat Width, NA = 30 in.

Figure SDC 7.8.3-1 Abutment Seat Width Requirements

N A   p / s  cr  sh  temp  eq  4

(in.)

(SDC 7.8.3-1)

where: NA

=

abutment seat width normal to the centerline of bearing. Note that for abutments skewed at an angle sk, the minimum seat width measured along the longitudinal axis of the bridge is NA/cos sk (in.)

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p/s

=

displacement attributed to pre-stress shortening (in.)

 cr  sh =  temp =

displacement attributed to creep and shrinkage (in.) displacement attributed to thermal expansion and contraction (in.)

Δeq

displacement demand, ΔD for the adjacent frame. Displacement of the abutment is assumed to be zero (in.)

=

The minimum seat width normal to the centerline of bearing as calculated above shall be not less than 30 in.

21.2.13

Hinge Seat Width Requirements For adjacent frames with ratio of fundamental periods of vibration of the less flexible and more flexible frames greater than or equal to 0.7, SDC Section 7.2.5.4 requires that enough hinge seat width be provided to accommodate the anticipated thermal movement (Δtemp), prestress shortening (Δp/s), creep and shrinkage (Δcr+sh), and the relative longitudinal earthquake displacement demand between the two frames (Δeq) - see Figure SDC 7.2.5.4-1. The minimum hinge seat width measured normal to the centerline of bent, N H is given by:



NH

where:  eq 

eq (iD)

= =



  p / s  cr  sh  temp  eq  4 in.   the larger of  or 24 (in.) 

     1 2 D

2 2 D

(SDC 7.2.5.4-1)

(SDC 7.2.5.4-2)

relative earthquake displacement demand at an expansion joint (in.) the larger earthquake displacement demand for each frame calculated by the global or stand-alone analysis (in.) NH

p/s+cr+sh + (Time-dependent terms)

eq temp

NH

4

 24

Figure SDC 7.2.5.4-1 Minimum Hinge Seat Width

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21.2.14

Abutment Shear Key Design

21.2.14.1

General According to SDC Section 7.8.4, abutment shear key force capacity, Fsk shall be determined as follows:

Fsk   (0.75V piles  Vww ) Fsk   Pdl 0.5    1

For Abutment on piles

(SDC 7.8.4-1)

For Abutment on Spread footing

(SDC 7.8.4-2) (SDC 7.8.4-3)

where:

V piles =

Sum of lateral capacity of the piles (kip)

Vww = Pdl =

Shear capacity of one wingwall (kip) Superstructure dead load reaction at the abutment plus the weight of the abutment and its footing (kip) factor that defines the range over which Fsk is allowed to vary

α

=

For abutments supported by a large number of piles, it is permitted to calculate the shear key capacity using the following equation, provided the value of Fsk is less than that furnished by SDC Equation 7.8.4-1:

Fsk  Pdlsup

(SDC 7.8.4-4)

where:

Pdlsup = 21.2.14.2

superstructure dead load reaction at the abutment (kip)

Abutment Shear Key Reinforcement The SDC provides two methods for designing abutment shear key reinforcement, namely, Isolated and Non-isolated methods. (1) Vertical Shear Key Reinforcement, Ask

Ask 

Fsk 1.8 f ye

Ask 

1 Fsk  0.4  Acv  1.4 f ye

 0.25 f ce' Acv   0.4 Acv  Fsk  min  1 . 5 A cv  

Chapter 21 – Seismic Design of Concrete Bridges

Isolated shear key Non-isolated shear key

(SDC 7.8.4.1A-1) (SDC 7.8.4.1A-2) (SDC 7.8.4.1A-3)

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Ask 

0.05Acv f ye

(SDC 7.8.4.1A-4)

where: Acv

=

area of concrete engaged in interface shear transfer (in.2)

In the above equations, f ye and f ce' have units of ksi, Acv and Ask are in in2, and Fsk is in kip. See SDC Figure 7.8.4.1-1 for placement of shear key reinforcement for both methods.

(2) Horizontal Reinforcement in the Stem Wall (Hanger Bars), Ash

Ash  (2.0) AskIso( provided)  iso (2.0) AskNon ( provided)  Ash  max Fsk f  ye

Isolated shear key

(SDC 7.8.4.1B-1)

Non-isolated shear key

(SDC 7.8.4.1B-2)

where:

AskIso( provided) =  iso AskNon ( provided) =

area of interface shear reinforcement provided in SDC Equation 7.8.4.1A-1(in.2) area of interface shear reinforcement provided in SDC Equation 7.8.4.1A-2 (in.2)

For the isolated key design method, the vertical shear key reinforcement, Ask should be positioned relative to the horizontal reinforcement, Ash to maintain a minimum length Lmin given by (see Figure SDC 7.8.4.1-1A):

Lmin,hooked  0.6(a  b)  ldh

(SDC 7.8.4.1B-3)

Lmin,headed  0.6(a  b)  3 in.

(SDC 7.8.4.1B-4)

where: a

=

b

=

ldh

=

vertical distance from the location of the applied force on the shear key to the top surface of the stem wall, taken as one-half the vertical length of the expansion joint filler plus the pad thickness (see Figure SDC 7.8.4.1-1(A)) vertical distance from the top surface of the stem wall to the centroid of the lowest layer of shear key horizontal reinforcement development length in tension of standard hooked bars as specified in AASHTO (2012)

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* Smooth construction joint is required at the shear key interfaces with the stemwall and backwall to effectively isolate the key except for specifically designed reinforcement. These interfaces should be trowel-finished smooth before application of a bond breaker such as construction paper. Form oil shall not be used as a bond breaker for this purpose.

(A) Isolatedy Shear Key

(B) Non-Isolated Shear Key NOTES: (a) Not all shear key bars shown (b) On high skews, use 2-inch expanded polystyrene with 1 inch expanded polystyrene over the 1-inch expansion joint filler to prevent binding on post-tensioned bridges.

Figure SDC 7.8.4.1-1 Abutment Shear Key Reinforcement Details

21.2.15

No-Splice Zone Requirements No splices in longitudinal column reinforcement are allowed in the plastic hinge regions (see SDC Section 7.6.3) of ductile members. These plastic hinge regions are called “No-Splice Zones,” and shall be detailed with enhanced lateral confinement and shown on the plans.

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In general, for seismic critical elements, no splices in longitudinal rebars are allowed if the rebar cage is less than 60 ft. long. Refer to SDC Section 8.1.1 for more provisions for “No-Splice Zones” in ductile members.

21.2.16

Seismic Design Procedure Flowchart 1. Select column size, column reinforcement, and bent cap width (SDC Sections 3.7.1, 3.8.1, 7.6.1, 8.2.5)

2. Perform cross-section analysis to determine column effective moment of inertia (Material properties - SDC Sections 3.2.6, 3.2.3) Ensure that dead load on column ~ 10 % column ultimate ' compressive capacity Ag f c

3. Check span configuration/balanced stiffness (SDC Section 7.1.1)

No Multi-frame bridge ?

Yes o 4. Check frame geometry (SDC Eq. 7.1.2-1)

5. Calculate minimum local displacement ductility capacity and demand (SDC Sections 3.1.3, 3.1.4, 3.1.4.1) Check that local displacement ductility capacity,  c  3

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6. Perform transverse pushover analysis (SDC Section 5.2.3, SDC Eq. 7.3.1.1-1) Check that: (a) displacement demand,  D < displacement capacity,  C (SDC Eq. 4.1.1-1) (b)  D  target value from SDC Section 2.2.4

7. Perform longitudinal pushover analysis (SDC Sections 5.2.3, 7.8.1) Check that: (a)  D <  C (b)  D  target value from SDC Section 2.2.4 8. Check P-  effects in transverse and longitudinal directions (SDC Eq. 4.2-1)

9. Check bent minimum lateral strength in transverse and longitudinal directions (SDC Section 3.5)

10. Perform column shear design in transverse and longitudinal directions (SDC Sections 3.6.1, 3.6.2, 3.6.3, 3.6.5, 4.3.1)

Are column bases pinned?

No

Yes No 11. Design column shear key (SDC Section 7.6.7)

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12. Check bent cap flexural and shear capacity (AASHTO Article 5.8, SDC Sect. 3.4, SDC Eq. 7.3.1.1-1)

13. Calculate column seismic load moments (SDC Sect. 4.3.2, MTD 20-6, SDC Eq. 4.3.1-1) col @ soffit @ soffit M eq  M 0col @ soffit  M dlcol @ soffit  M col ps





14. Distribute column seismic moments into the superstructure (S/S) to obtain S/S seismic demands (Perform right and left Pushover analyses) ; ; 15. Calculate S/S moment demands at location of interest (SDC Eq. 7.2.1.1-1, MTD 20-6) L R M DL  M dlL  M ps  M eqL ; M DR  M dlR  M ps  M eqR

16. Calculate S/S shear demands at location of interest (SDC Eq. 7.2.1.1-1, MTD 20-6) L L VD  VdlL  V ps  VeqL ; VDR  VdlR  VpsR  VeqR

Is Site PGA  0.6g ?

No

Yes o 17. Perform vertical acceleration analysis (SDC Sections 7.2.2 and 2.1.3)

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18. Calculate superstructure flexural and shear Capacity (MTD 20-6, AASHTO Article 5.8)

19. Design joint shear reinforcement (SDC Sections 7.4.2, 7.4.4, 7.4.4.3, 7.4.5.1) .)

Multi-frame bridge ? Yes

No

Yes 20. Determine minimum hinge seat width (SDC Section 7.2.5.4)

No Seat type abutments ? Yes o 21. Determine minimum abutment seat width (SDC Section 7.8.3)

22. Design abutment shear key reinforcement (SDC Section 7.8.4)

23. Check requirements for No-splice Zone (SDC Section 8.1.1, MTD 20-9)

END

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21.3

DESIGN EXAMPLE - THREE-SPAN CONTINUOUS CASTIN-PLACE CONCRETE BOX GIRDER BRIDGE

21.3.1

Bridge Data The three-span Prestress Reinforced Concrete Box Girder Bridge shown in Figure 21.3-1 will be used to illustrate the principles of seismic bridge design. The span lengths are 126 ft, 168 ft and 118 ft. The column height varies from 44 ft at Bent 2 to 47 ft at Bent 3. Both bents have a skew angle of 20 degrees. The columns are pinned at the bottom. The bridge ends are supported on seat-type abutments. Material Properties: Concrete: Reinforcing steel:

fc  4 ksi A706, f y  60 ksi ; Es  29,000 ksi ; f ye  68 ksi ;

fue  95 ksi

Bridge Site Conditions: This example bridge crosses a roadway and railroad tracks. Because of poor soil conditions, the footing is supported on piles. The ground motion at the bridge site is assumed to be: Soil Profile: Type C Magnitude: 8.0  0.25 Peak Ground Acceleration: 0.5g Figure 21.3-2 shows the assumed design spectrum. For more information on Design Spectrum development, refer to SDC Section 2.1.1 and Appendix B.

21.3.2

Design Requirements Perform seismic analysis and design in accordance with Caltrans SDC Version 1.7 (Caltrans 2013).

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Figure 21.3-1 General Plan (Bridge Design Academy Prototype Bridge) Chapter 21 – Seismic Design of Concrete Bridges

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Figure 21.3-2 Design Spectrum for Soil Profile C (M = 8.00.25)

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21.3.3

Step 1- Select Column Size, Column Reinforcement, and Bent Cap Width

21.3.3.1

Column size Given Ds  6.75 ft from the strength limit state design, we select a column width OK. (SDC 7.6.1-1) Dc  6.00 ft so that 0.70  Dc / Ds  0.89  1.00 .

21.3.3.2

Bent Cap Width

Bcap  Dc  2 = 6 + 2 = 8 ft 21.3.3.3

(SDC 7.4.2.1-1)

Column Longitudinal and Transverse Reinforcement

  As  0.015Ag  0.015 6.00122  0.0154071.5  61.07 in.2 4 Use: #14 bars for longitudinal reinforcement #8 hoops @ 5 in c/c for the plastic hinge region Maximum spacing of hoops = 5 in. < 8 in. < 6×1.693 = 10.2 in. < 72/5 = 14.4 in. OK. (SDC Section 8.2.5) 61.07  27.1 Number of #14 bars = 2.25 Let us use 26-#14 longitudinal bars (i.e., 1.44% of Ag ) 1.0  1.44  4.0 OK. (SDC 3.7.1-1/3.7.2-1) Assuming a concrete cover of 2 in. as specified in CA Amendment Table 5.12.3-1 for minimum concrete cover (Caltrans 2014). Diameter of longitudinal reinforcement loop (from centerline to centerline of longitudinal bars):  1.88  d M = 72  2(2)  2(1.13)  2   63.86 in.  2   dM  7.7 in. > 1.5(1.693) in. > 1.5 in.  Spacing of longitudinal bars = 26 OK. (AASHTO 5.10.3.1) Note: If the provided spacing turns out to be more that the maximum spacing allowed, then a smaller bar size can be used.

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21.3.4

Step 2 - Perform Cross-section Analysis

21.3.4.1

Calculate Dead Load Axial Force As a first step toward calculating effective section properties of the column, the dead load axial force at column top (location of potential plastic hinge) is calculated. These column axial forces are obtained from CTBridge output. It should also be noted that these loads do not include the weight of the integral bent cap. The CTBridge model has the regular superstructure cross-section with flared bottom slab instead of solid cap section. In this example, weight of the whole solid cap was added to the CTBridge results (conservative). As read from the CTBridge output results, the column dead load axial forces are: Column 1 1,489 1,425

Bent 2 (Pc) (kip) Bent 3 (Pc) (kip)

Average Bent Cap Length =

Column 2 1,494 1,453

 Deck Width  Soffit Width  1   2  cos(Skew Angle) 

=

 49.83  43.08  1    49.44ft 0   2  cos(20 ) 

Bent Cap Weight = 8(6.75)(49.44)(0.150)  400 kips Adding this bent cap weight, the total axial force in each column becomes:

Bent 2 (Pc) (kip) Bent 3 (Pc) (kip)

21.3.4.2

Column 1 1,689 1,625

Column 2 1,694 1,653

Check Column Dead Load Axial Force Ratio Using Column 2 of Bent 2 (worst case):

Chapter 21 – Seismic Design of Concrete Bridges

1694100%  10.4 % ~ 10% 4071.5 4 

OK.

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21.3.4.3

Material and Section Properties for Section Analysis Using xSECTION Program Expected compressive strength of concrete f ce'  1.3(4,000)  5,200 psi * > 5,000 psi

OK.

(SDC 3.2.6-3)

* The xSECTION input file was originally created with the value of f ce'  5.28 ksi. The resulting values of ductility parameters are not significantly different from the corresponding values obtained using f ce'  5.20 ksi. Therefore, the results with f ce'  5.28 ksi are retained.

Other concrete properties used are listed in SDC Section 3.2.6. The following values are used as input to xSECTION program:    

Column Diameter = 72.0 in. Concrete cover = 2 in. Main Reinforcement: #14 bars, total 26. Lateral Reinforcement: #8 hoops @ 5 in c/c, f ce'  5,200 psi * The program calculates the modulus of elasticity of concrete internally.

For Grade A706 bar reinforcing steel, 0.09 0.06

T ransverse steel Longitudinal steel



R  su 



Select Output for Bent 2 Column xSECTION run is shown in Appendix 21.3-1. Moment-Curvature ( M   ) diagram for Bent 2 Column is shown in Appendix 21.3-2. Bent 2 Column Axial Force, Pc = 1,694 kips. Bent 3 Column Axial Force, Pc = 1,653 kips.

  

From M-ϕ analysis results, cracked moment of inertia, Ie = 23.717 ft4 for Bent 2 columns (See Appendices 21.3-1 and 21.3-2). For Bent 3, Ie = 23.612 ft4.

21.3.5

Step 3 - Check Span Configuration/Balanced Stiffness

21.3.5.1

Bent 2 Stiffness

 331501.5 5,200  Ec  33wc 1.5 f c' ( psi)     4,372 ksi 1,000   k2e  (2)

(SDC 3.2.6-1)

 (3)(4,372)(23.717)(124 )  3EIe  ( 2 )    87.64 kip/in. L3 (44  (12))3  

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21.3.5.2

Bent 3 Stiffness  (3)(4,372)(23.612)(124 )  k3e  (2)    71.59 kip/in. (47(12))3   (2)(1,694) m2 = Total tributary mass at Bent 2 =  8.77 kip  s 2 /in. (32.2)(12) (2)(1,653) m3 = Total tributary mass at Bent 3 =  8.56 kip  s 2 /in. (32.2)(12)

kie 71.59   0.82  0.75  0.5 k ej 87.64

OK.

(SDC 7.1.1-1 and 7.1.1-3)

It is seen that the balanced stiffness criteria and span layout configuration are satisfied. Note that since this is a constant width bridge with only two bents and two columns in each bent, we only need to satisfy the more onerous of SDC Equations (7.1.1-1) and (7.1.1-3).

21.3.6

Step 4 - Check Frame Geometry Since this is a single-frame bridge, this step does not apply.

21.3.7

Step 5 – Calculate Minimum Local Displacement Ductility Capacity and Demand

21.3.7.1

Displacement Ductility Capacity (1) Bent 2 Columns L = 44 ft Y  0.000078rad/in. as read from the M   data listed in Appendix 21.3-1. Lp  0.08L  0.15 f ye dbl  0.3 f ye dbl

 0.08(528)  0.15(68)(1.693)  59.51in.  0.3(68)(1.693)  34.54in. OK. (SDC 7.6.2.1-1) 2 L  1 (SDC 3.1.3-2) Y   Y  (528) 2 (0.000078)  7.25 in.  3  3   Plastic curvature,  p  0.000747 rad/in. (See M   data shown in Appendices

21.3-1 and 21.3-2). Plastic rotation,  p  L p p  59.51 0.000747 0.044454 rad.

Chapter 21 – Seismic Design of Concrete Bridges

(SDC 3.1.3-4)

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Lp   59.51    0.044454 528  Plastic displacement,  p   p  L    22.15 in. 2  2    (SDC 3.1.3-8) Total Displacement Capacity, c  Y   p  7.25  22.15  29.40 in. (SDC 3.1.3-1)  c  29.40  Local displacement ductility capacity, c  =   4.1  3 Y  7.25  OK. (SDC Section 3.1.4.1) (2) Bent 3 Columns Similarly,  p = 24.93 in., Y = 8.27 in.

c  21.3.7.2

 c  33.20  =   4.0  3 Y  8.27 

OK.

Displacement Ductility Demand (1) Bent 2 The period of fundamental mode of vibration is as: m2 8.77 T2  2  2  1.99 sec. e 87.64 k2 From the Design spectrum shown in Figure 21.3-2, the value of spectral acceleration for T = 1.99 sec is read as: a2  0.36g

ma

8.77(0.36)(32.2)(12)  13.92 in. 87.64 ke 13.92  1.9  5 OK. (SDC Section 2.2.4) Displacement Demand ductility,  D  7.25 Displacement demand,  D 



(2) Bent 3 Similarly, for Bent 3, T3  2.17 sec. The longer period is expected because Bent 3 columns are longer. The corresponding value of spectral acceleration, a3  0.33g (Figure 21.3-2) Displacement demand,  D 

8.56(0.33)(32.2)(12)  15.25 in. 71.59

Displacement Demand ductility,  D 

Chapter 21 – Seismic Design of Concrete Bridges

15.25  1.8  5 8.27

OK.

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21.3.8

Step 6 – Perform Transverse Pushover Analysis

21.3.8.1

Modeling Figure 21.3-3 shows a schematic model of the frame in the transverse direction. Data used for the soil springs are shown in Appendix 21.3-3. The following values of column effective section properties for Bent 2 and idealized plastic moment capacity (under dead loads only) obtained from xSECTION output (see Appendix 21.3-1) are used as input in wFRAME program for pushover analysis. Pc (kip)

Mp (kip-ft)

Ie (ft4)

ϕy ( rad/in.)

ϕp ( rad/in.)

1,694

13,838

23.717

0.000078

0.000747

Appendices 21.3-4 and 21.3-5 show select portions of xSECTION output for the cap section for positive and negative bending, respectively. The following section properties are used for the wFRAME run: ve ve  55.57 ft 4 , I eff  48.94 ft 4 A  62.62 ft 2 *, I eff

*Note that per SDC Equation 7.3.1.1-1, the value of A (effective bent cap cross sectional area) would be 66.62 ft2. The value of 62.62 ft4 used is based on effective bent cap overhang width of 34 in. required by California Amendment Article 4.6.2.6.1 (Caltrans 2014). However, any errors introduced by using A = 62.62 ft4 instead of A = 66.62 ft4 would result in a conservative design. As the frame is pushed toward the right, the resulting overturning moment causes redistribution of the axial forces in the columns. This overturning causes an additional axial force on the front side column, which will experience additional compression. The column on the backside experiences the same value in tension, reducing the net axial load. Based on their behavior, these columns are usually known as compression and tension columns, respectively. At the instant the first plastic hinge forms (in this case at the top of the compression column), the following superstructure displacement and corresponding lateral force values are obtained from the wFRAME output (see Appendix 21.3-6):

 y  8.49 in. Corresponding lateral force = 0.171(3,382) = 578 kips, where, 3,382 kips is the total tributary weight on the bent. At this stage, the axial forces in tension and compression columns as read from the wFRAME analysis output are 907 kips and 2,474 kips, respectively.

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3.38

Rigid Links

35.80 7.72*

7.72* 34*

L OOSE Sand SAND: Loose N=10, , o, N = 10,ϕ=30 = O30 KK==2525PCI pci

3.28

8.20

Medium Dense Sand MEDIUM D ENSE SAND N = 20, ϕ = 33o, N=20, =33 degrees K = 150 pci K=150 pci

* Dimensions along the skewed bent line

Figure 21.3-3 Transverse Pushover Analysis Model These values can be quickly checked using simple hand calculations as described below: M overturning  57844  25,432 kip - ft Axial compression corresponding to M overturning , P  

Chapter 21 – Seismic Design of Concrete Bridges

25,432  748 kips 34

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The axial force in the compression column will increase to 1,694 + 748 = 2,242 kips. The tension column will see its axial compression drop to 1,694-748 = 946 kips. These values compare very well with the wFRAME results. The small differences are probably due to the presence of soil in the more realistic wFRAME model. Column section properties corresponding to the updated axial forces (i.e. with overturning) are obtained from new xSECTION runs and summarized in the table below (see Appendices 21.3-7 and 21.3-8 for select portions of the output for the compression and tension columns, respectively).

Column Type

Pc (kip)

Mp (kip-ft)

Ie (ft4)

ϕy ( rad/in.)

ϕp ( rad/in.)

Tension

907

12,636

21.496

0.000079

0.000836

Compression

2,474

14,964

25.572

0.000079

0.000682

Note that higher compression produces a higher value of Mp but a reduction in p. This trend occurs in all columns and is a reminder that Mp is not the only indicator of column performance. With updated values of Mp and Ie, we run a second iteration of the wFRAME program. As the frame is pushed laterally, the compression column yields at the top at a displacement y(1) = 8.79 inches. The tension column has not reached its capacity yet. See Appendix 21.3-9 for these results. At this stage, the column axial forces are read to be 880 kips and 2,502 kips for tension and compression columns, respectively. Since, the change in column axial load is now less than 5%, there is no need for further iteration. As the frame is pushed further, the already yielded compression column is able to undergo additional displacement because of its plastic hinge rotational capacity. As the bent is pushed further, the top of the tension column yields at a displacement, y(2) = 10.52 in (see Appendix 21.3-9). At this point the effective bent stiffness approaches zero and will not attract any additional force if pushed further. The bent, however, will be able to undergo additional displacement until the rotational capacity of one of the hinges is reached. The force-displacement relationship is shown in Appendix 21.3-10. The idealized yield y, which was calculated previously based upon the assumption that cap beam is infinitely rigid, is updated to 8.79 inches. The corresponding lateral force = 0.176(3,382) = 595 kips.

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21.3.8.2

Displacement Ductility Capacity The main purpose of the preliminary calculation for C was to size up the members and ensure that they meet the minimum local displacement ductility capacity of 3 before proceeding with the more realistic and comprehensive pushover analysis that includes the effects of bent cap flexibility. The displacement capacities for both columns are calculated as before (see Step 5) using updated values of ϕp, and summarized below: Tension Column

Compression Column

L = 44 ft, Lp = 59.51 in.

L = 44 ft, L p  59.51 in.

ϕp = 0.000836 rad/in.

ϕp = 0.000682 rad/in.

Δp = 24.79 in.

Δp = 20.22 in.

Δc = 10.52 + 24.79 = 35.31 in.

Δc = 8.79 + 20.22 = 29.01 in.

For bents having a large number of columns or more locations for potential hinging, tabulation of these results provides a quick way to determine the critical hinge. Hinge Location Compression Column Top Tension Column Top

Yield Displacement (in.)

Plastic Deformation (in.)

Total Displacement Capacity (in.)

8.79

20.22

29.01*

10.52

24.79

35.31

* Critical bent displacement capacity, C. 21.3.8.3

Displacement Ductility Demand (1) Bent 2 k2e 

Fy y

T  2



595 k  67.69 8.79 in

8.77  2.26 sec 67.69

From the Design Spectrum (DS) curve, the spectral acceleration a2 is read as 0.32g. The maximum seismic displacement demand is estimated as:

8.77  (0.32  32.2  12)  16.02 in. 67.69 16.02 D   1.82 < 5 8.79

D 

Chapter 21 – Seismic Design of Concrete Bridges

OK. (SDC Section 2.2.4)

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Also, D  16.02 in.  C  29.01 in.

OK. (SDC 4.1.1-1)

Note that the bent is forced well beyond its yield displacement but that collapse is prevented because of ductile capacity. This is what we expect of Caltrans “No Collapse” Performance Criteria. Based upon these checks one might conclude that the column is over designed for the anticipated seismic demand. However, as shown later, the P- effect controls the column flexural design. The above calculation was made assuming the bent stiffness equals the stiffness at first yield. This assumption is valid because the two hinges occurred close to each other (i.e., 8.79 in and 10.52 in). If this assumption is not valid, a more exact calculation may be carried out using idealized stiffness as follows (see Figure 21.34): Fy 640 k 2e    66.67 kip/in. y 9.6

T  2

8.77  2.28 sec . 66.67

a2 = 0.3g;

D 

D 

8.77(0.32)(32.2)(12)  16.27 in. 66.67

16.27  1.69  5 9.6

OK.

 D  16.27 in.  C  29.01in. OK.

640 595 640 Force (kip)

8.79 9.60

10.52

Displacement (in.)

Figure 21.3-4 Force – Displacement Relations

Chapter 21 – Seismic Design of Concrete Bridges

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(2) Bent 3 The same procedure is repeated to perform transverse pushover analysis for Bent 3. The results are summarized below: Tension Column L  47 ft, L p  62.39 in.

Compression Column L  47 ft, L p  62.39 in.

 p  0.000842rad/in.

 p  0.000685rad/in.

 p  27.99 in.

 p  22.77 in.

 c  11.48  27.99  39.47 in.

 c  9.71  22.77  32.48 in.*

* Critical bent displacement capacity, C. Seismic Demand ke3 

Fy y



0.180(3,278)  60.77 kip/in. 9.71

8.56  2.36 sec 60.77 From Design Spectrum, the spectral acceleration a3 is read as 0.31g. The period of vibration, T  2

8.56(0.31)(32.2)(12)  16.87 in. 60.77 16.87 D   1.74  5 9.71

D 

Also, D  16.87 in.  C  32.48 in.

21.3.9

Step 7- Perform Longitudinal Pushover Analysis

21.3.9.1

Abutment Soil Springs

OK. OK.

This bridge is supported on seat type abutments (see Figure 21.3-5 for effective abutment dimensions). The effective area is calculated as: Ae  hbwwbw  6.75(46.46)  313.6ft 2 ( wbw  (49.83  43.08) / 2  46.46ft )

h   6.75  Pw  Ae (5 ) bw   (313.6)(5)   1,924 kips  5.5   5.5 

Chapter 21 – Seismic Design of Concrete Bridges

(SDC 7.8.1-4)

(SDC 7.8.1-3)

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wbw

Figure 21.3-5

hbw

Effective Area of Seat Type Abutment

Using initial embankment fill stiffness,  kips/in.  Ki  50    ft  Initial abutment stiffness

(SDC 7.8.1-1)

 h   6.75  K abut  Ki w   50(46.46)   2,851 kip/in.  5.5   5.5 



(SDC 7.8.1-2)

F 1,924   0.67 in. (See Figure 21.3-6) K 2,851

effective     gap  0.67  2.60  3.27 in.  0.272ft See Appendix 21.3-11 for calculations for gap, the combined effect of thermal movement and anticipated shortening. Average contributory length is used in the calculation for gap. Abut Kinitial 

1,924  588 kip/in. 7,061 kip/ft 3.27

1,924 kip 588 kip/in.

2.6 in.

B

0.67 in.

Figure 21.3-6 Initial Abutment Stiffness Iteration

Chapter 21 – Seismic Design of Concrete Bridges

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This value is used as the starting abutment stiffness for the longitudinal pushover analysis. When the structure has reached its plastic limit state (i.e., when both bents 2 and 3 columns have yielded), the longitudinal bridge stiffness is calculated as follows: 0.38(8,430) klong   351 kip/in. 9.13 (See Appendix 21.3-12 for the force-deflection curve for Right Push). Mass, m 

T  2

W 8,430   21.82 kip  s 2 /in. g 32.2  12

m klong

 2

21.82  1.57 sec 351

Sa  0.48g F ma 21.82(0.48)(32.2)(12) D     11.53 in. K K 351

RA 

D  effective



11.53  3.53 3.27

Abut Abut  Kinitial  1.0  0.45( RA  2) Since 2 < R A < 4, K final

(SDC Section 7.8.1)

Abut K final  588(0.312)  183 kip/in.  2196kip/ft

The following stiffness values as shown in Figure 21.3-7 shall be used for all subsequent wFRAME longitudinal pushover analyses: K1 = 2,196 kip/ft and 1 = 0.272 ft K2 = 0 kip/ft and 2 = 1.0 ft

K2 K1

0.272 ft

1.0 ft

Figure 21.3-7 Final Abutment Stiffness Chapter 21 – Seismic Design of Concrete Bridges

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21.3.9.2

Displacement Ductility Capacity and Demand From the wFRAME results (see Appendix 21.3-13 for the force-displacement relationship for the right push), the yield displacements of Bent 2 and Bent 3 are:

Location Bent 2 Bent 3

Yield Displacement (Right Push) (in.) 8.86 9.11

Yield Displacement (Left Push) (in.) 8.36 9.84

The plastic deformation capacities for both Bent 2 and Bent 3 are exactly the same as calculated for the transverse bending for the case of gravity loading. This is because the longitudinal case has very little overturning to change the column axial loads.





p = 22.15 in. for Bent 2 and p = 24.93 in. for Bent 3.

(1) Bent 2

Min c 

c  8.86  22.15  =   3.5  3 Y  8.86 

OK.

(SDC Section 3.1.4)

(2) Bent 3

Min c 

c  9.84  24.93     3.5  3 y  9.84 

OK.

(SDC Section 3.1.4)

From wFRAME force-displacement relationship of Appendix 21.3-13, the bridge longitudinal stiffness is calculated when the first bent has yielded. 0.22(8,430) klong   209 kip/in. 8.86 T = 2.03 sec for which Sa = 0.35g 





D = 14.12 in.

This demand is the same at Bents 2 and 3 because the superstructure constrains the bents to move together. This might not be the case when the bridge has significant foundation flexibility that can result from rotational and/or translational foundation movements. 14.12 Max  D   1.7  5 (Bent 2) OK. (SDC Section 2.2.4) 8.36 14.12 Max  D   1.5  5 (Bent 3) OK. (SDC Section 2.2.4) 9.11

Chapter 21 – Seismic Design of Concrete Bridges

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21.3.10

Step 8 - Check P - Δ Effects

21.3.10.1

Transverse direction We have relatively heavily loaded tall columns. P-effects could be significant for this type of situation. (1) Bent 2 Columns Pdl = 1,694 kips, M p = 13,838 kip-ft, Maximum Seismic Displacement Δr = 16.02 in. Pdl  r 1,694(16.02)   0.16  0.20 13,838(12) M col p

OK.

(SDC 4.2-1)

(2) Bent 3 Columns Pdl = 1,653 kips, M p = 13,777 kip-ft, Maximum Seismic Displacement Δr = 16.87 in. Pdl  r 1,653(16.87)   0.17  0.20 13,777(12) M col p

OK.

(SDC 4.2-1)

Now we can see that although the selected column section has more than enough ductility capacity, the column sections meet the P- requirements only by a small margin. 21.3.10.2

Longitudinal Direction (1) Bent 2 Columns Pdl  r 1,694(14.12)   0.14  0.20 13,838(12) M col p

OK.

(SDC 4.2-1)

OK.

(SDC 4.2-1)

(2) Bent 3 Columns Pdl  r 1,653(14.12)   0.14  0.20 13,777(12) M col p

21.3.11

Step 9 - Check Bent Minimum Lateral Strength

21.3.11.1

Transverse direction From the force deflection data shown in Appendix 21.3-10,

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Minimum lateral strength per bent = 0.193,383  643kips  0.1(3,383)  338 kips

2.3.11.2

OK.

(SDC Section 3.5)

Longitudinal Direction From the force deflection data shown in Appendix 21.3-13, Minimum lateral strength per column =  8,430   8,430  0.22 OK.   927 kips  0.1   422 kips  2   2 

21.3.12

Step 10 - Perform Column Shear Design

21.3.12.1

Transverse Bending

(SDC Section 3.5)

(1) Bent 2

M 0  1.2M p  1.2(14,964)  17,957 kip  ft (includes overturning effects). Shear demand associated with overstrength moment is as: M 17,957 V0  0   408 kips L 44 Alternatively, from wFRAME output (see Appendix 21.3-9), the maximum column shear demand = 1.2(349)  419 kips. The presence of soil around the footing results in a slightly shorter effective column length, which in turn causes slightly higher column shear demand in the wFRAME output. Concrete Shear Capacity, Vc For #8 hoops @ 5 in o.c., Ab  0.79 in.2 , D'  72  2  2 

1.13 1.13   66.87 in. , s  5 in. 2 2

4 Ab  0.009451 D's f yh = 60 ksi

s 

(SDC 3.8.1-1)

s f yh  0.009451(60)  0.57  0.35

 Use  s f yh  0.35 ksi

(SDC Section 3.6.2)

Using the maximum value of the displacement ductility demand, d = 1.82 (see calculation for Bent 2 – Transverse pushover analysis), the shear capacity factor f1 is calculated as:

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f1 

 s f yh 0.150

 3.67   d 

0.35  3.67  1.82  4.18  3 0.150

 Use f1 = 3 f2  1 

Pc 1 2,000Ag

 Use f 2  1.11

(SDC 3.6.2-5)

88010  1.11  1.5   2 2,000 6  (12)  4 3

OK.

It is seen from the equations for concrete shear capacity, that the plastic hinge region is more critical as the capacity will be lower in this region. Furthermore, the shear capacity is reduced when the axial load is decreased. The controlling shear capacity will be found in the tension column. vc  f1 f 2 f c'  3(1.11) 4,000  211psi  4 4,000  253 psi

OK.

  Ae  0.8 6(12) 2  3,257in.2 4 Vc  vc Ae  211(3,257)  687,227 lb  687 kips Transverse Reinforcement Shear Capacity, Vs

 nAv f yh D'    0.79(60)(66.87)    Vs     996 kips  2 2 s 5    Maximum shear strength is as:

Vs, max  8 f c' Ae  8 4,0003,257 / 1,000  1,648 kips  996 kips Minimum shear reinforcement is as:  D's   Av, min  0.025  f yh     66.87(5)  2 2  0.025   0.14 in.  0.79 in.  60  Shear capacity

OK.

(SDC 3.6.5.1-1)

OK.

(SDC 3.6.5.2-1)

Vn  0.9Vc  Vs   0.9(687  996)  1,515 kips  V0  419 kips

Chapter 21 – Seismic Design of Concrete Bridges

OK.

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(2) Bent 3

V0 

M 0 1.2(14,893)   380 kips L 47

From the wFRAME analysis results, the maximum column shear demand = 1.2 × 340 = 408 kips. Going through a similar calculation as was done for Bent 2 columns, we determine that

Vn  0.9Vc  Vs   0.9(681 996)  1,509 kips  V0  408 kips 21.3.12.2

OK.

Longitudinal bending (1) Bent 2 V0 = 1.2Vp = 1.2(645/2) = 377 kips

This corresponds to the maximum shear value of Vp = 323 kips/column obtained from the wFRAME pushover analysis. For  D = 1.7, f 1 = 4.3 > 3. Use f 1 = 3. For dead load axial force, factor f 2 = 1.21 vc = 230 psi which gives Vc = 749 kips Vs = 996 kips as calculated before.

Vn  0.9(749  996)  1,571 kips  V0  307 kips

OK.

(2) Bent 3 V0 = 1.2Vp = 1.2(629/2) = 378 kips

This corresponds to the maximum shear value of Vp = 315 kips/column obtained from the wFRAME pushover analysis. For  D = 1.5, factor 1 = 4.5 > 3. Use f 1 = 3 For dead load axial force, factor f 2 = 1.20 

vc = 228 psi which gives Vc = 743 kips Vs = 996 kips as calculated earlier.

Vn  0.9(743  996)  1,565 kips  V0  378 kips

Chapter 21 – Seismic Design of Concrete Bridges

OK.

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21.3.13

Step 11- Design Column Shear Key

21.3.13.1

Determine Shear Key Reinforcement Since the net axial force on both columns of Bent 2 is compressive, the area of interface shear key, required Ask is given by

Ask 

1.2( Fsk  0.25P) fy

(SDC 7.6.7-1)

P = 815 kips (for column with the lowest axial load) – see Appendix 21.3-9 Shear force associated with column overstrength moment is as: Fsk = shear force associated with column overstrength moment

1.2349  419 kips For Bent 2 Fsk   1.2340  408 kips For Bent 3

See Step 10 – Perform Column Shear Design and Appendix 21.3-9. Therefore, Fsk = 419 kips

1.2419  0.25(815)  4.3 in.2  4 in.2 OK. 60 Provide 6#8 dowels in column key ( Ask , provided = 4.74 in.2 > 4.3 in.2 OK.) Ask 

Dowel Cage diameter: Preferred spacing of #8 bars = 4.25 in. (see BDD 13-20) Diameter of dowel cage = (6)(4.25)/ = 8.1 in. say 9 in. cage 21.3.13.2

Determine Concrete Area Engaged in Shear Transfer, Acv

Acv 

4.0(419)  419 in.2 4

4.0 Fsk = f c'

(SDC 7.6.7-3)

Acv  0.67Fsk  281 in.2

(SDC 7.6.7-4)

Per SDC Section 7.6.7, Acv must not be less than that required to meet the axial resistance requirements specified in AASHTO Article 5.7.4.4 (AASHTO 2012).



Pn   0.85 0.85 f c' ( Ag  Ast )  f y Ast



(AASHTO 5.7.4.4-2)

Using the largest axial load with overturning effects P = 2,567 kips (see Appendix 21.3-9) and  = 1 (seismic), we have:





Pn  1.00.85 0.854( Ag  4.74)  604.74  2,567 kips Ag = 809 in.2 > 419 in.2

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Therefore, Acv,reqd  809 in.2 Diameter of Acv 

8094  32 

in.

Use Acv diameter = 32 in. (see Figure 21.3-8)

9 in. Cage 6-#8 Dowel s Acv Diameter = 32 in.

Figure 21.3-8 Column Shear Key

21.3.14

Step 12 - Check Bent Cap Flexural and Shear Capacity

21.3.14.1

Check Bent Cap Flexural Capacity The design for strength limit states had resulted in the following main reinforcement for the bent cap: Top Reinforcement 22 - #11 rebars Bottom Reinforcement 24 - #11 rebars Ignoring the side face reinforcement, the positive and negative flexural capacity of the bent cap is estimated to be M+ve = 21,189 kip-ft and M-ve = 19,436 kip-ft. Appendices 21.3-4 and 21.3-5 show these values, which are based on when either the R concrete strain reaches 0.003 or the steel strain reaches  SU as required for capacity protected members (See SDC Section 3.4). The seismic flexural and shear demands in the bent cap are calculated corresponding to the column overstrength moment. These demands are obtained from

Chapter 21 – Seismic Design of Concrete Bridges

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a new wFRAME pushover analysis of Bent 2 with column moment capacity taken as Mo. As shown in Appendix 21.3-14 (right pushover), bent cap moment demands are: M Dve  14,350 kip  ft  M ve  21,189 kip  ft

OK.

M Dve  15,072 kip  ft  M ve  19,436 kip  ft

OK.

The associated shear demand obtained from the above pushover analysis, Vo = 2,009 kips. 21.3.14.2

Check Bent Cap Shear Capacity Nominal shear resistance of the bent cap, Vn is the lesser of: Vn  Vc  Vs + Vp

(AASHTO 5.8.3.3-1)

Vn  0.25 f c'bv dv + Vp

(AASHTO 5.8.3.3-2)

Vc  0.0316

(AASHTO 5.8.3.3-3)

and where:

Vs 

bv dv

f c' bv d v

Av f y d v cot

(AASHTO C5.8.3.3-1) s Vp = 0 (bent cap is not prestressed) = effective web width = 8 ft = 96 in. = effective shear depth = distance between the resultants of the tensile and compressive forces due to flexure, not to be taken less than the greater of 0.9d e or 0.72h (see AASHTO Article 5.8.2.9). 0.72 h = 0.72 (81) = 58.3 in. Assuming clear distance from cap bottom to main bottom bars = 5 in. d e = cap effective depth = 81-5-1.63/2 = 75.2 in. 0.9d e = 0.9(75.2) = 67.7 in. > 58.3 in. Therefore, d v, min = 67.7 in

Method 1 of AASHTO Article 5.8.3.4 (AASHTO 2012) is used to determine the values of β and  (the bent cap section is non-prestressed and the effect of any axial tension is assumed to be negligible).

 Use β = 2.0 and  = 45o per AASHTO Article 5.8.3.4.1. Vc  0.0316

f c' bv dv  0.0316(2)( 4 )(96)(67.7)  821 kips

Assuming 6-legged, #6 stirrups @ 7 in. o.c. transverse reinforcement (see Figure 21.3-19).

Chapter 21 – Seismic Design of Concrete Bridges

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Av f y d v cot

6(0.44)(60)(67.7)(cot 45)  1,532 kips s 7 Vn  Vc  Vs  821 1,532  2,353kips

Vs 



Vn  0.25 f c'bv dv  0.25(4)(96)(67.7)  6,499kips  2,353kips  Vn  2,353kips   Vn  0.9(2,353)  2,118kips  V0  2,009kips

OK.

21.3.15

Step 13 - Calculate Column Seismic Load Moments

21.3.15.1

Determine Dead Load, Additional Dead Load, and Prestress Secondary Moments at Column Tops/Deck Soffit For this bridge, the top of bent support results from CTBridge (Table 21.3-1) will need to be transformed to the consistent planar coordinate system (i.e., the plane formed by the centerline of the bridge and the vertical axis) to ensure consistency with wFRAME results and to account for the bridge skew. To do so, the following coordinate transformation (see Figure 21.3-9) will be applied to the top of column moments from CTBridge. Table 21.3-1 Top of Bent Column Moments (kip-ft) from CTBridge

Bent 2 3

Skew Mz (Degree) 20 -1,189 20 1,305

DL My

Mlong

Mz

ADL My

Mlong

Mz

91 -1

-1,148 1,227

-213 234

17 -1

-207 220

82 -127

Sec. PS My Mlong -371 287

204 -218

It is noted that the above values are for both columns in each bent. (1) Moment at Column top - Bent 2 

Dead load and additional dead load moments (Figure 21.3-10) col,bottom  0 kip - ft Column moment at base, M dl Column moment at deck soffit,

(CTBridge Output)

M dlcol,top @ jo int   1,148   207  1,355 kip - ft



Secondary Prestress Moments (Figure 21.3-11) ,bottom Column moment at base, M col  0 kip - ft ps

(CTBridge Output)

,top @ jo int Column moment at deck soffit, M col  204 kip - ft ps

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CL BRIDGE  (positive rotation) My

My

Mz Mz

 (positive skew)

Tx = Tx (out of page)

CL BENT

Column

Mz = Mz cos - My sin My = Mz sin + My cos Tx = Tx

(Longitudinal Moment) (Transverse Moment) (Torsional Moment)

Figure 21.3-9 Coordinate Transformation from Skewed to Unskewed Configuration Deck Soffit 30.8 kips 1,355 kip-ft Column

30.8 kips

Figure 21.3-10 Free Body Diagram Showing Equilibrium of Dead Loading at Bent 2 Chapter 21 – Seismic Design of Concrete Bridges

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204 kip-ft

Deck Soffit 4.6 kips

Column

4.6 kips

Figure 21.3-11 Free Body Diagram Showing Equilibrium of Secondary Prestress Forces at Bent 2 (2) Moment at Column Top - Bent 3 

Dead load and additional dead load moments

col,top @ jo int  0 kip - ft Column moment at base, M dl

(CTBridge Output)

Column moment at deck soffit,

M dlcol,top @ jo int   1,227   220  1,447 kip - ft



Secondary Prestress Moments

,bottom  0 kip  ft Column moment at base, M col ps

(CTBridge Output)

,top @ jo int  218 kip - ft Column Moment at deck soffit, M col ps

21.3.15.2

Determine Earthquake Moments in the Superstructure (1) Dead Load and Additional Dead Load Moments CTBridge output lists these moments at every 1/10th point of the span length and at the face of supports (see Table 21.3-2). (2) Secondary Prestress Moments CTBridge output lists these moments at every 1/10th point of the span length and at the face of supports (see Table 21.3-2).

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Table 21.3-2 Dead Load and Secondary Prestress Moments from CTBridge Output Location

Span 3

Span 2

Span 1

x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

x (ft) 1.5 12.6 25.2 37.8 50.4 63 75.6 88.2 100.8 113.4 123 129 142.8 159.6 176.4 193.2 210 226.8 243.6 260.4 277.2 291 297 305.8 317.6 329.4 341.2 353 364.8 376.6 388.4 400.2 410.5

Whole Superstructure Width M DL M ADL M PS (kip-ft) (kip-ft) (kip-ft) 619 114 647 7110 1275 1462 12158 2178 2272 14741 2640 3096 14857 2661 3956 12508 2240 4705 7693 1377 5617 412 74 6400 -9334 -1671 7911 -21553 -3857 8498 -32599 -5819 8672 -33654 -6009 8468 -17502 -3136 9516 -1955 -354 9005 9208 1645 8318 15989 2859 8281 18388 3289 8027 16406 2935 8072 10043 1795 7905 -699 -128 8355 -15820 -2835 8645 -31614 -5646 7554 -30429 -5434 7482 -20789 -3723 7275 -9854 -1766 6861 -1093 -197 5559 5506 986 4870 9943 1781 4085 12219 2189 3417 12333 2210 2669 10286 1844 1945 6077 1091 1230 637 117 529

Chapter 21 – Seismic Design of Concrete Bridges

Per Girder M DL M ADL M PS (kip-ft) (kip-ft) (kip-ft) 124 23 129 1422 255 292 2432 436 454 2948 528 619 2971 532 791 2502 448 941 1539 275 1123 82 15 1280 -1867 -334 1582 -4311 -771 1700 -6520 -1164 1734 -6731 -1202 1694 -3500 -627 1903 -391 -71 1801 1842 329 1664 3198 572 1656 3678 658 1605 3281 587 1614 2009 359 1581 -140 -26 1671 -3164 -567 1729 -6323 -1129 1511 -6086 -1087 1496 -4158 -745 1455 -1971 -353 1372 -219 -39 1112 1101 197 974 1989 356 817 2444 438 683 2467 442 534 2057 369 389 1215 218 246 127 23 106

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(3) Case 1 Earthquake Loading: Bridge moves from Abutment 1 towards Abutment 4 As shown in Figure 21.3-12, such loading results in positive moments in the columns according to the sign convention used here.

Abut 1

Abut 4

Bent 2 Bent 3 Figure 21.3-12 Seismic Loading Case “1” Producing Positive Moments in Columns As calculated previously, the columns have already been “pre-loaded” by: col @ soffit @ soffit M dl  M col  (1,355)  (204)  1,151kip - ft (Bent 2) ps col @ soffit @ soffit M dl  M col  (1,447)  (218)  1229 kip - ft (Bent 3) ps

Column moment generated by seismic loading at column soffit is: col @ soffit @ soffit @ soffit M eq  1.2M col (M dlcol  M col ) p ps

 1.2(2)(13,838)  (1,355)  0)  34,566 kip - ft (Bent 2)

It should be noted that the secondary prestress moment is neglected because doing so results in increased seismic demand on the column and hence in the superstructure. Figure 21.3-13 schematically explains this superposition approach. col @ soffit M eq  1.2(2)(13,777)  (1,447  218)  31,835 kip - ft (Bent 3)

It should be noted that for Bent 3, the effect of secondary prestress moments is included because doing so results in increased seismic moment in the columns and hence in the superstructure.

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30.8 kips

4.6 kips

Deck Soffit 1,355 kip-ft Column

n

+

S o f f i t

204 kip-ft col M eq

+ 4.6 kips

30.8 kips Dead Load and Additional Dead Load and Dead Load Additional Dead Load State

col Veq

col Veq

Secondary Prestress State Secondary Prestress

Seismic State

State 755 kips

M 0col  1.2M Pcol  33,211kip - ft

755 kips Collapse Limit State

Figure 21.3-13 Superposition of Column Forces at Bent 2 for Loading Case “1” (4) Case 2 Earthquake Loading: Bridge moves from Abutment 4 towards Abutment 1 As shown in Figure 21.3-14, such loading results in negative moments in the columns according to our sign convention.

Abut 4

Abut 1

Bent 2

Bent 3 Figure 21.3-14 Seismic Loading Case “2” Producing Negative Moments in Columns

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Bent 2 col @ soffit col col M eq  1.2M col p  ( M dl  M ps )

 1.2(2)(13,838)  (1,355  204)  32,060 kip  ft

Bent 3 col @ soffit col col M eq  1.2M col p  ( M dl  M ps )  1.2(2)(13,777)  (1,447  0)  34,512 kip - ft

Figure 21.3-15 schematically shows the Free Body Diagram at Bent 2 for this seismic loading case.

30.8 kips

4.6 kips

Deck Soffit

Veqcol

204 kip-ft

1,355 kip-ft

col M eq

Column

+

+ + 4.6 kips

30.8 kips Dead Load and Additional Dead Load and Dead Load

Secondary Prestress State Secondary Prestress

col Veq

Seismic State

State

Additional Dead Load

755 kips

State

M 0col  1.2M Pcol  33,211 kip - ft

= 755 kips Limit State

Figure 21.3-15 Superposition of Column Forces at Bent 2 for Loading Case “2”

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21.3.16

Step 14 - Distribute M eqcol @ soffit into the Superstructure The static non-linear “push-over” frame analysis program wFRAME is used to col @ soffit distribute the column earthquake moments M eq into the superstructure. Note the difference in sign convention between the wFRAME model and the one adopted here. Therefore, for the input file, the positive column earthquake moments corresponding to “Case 1” loading are used as negative column moment capacities for pushover analysis while the negative column earthquake moments corresponding to “Case 2” are modeled as positive column moment capacities. Also, the superstructure dead load is removed from the wFRAME model. Appendix 21.3-15 shows portions of the output file for Case 1 (i.e., right push). Table 21.3-3 lists the distribution of earthquake moments in the superstructure as obtained from these pushover analyses.

21.3.17

Step 15 - Calculate Superstructure Seismic Moment Demand at Location of Interest Let us calculate superstructure moment demand at the face of the cap on each side of the column. (1) Example Calculation - Bent 2: Left and Right Faces of Bent Cap The effective section width is: beff = Dc + 2Ds = 6.00 + 2(6.75) = 19.50 ft.

(SDC 7.2.1.1-1)

Based on the column location and the girder spacing, it can easily be concluded that the girder aligned along the centerline of the bridge lies outside the effective width. Therefore, at the face of bent cap, four girders are within the effective section. All five girders fall within the effective width for all the other tenth point locations (see Table 21.3-4). Note that the per-girder values used below have previously been listed in Table 21.3-2. Case 1

M dlL   6,520   1,164(4 )  30,736 kip - ft

M dlR   6,731   1,202(4 )  31,732 kip - ft L M ps   1,7344   6,936 kip - ft

R M ps   1,6944   6,776 kip - ft L M eq  15,015 kip - ft (see Table 21.3-3) R M eq  21,135 kip - ft (see Table 21.3-3)

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The superstructure moment demand is then calculated as: L M DL  M dlL  M ps  M eqL = (-30,736) + (6,936*) + (-15,015) = -45,751 kip - ft R M DR  M dlR  M ps  M eqR = (-31,732) + (6,776) + (21,135) = -3,821 kip - ft

Table 21.3-4 lists these superstructure seismic moment demands. Case 2 L M eq  13,201kip - ft;

M eqR = -20,299 kip - ft

M DL = (-30,736) + (6,936) + (13,201) = -10,599 kip - ft M DR = (-31,732) + (6,776*) + (-20,295) = -52,027 kip - ft *

The prestressing secondary effect is ignored as doing so results in a conservatively higher seismic demand in the superstructure. (2) Bent 3 Similarly, we obtain the following:  49,702 kip - ft Case 1 M DL     3,001kip - ft Case 2   9,434 kip - ft Case 1 M DR    43,915 kip - ft Case 2

Seismic moment demands along the superstructure length have been summarized in the form of moment envelope values (see Table 21.3-4).

M positive  M EQ,max  M DL  M ADL  M *ps M negative  M EQ,min  M DL  M ADL  M *ps* * Only **

include Mps when it maximizes Mpositive Only include Mps when it minimizes Mnegative

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Table 21.3-3 Earthquake Moments from wFRAME Output Location

Span 3

Span 2

Span 1

0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0 0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0 0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0

M EQ (kip-ft) Standard Convention wFRAME Convention Case 1 Case 2 Case 1 Case 2 0 0 0 0 -183 161 -1538 1352 -3076 2705 -4614 4057 -6152 5409 -7691 6761 -9229 8114 -10767 9466 -12305 10818 -13843 12170 -15015 13201 -15381 13523 -15381 13523 -21895 21055 21895 -21055 21135 -20295 17640 -16798 13385 -12540 9131 -8282 4876 -4024 621 234 -3634 4492 -7889 8750 -12144 13008 -16399 17266 -19894 20763 -20653 21524 -20653 21524 -13620 15621 13620 -15621 13274 -15223 12258 -14059 10896 -12496 9534 -10934 8172 -9372 6810 -7810 5448 -6248 4086 -4686 2724 -3124 1362 -1562 173 -199 0 0 0 0

Start Node

End Node wFRAME Positive Convention

Chapter 21 – Seismic Design of Concrete Bridges

Start Node

End Node Standard Positive Convention

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Span 1

Span 2

Chapter 21 – Seismic Design of Concrete Bridges

Span 3

x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

Location x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5

No. of Girders in Effective Section 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 (kip-ft) 496 7110 12158 14741 14857 12508 7693 412 -9334 -21553 -26079 -26923 -17502 -1955 9208 15989 18388 16406 10043 -699 -15820 -25291 -24344 -20789 -9854 -1093 5506 9943 12219 12333 10286 6077 509

M DL (kip-ft) 91 1275 2178 2640 2661 2240 1377 74 -1671 -3857 -4656 -4807 -3136 -354 1645 2859 3289 2935 1795 -128 -2835 -4517 -4347 -3723 -1766 -197 986 1781 2189 2210 1844 1091 94

M ADL (kip-ft) 517 1462 2272 3096 3956 4705 5617 6400 7911 8498 6937 6774 9516 9005 8318 8281 8027 8072 7905 8355 8645 6043 5986 7275 6861 5559 4870 4085 3417 2669 1945 1230 423

M PS (kip-ft) -183 -1538 -3076 -4614 -6152 -7691 -9229 -10767 -12305 -13843 -15015 21135 17640 13385 9131 4876 621 -3634 -7889 -12144 -16399 -19894 13274 12258 10896 9534 8172 6810 5448 4086 2724 1362 173

M EQ

Case 1

(kip-ft) 161 1352 2705 4057 5409 6761 8114 9466 10818 12170 13201 -20295 -16798 -12540 -8282 -4024 234 4492 8750 13008 17266 20763 -15223 -14059 -12496 -10934 -9372 -7810 -6248 -4686 -3124 -1562 -199

M EQ

Case 2

Table 21.3-4 Moment Demand Envelope

(kip-ft) 921 8309 13532 15862 15321 11762 5459 -3881 -15399 -30755 -38815 -3821 6518 20082 28301 32005 30324 23779 11854 -4616 -26409 -43658 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199

M postive (kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -10597 -2998 11077 19984 23724 22297 15707 3950 -12970 -35054 -49702 -15418 -12254 -724 8244 14663 18534 19856 18629 14854 8530 776

M negative

Case 1

(kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -45251 -27920 -5843 10889 23105 29938 31905 28493 20536 7256 -3001 -37931 -31296 -17255 -6665 1988 7999 11576 12526 10950 6836 827

M positive (kip-ft) 748 9737 17041 21438 22927 21509 17184 9952 -187 -13240 -17535 -52027 -37436 -14848 2571 14824 21911 23833 20588 12181 -1390 -9045 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404

M negative

Case 2

(kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -3821 6518 20082 28301 32005 30324 31905 28493 20536 7256 -3001 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199

M positive

(kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -52027 -37436 -14848 2571 14824 21911 15707 3950 -12970 -35054 -49702 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404

M negative

Envelope

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21.3.18

Step 16 - Calculate Superstructure Seismic Shear Demand at Location of Interest Values of shear forces due to dead load, additional dead load, and secondary prestress, as read from CTBridge output, are listed in Table 21.3-5.

Superstructure Seismic Shear Forces due to Seismic Moments, Veq Span 1, Case 1 (1) Seismic Moment at Abutment 1, M eq  0 kip - ft ( 2) Seismic Moment at Bent 2 M eq  15,381kip - ft

Shear force in Span 1, Veq 

M

( 2) eq

(1)  M eq



Length of Span 1

 15,381 0  122 kips

=

126

Span 1, Case 2 (1) Seismic Moment at Abutment 1, M eq  0 kip - ft ( 2) Seismic Moment at Bent 2, M eq  13,523 kip - ft

Shear force in Span 1, Veq 

M

( 2) eq

(1)  M eq



=

13,523 0  107kips

Length of Span 1 126 Similarly, the seismic shear forces for the remaining spans are calculated to be:  253 kips Case 1 Span 2, Veq    253 kips Case 2

 115 kips Case 1 Span 3, Veq    133 kips Case 2

Table 21.3-6 lists these values. Table 21.3-7 lists the maximum shear demands summarized as a shear envelope. * V positive  VEQ,max  VDL  VADL  V ps ** Vnegative  VEQ,min  VDL  VADL  V ps

Vmax  Greater of Absolute (V posittive ) or Absolute(Vnegative ) * **

Only include VPS when it maximizes Vpositive Only include VPS when it minimizes Vnegative

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Table 21.3-5 Dead Load and Secondary Prestress Shears Forces from CTBridge Output

Span 3

Span 2

Span 1

Location x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5

Whole Superstructure Width V DL V ADL V PS (kip) (kip) (kip) 671 120 79 498 89 78 303 54 76 107 19 76 -89 -16 75 -284 -51 75 -480 -86 75 -675 -121 75 -871 -156 75 -1070 -191 30 -1232 -218 134 1287 227 -44 1056 189 -22 795 142 2 534 96 2 273 49 2 13 2 2 -248 -45 1 -509 -91 1 -770 -138 1 -1031 -185 -28 -1261 -223 37 1171 207 -118 1021 182 -69 834 149 -48 651 117 -48 468 84 -48 284 51 -49 101 18 -48 -82 -15 -48 -265 -47 -48 -448 -80 -48 -608 -109 -68

Chapter 21 – Seismic Design of Concrete Bridges

V DL (kip) 134 100 61 21 -18 -57 -96 -135 -174 -214 -246 257 211 159 107 55 3 -50 -102 -154 -206 -252 234 204 167 130 94 57 20 -16 -53 -90 -122

Per Girder V ADL (kip) 24 18 11 4 -3 -10 -17 -24 -31 -38 -44 45 38 28 19 10 0 -9 -18 -28 -37 -45 41 36 30 23 17 10 4 -3 -9 -16 -22

V PS (kip) 16 16 15 15 15 15 15 15 15 6 27 -9 -4 0 0 0 0 0 0 0 -6 7 -24 -14 -10 -10 -10 -10 -10 -10 -10 -10 -14

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Table 21.3-6 Earthquake Shear Forces from wFRAME Output

Span 3

Span 2

Span 1

Location 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1

0.0 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 126.0 126.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 294.0 294.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 412.0

Start Node

End Node wFRAME Positive Convention

Chapter 21 – Seismic Design of Concrete Bridges

V EQ (kip) Standard Convention wFRAME Convention Case 1 Case 2 Case 1 Case 2 -122 107 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 -122 107 -122 107 -253 253 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 -253 253 -253 253 -115 133 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 -115 133 -115 133

Start Node

End Node Standard Positive Convention

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Span 1

Span 2

Chapter 21 – Seismic Design of Concrete Bridges

Span 3

x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

Location x (ft) 12.6 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 5 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4

No. of Girders in Effective Section V DL (kip) 498 498 303 107 -89 -284 -480 -675 -871 -1070 -986 1029 1056 795 534 273 13 -248 -509 -770 -1031 -1009 937 1021 834 651 468 284 101 -82 -265 -448 -486

(kip) 89 89 54 19 -16 -51 -86 -121 -156 -191 -174 182 189 142 96 49 2 -45 -91 -138 -185 -178 165 182 149 117 84 51 18 -15 -47 -80 -87

V ADL

V PS (kip) 78 78 76 76 75 75 75 75 75 30 107 -35 -22 2 2 2 2 1 1 1 -28 30 -94 -69 -48 -48 -48 -49 -48 -48 -48 -48 -54

(kip) -122 -122 -122 -122 -122 -122 -122 -122 -122 -122 -122 -253 -253 -253 -253 -253 -253 -253 -253 -253 -253 -253 -115 -115 -115 -115 -115 -115 -115 -115 -115 -115 -115

V EQ

Case 1

(kip) 107 107 107 107 107 107 107 107 107 107 107 253 253 253 253 253 253 253 253 253 253 253 133 133 133 133 133 133 133 133 133 133 133

V EQ

Case 2

(kip) 543 543 311 80 -151 -382 -613 -843 -1075 -1354 -1175 958 992 686 378 71 -237 -544 -852 -1160 -1469 -1411 987 1088 868 652 436 220 4 -212 -428 -644 -689

V positive (kip) 465 465 235 4 -227 -457 -688 -918 -1149 -1383 -1282 923 971 684 377 69 -238 -546 -853 -1161 -1496 -1440 893 1020 820 604 388 172 -44 -259 -476 -692 -743

V negative

Case 1

Table 21.3-7 Shear Demand Envelope

(kip) 772 772 540 309 78 -153 -384 -614 -846 -1125 -946 1464 1498 1192 884 577 269 -38 -346 -654 -963 -905 1234 1336 1116 900 684 468 252 36 -180 -396 -441

V positive (kip) 694 694 464 233 3 -228 -459 -689 -920 -1154 -1053 1429 1477 1190 883 575 268 -40 -347 -655 -990 -934 1141 1267 1068 852 636 420 204 -11 -228 -444 -495

V negative

Case 2

(kip) 772 772 540 309 78 -153 -384 -614 -846 -1125 -946 1464 1498 1192 884 577 269 -38 -346 -654 -963 -905 1234 1336 1116 900 684 468 252 36 -180 -396 -441

V positive

(kip) 465 465 235 4 -227 -457 -688 -918 -1149 -1383 -1282 923 971 684 377 69 -238 -546 -853 -1161 -1496 -1440 893 1020 820 604 388 172 -44 -259 -476 -692 -743

V negative

Envelope

(kip) 772 772 540 309 227 457 688 918 1149 1383 1282 1464 1498 1192 884 577 269 546 853 1161 1496 1440 1234 1336 1116 900 684 468 252 259 476 692 743

V max

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21.3.19

Step 17 - Perform Vertical Acceleration Analysis Since the site PRA = 0.5g < 0.6g, vertical acceleration analysis is not required.

21.3.20

Step 18 - Calculate Superstructure Flexural and Shear Capacity

21.3.20.1

Superstructure Flexural Capacity Table 21.3-8 lists the data that will be used to calculate the flexural section capacity using the computer program PSSECx. Symbols in Table 21.38 are shown in Figure 21.3-16. Appendix 21.3-16 lists the PSSECx input for the superstructure section that lies just to the left of Bent 2. The model is shown in Appendix 21.3-17. The results for negative capacity calculations are shown in Appendix 21.3-18. The limiting condition for flexural capacity in this case was the steel reaching its maximum allowable strain.

Figure 21.3-16 Typical Superstructure Cross Section PSSECx was run repeatedly to calculate superstructure flexural capacities at various points along the span length. Table 21.3-9 lists these capacities and also compares them with the maximum moment demands. As can be seen from these results, the superstructure has sufficient flexural capacity to meet the anticipated seismic demands. The worst D/C ratio of 0.63 suggests overdesign. If such case is found across a broad spectrum of various Caltrans bridges, perhaps the requirement of #8 spaced 12 in. may be revised in the future.

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Table 21.3-8 Section Flexural Capacity Calculation Data

Span 3

Span 2

Span 1

Location

No. No. Eccentricity Girders Girders in Effective e ps Section (in.) 5 4 -2.6628 5 5 -14.9760 5 5 -25.1328 5 5 -31.2264 5 5 -33.2568 5 5 -31.4076 5 5 -25.8576 5 5 -16.6068 5 5 -3.6576 5 5 14.9160 5 4 25.4412 5 4 25.6116 5 5 12.0432 5 5 -8.2824 5 5 -22.1568 5 5 -30.4824 5 5 -33.2568 5 5 -30.4824 5 5 -22.1568 5 5 -8.2824 5 5 12.0432 5 4 25.6116 5 4 25.3668 5 5 15.1068 5 5 -3.6576 5 5 -16.6068 5 5 -25.8576 5 5 -31.4076 5 5 -33.2568 5 5 -31.2264 5 5 -25.1328 5 5 -14.9760 5 4 -2.7900

PS Force After All Losses

x/L x (ft) (kip) Support 1.5 7439 0.1 12.6 7508 0.2 25.2 7582 0.3 37.8 7650 0.4 50.4 7712 0.5 63.0 7766 0.6 75.6 7814 0.7 88.2 7859 0.8 100.8 7839 0.9 113.4 7765 Support 123.0 7697 Support 129.0 7595 0.1 142.8 7413 0.2 159.6 7370 0.3 176.4 7327 0.4 193.2 7272 0.5 210.0 7212 0.6 226.8 7148 0.7 243.6 7079 0.8 260.4 6999 0.9 277.2 6922 Support 291.0 6844 Support 297.0 6742 0.1 305.8 6572 0.2 317.6 6545 0.3 329.4 6522 0.4 341.2 6484 0.5 353.0 6443 0.6 364.8 6398 0.7 376.6 6345 0.8 388.4 6287 0.9 400.2 6225 Support 410.5 6174 P jack = 9,689 kips * Area of mild steel based on minimum seismic requirement only

For Effective Section Area of Distance to PS Force Area of Top Mild Top Mild After All PS Steel* Steel Losses A ps A st,top y st,top (kip) 5952 7508 7582 7650 7712 7766 7814 7859 7839 7765 6157 6076 7413 7370 7327 7272 7212 7148 7079 6999 6922 5475 5393 6572 6545 6522 6484 6443 6398 6345 6287 6225 4940

(in.2) 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28

(in.2) 8.00 8.00 8.00 8.00 8.00 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 8.00 8.00 8.00 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 8.00 8.00 8.00 8.00 8.00

(in.) 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80

Area of Bottom Mild Steel* A st,bot (in.2) 6.00 6.00 6.00 6.00 6.00 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 6.00 6.00 6.00 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 6.00 6.00 6.00 6.00 6.00

Distance to Bottom Mild Steel y st,bot (in.) -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13

(Remaining limit state requirements need to be satisfied, A st,top = 56.6 in.2 at right face of Bent 2

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Table 21.3-9 Section Flexural Capacity Calculation Data Moment Demand

Span 3

Span 2

Span 1

Location x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5

M positive (kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -3821 6518 20082 28301 32005 30324 31905 28493 20536 7256 -3001 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199

M negative (kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -52027 -37436 -14848 2571 14824 21911 15707 3950 -12970 -35054 -49702 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404

Chapter 21 – Seismic Design of Concrete Bridges

Moment Capacity M positive (kip-ft) 34530 54933 65503 72025 73892 86109 81016 71684 58705 38646 26587 26432 41672 63619 77024 71217 73881 71218 77020 63615 41623 26344 26540 38316 58628 71666 80998 86086 74193 72006 65484 54917 34611

M negative (kip-ft) -39444 -34040 -23133 -16558 -14352 -36067 -41879 -51573 -65648 -85898 -81787 -81933 -82802 -60763 -45653 -17311 -14256 -17293 -45591 -60698 -82794 -81924 -81708 -86086 -65419 -51881 -41529 -35585 -13993 -16314 -22996 -34070 -39346

D/C Ratio Postive Moment 0.04 0.20 0.29 0.34 0.36 0.30 0.28 0.23 0.13 0.00 0.00 0.00 0.16 0.32 0.37 0.45 0.41 0.45 0.37 0.32 0.17 0.00 0.00 0.00 0.10 0.19 0.24 0.26 0.31 0.30 0.26 0.18 0.03

Negative Moment 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.36 0.46 0.56 0.63 0.45 0.24 0.00 0.00 0.00 0.00 0.00 0.21 0.42 0.61 0.54 0.45 0.37 0.24 0.07 0.00 0.00 0.00 0.00 0.00 0.00

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21.3.20.2

Superstructure Shear Capacity As shown in Table 21.3-10, seismic shear demands do not control as they are less than the demands from the controlling limit state (i.e. Strength I, Strength II, etc.) calculated using CTBridge. Therefore, the superstructure has sufficient shear capacity to resist seismic demands. Table 21.3-10 Shear Demand vs. Capacity

Span 3

Span 2

Span 1

Location

Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support

1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5

Shear Demand V max 803 772 540 309 227 457 688 918 1149 1383 1282 1464 1498 1192 884 577 269 546 853 1161 1496 1440 1234 1336 1116 900 684 468 252 259 476 692 743

Chapter 21 – Seismic Design of Concrete Bridges

Shear Capacity = Strength Shear Demand φV n = V u, strength 2851 2317 1687 1101 681 1207 1782 2341 2901 3596 3966 4378 3759 2961 2160 1399 686 1375 2139 2942 3792 4367 3760 3388 2817 2312 1774 1238 738 1000 1548 2138 2653

D/C Ratio D/C 0.28 0.33 0.32 0.28 0.33 0.38 0.39 0.39 0.40 0.38 0.32 0.33 0.40 0.40 0.41 0.41 0.39 0.40 0.40 0.39 0.39 0.33 0.33 0.39 0.40 0.39 0.39 0.38 0.34 0.26 0.31 0.32 0.28

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21.3.21

Step 19 - Design Joint Shear Reinforcement Figure 21.3-17 shows the bent cap-to-column joint.

Bent Cap Main Top Reinforcement: 22 #11 Bent Cap Main Bottom Reinforcement: 24 #11

3.06

5.32

Basic Development length, ldb = 58.5 8.38

S =2.92

Dc =6.00

14.3

Dimensions along skew direction

Figure 21.3-17 Bent Cap-to-Column Joint Cap beam short stub length, S = 14.3 – 8.38 – 3 = 2.92 ft < Dc = 6 ft (SDC 7.4.3-1). Therefore the joint will be designed as a knee joint in the transverse direction and a T joint in the longitudinal direction. 21.3.21.1

Transverse Direction (Knee Joint Design)

6.00  3.0 ft 2 Therefore, the joint is classified as Case 1 Knee joint. S  2.92 ft 

Chapter 21 – Seismic Design of Concrete Bridges

(SDC 7.4.5.1-1)

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(1) Closing Failure Mode - Bent 2 Knee Joint Given: Superstructure depth, Ds = 6.75 ft Column diameter, Dc = 6 ft, Concrete cover = 2 in. Column reinforcement:  Main reinforcement anchored into cap beam: #14 bars, total 26 giving Ast = 58.50 in.2  Transverse reinforcement: #8 hoops spaced at 5 in. c/c. Column main reinforcement embedment length into the bent cap, lac, provided  66 in. From the xSECTION analysis of Bent 2 with overturning effects (see Appendix 21.3-7): Column plastic moment, Mp = 14,964 kip-ft Column axial force (including the effect of overturning), Pc = 2,474 kips Cap Beam main reinforcement: top: #11 bars, total 2 and bottom: #11 bars, total 24. Calculate principal stresses, p t and p c

Vertical Shear Stress,  jv Tc  1.2 (2862) kips = 3,434 kips (Using xSECTION results of Appendix 21.3-7)

A jv  l ac Bcap  6696  6,336 in.2

(SDC 7.4.4.1-4)

Tc 3,434   0.542 ksi A jv 6336

(SDC 7.4.4.1-3)

 jv 

Normal Stress (Vertical), f v

fv 

Pc Pc 2,474    0.168 ksi A jh Dc  Ds Bcap 6.00  6.75(8.00)(144) (SDC 7.4.4.1-5)

Normal Stress (Horizontal) Assume Pb = 0 since no prestressing is specifically designed to provide horizontal joint compression. Therefore, horizontal normal stress fh = (Pb / BcapDs) = 0.

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pt 

fh  fv 



2

2

 f  fv    v2   h jv   2  

0.00  0.168 

2  0.464 ksi

2

 0.00  0.168  2    0.542 (+ for joint in tension) 2  

(SDC 7.4.4.1-1)

pc 

0.00  0.168 

2  0.632 ksi

2

 0.00  0.168  2    0.542 (+ for joint in compression) 2  

(SDC 7.4.4.1-2) Check Joint Size Adequacy Principal compression, pc = 0.632 ksi  [0.25 f c' = 0.25 (4.0) = 1 ksi] OK (SDC 7.4.2-1) Principal tension, pt = 0.464 ksi  [12

fc'  12 4000 / 1000= 0.76 ksi] OK (SDC 7.4.2-2) Check the Need for Additional Joint Reinforcement Since pt = 0.464 ksi > [3.5 f c'  3.5 4000 / 1000 = 0.221 ksi], additional joint reinforcement is required (see SDC Section 7.4.4.2). Similar calculations can be performed for Bent 3. (2) Opening Failure Mode - Bent 2 Knee Joint From wFRAME push-over analysis results (see Appendix 21.3-6), Column axial force (including the effect of overturning), Pc = 907 kips* Column plastic moment, Mp = 12,636 kip-ft* * These values were obtained from xSECTION analysis of Bent 2 with overturning effects (see Appendix 21.3-8) Tc  1.2 (3,148) kips = 3,778 kips using xSECTION results. Ajv = 66 (96) = 6,336 in.2

 jv 

3,778  0.596 ksi 6,336

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fv 

Pc Pc 907    0.062 ksi A jh Dc  Ds  Bcap 6.00  6.75 8.00 144

fh = 0 (since Pb = 0)

pt 

0.00  0.062 

2  0.00  0.062  2    0.596  0.566 ksi 2  

pc 

0.00  0.062 

 0.00  0.062  2    0.596  0.628 ksi 2  

2

2

2

Check Joint Size Adequacy Principal compression, pc = 0.628 ksi < [0.25  4.0 = 1 ksi] Principal tension, [ p t = 0.566 ksi] < [ 12 4000 / 1000= 0.760 ksi]

OK OK

Check the Need for Additional Joint Reinforcement Since pt = 0.566 ksi > [ 3.5 4000 / 1000 = 0.221 ksi], additional joint reinforcement is required. Based upon joint stress condition evaluation for both closing and opening modes of failure, the joint needs additional joint reinforcement. Refer to Figure 21.3-18 for regions of additional joint shear reinforcement.

Figure 21.3-18 Regions of Additional Joint Shear Reinforcement

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Joint Shear Requirement  Bent Cap Top and Bottom Flexural Reinforcement, AsU  Bar (Refer to Figure 21.3-19)

AsU  Bar required = 0.33 Ast = 0.33 (58.5) = 19.3 in.2

(SDC 7.4.5.1-3)

The bent cap reinforcement based upon service and seismic loading consists of: Top Reinforcement #11, total 22 bars giving Ast = 34.32 in.2 Bottom Reinforcement #11, total 24 bars giving Ast = 37.44 in.2 AsU  Bar provided = 12 (1.56) = 18.72 in.2 (within 4 % of 19.3 in.2) Say OK See Figure 21.3-19 for the rebar layout.

Figure 21.3-19 Location of Joint Shear Reinforcement (Elevation View)



Vertical Stirrups in Joint Region Asjv required = 0.2 Ast = 0.20 (58.5) = 11.7 in.2

(SDC 7.4.5.1-4)

Provide 5 sets of 6-legged, #6 stirrups so that

Asjv provided = (6 legs)(5 sets)(0.44) = 13.2 in.2 > 11.7 in.2

Chapter 21 – Seismic Design of Concrete Bridges

OK

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Place stirrups transversely within a distance Dc = 72 inches extending from either side of the column centerline. These vertical stirrups are shown in Figure 21.3-19 and also in Figure 21.3-20.



Horizontal Stirrups in Joint Region Asjh required = 0.1 Ast = 0.1 (58.5) = 5.85 in.2

(SDC 7.4.5.1-5)

As shown in Figure 21.3-19, provide 3-legged #6 stirrups, total 14 sets

Asjh provided = (3 legs)(14 sets)(0.44) = 18.48 in.2 > 5.85 in.2 Placed within a distance Dc = 72 in. extending from either side of the column centerline as shown in Figure 21.3-19.

Figure 21.3-20 Location of Vertical Stirrups, Asjv 

Horizontal Side Reinforcement top 0.1  Acap  Assf   or 0.1  Abot cap 

(SDC 7.4.5.1-6)

top Acap = 34.32 in.2 bot Acap = 37.44 in.2

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Assf

0.1 (34.32)  3.43 in.2   or 0.1 (37.44)  3.74 in.2 

 Assf = 3.74 in.2

As shown in Figures 21.3-21 and 21.3-22, provide #6, total 5 (“U” shaped), giving:

Assf provided = (2 legs)(5 bars) (0.44)= 4.4 in.2 > 3.74 in.2 

Horizontal Cap End Ties Asjhc  0.33Asu bar  0.33(19.3)  6.37 in.2

(SDC 7.4.5.1-7)

2 2 Provide #8, total 10 ( Asjhc , provided  10(0.79)  7.9 in.  6.37 in. )

OK

See SDC Figures 7.4.5.1-2, 7.4.5.1-3, and 7.4.5.1-5 for placement of Asjhc 

J-Dowels Strictly following SDC guidelines, there is no need for J-Dowels for this bridge. Let us provide it anyway.

Asj bar = 0.08 Ast = 0.08 (58.5) = 4.68 in.2 Use 16, #5 J-Dowels.

(SDC 7.4.5.1-8)

Asj bar provided = (16 bars)(0.31) = 4.96 in.2 > 4.68 in.2 These dowels are placed within the rectangle defined by Dc on either side of the column centerline and the cap width. They are shown in Figures 21.3-21 and 21.3-22. 

Check Transverse Reinforcement Minimum reinforcement ratio of transverse reinforcement (hoops)

  Ast   0.76 58.5   0.00936  72(66)   Dclac, provided      (SDC 7.4.5.1-9)

 s, required  0.76 

Column transverse reinforcement that extends into the joint region consists of #8 hoops at 5 in. spacing.

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s,



provided



4 Ab 4(0.79)   0.0095  0.00936 OK D' s  1.13   72  2(2)  2  5  2  

Check Anchorage for Main Column Reinforcement lac, required = 24dbl = 24 (1.69) = 40.6 in. < [lac, provided = 66 in.] OK (SDC 8.2.1-1)

Figure 21.3-21 Joint Reinforcement Within the Column Region

Figure 21.3-22 Joint Reinforcement Outside the Column Region Chapter 21 – Seismic Design of Concrete Bridges

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21.3.21.2

Longitudinal Direction (T-Joint) Let us calculate joint stresses for the tension column, which will provide higher value of principal tensile stress (generally more critical than principal compressive stress). Column plastic moment, Mp = 13,838 kip-ft* Column axial force, Pc = 1,694 kips* Cap beam main reinforcement: top: #11 bars, total 22 and bottom: #11 bars, total 24. * These values were obtained from the xSECTION analysis of Bent 2 columns without overturning effects (see Appendix 21.3-1). (1) Calculate Principal Stresses, pt and pc

Tc  1.2(2,948) kips = 3,538 kips using xSECTION results Ajv = lac Bcap = 66 (96) = 6,336 in.2

Vertical Shear Stress  jv 

Tc 3,538   0.558 ksi A jv 6,336

Normal Stress (Vertical) fv 

Pc Pc 1,694    0.115 ksi A jh Dc  Ds Bcap 6.00  6.75(8.00)(144)

Assume Pb = 0 since no prestressing is specifically designed to provide horizontal joint compression. Therefore, horizontal normal stress fh = (Pb/BcapDs) = 0.

pt 

0.00  0.115 

 0.00  0.115  2    0.558  0.503 ksi (i.e., tension) 2  

pc 

0.00  0.115 

 0.00  0.115  2    0.558  0.618 ksi (i.e., compression) 2  

2

2

2

2

Check Joint Size Adequacy Principal compression, pc = 0.618 ksi  [0.25 (4.0) = 1 ksi]

OK

Principal tension, pt = 0.503 ksi  [ 12 4000 / 1000 = 0.760 ksi]

OK

Chapter 21 – Seismic Design of Concrete Bridges

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Check the Need for Additional Joint Reinforcement pt = 0.503 ksi > [ 3.5 4000 / 1000 = 0.221 ksi], therefore additional joint reinforcement is required. The horizontal stirrups, cap beam u-bar requirements, continuous cap side face reinforcement, J-dowels, minimum transverse reinforcement, and column reinforcement anchorage provided for transverse bending will also satisfy the joint shear requirements for longitudinal bending. The only additional joint reinforcement requirement that needs to be satisfied for longitudinal bending is to provide vertical stirrups in Regions 1 and 2 of Figure 21.3-18. 

Vertical Stirrups in Joint Region – Regions 1 and 2 of Figure 21.3-18 Asjv required = 0.2 Ast = 0.2 (58.5) = 11.7 in.2 Provide: total 14 sets of 2-legged #6 stirrups or ties on each side of the column. Asjv provided = (2 legs)(14 sets)(0.44) = 12.32 in.2 > 11.7 in.2 OK As shown in Figures 21.3-19 and 21.3-20, these vertical stirrups and ties are placed transversely within a distance Dc extending from either side of the column centerline. Note that in the overlapping portions of regions 1 and 2 with region 3, the outside two legs of the 6-legged vertical stirrups provided for transverse bending are also counted toward the two legs of the vertical stirrups required for the longitudinal bending.

21.3.22

Step 20 - Determine Minimum Hinge Seat Width This bridge is not a multi-frame bridge. Therefore this step does not apply.

21.3.23

Step 21 - Determine Minimum Abutment Seat Width Minimum required abutment seat width, NA = 30 in. NA  p/s + cr+sh +temp +eq + 4 (in.)

(SDC Section 7.8.3) (SDC 7.8.3-1)

The combined effect of p/s, cr+sh, and temp, is calculated as 2.6 inches (see Joint Movement Calculation form - Appendix 21.3-11). The maximum seismic demand along the longitudinal direction of the bridge is calculated in a conservative way assuming that maximum longitudinal and transverse (along the bent line) demand displacements occur simultaneously, i.e., Chapter 21 – Seismic Design of Concrete Bridges

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 

 eq,longitudinal = 14.12 + 16.87 sin (20) = 19.89 in. NA, required (normal to centerline of bearing) = (19.89 + 2.6) cos (20) + 4 = 25.13 in. < 30 in. Provide abutment seat width NA = 36 in. > 30 in.

21.3.24

OK

Step 22 - Design Abutment Shear Key Reinforcement Shear key force capacity, Fsk   (0.75 Vpiles  Vww )

(SDC 7.8.4-1)

We shall assume the following information to be available from the abutment foundation and wingwall design:    

14 piles for the abutment, and 40 k/pile as ultimate shear capacity of the pile (see MTD 5-1) f c = 3.6 ksi Wingwall thickness = 12 in. Wingwall height to top of abutment footing = 14 ft (i.e., 6.75 ft Superstructure depth + 7.25 ft abutment stem height) V piles = 14 (40) = 560 kips Using Method 1 of AASHTO Article 5.8.3.4, the shear capacity of one wingwall, Vww may conservatively be estimated as follows: Effective shear depth d v = 0.72 (12 in) = 8.64 in. Effective width bv = [6.75  1 (7.25)](12) in = 110 in. 3 Vww = 0.0316 fcbv dv = 0.0316(2) 3.6 (110)(8.64) = 114 kips

Assuming   0.75, Fsk  0.75 0.75(560)  114  401 kips We shall use the Isolated Shear Key Method for this example. Vertical shear key reinforcement: F 401 (SDC 7.8.4.1A-1) Ask  sk   3.28 in.2 1.8 f ye 1.8(68) Provide 8 #6 – bundle bars as shown in Figure SDC 7.8.4.1-1A ( Ask , provided = 3.52 in.2 > 3.28 in.2) Hanger bars, Ash  2.0 AskIso( provided) = 2 (3.52) = 7.04 in.2 Provide 5 #11 hooked bars ( Ash, provided = 7.8 in.2 > 7.04 in.2)

Chapter 21 – Seismic Design of Concrete Bridges

OK

(SDC 7.8.4.1B-1) OK

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Place the vertical shear key bars, Ask at least Lmin from the end of the lowest layer of the hanger bars, where

Lmin,hooked  0.6(a  b)  ldh (SDC 7.8.4.1B-3) Assuming 5-inch thick bearing pads and 12 in. vertical height of expansion joint filler (see SDC Figure 7.8.4.1-1A), a = (Bearing pad thichness + 6 in.) = 11 in. Assuming 2 in cover and #4 distribution bars for the hanger bars, b = 2 + 0.56 + 0.5 (1.63) = 3.4 in. of dimension “b”) ldh 

(see SDC Figure 7.8.4.1-1A for definition

38db 38(1.41)   28.2 in. fc 3.6

Lmin,hooked  0.6(a  b)  ldh  0.6(11 3.4)  28.2  37 in. Place vertical shear key bars Ask 40 in. from the hooked ends of the hanger bars Ash .

21.3.25

Step 23 - Check Requirements for No-splice Zone For this bridge, only columns have been designated as “seismic critical” elements. Maximum length of column rebar can be estimated as Lmax = 47.00 + 5.5 = 52.5 ft < 60.00 ft Therefore, we will specify on the plans that no splices will be permitted for column main reinforcement. The superstructure rebars, however, will need to be spliced with Service Splice per MTD 20-9 (Caltrans 2001b).

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APPENDIX 21.3-1 Output from xSECTION 04/17/2006, 11:45 ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY

Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050

--------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313

W 148 148

Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber No. type 1 2 2 2 3 2 4 2 5 2 6 2

xc in 31.93 31.00 28.27 23.90 18.14 11.32

yc in 0.00 7.64 14.84 21.17 26.28 29.86

area in^2 2.25 2.25 2.25 2.25 2.25 2.25

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7 ... ...

2

3.85

31.70

2.25

25 26

2 2

28.27 31.00

-14.84 -7.64

2.25 2.25

Force Equilibrium Condition of the x-section: Max. Conc. Strain step epscmax 0 0.00000 1 0.00029 2 0.00032 ... ... 25 0.00322 26 0.00356 27 0.00394 28 0.00435 29 0.00481 30 0.00532 31 0.00588 32 0.00650 33 0.00718 34 0.00794 35 0.00878 36 0.00971 37 0.01073 38 0.01186 39 0.01312 40 0.01450

Max. Neutral Steel Axis Strain Conc. in. Tens. Comp. 0.00 0.0000 0 -12.30 -0.0001 1570 -9.09 -0.0002 1585 16.99 17.39 17.67 17.91 18.07 18.11 18.15 18.21 18.27 18.30 18.33 18.34 18.34 18.34 18.38 18.41

-0.0083 -0.0094 -0.0106 -0.0119 -0.0134 -0.0148 -0.0164 -0.0183 -0.0203 -0.0225 -0.0249 -0.0275 -0.0304 -0.0336 -0.0373 -0.0414

3210 3249 3309 3361 3388 3413 3461 3515 3570 3630 3686 3743 3792 3834 3847 3857

Steel force Comp. Tens. 0 0 174 -49 182 -73 889 929 952 978 1008 1037 1048 1060 1072 1087 1103 1122 1148 1181 1217 1256

-2406 -2483 -2568 -2646 -2703 -2756 -2816 -2881 -2948 -3021 -3096 -3171 -3246 -3321 -3371 -3420

P/S force 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Net Curvature Moment force rad/in (K-ft) 0.00 0.000000 0 1.52 0.000006 2588 0.95 0.000007 2843 -0.96 0.66 -1.69 -1.26 -0.57 0.59 -0.56 -0.42 -0.93 1.38 -1.20 -0.61 0.07 -0.67 -0.48 -1.66

0.000170 0.000192 0.000215 0.000241 0.000269 0.000298 0.000330 0.000366 0.000406 0.000449 0.000497 0.000550 0.000608 0.000672 0.000745 0.000825

12508 12718 12926 13129 13267 13362 13495 13660 13834 14017 14194 14368 14536 14695 14841 14976

First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000054 Moment (ft-k) = 9537

-3.85, -31.70

Cross Section Information: Axial Load on Section (kips) = 1694 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 23.717 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID

Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000155 Strain @ 0.004 0.000219 Strain @ 0.005 0.000279 CALTRANS 0.00720 0.000407 [email protected] 0.000270

Moment (K-ft) 12388 12957 13302 13838 13271

Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000070 12388 Mn 0.000755 0.000073 12957 Mn 0.000752 0.000075 13302 Mn 0.000750 0.000078 13838 Mp 0.000747 0.000075 13271 Mn 0.000750

Chapter 21 – Seismic Design of Concrete Bridges

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APPENDIX 21.3-2 Moment – Curvature Relationship

Chapter 21 – Seismic Design of Concrete Bridges

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B

APPENDIX 21.3-3 Soil Spring Data

p-y Data - Bent 2 (Location 1) 250

Stiffness (lbs/in)

200 150 100 50 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

3.0

3.5

4.0

4.5

Displacement (in)

p-y Data - Bent 2 (Location 2) 1400

Stiffness (lbs/in)

1200 1000 800 600 400 200 0 0.0

0.5

1.0

1.5

2.0

2.5

Displacement (in)

Chapter 21 – Seismic Design of Concrete Bridges

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B

p-y Data - Bent 2 (Location 3) 1600

Stiffness (lbs/in)

1400 1200 1000 800 600 400 200 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

3.0

3.5

4.0

4.5

Displacement (in)

p-y Data - Bent 2 (Location 4) 5000 4500

Stiffness (lbs/in)

4000 3500 3000 2500 2000 1500 1000 500 0 0.0

0.5

1.0

1.5

2.0

2.5

Displacement (in)

Chapter 21 – Seismic Design of Concrete Bridges

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APPENDIX 21.3-4 Bent Cap – Positive Bending Section Capacities – Select Output 05/16/2006, 10:17 ************************************************************ * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 …………………………………………………………………………………………………….. …………………………………………………………………………………………………….. Concrete Type Information: ----------strains-------- --------strength-------Type e0 e2 ecc eu f0 f2 fcc fu E 1 0.0020 0.0040 0.0027 0.0115 5.00 5.01 5.35 2.63 4200 2 0.0020 0.0040 0.0020 0.0050 5.00 3.52 5.00 2.50 4200

W 148 148

Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 ………………………………………………………………………………………………………….. First Yield of Rebar Information (not Idealized): Rebar Number 1 Coordinates X and Y (global in.) -44.80, -35.49 Yield strain = 0.00230 Curvature (rad/in)= 0.000037 Moment (ft-k) = 14873 Cross Section Information: Axial Load on Section (kips) = 1 Percentage of Main steel in Cross Section = 0.80 Concrete modulus used in Idealization (ksi) = 4200 Cracked Moment of Inertia (ft^4) = 55.568 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve Idealized Values =============== ============================= Method Conc. Yield symbol Plastic ID | Strain Curv. Moment | Curv. Moment for Curv. | in/in rad/in (K-ft) | rad/in (K-ft) moment rad/in Strain @ 0.003 0.000520 21189 0.000053 21189 Mn 0.000665 Strain @ 0.004 0.000684 21635 0.000054 21635 Mn 0.000664 Strain @ 0.005 0.000000 0 0.000000 0 Mn 0.000718 CALTRANS 0.00187 0.000306 19484 0.000048 19484 Mp 0.000669 [email protected] 0.000184 17426 0.000043 17426 Mn 0.000674

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APPENDIX 21.3-5 Bent Cap – Negative Bending Section Capacities – Select Output 05/15/2006, 08:26 ************************************************************ * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 ……………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………….. Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0027 0.0115 2 0.0020 0.0040 0.0020 0.0050

--------strength-------f0 f2 fcc fu E 5.00 5.01 5.35 2.63 4200 5.00 3.52 5.00 2.50 4200

W 148 148

Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 First Yield of Rebar Information (not Idealized): Rebar Number 25 Coordinates X and Y (global in.) 44.80, -34.49 Yield strain = 0.00230 Curvature (rad/in)= 0.000037 Moment (ft-k) = 13030 Cross Section Information: Axial Load on Section (kips) = 1 Percentage of Main steel in Cross Section = 0.80 Concrete modulus used in Idealization (ksi) = 4200 Cracked Moment of Inertia (ft^4) = 48.938 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve Idealized Values =============== ============================= Method Conc. Yield symbol Plastic ID | Strain Curv. Moment | Curv. Moment for Curv. | in/in rad/in (K-ft) | rad/in (K-ft) moment rad/in Strain @ 0.003 0.000593 19436 0.000055 19436 Mn 0.000563 Strain @ 0.004 0.000000 0 0.000000 0 Mn 0.000618 Strain @ 0.005 0.000000 0 0.000000 0 Mn 0.000618 CALTRANS 0.00159 0.000282 17307 0.000049 17307 Mp 0.000569 [email protected] 0.000183 15735 0.000044 15735 Mn 0.000573

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APPENDIX 21.3-6 wFRAME Output File 05/15/2006, 07:47 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r 41.72 -41.22 s 1.4e+002 r r 41.72 -43.27 s 4.1e+002 r r 41.72 -45.32 s 6.7e+002 r r 41.72 -47.37 f 0.0000 f 0.0000 r 44.72 0.00 r r r 49.44 0.00 r r r

Spring Information at node points: k's = k/ft or ft-k/rad.; node spring k1

d's = ft or rad. d1 k2 d2

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# 8 9 10 22 23 24

name P01X01 P01X02 P01X03 P02X01 P02X02 P02X03

136.37 414.83 665.70 136.37 414.83 665.70

0.149 0.105 0.106 0.149 0.105 0.106

0.00 0.00 0.00 0.00 0.00 0.00

1.000 1.000 1.000 1.000 1.000 1.000

0.00 0.00 0.00 0.00 0.00 0.00

1000.000 1000.000 1000.000 1000.000 1000.000 1000.000

Structural Setup: Spans= 3, Columns= 2, Piles= 2, Link Beams= 0 Element Information: element nodes depth # name fix i j L d area Ei Ef 1 S01-01 rn 1 2 4.72 6.8 62.6 629528 62953 2 S01-02 rn 2 3 3.00 6.8 62.6 629528 62953 3 C01-01 rn 3 4 3.38 6.0 28.3 629528 62953 4 C01-02 rn 4 5 11.93 6.0 28.3 629528 62953 5 C01-03 rn 5 6 11.93 6.0 28.3 629528 62953 6 C01-04 rn 6 7 11.93 6.0 28.3 629528 62953 7 P01-01 rn 7 8 2.05 6.0 28.3 629528 62953 8 P01-02 rn 8 9 2.05 6.0 28.3 629528 62953 9 P01-03 rn 9 10 2.05 6.0 28.3 629528 62953 10 P01-04 rn 10 11 2.05 6.0 28.3 629528 62953 11 S02-01 rn 3 12 3.00 6.8 62.6 629528 62953 12 S02-02 rn 12 13 7.00 6.8 62.6 629528 62953 13 S02-03 rn 13 14 7.00 6.8 62.6 629528 62953 14 S02-04 rn 14 15 7.00 6.8 62.6 629528 62953 15 S02-05 rn 15 16 7.00 6.8 62.6 629528 62953 16 S02-06 rn 16 17 3.00 6.8 62.6 629528 62953 17 C02-01 rn 17 18 3.38 6.0 28.3 629528 62953 18 C02-02 rn 18 19 11.93 6.0 28.3 629528 62953 19 C02-03 rn 19 20 11.93 6.0 28.3 629528 62953 20 C02-04 rn 20 21 11.93 6.0 28.3 629528 62953 21 P02-01 rn 21 22 2.05 6.0 28.3 629528 62953 22 P02-02 rn 22 23 2.05 6.0 28.3 629528 62953 23 P02-03 rn 23 24 2.05 6.0 28.3 629528 62953 24 P02-04 rn 24 25 2.05 6.0 28.3 629528 62953 25 S03-01 rn 17 26 3.00 6.8 62.6 629528 62953 26 S03-02 rn 26 27 4.72 6.8 62.6 629528 62953 bandwidth of the problem = 10 Number of rows and columns in strage = 81 x 30 Cumulative Results of analysis at end of stage

Icr 52.25 52.25 47.44 23.72 23.72 23.72 23.72 23.72 23.72 23.72 52.25 52.25 52.25 52.25 52.25 52.25 47.44 23.72 23.72 23.72 23.72 23.72 23.72 23.72 52.25 52.25

q -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 -68.40 -68.40 -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40

Mpp 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 27676 27676 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676

Mpn 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 27676 27676 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676

tol status 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e

0

Plastic Action at: Element/ Stage/ Code/ node# name 1 2 3 4 5 6 7 8 9 10 11 12 13

S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02

Lat. Force *g (DL= 3381.7)

---------- GLOBAL Displ.x Displ.y 0.00001 0.00633 0.00001 -0.00014 0.00001 -0.00450 -0.00484 -0.00418 -0.01446 -0.00304 -0.01376 -0.00191 -0.00662 -0.00078 -0.00503 -0.00058 -0.00338 -0.00039 -0.00170 -0.00019 0.00000 0.00000 0.00001 -0.00941 0.00000 -0.02023

/ Deflection / (in)

--------Rotation -0.00136 -0.00140 -0.00152 -0.00135 -0.00032 0.00038 0.00076 0.00079 0.00081 0.00083 0.00083 -0.00170 -0.00121

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14 15 16 17 18 19 20 21 22 23 24 25 26 27

S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

-0.00001 -0.00001 -0.00002 -0.00002 0.00483 0.01445 0.01375 0.00662 0.00503 0.00338 0.00170 0.00000 -0.00002 -0.00002

element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rn 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn

-0.02467 -0.02023 -0.00941 -0.00450 -0.00417 -0.00304 -0.00191 -0.00078 -0.00058 -0.00039 -0.00019 0.00000 -0.00014 0.00633

0.00000 0.00121 0.00170 0.00152 0.00135 0.00032 -0.00038 -0.00076 -0.00079 -0.00081 -0.00083 -0.00083 0.00140 0.00136

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.00001 0.00633 -0.00136 0.00 0.00 0.00 2 0.00001 -0.00014 -0.00140 0.00 322.85 -761.94 2 0.00001 -0.00014 -0.00140 0.00 -322.85 761.93 3 0.00001 -0.00450 -0.00152 0.00 528.05 -2038.28 3 0.00450 0.00001 -0.00152 1690.85 -34.15 -1605.41 4 0.00418 -0.00484 -0.00135 -1690.85 34.15 1489.98 4 0.00418 -0.00484 -0.00135 1690.85 -34.15 -1489.98 5 0.00304 -0.01446 -0.00032 -1690.85 34.15 1082.57 5 0.00304 -0.01446 -0.00032 1690.85 -34.15 -1082.57 6 0.00191 -0.01376 0.00038 -1690.85 34.15 675.15 6 0.00191 -0.01376 0.00038 1690.85 -34.15 -675.15 7 0.00078 -0.00662 0.00076 -1690.85 34.15 267.73 7 0.00078 -0.00662 0.00076 1690.85 -34.14 -267.71 8 0.00058 -0.00503 0.00079 -1690.85 34.14 197.70 8 0.00058 -0.00503 0.00079 1690.85 -33.46 -197.70 9 0.00039 -0.00338 0.00081 -1690.85 33.46 129.10 9 0.00039 -0.00338 0.00081 1690.85 -32.05 -129.10 10 0.00019 -0.00170 0.00083 -1690.85 32.05 63.39 10 0.00019 -0.00170 0.00083 1690.85 -30.92 -63.40 11 0.00000 0.00000 0.00083 -1690.85 30.92 0.00 3 0.00001 -0.00450 -0.00152 34.15 1162.80 3643.68 12 0.00001 -0.00941 -0.00170 -34.15 -957.60 -463.07 12 0.00001 -0.00941 -0.00170 34.15 957.61 463.06 13 0.00000 -0.02023 -0.00121 -34.15 -478.81 4564.38 13 0.00000 -0.02023 -0.00121 34.15 478.81 -4564.38 14 -0.00001 -0.02467 0.00000 -34.15 -0.01 6240.23 14 -0.00001 -0.02467 0.00000 34.15 0.01 -6240.23 15 -0.00001 -0.02023 0.00121 -34.15 478.79 4564.47 15 -0.00001 -0.02023 0.00121 34.15 -478.80 -4564.46 16 -0.00002 -0.00941 0.00170 -34.15 957.60 -462.91 16 -0.00002 -0.00941 0.00170 34.15 -957.59 462.89 17 -0.00002 -0.00450 0.00152 -34.15 1162.79 -3643.48 17 0.00450 -0.00002 0.00152 1690.83 34.15 1605.20 18 0.00417 0.00483 0.00135 -1690.83 -34.15 -1489.77 18 0.00417 0.00483 0.00135 1690.83 34.15 1489.77 19 0.00304 0.01445 0.00032 -1690.83 -34.15 -1082.42 19 0.00304 0.01445 0.00032 1690.83 34.15 1082.42 20 0.00191 0.01375 -0.00038 -1690.83 -34.15 -675.06 20 0.00191 0.01375 -0.00038 1690.83 34.15 675.06 21 0.00078 0.00662 -0.00076 -1690.83 -34.15 -267.70 21 0.00078 0.00662 -0.00076 1690.83 34.14 267.71 22 0.00058 0.00503 -0.00079 -1690.83 -34.14 -197.71 22 0.00058 0.00503 -0.00079 1690.83 33.46 197.71 23 0.00039 0.00338 -0.00081 -1690.83 -33.46 -129.11 23 0.00039 0.00338 -0.00081 1690.83 32.06 129.12 24 0.00019 0.00170 -0.00083 -1690.83 -32.06 -63.39 24 0.00019 0.00170 -0.00083 1690.83 30.93 63.40 25 0.00000 0.00000 -0.00083 -1690.83 -30.93 0.00 17 -0.00002 -0.00450 0.00152 0.00 528.05 2038.28

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26 S03-02 rn

26 -0.00002 -0.00014 26 -0.00002 -0.00014 27 -0.00002 0.00633

0.00140 0.00140 0.00136

Cumulative Results of analysis at end of stage

0.00 0.00 0.00

-322.85 322.85 0.00

-761.92 761.93 0.00

1

Plastic Action at: Element/ Stage/ Code/ C02-02 1 rs node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

---------- GLOBAL Displ.x Displ.y 0.70748 0.02708 0.70748 0.00919 0.70747 -0.00241 0.69197 -0.00224 0.58279 -0.00163 0.39898 -0.00103 0.16941 -0.00042 0.12746 -0.00031 0.08515 -0.00021 0.04263 -0.00010 0.00000 0.00000 0.70749 -0.01286 0.70750 -0.02604 0.70750 -0.02467 0.70749 -0.01442 0.70746 -0.00595 0.70744 -0.00658 0.70163 -0.00611 0.61168 -0.00445 0.42648 -0.00280 0.18265 -0.00114 0.13751 -0.00085 0.09192 -0.00057 0.04603 -0.00028 0.00000 0.00000 0.70745 -0.00948 0.70745 -0.01442

element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn

Lat. Force *g (DL= 3381.7) 0.1712

/ Deflection / (in) 8.4898

--------Rotation -0.00378 -0.00382 -0.00394 -0.00522 -0.01268 -0.01773 -0.02035 -0.02056 -0.02070 -0.02078 -0.02080 -0.00301 -0.00077 0.00103 0.00165 0.00039 -0.00090 -0.00252 -0.01204 -0.01849 -0.02187 -0.02214 -0.02233 -0.02243 -0.02246 -0.00102 -0.00106

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.70748 0.02708 -0.00378 27.69 -0.01 -0.01 2 0.70748 0.00919 -0.00382 -27.69 322.85 -761.97 2 0.70748 0.00919 -0.00382 71.78 -322.85 761.95 3 0.70747 -0.00241 -0.00394 -71.78 528.05 -2038.29 3 0.00241 0.70747 -0.00394 907.25 253.50 11715.99 4 0.00224 0.69197 -0.00522 -907.25 -253.50 -10859.41 4 0.00224 0.69197 -0.00522 907.18 253.90 10859.10 5 0.00163 0.58279 -0.01268 -907.18 -253.90 -7830.08 5 0.00163 0.58279 -0.01268 907.18 253.90 7830.12 6 0.00103 0.39898 -0.01773 -907.18 -253.90 -4801.02 6 0.00103 0.39898 -0.01773 907.18 253.90 4801.03 7 0.00042 0.16941 -0.02035 -907.18 -253.90 -1771.96 7 0.00042 0.16941 -0.02035 907.18 253.53 1771.40 8 0.00031 0.12746 -0.02056 -907.18 -253.53 -1250.91 8 0.00031 0.12746 -0.02056 907.18 236.71 1251.16 9 0.00021 0.08515 -0.02070 -907.18 -236.71 -765.72 9 0.00021 0.08515 -0.02070 907.18 201.31 766.32 10 0.00010 0.04263 -0.02078 -907.18 -201.31 -353.62 10 0.00010 0.04263 -0.02078 907.18 172.67 354.11 11 0.00000 0.00000 -0.02080 -907.18 -172.67 0.01 3 0.70747 -0.00241 -0.00394 -147.11 379.20 -9677.96 12 0.70749 -0.01286 -0.00301 147.11 -174.00 10507.76 12 0.70749 -0.01286 -0.00301 -88.03 173.93 -10507.76

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13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn

13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27

0.70750 0.70750 0.70750 0.70750 0.70749 0.70749 0.70746 0.70746 0.70744 0.00658 0.00611 0.00611 0.00445 0.00445 0.00280 0.00280 0.00114 0.00114 0.00085 0.00085 0.00057 0.00057 0.00028 0.00028 0.00000 0.70744 0.70745 0.70745 0.70745

-0.02604 -0.02604 -0.02467 -0.02467 -0.01442 -0.01442 -0.00595 -0.00595 -0.00658 0.70744 0.70163 0.70163 0.61168 0.61168 0.42648 0.42648 0.18265 0.18265 0.13751 0.13751 0.09192 0.09192 0.04603 0.04603 0.00000 -0.00658 -0.00948 -0.00948 -0.01442

Chapter 21 – Seismic Design of Concrete Bridges

-0.00077 -0.00077 0.00103 0.00103 0.00165 0.00165 0.00039 0.00039 -0.00090 -0.00090 -0.00252 -0.00252 -0.01204 -0.01204 -0.01849 -0.01849 -0.02187 -0.02187 -0.02214 -0.02214 -0.02233 -0.02233 -0.02243 -0.02243 -0.02246 -0.00090 -0.00102 -0.00102 -0.00106

88.03 -5.78 5.78 76.09 -76.09 158.08 -158.08 216.23 -216.23 2474.51 -2474.51 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 -72.95 72.95 -28.11 28.11

304.87 -304.87 783.67 -783.67 1262.47 -1262.47 1741.27 -1741.27 1946.47 322.57 -322.57 322.17 -322.17 322.18 -322.18 322.18 -322.18 322.17 -322.17 303.41 -303.41 265.35 -265.35 234.74 -234.74 528.06 -322.86 322.85 0.00

10049.47 -10049.47 6239.57 -6239.57 -921.93 921.94 -11435.05 11435.04 -16966.67 14926.95 -13838.61 13838.00 -9994.56 9994.59 -6150.91 6150.92 -2307.32 2307.40 -1646.88 1647.21 -1024.71 1024.96 -481.13 481.23 0.07 2038.30 -761.92 761.91 0.00

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APPENDIX 21.3-7 Select Output from xSECTION, Compression Column ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050

--------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313

W 148 148

Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber xc yc No. type in in 1 2 31.93 0.00 2 2 31.00 7.64 ... 25 2 28.27 -14.84 26 2 31.00 -7.64

area in^2 2.25 2.25 2.25 2.25

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Force Equilibrium Condition of the x-section: Max. Conc. Strain step epscmax 0 0.00000 1 0.00029 ... 24 0.00291 25 0.00322 26 0.00356 27 0.00394 28 0.00435 29 0.00481 30 0.00532 31 0.00588 32 0.00650 33 0.00718 34 0.00794 35 0.00878 36 0.00971 37 0.01073 38 0.01186 39 0.01312 40 0.01450

Neutral Axis in. 0.00 -29.19 14.11 14.74 15.28 15.73 16.07 16.24 16.23 16.38 16.52 16.66 16.77 16.86 16.91 16.97 16.96 16.95 16.91

Max. Steel Strain Conc. Tens. Comp. 0.0000 0 0.0000 2256

-0.0061 -0.0070 -0.0081 -0.0092 -0.0104 -0.0117 -0.0129 -0.0144 -0.0161 -0.0180 -0.0200 -0.0223 -0.0248 -0.0275 -0.0304 -0.0335 -0.0370

3742 3778 3813 3856 3904 3950 4008 4043 4089 4135 4180 4226 4271 4310 4366 4415 4458

Steel force Comp. Tens. 0 0 222 -2 926 963 991 1018 1049 1075 1092 1106 1121 1137 1156 1177 1201 1231 1242 1255 1269

-2194 -2268 -2330 -2399 -2478 -2552 -2623 -2675 -2734 -2797 -2862 -2928 -2997 -3069 -3132 -3195 -3255

P/S force 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Net Curvature Moment force rad/in (K-ft) 0.00 0.000000 0 1.88 0.000004 2346 0.52 -1.28 0.34 0.71 0.63 -0.48 1.90 -0.34 1.91 0.76 0.35 1.07 0.93 -2.02 1.47 0.47 -1.79

0.000133 0.000152 0.000172 0.000194 0.000219 0.000244 0.000269 0.000300 0.000334 0.000372 0.000414 0.000459 0.000509 0.000565 0.000624 0.000689 0.000761

13492 13683 13834 14012 14204 14332 14424 14544 14706 14879 15055 15231 15403 15573 15730 15869 15987

First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000057 Moment (ft-k) = 10802

-3.85, -31.70

Cross Section Information: Axial Load on Section (kips) = 2474 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 25.572 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID

Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000138 Strain @ 0.004 0.000198 Strain @ 0.005 0.000253 CALTRANS 0.00755 0.000392 [email protected] 0.000283

Moment (K-ft) 13546 14042 14366 14964 14479

Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000071 13546 Mn 0.000689 0.000074 14042 Mn 0.000687 0.000075 14366 Mn 0.000685 0.000079 14964 Mp 0.000682 0.000076 14479 Mn 0.000685

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APPENDIX 21.3-8 Select Output from xSECTION, Tension Column 05/10/2006, 07:43 ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050

--------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313

W 148 148

Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber xc yc area No. type in in in^2 1 2 31.93 0.00 2.25 2 2 31.00 7.64 2.25 ................................ ................................ 14 2 -31.93 0.00 2.25 15 2 -31.00 -7.64 2.25 16 2 -28.27 -14.84 2.25 17 2 -23.90 -21.17 2.25 18 2 -18.14 -26.28 2.25 19 2 -11.32 -29.86 2.25

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20 21 22 23 24 25 26

2 2 2 2 2 2 2

-3.85 3.85 11.32 18.14 23.90 28.27 31.00

-31.70 -31.70 -29.85 -26.28 -21.17 -14.84 -7.64

2.25 2.25 2.25 2.25 2.25 2.25 2.25

Force Equilibrium Condition of the x-section: Max. Max. Conc. Neutral Steel Steel Strain Axis Strain Conc. force P/S Net Curvature Moment step epscmax in. Tens. Comp. Comp. Tens. force force rad/in (K-ft) 0 0.00000 0.00 0.0000 0 0 0 0 0.00 0.000000 0 1 0.00029 2.87 -0.0003 949 131 -173 0 -0.82 0.000009 2393 ............................................................................... ............................................................................... 27 0.00394 19.79 -0.0125 2770 862 -2726 0 -0.83 0.000243 11770 28 0.00435 19.97 -0.0140 2820 878 -2792 0 -0.67 0.000272 11964 29 0.00481 20.02 -0.0156 2862 903 -2859 0 -0.64 0.000301 12123 30 0.00532 19.99 -0.0172 2893 936 -2922 0 -0.21 0.000333 12243 31 0.00588 20.03 -0.0191 2927 973 -2993 0 0.00 0.000368 12404 32 0.00650 20.00 -0.0210 2993 980 -3067 0 -0.17 0.000407 12576 33 0.00718 19.97 -0.0232 3066 989 -3148 0 0.19 0.000449 12758 34 0.00794 20.02 -0.0257 3114 993 -3201 0 -0.80 0.000498 12933 35 0.00878 20.05 -0.0285 3160 999 -3252 0 -0.36 0.000551 13102 36 0.00971 20.08 -0.0316 3203 1005 -3302 0 -0.67 0.000611 13262 37 0.01073 20.10 -0.0350 3245 1016 -3355 0 -0.91 0.000676 13418 38 0.01186 20.10 -0.0387 3280 1032 -3405 0 0.08 0.000747 13563 39 0.01312 20.12 -0.0429 3308 1052 -3453 0 -0.24 0.000827 13700 40 0.01450 20.13 -0.0474 3326 1077 -3496 0 0.08 0.000915 13815 First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000051 Moment (ft-k) = 8190

-3.85, -31.70

Cross Section Information: Axial Load on Section (kips) = 907 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 21.496 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID

Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000176 Strain @ 0.004 0.000248 Strain @ 0.005 0.000313 CALTRANS 0.00673 0.000421 [email protected] 0.000256

Moment (K-ft) 11159 11800 12168 12636 11855

Chapter 21 – Seismic Design of Concrete Bridges

Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000070 11159 Mn 0.000845 0.000074 11800 Mn 0.000841 0.000076 12168 Mn 0.000839 0.000079 12636 Mp 0.000836 0.000074 11855 Mn 0.000841

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APPENDIX 21.3-9 wFRAME, Output File 05/15/2006, 08:02 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r 41.72 -41.22 s 1.4e+002 r r 41.72 -43.27 s 4.1e+002 r r 41.72 -45.32 s 6.7e+002 r r 41.72 -47.37 f 0.0000 f 0.0000 r 44.72 0.00 r r r 49.44 0.00 r r r

Spring Information at node points: k's = k/ft or ft-k/rad.;

d's = ft or rad.

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node # 8 9 10 22 23 24

spring name P01X01 P01X02 P01X03 P02X01 P02X02 P02X03

k1

d1

136.37 414.83 665.70 136.37 414.83 665.70

k2

d2

0.149 0.105 0.106 0.149 0.105 0.106

0.00 0.00 0.00 0.00 0.00 0.00

1.000 1.000 1.000 1.000 1.000 1.000

0.00 0.00 0.00 0.00 0.00 0.00

1000.000 1000.000 1000.000 1000.000 1000.000 1000.000

Structural Setup: Spans= 3, Columns= 2, Piles= 2, Link Beams= 0 Element Information: element nodes depth # name fix i j L d area Ei Ef 1 S01-01 rn 1 2 4.72 6.8 62.6 629528 62953 2 S01-02 rn 2 3 3.00 6.8 62.6 629528 62953 3 C01-01 rn 3 4 3.38 6.0 28.3 629528 62953 4 C01-02 rn 4 5 11.93 6.0 28.3 629528 62953 5 C01-03 rn 5 6 11.93 6.0 28.3 629528 62953 6 C01-04 rn 6 7 11.93 6.0 28.3 629528 62953 7 P01-01 rn 7 8 2.05 6.0 28.3 629528 62953 8 P01-02 rn 8 9 2.05 6.0 28.3 629528 62953 9 P01-03 rn 9 10 2.05 6.0 28.3 629528 62953 10 P01-04 rn 10 11 2.05 6.0 28.3 629528 62953 11 S02-01 rn 3 12 3.00 6.8 62.6 629528 62953 12 S02-02 rn 12 13 7.00 6.8 62.6 629528 62953 13 S02-03 rn 13 14 7.00 6.8 62.6 629528 62953 14 S02-04 rn 14 15 7.00 6.8 62.6 629528 62953 15 S02-05 rn 15 16 7.00 6.8 62.6 629528 62953 16 S02-06 rn 16 17 3.00 6.8 62.6 629528 62953 17 C02-01 rn 17 18 3.38 6.0 28.3 629528 62953 18 C02-02 rn 18 19 11.93 6.0 28.3 629528 62953 19 C02-03 rn 19 20 11.93 6.0 28.3 629528 62953 20 C02-04 rn 20 21 11.93 6.0 28.3 629528 62953 21 P02-01 rn 21 22 2.05 6.0 28.3 629528 62953 22 P02-02 rn 22 23 2.05 6.0 28.3 629528 62953 23 P02-03 rn 23 24 2.05 6.0 28.3 629528 62953 24 P02-04 rn 24 25 2.05 6.0 28.3 629528 62953 25 S03-01 rn 17 26 3.00 6.8 62.6 629528 62953 26 S03-02 rn 26 27 4.72 6.8 62.6 629528 62953 bandwidth of the problem = 10 Number of rows and columns in strage = 81 x 30 Cumulative Results of analysis at end of stage

Icr 52.25 52.25 43.00 21.50 21.50 21.50 21.50 21.50 21.50 21.50 52.25 52.25 52.25 52.25 52.25 52.25 51.14 25.57 25.57 25.57 25.57 25.57 25.57 25.57 52.25 52.25

q -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 -68.40 -68.40 -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40

Mpp 29928 29928 29928 12636 12636 12636 12636 12636 12636 12636 29928 29928 29928 29928 29928 29928 29928 14964 14964 14964 14964 14964 14964 14964 29928 29928

Mpn 29928 29928 29928 12636 12636 12636 12636 12636 12636 12636 29928 29928 29928 29928 29928 29928 29928 14964 14964 14964 14964 14964 14964 14964 29928 29928

tol status 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e

0

Plastic Action at: Element/ Stage/ Code/

Lat. Force *g (DL= 3381.7)

/ Deflection / (in)

node# name

---------- GLOBAL --------Displ.x Displ.y Rotation 1 S01.00 -0.00603 0.00640 -0.00137 2 S01.01 -0.00603 -0.00012 -0.00141 ………………………………………………………………………………………………. …………………………………………………………………………………………….. 25 P02.04 0.00000 0.00000 -0.00064 26 S03.01 -0.00606 -0.00012 0.00141 27 S03.02 -0.00606 0.00639 0.00137

element node # name fix 1 S01-01 rn 2 S01-02 rn

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 -0.00603 0.00640 -0.00137 0.00 0.00 0.00 2 -0.00603 -0.00012 -0.00141 0.00 322.85 -761.94 2 -0.00603 -0.00012 -0.00141 0.01 -322.84 761.93 3 -0.00603 -0.00450 -0.00153 -0.01 528.04 -2038.28

Chapter 21 – Seismic Design of Concrete Bridges

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25 S03-01 rn 26 S03-02 rn

17 26 26 27

-0.00606 -0.00450 -0.00606 -0.00012 -0.00606 -0.00012 -0.00606 0.00639

0.00153 0.00141 0.00141 0.00137

Cumulative Results of analysis at end of stage

0.00 0.00 0.00 0.00

528.05 -322.85 322.85 0.00

2038.27 -761.93 761.93 0.00

1

Plastic Action at: Element/ Stage/ Code/ rs 1 C02-02 node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

Lat. Force *g (DL= 3381.7) 0.1760

---------- GLOBAL Displ.x Displ.y 0.73219 0.02482 0.73218 0.00836 0.73218 -0.00234 0.71753 -0.00217 0.60724 -0.00158 0.41672 -0.00099 0.17709 -0.00040 0.13324 -0.00030 0.08902 -0.00020 0.04456 -0.00010 0.00000 0.00000 0.73219 -0.01192 0.73220 -0.02352 0.73220 -0.02138 0.73218 -0.01148 0.73215 -0.00478 0.73213 -0.00665 0.72472 -0.00618 0.62890 -0.00450 0.43749 -0.00283 0.18720 -0.00115 0.14093 -0.00086 0.09420 -0.00058 0.04717 -0.00029 0.00000 0.00000 0.73214 -0.01097 0.73214 -0.01813

node element name fix # 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn

/ Deflection (in) / 8.7862

--------Rotation -0.00348 -0.00351 -0.00364 -0.00501 -0.01304 -0.01846 -0.02127 -0.02149 -0.02164 -0.02172 -0.02175 -0.00274 -0.00059 0.00106 0.00151 0.00003 -0.00137 -0.00300 -0.01255 -0.01903 -0.02242 -0.02270 -0.02288 -0.02299 -0.02302 -0.00149 -0.00153

-------- local ----------- ------ element ---------moment shear displ.x displ.y rotation axial 0.01 -0.01 29.06 1 0.73219 0.02482 -0.00348 -761.97 322.86 -29.06 2 0.73218 0.00836 -0.00351 761.94 -322.85 74.05 2 0.73218 0.00836 -0.00351 528.05 -2038.29 -74.05 3 0.73218 -0.00234 -0.00364 248.18 11431.10 879.83 3 0.00234 0.73218 -0.00364 -248.18 -10591.82 -879.83 4 0.00217 0.71753 -0.00501 248.19 10591.19 879.86 4 0.00217 0.71753 -0.00501 -248.19 -7630.29 -879.86 5 0.00158 0.60724 -0.01304 7630.31 248.21 879.86 5 0.00158 0.60724 -0.01304 -248.21 -4669.24 -879.86 6 0.00099 0.41672 -0.01846 4669.30 248.20 879.86 6 0.00099 0.41672 -0.01846 -248.20 -1708.25 -879.86 7 0.00040 0.17709 -0.02127 1708.95 248.12 879.86 7 0.00040 0.17709 -0.02127 -248.12 -1200.10 -879.86 8 0.00030 0.13324 -0.02149 1200.16 230.01 879.86 8 0.00030 0.13324 -0.02149 -728.78 -230.01 -879.86 9 0.00020 0.08902 -0.02164 729.12 192.70 879.86 9 0.00020 0.08902 -0.02164 -334.12 -192.70 -879.86 10 0.00010 0.04456 -0.02172 334.16 163.00 879.86 10 0.00010 0.04456 -0.02172 -0.02 -163.00 -879.86 11 0.00000 0.00000 -0.02175 351.79 -9393.48 -137.94 3 0.73218 -0.00234 -0.00364 -146.59 10141.05 137.94 12 0.73219 -0.01192 -0.00274

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12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn

12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27

0.73219 0.73220 0.73220 0.73220 0.73220 0.73218 0.73218 0.73215 0.73215 0.73213 0.00665 0.00618 0.00618 0.00450 0.00450 0.00283 0.00283 0.00115 0.00115 0.00086 0.00086 0.00058 0.00058 0.00029 0.00029 0.00000 0.73213 0.73214 0.73214 0.73214

-0.01192 -0.02352 -0.02352 -0.02138 -0.02138 -0.01148 -0.01148 -0.00478 -0.00478 -0.00665 0.73213 0.72472 0.72472 0.62890 0.62890 0.43749 0.43749 0.18720 0.18720 0.14093 0.14093 0.09420 0.09420 0.04717 0.04717 0.00000 -0.00665 -0.01097 -0.01097 -0.01813

-0.00274 -0.00059 -0.00059 0.00106 0.00106 0.00151 0.00151 0.00003 0.00003 -0.00137 -0.00137 -0.00300 -0.00300 -0.01255 -0.01255 -0.01903 -0.01903 -0.02242 -0.02242 -0.02270 -0.02270 -0.02288 -0.02288 -0.02299 -0.02299 -0.02302 -0.00137 -0.00149 -0.00149 -0.00153

-78.08 78.08 6.47 -6.47 91.08 -91.08 175.49 -175.49 236.81 -236.81 2501.90 -2501.90 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 -74.72 74.72 -28.84 28.84

146.67 332.13 -332.13 810.93 -810.93 1289.73 -1289.73 1768.53 -1768.53 1973.73 348.38 -348.38 348.05 -348.05 348.04 -348.04 348.03 -348.03 347.72 -347.72 328.53 -328.53 289.56 -289.56 257.88 -257.88 528.17 -322.97 322.85 0.00

-10141.02 9491.90 -9491.90 5491.19 -5491.19 -1861.15 1861.14 -12565.08 12565.06 -18178.47 16141.41 -14963.38 14964.00 -10811.87 10811.93 -6659.75 6659.80 -2507.76 2507.71 -1795.38 1795.20 -1121.56 1122.16 -528.43 528.77 -0.07 2038.64 -761.94 761.91 0.01

…………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… Cumulative Results of analysis at end of stage

6

Plastic Action at: Element/ Stage/ Code/ C02-02 1 rs P02X01 2 2 P01X01 3 2 P02X02 4 2 P01X02 5 2 C01-02 6 rs node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05

Lat. Force *g (DL= 3381.7) 0.1760 0.1818 0.1847 0.1875 0.1891 0.1903

---------- GLOBAL Displ.x Displ.y 0.87699 0.03093 0.87698 0.01084 0.87698 -0.00217 0.85928 -0.00201 0.72688 -0.00147 0.49873 -0.00092 0.21194 -0.00038 0.15947 -0.00028 0.10654 -0.00019 0.05333 -0.00009 0.00000 0.00000 0.87699 -0.01377 0.87701 -0.02802 0.87702 -0.02622 0.87700 -0.01502 0.87697 -0.00605

Chapter 21 – Seismic Design of Concrete Bridges

/ Deflection / (in) 8.7862 9.4798 9.8322 10.1774 10.3724 10.5239

--------Rotation -0.00425 -0.00428 -0.00441 -0.00605 -0.01563 -0.02210 -0.02546 -0.02572 -0.02590 -0.02600 -0.02603 -0.00332 -0.00079 0.00115 0.00178 0.00039

21-117

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17 18 19 20 21 22 23 24 25 26 27

S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

0.87696 0.87079 0.73519 0.50407 0.21424 0.16121 0.10771 0.05392 0.00000 0.87696 0.87697

element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rs 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn

-0.00682 -0.00634 -0.00462 -0.00290 -0.00118 -0.00089 -0.00059 -0.00030 0.00000 -0.01003 -0.01545

-0.00100 -0.00263 -0.01588 -0.02235 -0.02573 -0.02600 -0.02618 -0.02628 -0.02631 -0.00112 -0.00116

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.87699 0.03093 -0.00425 31.37 -0.01 0.00 2 0.87698 0.01084 -0.00428 -31.37 322.86 -761.98 2 0.87698 0.01084 -0.00428 79.93 -322.85 761.94 3 0.87698 -0.00217 -0.00441 -79.93 528.05 -2038.30 3 0.00217 0.87698 -0.00441 814.86 295.87 13637.29 4 0.00201 0.85928 -0.00605 -814.86 -295.87 -12636.74 4 0.00201 0.85928 -0.00605 814.88 295.88 12636.00 5 0.00147 0.72688 -0.01563 -814.88 -295.88 -9106.20 5 0.00147 0.72688 -0.01563 814.88 295.89 9106.22 6 0.00092 0.49873 -0.02210 -814.88 -295.89 -5576.25 6 0.00092 0.49873 -0.02210 814.88 295.88 5576.32 7 0.00038 0.21194 -0.02546 -814.88 -295.88 -2046.37 7 0.00038 0.21194 -0.02546 814.88 295.73 2047.11 8 0.00028 0.15947 -0.02572 -814.88 -295.73 -1440.69 8 0.00028 0.15947 -0.02572 814.88 275.51 1440.64 9 0.00019 0.10654 -0.02590 -814.88 -275.51 -876.00 9 0.00019 0.10654 -0.02590 814.88 231.57 876.37 10 0.00009 0.05333 -0.02600 -814.88 -231.57 -401.75 10 0.00009 0.05333 -0.02600 814.88 196.02 401.81 11 0.00000 0.00000 -0.02603 -814.88 -196.02 -0.02 3 0.87698 -0.00217 -0.00441 -176.75 286.81 -11599.76 12 0.87699 -0.01377 -0.00332 176.75 -81.61 12152.41 12 0.87699 -0.01377 -0.00332 -112.03 81.70 -12152.37 13 0.87701 -0.02802 -0.00079 112.03 397.10 11048.49 13 0.87701 -0.02802 -0.00079 -20.60 -397.10 -11048.48 14 0.87702 -0.02622 0.00115 20.60 875.90 6593.00 14 0.87702 -0.02622 0.00115 70.88 -875.90 -6593.00 15 0.87700 -0.01502 0.00178 -70.88 1354.70 -1214.09 15 0.87700 -0.01502 0.00178 162.24 -1354.70 1214.09 16 0.87697 -0.00605 0.00039 -162.24 1833.50 -12372.80 16 0.87697 -0.00605 0.00039 228.60 -1833.50 12372.78 17 0.87696 -0.00682 -0.00100 -228.60 2038.70 -18181.08 17 0.00682 0.87696 -0.00100 2566.87 349.18 16143.98 18 0.00634 0.87079 -0.00263 -2566.87 -349.18 -14963.33 18 0.00634 0.87079 -0.00634 2566.81 348.82 14964.00 19 0.00462 0.73519 -0.01588 -2566.81 -348.82 -10802.60 19 0.00462 0.73519 -0.01588 2566.81 348.82 10802.66 20 0.00290 0.50407 -0.02235 -2566.81 -348.82 -6641.19 20 0.00290 0.50407 -0.02235 2566.81 348.81 6641.24 21 0.00118 0.21424 -0.02573 -2566.81 -348.81 -2479.92 21 0.00118 0.21424 -0.02573 2566.81 348.49 2479.88 22 0.00089 0.16121 -0.02600 -2566.81 -348.49 -1765.94 22 0.00089 0.16121 -0.02600 2566.81 328.24 1765.77 23 0.00059 0.10771 -0.02618 -2566.81 -328.24 -1092.73 23 0.00059 0.10771 -0.02618 2566.81 284.77 1093.33 24 0.00030 0.05392 -0.02628 -2566.81 -284.77 -509.41 24 0.00030 0.05392 -0.02628 2566.81 248.60 509.76 25 0.00000 0.00000 -0.02631 -2566.81 -248.60 -0.07 17 0.87696 -0.00682 -0.00100 -80.75 528.17 2038.65 26 0.87696 -0.01003 -0.00112 80.75 -322.97 -761.94 26 0.87696 -0.01003 -0.00112 -31.13 322.85 761.91 27 0.87697 -0.01545 -0.00116 31.13 0.00 0.01

Chapter 21 – Seismic Design of Concrete Bridges

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APPENDIX 21.3-10 Force – Displacement Relationship, Bent 2, Right Push with Overturning

Chapter 21 – Seismic Design of Concrete Bridges

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

B

APPENDIX 21.3-11 Joint Movement Calculation

STATE OF CALIFORNIA.DEPARTMENT OF TRANSPORTATION a

JOINT MOVEMENTS CALCULATIONS

Note: Specific instructions are included as footnotes.

DS-D-0129(Rev.5/93) EA

DISTRICT

910076

COUNTY

59

ROUTE

PM (KP)

999

99

ES

BRIDGE NAME AND NUMBER Prototype Bridge

TYPE OF STRUCTURE

TYPE ABUTMENT

TYPE EXPANSION(2" elasto pads, etc.)

CIP/PS BOX GIRDER

Seat

Elastomeric Bearing Pads

(1) TEMPERATURE EXTREMES(from Preliminary Rerport) Type Of Structure

ANTICIPATED SHORTENING

(inches/100 feet)

(inches/100 feet)

(3)MOVEMENT FACTOR (inches/100 feet)

o

Steel

Range(

o

F)(0.0000065X1200) =

+

0

o

Concrete (Conventional)

Range(

o

F)(0.0000060X1200) =

+

0.06

Concrete(Pretensioned)

Range(

o

F)(0.0000060X1200) =

+

0.12

g

=

Concrete(Post Tensioned)

Range(

+

0.63

g

=

MAXIMUM

110 F

- MINIMUM

23 F

o

= Range

(2)THERMAL MOVEMENT

87 F

ITEM(1) DESIGNER

o

87 F)(0.0000060X1200) = DATE

0.6264

= =

1.26

ITEM(2)CHECKED BY

DESIGNER

DATE

CHECKER

To be filled in by Office of Structures Design

b

To be filled in by SR

c

Date:

Seal Width Limits

Location

d

Groove (saw cut) Width or Installation Width

Skew

(4)

Calculated

M.R.

Seal Type

(degrees)

Contributing

Movement

(inches)

A,B,

Catalog

W1

(5) W2

Structure

Do not

Length

(inches)

(Round up

(Others)

Number

(inches)

(inches)

Temperature

use in

(feet)

(3)X(4)/100

to 1/2")

or

Maximum

Min.@Max.

( F)

calculation

Open Joint

Abut 1

0

202

2.53

2.50

Joint Seal Assembly(strip seal)

Abut 4

0

210

2.64

3.00

Joint Seal Assembly(strip seal)

Temperature

o

f

e

(6)Adjust from

Width at

Maximum Temp.

Temp. Listed

(inches)

(inches)

 /(1)X(2)X(4)/100

see XS-12-59

Anticipated Shortening =

1.26  202  210     2.60 in. 100  2 

Chapter 21 – Seismic Design of Concrete Bridges

21-120

w=(5)+(6)

BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

APPENDIX 21.3-12 wFRAME Longitudinal Push Over – Force/Displacement Relationship, Right Push

Chapter 21 – Seismic Design of Concrete Bridges

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APPENDIX 21.3-13 wFRAME Longitudinal Push Over – Force vs. Displacement Relationship

Chapter 21 – Seismic Design of Concrete Bridges

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APPENDIX 21.3-14 Cap Beam – Seismic Moment and Shear Demands 05/15/2006, 15:50 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04

coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r

Cumulative Results of analysis at end of stage

6

Plastic Action at: Element/ Stage/ Code/ P02X01 1 2 P01X01 2 2 P02X02 3 2 P01X02 4 2 C02-02 5 rs

Lat. Force *g (DL= 3381.7) 0.1863 0.1958 0.1966 0.2059 0.2147

Chapter 21 – Seismic Design of Concrete Bridges

/ Deflection / (in) 9.3076 9.7870 9.8292 10.3036 10.7599

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BRIDGE DESIGN PRACTICE ● FEBRUARY 2015

C01-02

6

node# name

rs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

0.2275

S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02

---------- GLOBAL Displ.x Displ.y 1.03149 0.03456 1.03149 0.01254 1.03148 -0.00170 1.01183 -0.00158 0.85856 -0.00115 0.59020 -0.00072 0.25113 -0.00029 0.18897 -0.00022 0.12627 -0.00015 0.06321 -0.00007 0.00000 0.00000 1.03150 -0.01419 1.03152 -0.02840 1.03153 -0.02512 1.03151 -0.01281 1.03148 -0.00492 1.03145 -0.00729 1.02247 -0.00677 0.86615 -0.00493 0.59507 -0.00310 0.25323 -0.00126 0.19057 -0.00095 0.12734 -0.00063 0.06376 -0.00032 0.00000 0.00000 1.03146 -0.01250 1.03147 -0.02108

element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rs 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn

12.3779

--------Rotation -0.00466 -0.00469 -0.00482 -0.00679 -0.01829 -0.02608 -0.03015 -0.03047 -0.03069 -0.03081 -0.03085 -0.00350 -0.00064 0.00137 0.00182 -0.00001 -0.00167 -0.00363 -0.01853 -0.02630 -0.03039 -0.03072 -0.03095 -0.03107 -0.03111 -0.00179 -0.00183

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 1.03149 0.03456 -0.00466 37.40 -0.01 0.01 2 1.03149 0.01254 -0.00469 -37.40 322.86 -761.98 2 1.03149 0.01254 -0.00469 95.50 -322.85 761.94 3 1.03148 -0.00170 -0.00482 -95.50 528.05 -2038.30 3 0.00170 1.03148 -0.00482 639.94 353.65 16359.82 4 0.00158 1.01183 -0.00679 -639.94 -353.65 -15163.66 4 0.00158 1.01183 -0.00679 639.98 353.64 15163.00 5 0.00115 0.85856 -0.01829 -639.98 -353.64 -10944.11 5 0.00115 0.85856 -0.01829 639.98 353.66 10944.14 6 0.00072 0.59020 -0.02608 -639.98 -353.66 -6725.06 6 0.00072 0.59020 -0.02608 639.98 353.64 6725.13 7 0.00029 0.25113 -0.03015 -639.98 -353.64 -2506.10 7 0.00029 0.25113 -0.03015 639.98 353.51 2506.99 8 0.00022 0.18897 -0.03047 -639.98 -353.51 -1782.15 8 0.00022 0.18897 -0.03047 639.98 333.29 1782.11 9 0.00015 0.12627 -0.03069 -639.98 -333.29 -1099.10 9 0.00015 0.12627 -0.03069 639.98 289.29 1099.54 10 0.00007 0.06321 -0.03081 -639.98 -289.29 -506.62 10 0.00007 0.06321 -0.03081 639.98 247.19 506.68 11 0.00000 0.00000 -0.03085 -639.98 -247.19 0.00 3 1.03148 -0.00170 -0.00482 -211.27 111.89 -14322.39 12 1.03150 -0.01419 -0.00350 211.27 93.31 14350.28 12 1.03150 -0.01419 -0.00350 -133.82 -93.18 -14350.24 13 1.03152 -0.02840 -0.00064 133.82 571.98 12022.15 13 1.03152 -0.02840 -0.00064 -24.49 -571.98 -12022.14 14 1.03153 -0.02512 0.00137 24.49 1050.78 6342.45 14 1.03153 -0.02512 0.00137 84.84 -1050.79 -6342.45 15 1.03151 -0.01281 0.00182 -84.84 1529.59 -2688.86 15 1.03151 -0.01281 0.00182 194.14 -1529.59 2688.86 16 1.03148 -0.00492 -0.00001 -194.14 2008.39 -15071.78 16 1.03148 -0.00492 -0.00001 273.35 -2008.39 15071.76 17 1.03145 -0.00729 -0.00167 -273.35 2213.59 -21404.73

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17 C02-01 rn

17 18

0.00729 0.00677

18 C02-02 rs

18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27

0.00677 0.00493 0.00493 0.00310 0.00310 0.00126 0.00126 0.00095 0.00095 0.00063 0.00063 0.00032 0.00032 0.00000 1.03145 1.03146 1.03146 1.03147

19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn

1.03145 -0.00167 1.02247 -0.00363 1.02247 0.86615 0.86615 0.59507 0.59507 0.25323 0.25323 0.19057 0.19057 0.12734 0.12734 0.06376 0.06376 0.00000 -0.00729 -0.01250 -0.01250 -0.02108

Chapter 21 – Seismic Design of Concrete Bridges

-0.00706 -0.01853 -0.01853 -0.02630 -0.02630 -0.03039 -0.03039 -0.03072 -0.03072 -0.03095 -0.03095 -0.03107 -0.03107 -0.03111 -0.00167 -0.00179 -0.00179 -0.00183

2741.74 -2741.74

417.33 19367.77 -417.33 -17956.42

2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 -96.61 96.61 -37.19 37.19

417.18 17957.00 -417.18 -12980.05 417.18 12980.12 -417.18 -8003.12 417.17 8003.17 -417.17 -3026.31 416.81 3026.30 -416.81 -2172.40 396.55 2172.30 -396.55 -1359.19 353.00 1359.84 -353.00 -635.98 310.33 636.35 -310.33 -0.06 528.16 2038.61 -322.96 -761.93 322.85 761.91 0.00 0.01

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APPENDIX 21.3-15 wFRAME Select Output File – To Determine Superstructure Forces due to Column Hinging, Case 1 10/26/2013, 09:39 Design Academy Example No: 1 (Superstructure Right Push) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

name S01.00 S01.01 C01.01 P01.01 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 S02.07 S02.08 S02.09 S02.10 S02.11 S02.12 S02.13 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 S03.03 S03.04 S03.05 S03.06 S03.07 S03.08 S03.09

coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 2.00 0.00 r r r 2.00 -1.00 r r r 2.00 -2.00 f 0.0000 f 0.0000 r 12.57 0.00 r r r 23.14 0.00 r r r 33.71 0.00 r r r 44.28 0.00 r r r 54.85 0.00 r r r 65.42 0.00 r r r 75.99 0.00 r r r 86.56 0.00 r r r 97.13 0.00 r r r 107.70 0.00 r r r 115.70 0.00 r r r 123.70 0.00 r r r 127.96 0.00 r r r 127.96 -3.38 r r r 127.96 -15.31 r r r 127.96 -27.24 r r r 127.96 -39.17 r r r 127.96 -41.22 s 2.7e+002 r r 127.96 -43.27 s 8.3e+002 r r 127.96 -45.32 s 1.3e+003 r r 127.96 -47.37 f 0.0000 f 0.0000 r 132.22 0.00 r r r 140.22 0.00 r r r 148.22 0.00 r r r 160.97 0.00 r r r 173.72 0.00 r r r 186.47 0.00 r r r 199.22 0.00 r r r 211.97 0.00 r r r 224.72 0.00 r r r

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35 S03.10 237.47 36 S03.11 250.22 37 S03.12 262.97 38 S03.13 275.72 39 S03.14 283.72 40 S03.15 291.72 41 S03.16 295.98 42 C03.01 295.98 43 C03.02 295.98 44 C03.03 295.98 45 C03.04 295.98 46 P03.01 295.98 47 P03.02 295.98 48 P03.03 295.98 49 P03.04 295.98 50 P03.05 295.98 51 S04.01 300.24 52 S04.02 308.24 53 S04.03 316.24 54 S04.04 326.01 55 S04.05 335.78 56 S04.06 345.55 57 S04.07 355.32 58 S04.08 365.09 59 S04.09 374.86 60 S04.10 384.63 61 S04.11 394.40 62 S04.12 404.17 63 S04.13 413.94 64 C04.01 413.94 65 P04.01 413.94 66 S05.01 415.94

0.00 r 0.00 0.00 0.00 0.00 0.00 0.00 -3.38 -15.33 -27.28 -39.23 -41.46 -43.69 -45.92 -48.15 -50.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00 -2.00 0.00

r r r r r r r r r r r s s s s f r r r r r r r r r r r r r r f s

r r r r r r r r r r r

r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

3.2e+002 r 9.2e+002 r 1.5e+003 r 2e+003 r 0.0000 f 0.0000 r r r r r r r r r r r r r r 0.0000 f 0.0000 7.1e+003 r

Spring Information at node points: k's = k/ft or ft-k/rad.; d's = ft or rad. node spring k1 d1 k2 d2 # name 22 P02X01 272.74 0.149 0.00 23 P02X02 828.36 0.105 0.00 24 P02X03 1326.91 0.106 0.00 46 P03X01 317.46 0.149 0.00 47 P03X02 919.77 0.110 0.00 48 P03X03 1476.21 0.109 0.00 49 P03X04 2038.47 0.109 0.00 66 S05X01 7061.00 0.272 0.00

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000

Structural Setup: Spans= 5, Columns= 4, Piles= 4, Link Beams= 0 Element Information: element nodes # name fix i j 1 S01-01 rn 1 2 2 C01-01 rs 2 3 3 P01-01 rn 3 4 4 S02-01 rn 2 5 5 S02-02 rn 5 6 6 S02-03 rn 6 7 7 S02-04 rn 7 8 8 S02-05 rn 8 9 9 S02-06 rn 9 10 10 S02-07 rn 10 11 11 S02-08 rn 11 12 12 S02-09 rn 12 13 13 S02-10 rn 13 14 14 S02-11 rn 14 15 15 S02-12 rn 15 16

depth L d area Ei Ef 2.00 6.8 103.5 629528 60480 1.00 6.0 56.5 629528 62107 1.00 6.0 56.5 629528 62107 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 8.00 6.8 109.6 629528 60480 8.00 6.8 109.6 629528 60480

Chapter 21 – Seismic Design of Concrete Bridges

Icr 826.75 94.88 47.44 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 778.93 778.93

q Mpp Mpn tol status -0.01 99999 99999 0.02 e 0.00 99999 99999 0.02 e 0.00 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e

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16 S02-13 rn 16 17 4.26 6.8 115.6 629528 60480 826.75 -0.01 17 C02-01 rn 17 18 3.38 6.0 56.5 629528 62107 94.88 0.00 18 C02-02 rn 18 19 11.93 6.0 56.5 629528 62107 47.44 0.00 19 C02-03 rn 19 20 11.93 6.0 56.5 629528 62107 47.44 0.00 20 C02-04 rn 20 21 11.93 6.0 56.5 629528 62107 47.44 0.00 21 P02-01 rn 21 22 2.05 6.0 56.5 629528 62107 47.44 0.00 22 P02-02 rn 22 23 2.05 6.0 56.5 629528 62107 47.44 0.00 23 P02-03 rn 23 24 2.05 6.0 56.5 629528 62107 47.44 0.00 24 P02-04 re 24 25 2.05 6.0 56.5 629528 62107 47.44 0.00 25 S03-01 rn 17 26 4.26 6.8 115.6 629528 60480 826.75 -0.01 26 S03-02 rn 26 27 8.00 6.8 109.6 629528 60480 778.93 -0.01 27 S03-03 rn 27 28 8.00 6.8 109.6 629528 60480 778.93 -0.01 28 S03-04 rn 28 29 12.75 6.8 103.5 629528 60480 731.10 -0.01 29 S03-05 rn 29 30 12.75 6.8 103.5 629528 60480 731.10 -0.01 30 S03-06 rn 30 31 12.75 6.8 103.5 629528 60480 731.10 -0.01 31 S03-07 rn 31 32 12.75 6.8 103.5 629528 60480 731.10 -0.01 32 S03-08 rn 32 33 12.75 6.8 103.5 629528 60480 731.10 -0.01 33 S03-09 rn 33 34 12.75 6.8 103.5 629528 60480 731.10 -0.01 34 S03-10 rn 34 35 12.75 6.8 103.5 629528 60480 731.10 -0.01 35 S03-11 rn 35 36 12.75 6.8 103.5 629528 60480 731.10 -0.01 36 S03-12 rn 36 37 12.75 6.8 103.5 629528 60480 731.10 -0.01 37 S03-13 rn 37 38 12.75 6.8 103.5 629528 60480 731.10 -0.01 38 S03-14 rn 38 39 8.00 6.8 109.6 629528 60480 778.93 -0.01 39 S03-15 rn 39 40 8.00 6.8 109.6 629528 60480 778.93 -0.01 40 S03-16 rn 40 41 4.26 6.8 115.6 629528 60480 826.75 -0.01 41 C03-01 rn 41 42 3.38 6.0 56.5 629528 62107 94.44 0.00 42 C03-02 rn 42 43 11.95 6.0 56.5 629528 62107 47.22 0.00 43 C03-03 rn 43 44 11.95 6.0 56.5 629528 62107 47.22 0.00 44 C03-04 rn 44 45 11.95 6.0 56.5 629528 62107 47.22 0.00 45 P03-01 rn 45 46 2.23 6.0 56.5 629528 62107 47.22 0.00 46 P03-02 rn 46 47 2.23 6.0 56.5 629528 62107 47.22 0.00 47 P03-03 rn 47 48 2.23 6.0 56.5 629528 62107 47.22 0.00 48 P03-04 rn 48 49 2.23 6.0 56.5 629528 62107 47.22 0.00 49 P03-05 re 49 50 2.23 6.0 56.5 629528 62107 47.22 0.00 50 S04-01 rn 41 51 4.26 6.8 115.6 629528 60480 826.75 -0.01 51 S04-02 rn 51 52 8.00 6.8 109.6 629528 60480 778.93 -0.01 52 S04-03 rn 52 53 8.00 6.8 109.6 629528 60480 778.93 -0.01 53 S04-04 rn 53 54 9.77 6.8 103.5 629528 60480 731.10 -0.01 54 S04-05 rn 54 55 9.77 6.8 103.5 629528 60480 731.10 -0.01 55 S04-06 rn 55 56 9.77 6.8 103.5 629528 60480 731.10 -0.01 56 S04-07 rn 56 57 9.77 6.8 103.5 629528 60480 731.10 -0.01 57 S04-08 rn 57 58 9.77 6.8 103.5 629528 60480 731.10 -0.01 58 S04-09 rn 58 59 9.77 6.8 103.5 629528 60480 731.10 -0.01 59 S04-10 rn 59 60 9.77 6.8 103.5 629528 60480 731.10 -0.01 60 S04-11 rn 60 61 9.77 6.8 103.5 629528 60480 731.10 -0.01 61 S04-12 rn 61 62 9.77 6.8 103.5 629528 60480 731.10 -0.01 62 S04-13 rn 62 63 9.77 6.8 103.5 629528 60480 731.10 -0.01 63 C04-01 rs 63 64 1.00 6.0 56.5 629528 62107 94.44 0.00 64 P04-01 rn 64 65 1.00 6.0 56.5 629528 62107 47.22 0.00 65 S05-01 rn 63 66 2.00 6.8 103.5 629528 60480 826.75 -0.01 bandwidth of the problem = 11 Number of rows and columns in strage = 198 x 33 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------Cumulative Results of analysis at end of stage 8

99999 99999 32060 32060 32060 32060 32060 32060 32060 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 34512 34512 34512 34512 34512 34512 34512 34512 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999

99999 0.02 e 99999 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e

Plastic Action at: Element/ Stage/ Code/ S05X01 1 2 P03X02 2 2 P03X01 3 2 P02X01 4 2 P02X02 5 2 P03X03 6 2 C02-02 7 rs C03-02 8 rs

Lat. Force *g (DL= 4.2) 574.4611 693.4553 697.6155 773.7037 790.2726 800.7841 823.1016 826.2495

Chapter 21 – Seismic Design of Concrete Bridges

/ Deflection / (in) 3.3364 6.8393 6.9630 9.2504 9.7509 10.0718 10.7641 10.9793

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node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

S01.00 S01.01 C01.01 P01.01 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 S02.07 S02.08 S02.09 S02.10 S02.11 S02.12 S02.13 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 S03.03 S03.04 S03.05 S03.06 S03.07 S03.08 S03.09 S03.10 S03.11 S03.12 S03.13 S03.14 S03.15 S03.16 C03.01 C03.02 C03.03 C03.04 P03.01 P03.02 P03.03 P03.04 P03.05 S04.01 S04.02 S04.03 S04.04 S04.05 S04.06 S04.07 S04.08 S04.09 S04.10 S04.11 S04.12 S04.13 C04.01 P04.01 S05.01

---------- GLOBAL Displ.x Displ.y 0.91494 -0.00139 0.91494 0.00001 0.45747 0.00000 0.00000 0.00000 0.91493 0.00733 0.91490 0.01435 0.91487 0.02074 0.91481 0.02619 0.91474 0.03040 0.91466 0.03304 0.91457 0.03381 0.91446 0.03238 0.91433 0.02846 0.91419 0.02172 0.91409 0.01461 0.91397 0.00569 0.91391 0.00017 0.90584 0.00016 0.78568 0.00012 0.54645 0.00007 0.23382 0.00003 0.17604 0.00002 0.11766 0.00001 0.05892 0.00001 0.00000 0.00000 0.91389 -0.00523 0.91386 -0.01338 0.91381 -0.01907 0.91372 -0.02355 0.91360 -0.02325 0.91347 -0.01929 0.91332 -0.01282 0.91314 -0.00498 0.91294 0.00309 0.91273 0.01025 0.91249 0.01537 0.91223 0.01729 0.91195 0.01487 0.91178 0.01070 0.91160 0.00425 0.91150 -0.00020 0.90448 -0.00018 0.79901 -0.00014 0.58202 -0.00009 0.29496 -0.00004 0.23698 -0.00003 0.17826 -0.00003 0.11906 -0.00002 0.05959 -0.00001 0.00000 0.00000 0.91145 -0.00476 0.91133 -0.01206 0.91121 -0.01776 0.91104 -0.02267 0.91086 -0.02546 0.91067 -0.02639 0.91047 -0.02567 0.91025 -0.02355 0.91002 -0.02025 0.90978 -0.01601 0.90953 -0.01107 0.90926 -0.00565 0.90898 -0.00001 0.45449 0.00000 0.00000 0.00000 0.90892 0.00116

--------Rotation 0.00070 0.00070 -0.45747 -0.45747 0.00068 0.00064 0.00057 0.00046 0.00033 0.00017 -0.00003 -0.00025 -0.00050 -0.00078 -0.00100 -0.00123 -0.00136 -0.00339 -0.01570 -0.02377 -0.02801 -0.02835 -0.02858 -0.02871 0.00000 -0.00118 -0.00086 -0.00057 -0.00015 0.00018 0.00042 0.00058 0.00064 0.00061 0.00050 0.00029 0.00000 -0.00039 -0.00066 -0.00096 -0.00113 -0.00301 -0.01407 -0.02167 -0.02580 -0.02619 -0.02646 -0.02662 -0.02671 0.00000 -0.00102 -0.00081 -0.00062 -0.00039 -0.00019 -0.00001 0.00015 0.00028 0.00039 0.00047 0.00053 0.00057 0.00058 -0.45449 -0.45449 0.00058

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element node # name fix 1 S01-01 rn 2 C01-01 rs 3 P01-01 rn 4 S02-01 rn 5 S02-02 rn 6 S02-03 rn 7 S02-04 rn 8 S02-05 rn 9 S02-06 rn 10 S02-07 rn 11 S02-08 rn 12 S02-09 rn 13 S02-10 rn 14 S02-11 rn 15 S02-12 rn 16 S02-13 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 re 25 S03-01 rn 26 S03-02 rn 27 S03-03 rn 28 S03-04 rn 29 S03-05 rn 30 S03-06 rn 31 S03-07 rn 32 S03-08 rn 33 S03-09 rn

-------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.91494 -0.00139 0.00070 7.84 -0.01 -0.02 2 0.91494 0.00001 0.00070 -7.84 0.03 -0.12 2 -0.00001 0.91494 -0.45747 -121.46 -6.52 -0.08 3 0.00000 0.45747 -0.45747 121.46 6.52 -7.69 3 0.00000 0.45747 -0.45747 -121.46 2.15 2.75 4 0.00000 0.00000 -0.45747 121.46 -2.15 -0.30 2 0.91494 0.00001 0.00070 66.87 -121.49 0.11 5 0.91493 0.00733 0.00068 -66.87 121.59 -1284.78 5 0.91493 0.00733 0.00068 154.24 -121.59 1284.77 6 0.91490 0.01435 0.00064 -154.24 121.70 -2570.58 6 0.91490 0.01435 0.00064 241.60 -121.69 2570.61 7 0.91487 0.02074 0.00057 -241.60 121.80 -3857.45 7 0.91487 0.02074 0.00057 328.86 -121.79 3857.49 8 0.91481 0.02619 0.00046 -328.86 121.90 -5145.39 8 0.91481 0.02619 0.00046 416.22 -121.90 5145.38 9 0.91474 0.03040 0.00033 -416.22 122.01 -6434.50 9 0.91474 0.03040 0.00033 503.69 -122.01 6434.51 10 0.91466 0.03304 0.00017 -503.69 122.12 -7724.70 10 0.91466 0.03304 0.00017 590.80 -122.12 7724.80 11 0.91457 0.03381 -0.00003 -590.80 122.23 -9016.16 11 0.91457 0.03381 -0.00003 678.33 -122.23 9016.11 12 0.91446 0.03238 -0.00025 -678.33 122.33 -10308.64 12 0.91446 0.03238 -0.00025 765.50 -122.32 10308.56 13 0.91433 0.02846 -0.00050 -765.50 122.43 -11602.05 13 0.91433 0.02846 -0.00050 852.83 -122.42 11602.07 14 0.91419 0.02172 -0.00078 -852.83 122.53 -12896.67 14 0.91419 0.02172 -0.00078 929.37 -122.52 12896.78 15 0.91409 0.01461 -0.00100 -929.37 122.60 -13877.32 15 0.91409 0.01461 -0.00100 995.44 -122.61 13877.38 16 0.91397 0.00569 -0.00123 -995.44 122.69 -14858.62 16 0.91397 0.00569 -0.00123 1045.39 -122.66 14858.62 17 0.91391 0.00017 -0.00136 -1045.39 122.71 -15381.25 17 -0.00017 0.91391 -0.00136 -129.65 802.62 37277.89 18 -0.00016 0.90584 -0.00339 129.65 -802.62 -34564.80 18 -0.00016 0.90584 -0.00381 -129.70 803.15 34566.00 19 -0.00012 0.78568 -0.01570 129.70 -803.15 -24984.42 19 -0.00012 0.78568 -0.01570 -129.70 803.20 24984.50 20 -0.00007 0.54645 -0.02377 129.70 -803.20 -15402.33 20 -0.00007 0.54645 -0.02377 -129.70 803.19 15402.42 21 -0.00003 0.23382 -0.02801 129.70 -803.19 -5820.25 21 -0.00003 0.23382 -0.02801 -129.70 802.91 5819.92 22 -0.00002 0.17604 -0.02835 129.70 -802.91 -4173.36 22 -0.00002 0.17604 -0.02835 -129.70 763.19 4173.73 23 -0.00001 0.11766 -0.02858 129.70 -763.19 -2608.48 23 -0.00001 0.11766 -0.02858 -129.70 675.43 2608.88 24 -0.00001 0.05892 -0.02871 129.70 -675.43 -1224.02 24 -0.00001 0.05892 -0.02871 -129.70 596.96 1223.94 25 0.00000 0.00000 -0.02876 129.70 -596.96 0.19 17 0.91391 0.00017 -0.00136 274.71 -252.37 -21895.31 26 0.91389 -0.00523 -0.00118 -274.71 252.41 20820.07 26 0.91389 -0.00523 -0.00118 323.11 -252.43 -20820.31 27 0.91386 -0.01338 -0.00086 -323.11 252.51 18800.50 27 0.91386 -0.01338 -0.00086 389.44 -252.52 -18800.52 28 0.91381 -0.01907 -0.00057 -389.44 252.60 16780.05 28 0.91381 -0.01907 -0.00057 475.44 -252.60 -16780.07 29 0.91372 -0.02355 -0.00015 -475.44 252.73 13558.60 29 0.91372 -0.02355 -0.00015 580.52 -252.73 -13558.60 30 0.91360 -0.02325 0.00018 -580.52 252.86 10335.48 30 0.91360 -0.02325 0.00018 685.74 -252.86 -10335.48 31 0.91347 -0.01929 0.00042 -685.74 252.98 7110.72 31 0.91347 -0.01929 0.00042 791.29 -252.99 -7110.76 32 0.91332 -0.01282 0.00058 -791.29 253.11 3884.34 32 0.91332 -0.01282 0.00058 896.76 -253.11 -3884.34 33 0.91314 -0.00498 0.00064 -896.76 253.24 656.30 33 0.91314 -0.00498 0.00064 1001.66 -253.24 -656.33 34 0.91294 0.00309 0.00061 -1001.66 253.37 -2573.29

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34 S03-10 rn 35 S03-11 rn 36 S03-12 rn 37 S03-13 rn 38 S03-14 rn 39 S03-15 rn 40 S03-16 rn 41 C03-01 rn 42 C03-02 rs 43 C03-03 rn 44 C03-04 rn 45 P03-01 rn 46 P03-02 rn 47 P03-03 rn 48 P03-04 rn 49 P03-05 re 50 S04-01 rn 51 S04-02 rn 52 S04-03 rn 53 S04-04 rn 54 S04-05 rn 55 S04-06 rn 56 S04-07 rn 57 S04-08 rn 58 S04-09 rn 59 S04-10 rn 60 S04-11 rn 61 S04-12 rn 62 S04-13 rn 63 C04-01 rs 64 P04-01 rn 65 S05-01 rn

34 0.91294 35 0.91273 35 0.91273 36 0.91249 36 0.91249 37 0.91223 37 0.91223 38 0.91195 38 0.91195 39 0.91178 39 0.91178 40 0.91160 40 0.91160 41 0.91150 41 0.00020 42 0.00018 42 0.00018 43 0.00014 43 0.00014 44 0.00009 44 0.00009 45 0.00004 45 0.00004 46 0.00003 46 0.00003 47 0.00003 47 0.00003 48 0.00002 48 0.00002 49 0.00001 49 0.00001 50 0.00000 41 0.91150 51 0.91145 51 0.91145 52 0.91133 52 0.91133 53 0.91121 53 0.91121 54 0.91104 54 0.91104 55 0.91086 55 0.91086 56 0.91067 56 0.91067 57 0.91047 57 0.91047 58 0.91025 58 0.91025 59 0.91002 59 0.91002 60 0.90978 60 0.90978 61 0.90953 61 0.90953 62 0.90926 62 0.90926 63 0.90898 63 0.00001 64 0.00000 64 0.00000 65 0.00000 63 0.90898 66 0.90892

0.00309 0.01025 0.01025 0.01537 0.01537 0.01729 0.01729 0.01487 0.01487 0.01070 0.01070 0.00425 0.00425 -0.00020 0.91150 0.90448 0.90448 0.79901 0.79901 0.58202 0.58202 0.29496 0.29496 0.23698 0.23698 0.17826 0.17826 0.11906 0.11906 0.05959 0.05959 0.00000 -0.00020 -0.00476 -0.00476 -0.01206 -0.01206 -0.01776 -0.01776 -0.02267 -0.02267 -0.02546 -0.02546 -0.02639 -0.02639 -0.02567 -0.02567 -0.02355 -0.02355 -0.02025 -0.02025 -0.01601 -0.01601 -0.01107 -0.01107 -0.00565 -0.00565 -0.00001 0.90898 0.45449 0.45449 0.00000 -0.00001 0.00116

Chapter 21 – Seismic Design of Concrete Bridges

0.00061 0.00050 0.00050 0.00029 0.00029 0.00000 0.00000 -0.00039 -0.00039 -0.00066 -0.00066 -0.00096 -0.00096 -0.00113 -0.00113 -0.00301 -0.00301 -0.01407 -0.01407 -0.02167 -0.02167 -0.02580 -0.02580 -0.02619 -0.02619 -0.02646 -0.02646 -0.02662 -0.02662 -0.02671 -0.02671 -0.02673 -0.00113 -0.00102 -0.00102 -0.00081 -0.00081 -0.00062 -0.00062 -0.00039 -0.00039 -0.00019 -0.00019 -0.00001 -0.00001 0.00015 0.00015 0.00028 0.00028 0.00039 0.00039 0.00047 0.00047 0.00053 0.00053 0.00057 0.00057 0.00058 -0.45449 -0.45449 -0.45449 -0.45449 0.00058 0.00058

1106.93 -1106.93 1211.99 -1211.99 1317.09 -1317.09 1422.49 -1422.49 1508.34 -1508.34 1574.11 -1574.11 1624.06 -1624.06 139.21 -139.21 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 935.80 -935.80 983.99 -983.99 1049.77 -1049.77 1122.99 -1122.99 1203.85 -1203.85 1284.17 -1284.17 1364.86 -1364.86 1445.46 -1445.46 1526.44 -1526.44 1607.02 -1607.02 1687.98 -1687.98 1768.74 -1768.74 1849.31 -1849.31 116.10 -116.10 116.10 -116.10 1913.55 -1913.55

-253.37 253.49 -253.49 253.62 -253.62 253.75 -253.74 253.87 -253.87 253.95 -253.95 254.03 -254.03 254.07 721.18 -721.18 721.63 -721.63 721.60 -721.60 721.60 -721.60 721.93 -721.93 675.41 -675.41 573.88 -573.88 412.94 -412.94 291.44 -291.44 -114.87 114.91 -114.90 114.98 -114.98 115.06 -115.04 115.14 -115.16 115.26 -115.24 115.34 -115.36 115.46 -115.47 115.56 -115.57 115.67 -115.68 115.78 -115.78 115.88 -115.89 115.99 -115.99 116.09 3.57 -3.57 -8.59 8.59 0.03 -0.01

2573.29 -5804.54 5804.56 -9037.39 9037.42 -12271.85 12271.87 -15507.92 15507.96 -17539.28 17539.29 -19571.26 19571.27 -20653.43 34272.36 -31834.75 31835.00 -23211.55 23211.56 -14588.34 14588.36 -5965.18 5964.92 -4354.71 4356.28 -2849.53 2849.93 -1570.10 1570.30 -649.82 649.96 0.02 -13619.84 13130.47 -13130.79 12211.27 -12211.27 11291.09 -11291.09 10166.65 -10166.64 9041.13 -9041.08 7914.66 -7914.64 6787.06 -6787.07 5658.43 -5658.45 4528.83 -4528.86 3398.19 -3398.23 2266.51 -2266.52 1133.80 -1133.79 0.10 -4.40 -2.28 -4.71 -3.97 -0.06 0.02

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APPENDIX 21.3-16 PSSECx Input File PSSEC300,_OCT_26_2005 Bridge Design Academy - Prototype Supestructure Capacity S1 1.0NEG Number of different types of concrete 1 For each concrete type input: Type number; Model code= 0 simple(unconfined/confined), 1 Mander's (unconfined) strength f'c0 (ksi), strain ec0, strength fcu (ksi), ult. strain ecu, conc. density 1 1 5.200 .002 0.5 0.0025 150 Number of different types of P/S steel 1 For each type, 1st line for tensile parameters,2nd line for cmpressive parameters type#;E;fy;strain hard. factor;fu;ult. strain;PS-code: 0 tendons, 1 otherwise E;fy;strain hard. factor;fu;ult. strain 1 28500 245 2 270 0.030 0 0 0 0 0 0 Number of different types of mild steel 1 For each steel type input: Type number;Model code= 0 simple, 1 complex E(ksi);fy(ksi);strain hard. factor;fu(ksi);ultimate strain 1 1 29000 68 6.41 95 0.09 Number of Conc. Subsections 1 For each Subsec.:Subsection #,Section shape type, Concrete type, No. of fibers Subsec. Dim.(in):(See Manual for input parameters.) Subsec. Dim.(in):(See Manual for input parameters.) Global coord. of the center of Subsec.: Xg, Yg 1 I-shaped, 1 200 706.0 48.0 517.0 81.0 9.125 8.25 0 -5.26 Number of P/S steel groups 1 For each group:group#;P/S type;x-coord.(in);y-coord.(in);area(in^2);P/S force 1 1 0 25.4412 38.28 6157 Number of mild steel rebar cages (rebar distributed around the perimeter) 0 cage#;steel type;cage shape;#of bars;x(in) of 1st bar(y=0);area(in^2)of bar Number of mild steel groups (no logical pattern for distribution) 2 n group#;steel type;x-coord.(in); y-coord.(in); area(in^2) 1 1 0 31.80 47.40 2 1 0 -42.13 34.76 Non P/S Axial load on mid-depth of section (Kips)(+ sign=compression) 0 Numerical Computation Factor (1 to 10) 5 Computer Graphics Card identifier: 0 none; 2 CGA; 3 Hercules; 9 EGA; 12 VGA 12 Output control: 0 short; 1 long output 1 X-Sec. plot control (0=no plot, 1=each stage, 2=every iteration of each step) 0 Analysis Control: p - Positive moment, n - Negative moment n

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APPENDIX 21.3-17 PSSECx Model for Superstructure

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APPENDIX 21.3-18 Partial Output from PSSECx Run 05-15-2006 ********

SECx

********

DUCTILITY and STRENGTH of Rectangular, T-, I-, Hammer, Octagonal, Circular, Ring, and Hollowed shaped Prestressed and Reinforced Concrete Sections using fiber models Ver. 3.00, OCT-26-2005 Copyright (C) 2005 By Mark Seyed and Don Lee. This program should not be distributed under any condition. This release is for demo ONLY (beta testing is not complete). Caltrans or the author make no expressed or implied warranty of any kind with regard to this program. In no event shall the author or Caltrans be held liable for incidental or consequential damages arising out of the use of this program. JOB TITLE:

Bridge Design Academy - Prototype Supestructure Capacity S1 1.0NEG

Concrete Data, Complex Model, Mander's unconfined Concrete Type Compressive Strength (max.) Strain at max. Strength Strength at Ultimate Strain Ultimate strain Unit Weight (pcf)

(ksi) (ksi)

= 1 = 5.200 = .00200 = 0.000 = .00500 = 150.00

Prestressing Steel Data Material No.

Yield Strain

Hardening Strain

Yield Ultimate Modulus Stress Stress of Elasticity ksi ksi ksi 0.03000 245.10 270.00 28500.00 Tensile prop. 0.00000 0.00 0.00 0.00 Compressive prop. 1 is 7-wire and Low-Relaxation Tendon

1

Ultimate Strain

0.00860 0.00860 0.00000 0.00000 Prestress element type # with 270 ksi strands. ( Refer to PCI Design Handbook 4th Edition.) Mild Steel Reinforcing Data Material No. 1

Yield Strain

0.00234

Hardening Strain 0.01503

Ultimate Strain 0.09000

Yield Stress ksi 68.00

Ultimate Stress ksi 95.00

Rectangular, T-, or I-shaped section information Depth of Section Top Flange width Top Flange thickness Bot Flange width Bot Flange thickness Web thickness

(in.) (in.) (in.) (in.) (in.) (in.)

= = = = = =

81.00 706.00 9.13 517.00 8.25 48.00

Concrete fiber information Fiber Material x y # # (in) (in) 1 1.0 0.00 -45.56 2 1.0 0.00 -45.17 ……………………………………………… ……………………………………………… ……………………………………………… 197 1.0 0.00 33.79 199 1.0 0.00 34.62 200 1.0 0.00 35.03

area (in^2) 203.11 203.11

292.83 292.83 292.83

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Prestressing Steel Fiber Data Fiber No. 1

Material No. 1

x (in)

y (in)

0.00

25.44

area (in^2) 38.28

P/S force Kips

6157.00

Total P/S force on the section = 6157.0 kips Total moment due to P/S about point (0, 0) = 13053.5 ft-kip Mild Steel Fiber Data Fiber No. 1 2

Material No. 1 1

x (in) 0.00 0.00

y (in) 31.80 -42.13

area (in^2) 47.40 34.76

Axial load at mid-depth of section (kip)(positive means compression) =

0.0

*************************************************** * Analysis Results --Negative Moment Capacity * *************************************************** Initial state due to P/S without non-P/S axial force: N.A. Loc. Curvature Conc. Strain @ max. compressed fiber -41.50 0.0000023 0.00017950 Undeformed P/S element position w.r.t. reference plane P/S Fiber Loc.(y) Undef. pos. Conc. Strain @ same loc. 1 25.44 -0.0058006 -0.0001570 Force Equilibrium Condition of the x-section: Max. Max. Conc. Neutral Steel Strain Axis Strain Conc. step epscmax in. Tens. Comp. 0 -.00001 -41.50 -.00000 5923. 1 -.00001 -42.26 0.00000 5923. 2 -.00001 -43.05 0.00000 5923. 3 -.00000 -43.86 0.00000 5923. 4 -.00000 -44.70 0.00000 5924. 5 0.00000 -45.56 0.00000 5925. 6 0.00010 9055.25 0.00000 5983. 7 0.00011 362.50 0.00000 5990. 8 0.00013 174.76 0.00000 5997. 9 0.00014 110.77 0.00000 6006. 10 0.00016 78.67 0.00000 6017. 11 0.00018 59.45 0.00000 6028. 12 0.00020 46.72 0.00000 6041. 13 0.00022 37.74 0.00000 6055. 14 0.00025 29.97 -.00001 6079. 15 0.00028 14.77 -.00008 6224. 16 0.00032 2.23 -.00020 6470. 17 0.00035 -6.69 -.00035 6806. 18 0.00040 -12.96 -.00055 7231. 19 0.00045 -17.40 -.00078 7745. 20 0.00050 -20.61 -.00105 8346. 21 0.00056 -22.97 -.00136 9034. 22 0.00063 -24.75 -.00171 9811. 23 0.00071 -26.11 -.00210 10680. 24 0.00079 -27.79 -.00266 11498. 25 0.00089 -30.09 -.00356 12094. 26 0.00100 -32.67 -.00499 12356. 27 0.00112 -34.65 -.00682 12476. 28 0.00126 -36.06 -.00897 12528.

Chapter 21 – Seismic Design of Concrete Bridges

Steel force P/S Comp. Tens. force 236. -1. -6157. 235. 0. -6158. 236. 0. -6158. 236. 0. -6159. 237. 0. -6160. 237. 0. -6161. 237. 0. -6220. 237. 0. -6227. 237. 0. -6235. 237. 0. -6244. 237. 0. -6254. 238. 0. -6265. 238. 0. -6278. 238. 0. -6292. 242. -8. -6312. 268. -109. -6383. 296. -269. -6496. 326. -483. -6648. 359. -751. -6840. 395. -1072. -7069. 435. -1445. -7336. 480. -1872. -7642. 530. -2354. -7987. 587. -2893. -8372. 645. -3223. -8920. 698. -3223. -9568. 739. -3223. -9872. 774. -3223.-10027. 809. -3223.-10115.

Net Curvature Moment force in/in (K-ft) -0.8 0.000002 -4. -0.4 0.000002 -147. -0.5 0.000002 -307. 0.3 0.000002 -486. 0.2 0.000002 -683. -0.7 0.000002 -899. -0.4 -.000000 -13142. -0.3 -.000000 -14634. 0.8 -.000001 -16309. 0.9 -.000001 -18186. -0.1 -.000001 -20287. -0.7 -.000002 -22643. -0.3 -.000002 -25286. -1.0 -.000003 -28243. -0.1 -.000003 -31443. 0.4 -.000005 -33995. -0.7 -.000007 -36442. -0.9 -.000009 -39119. 0.5 -.000012 -42153. 0.4 -.000016 -45615. 0.3 -.000020 -49549. -0.4 -.000025 -53987. -0.2 -.000030 -58960. -1.0 -.000036 -64494. 0.4 -.000045 -69683. -0.5 -.000058 -73488. 0.3 -.000077 -75410. 0.9 -.000103 -76502. 0.1 -.000132 -77210.

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29 30 31 32 33 34 35 36 37 38 39 40

0.00141 0.00158 0.00178 0.00199 0.00223 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

-37.05 -37.76 -38.26 -38.64 -38.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-.01141 -.01411 -.01704 -.02027 -.02388 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

12543. 12533. 12639. 12782. 12893. 0. 0. 0. 0. 0. 0. 0.

848. 893. 948. 1012. 1084. 0. 0. 0. 0. 0. 0. 0.

-3223.-10168. -3223.-10204. -3360.-10228. -3546.-10247. -3716.-10261. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0.6 0.9 0.9 -0.3 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0

-77720. -78121. -79244. -80600. -81787. 0. 0. 0. 0. 0. 0. 0.

-.000166 -.000203 -.000243 -.000288 -.000338 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Prestress Tendon Strain on the x-section: Max. Neutral P/S Steel Conc. Axis Strain No. Strain in. step epscmax 0 -.00001 -41.50 1 -.005644 1 -.00001 -42.26 1 -.005644 2 -.00001 -43.05 1 -.005645 3 -.00000 -43.86 1 -.005646 4 -.00000 -44.70 1 -.005647 5 0.00000 -45.56 1 -.005648 6 0.00010 9055.25 1 -.005701 7 0.00011 362.50 1 -.005708 ……………………………………………… ……………………………………………… ……………………………………………… 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.00063 0.00071 0.00079 0.00089 0.00100 0.00112 0.00126 0.00141 0.00158 0.00178 0.00199 0.00223 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

-24.75 -26.11 -27.79 -30.09 -32.67 -34.65 -36.06 -37.05 -37.76 -38.26 -38.64 -38.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Strain No. Strain

No. Strain

No. Strain

No. Strain

-.007321 -.007674 -.008176 -.008996 -.010301 -.011970 -.013932 -.016152 -.018616 -.021292 -.024243 -.027534 -.005801 -.005801 -.005801 -.005801 -.005801 -.005801 -.005801

Recommended value of 'effective moment of inertia' based on initial slope of moment-curvature diagram (ft^4) = 211.8303 Yield pt. is defined as the First mild steel yields. 23 and The first mild steel yields between the following Steps: The computation of mild steel yield point IS within 2% tolerance. 24 and The first P/S steel yields between the following Steps: The computation of P/S steel yield point IS NOT within 2% tolerance. Yield Nominal Ultimate

24 25

Moments (ft-K) Curvature(rad/in) 67871 0.000040 See force equilibrium table at concrete strain of .003 0 0.000000

End

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NOTATION AASHTO

=

AASHTO LRFD Bridge Design Specifications with Interims and California Amendments

Ab

= area of individual reinforcing steel bar (in.2)

top Acap

= area of bent cap top flexural steel (in.2)

bot Acap

= area of bent cap bottom flexural steel (in.2)

Acv

=

area of concrete engaged in interface shear transfer (in.2)

Ae

=

effective shear area; effective abutment wall area (in.2)

Ag

=

gross cross section area (in.2)

Ajh

=

effective horizontal area of a moment resisting joint (in.2)

Ajv

=

effective vertical area for a moment resisting joint (in.2)

Aps

=

prestressing steel area (in.2)

As

=

area of supplemental non-prestressed tension reinforcement (in.2)

Asjh

=

area of horizontal joint shear reinforcement required at moment resisting joints (in.2)

Asjhc

=

total area of horizontal ties placed at the end of the bent cap in Case 1 Knee joints (in.2)

Asjv

=

area of vertical joint shear reinforcement required at moment resisting joints (in.2)

Asj bar

= area of vertical “J” bar reinforcement required at moment resisting joints with a skew angle > 20 (in.2)

Assf

=

area of bent cap side face steel required at moment resisting joints (in.2)

Ask

=

area of interface shear reinforcement crossing the shear plane (Vertical shear key reinforcement) (in.2)

Ast ,max

= maximum longitudinal reinforcement area (in.2)

Ast ,min

= minimum longitudinal reinforcement area (in.2)

Ast

=

total area of column longitudinal reinforcement anchored in the joint; total area of column/pier wall longitudinal reinforcement (in.2)

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Asu bar

=

Ash

= area of horizontal shear key reinforcement (hanger bars) (in.2)

area of bent cap top and bottom reinforcement bent in the form of “U” bars in Knee joints (in.2)

AskIso( provided) = area of interface shear reinforcement provided for isolated shear key (in.2) iso AskNon area of interface shear reinforcement provided for non-isolated shear key ( provided) =

(in.2) Av

=

area of shear reinforcement perpendicular to flexural tension reinforcement (in.2)

Bcap

= bent cap width (in.)

BDD

= Caltrans Bridge Design Details

Beff

= effective width of the superstructure for resisting longitudinal seismic moments (in.)

Dc

= column cross sectional dimension in the direction of interest (in.)

Dftg

= depth of footing (in.)

Ds

= depth of superstructure at the bent cap (in.)

DSH

=

D'

= cross-sectional dimension of confined concrete core measured between the centerline of the peripheral hoop or spiral (in.)

Dc'

= confined column cross-section dimension, measured out to out of ties, in the direction parallel to the axis of bending (in.)

Ec

= modulus of elasticity of concrete (ksi)

EDA

=

Elastic Dynamic Analysis

ESA

=

Elastic Static Analysis

Fsk

=

abutment shear key force capacity; Shear force associated with column

Design Seismic Hazards

overstrength moment, including overturning effects (ksi) Ieff , I e

=

effective moment of inertia for computing member stiffness (in.4)

ISA

=

Inelastic Static Analysis

K abut

=

abutment backwall stiffness (kip/in./ft)

Keff

=

effective abutment backwall stiffness (kip/in./ft)

Ki

=

Initial abutment backwall stiffness (kip/in./ft)

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L

= member length from the point of maximum moment to the point of contraflexure (in); length of bridge deck between adjacent expansion joints

Lmin,headed = minimum horizontal length from the end of the lowest layer of headed hanger bar to the intersection with the shear key vertical reinforcement (in.)

Lmin,hooked

= minimum horizontal length from the end of the lowest layer of hanger bar hooks to the intersection with the shear key vertical reinforcement (in.)

Lp

= equivalent analytical plastic hinge length (in.)

Mdl

= moment attributed to dead load (kip-ft)

M eqcol

=

column moment when coupled with any existing Mdl & Mp/s will equal the column’s overstrength moment capacity, Mocol (kip-ft)

M eqR , L

= portion of Meqcol distributed to the left or right adjacent superstructure spans (kip-ft)

Mn

=

nominal moment capacity based on the nominal concrete and steel strengths when the concrete strain reaches 0.003 (kip-ft)

Mne

=

nominal moment capacity based on the expected material properties and a concrete strain, c = 0.003 (kip-ft)

sup R , L M ne

=

expected nominal moment capacity of the right and left superstructure spans utilizing expected material properties (kip-ft)

M ocol

=

column overstrength moment (kip-ft)

Mpcol

=

Idealized plastic moment capacity of a column calculated by M- analysis (kip-ft)

Mp/s

=

moment attributed to secondary prestress effects (kip-ft)

My

=

Moment capacity of a ductile component corresponding to the first reinforcing bar yielding (kip-ft)

M-

=

moment curvature analysis

MTD

=

Caltrans Memo To Designers

NH

=

minimum hinge seat width normal to the centerline of bent (in.)

NA

=

abutment support width normal to centerline of bearing (in.)

P

=

absolute value of the net axial force normal to the shear plane (kip)

Pb

=

beam axial force at the center of the joint including prestressing (kip)

Pc

=

column axial force including the effects of overturning (kip)

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Pdia

=

passive pressure force resisting movement at diaphragm abutment (ksf)

Pdl

=

superstructure dead load reaction at the abutment plus weight of the abutment and its footing (kip)

Pdlsup

=

superstructure axial load resultant at the abutment (kip)

P/S

=

prestressed concrete; prestressing strand

Pjack

=

total prestress jacking force (kip)

Pn

=

nominal axial resistance (kip)

Pbw

=

passive pressure force resisting movement at seat abutment (ksf)

RA

=

abutment displacement coefficient

S

=

cap beam short stub length (ft)

SDC

=

Seismic Design Criteria

T

=

natural period of vibration, (seconds), T = 2 m k

Tc

=

total tensile force in column longitudinal reinforcement associated with Mocol (kip)

Ti

=

natural period of the stiffer frame (sec.)

Tj

= natural period of the more flexible frame (sec.)

Vc

=

nominal shear strength provided by concrete (kip)

Vn

=

nominal shear strength (kip)

Vo

=

overstrength shear associated with the overstrength moment Mo (kip)

Vocol

=

column overstrength shear, typically defined as Mocol /L (kip)

V pcol Vs

= =

column plastic shear, typically defined as Mpcol/L (kip) nominal shear strength provided by shear reinforcement (kip)

Vww

=

shear capacity of one wingwall (kip)

a

=

demand spectral acceleration

bv

=

effective web width taken as the minimum web width within the shear depth

d v (in.) dbl

=

nominal bar diameter of longitudinal column reinforcement (in.2)

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dv

=

effective shear depth defined as the distance between resultants of tensile and compressive forces due to flexural, but need not be taken less than 0.9d e or 0.72h (in.)

fh

=

average normal stress in the horizontal direction within a moment resisting joint (ksi)

fv

=

average normal stress in the vertical direction within a moment resisting joint (ksi)

fy

=

nominal yield stress for A706 reinforcement (ksi)

fye

=

expected yield stress for A706 reinforcement (ksi)

fyh

=

nominal yield stress of transverse column reinforcement, hoops/spirals (ksi)

f c'

=

compressive strength of unconfined concrete (psi)

f cc'

=

confined compression strength of concrete (psi)

f ce'

=

expected compressive strength of unconfined concrete (psi)

f1 , f 2

= concrete shear factors for ductile members

g

=

acceleration due to gravity, 32.2 ft sec2

h

=

distance from the center of gravity of the tensile force to the center of gravity of the compressive force of the column section (in.)

hdia

=

backwall height for diaphragm abutment (in.)

hbw

=

backwall height for seat abutment (in.)

k ie , k ej

=

smaller and larger effective bent or column stiffness, respectively (kip/in.)

lac

=

minimum length of column longitudinal reinforcement extension into the bent cap (in.)

lac,provided

=

actual length of column longitudinal reinforcement embedded into the bent cap (in.)

ld

=

development length of the main reinforcement (in.)

l dh

=

development length in tension of standard hooked bars (in.)

mi

=

tributary mass of column or bent i, m = W/g (kip-sec2/ft)

mj

=

tributary mass of column or bent j, m = W/g (kip-sec2/ft)

pbw

=

maximum abutment backwall soil pressure (ksf)

pc

=

nominal principal compression stress in a joint (psi)

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pt

=

nominal principal tension stress in a joint (psi)

s

=

spacing of shear/transverse reinforcement (in.)

t

= top or bottom slab thickness (in.)

vjv

=

nominal vertical shear stress in a moment resisting joint (psi)

vc

=

permissible shear stress carried by concrete (psi)

w

=

width of the backwall or diaphragm, as appropriate (in.)



= factor defining the range over which Fsk is allowed to vary



= factor indicating ability of diagonally cracked concrete to transmit tension and shear

 suR

=

reduced ultimate tensile strain for A706 reinforcement

c

=

local member displacement capacity (in.)

col

=

displacement attributed to the elastic and plastic deformation of the column (in.)

C

=

global displacement capacity (in.)

cr+sh

= displacement due to creep and shrinkage (in.)

d

=

local member displacement demand (in.)

D

=

global system displacement (in.)

 eff

= effective longitudinal abutment displacement at idealized yield (in.)

eq

=

relative longitudinal displacement demand at an expansion joint due to earthquake (in.)

p

=

idealized plastic displacement capacity due to rotation of the plastic hinge (in.)

p/s

=

displacement due to prestress shortening (in.)

r

=

relative lateral offset between the point of contra-flexure and the base of the plastic hinge (in.)

tem

=

displacement due to temperature variation (in.)

Y

=

idealized yield displacement of the subsystem at the formation of the plastic hinge (in.)

Y(i)

=

idealized yield displacement of the subsystem at the formation of plastic hinge (i) (in.)

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col Y

=

idealized yield displacement of a column at the formation of the plastic hinge (in.)



=

angle of inclination of diagonal compressive stresses (radians)

p

=

plastic rotation capacity (radians)

 sk

=

skew angle (degree)

s

=

amount of transverse reinforcement expressed as volumetric ratio



=

resistance factor

p

=

idealized plastic curvature (1/in.)

u

=

ultimate curvature capacity (1/in.)

y

=

yield curvature corresponding to the first yield of the reinforcement in a ductile component (1/in.)

Y

=

idealized yield curvature (1/in.)

d

=

local displacement ductility demand

D

=

global displacement ductility demand

c

=

local displacement ductility capacity

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REFERENCES 1. AASHTO, (2012). “AASHTO LRFD Bridge Design Specifications,” U.S. Customary Units 2012 (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 2. Caltrans, (2014). California Amendments to the AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2013). “Seismic Design Criteria,” Version 1.7, California Department of Transportation, Sacramento, CA. 4. Caltrans (2009). Bridge Memo To Designers 6-1 Column Analysis Consideration, California Department of Transportation, Sacramento, CA, February 2009. 5. Caltrans, (2007). CTBridge Help System, Version 1.3 (Online), Caltrans Bridge Analysis and Design, California Department of Transportation, Sacramento, CA. 6. Caltrans, (2001a). Bridge Memo To Designers 20-6 Seismic Strength of Concrete Bridge Superstructures, California Department of Transportation, Sacramento, CA. 7. Caltrans, (2001b). Memo To Designers 20-9 Splices in Bar Reinforcing Steel, California Department of Transportation, Sacramento, CA, August 2001. 8. CSI, (1976-2007). SAP2000 Advanced 11.0.8, Static and Dynamic Finite Element Analysis of Structures, Computers and Structures, Inc., Berkeley, CA. 9. Mahan, M., (2006). User’s Manual for “xSECTION,” Cross Section Analysis Program, Version 4.00, California Department of Transportation, Sacramento, CA. 10. Mahan, M., (1995). User’s Manual for “wFRAME,” 2-D Push Analysis Program, Version 1.13, California Department of Transportation, Sacramento, CA.

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