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ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 107-S28

Relationship between Strain, Curvature, and Drift in Reinforced Concrete Moment Frames in Support of Performance-Based Seismic Design by Aidcer L. Vidot-Vega and Mervyn J. Kowalsky Through the use of moment-curvature analyses, trends between strain and curvature for rectangular reinforced concrete (RC) sections are explored. Variables in the analyses included section dimensions, axial load ratio (ALR) and longitudinal steel ratio (ρlong). Curvature-strain relationships that depend on ALR and ρlong are developed from the results. These expressions are subsequently used to develop equations to compute interstory drift as a function of material strains for RC moment frames. The resultant equations can be used in performance-based design approaches such as direct displacement-based seismic design to compute target drifts and system displacements based on concrete and reinforcing steel strains. The interstory drift equations were correlated against 54 frame building analyses using a fiber-based analysis program. The drift expressions are shown to be accurate to within ±12% and ±20% for steel tension and concrete compression strain, respectively. Lastly, the implications of the proposed expressions on current code-based drift and ductility limits are explored. Keywords: limit states; performance-based design; reinforced concrete frames.

INTRODUCTION Performance-based seismic design has been the subject of significant research activity among the earthquake engineering community for over two decades. According to the Structural Engineering Association of California (SEAOC) Vision 2000 document (1995), performance-based seismic engineering (PBSE) consists of a “set of engineering procedures for the design and construction of structures to achieve predictable levels of performance in response to specified level of hazards (design earthquakes), within definable levels of reliability.” This is accomplished by the definition of performance objectives that are selected by the owner and engineer prior to the design. The Direct Displacement-based design (DDBD) method has evolved as a way to implement PBSE in a direct manner (Priestley et al. 2007). In general, PBSE relies on the identification of structural performance on the basis of limit states that are often defined on the basis of drift or displacement. The relationship of these deformation quantities with material strains becomes important because damage is often assumed to be well correlated with concrete compression and steel tension strain levels. This study has three main objectives. The first objective is the development of dimensionless curvature relationships as a function of concrete compression and steel tension strains for rectangular reinforced concrete (RC) sections typically used for RC building frames. The impact of variables, such as longitudinal and transverse steel ratios, reinforcement layouts, and strain levels on curvature, are explored. The second goal is to study the relationship between interstory drift, which is a typical parameter used to define target ACI Structural Journal/May-June 2010

displacements, and material strains in beam plastic hinges. Equations are proposed through the use of fiber-based analysis with OpenSees (McKenna et a1. 2000) of moment frame structures. Lastly, this paper investigates the impacts of the proposed equations on current code-based drift and ductility limits. RESEARCH SIGNIFICANCE Several researchers have defined expressions for curvatures and drifts that are based on material strains. Priestley and Kowalsky (1998) developed dimensionless yield and serviceability curvatures for structural wall buildings. Also, Kowalsky (2000) explored the use of moment-curvature analysis to develop dimensionless curvature relationships based on strains for circular RC bridge columns to be used in displacement-based design methods. Such expressions for RC moment frames, however, are not available in the literature. The proposed equations are aimed to fill the gap between current engineering practice and PBSE methodologies. They have potential use in numerous PBSE applications including direct displacement-based design. PRELIMINARY ANALYSES FOR CURVATURE-STRAIN RELATIONSHIPS It is clear that longitudinal steel ratio, section geometry, material properties, and axial load impact the moment curvature response of RC sections. It is also clear that transverse steel ratio and longitudinal bar arrangement will affect the momentcurvature response. It is perhaps not as straightforward to determine how each of these variables affects the relationship between strain and curvature, however. Instead of attempting to analyze all of these variables in a factorial manner that would lead to more analyses than are likely needed, an incremental approach was used. Because longitudinal steel ratio and section geometry clearly impact the relationship between strain and curvature and small variations in material properties can be normalized based on past experience (Priestley and Kowalsky 2000; Kowalsky 2000), initial analysis was conducted to study the effect of 1) arrangement of reinforcing bars (six different arrangements, as shown in Fig. 1(a)); and 2) volumetric ratio of transverse steel (Fig. 1(b)), ranging from 0.3 to 1.0%. The initial analysis was conducted on a “baseline” section with constant longitudinal steel ratio (2%) and dimensions ACI Structural Journal, V. 107, No. 3, May-June 2010. MS No. S-2008-260.R4 received September 1, 2009, and reviewed under Institute publication policies. Copyright © 2010, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the MarchApril 2011 ACI Structural Journal if the discussion is received by November 1, 2010.

291

(20 x 30 in. [500 x 750 mm]). The longitudinal reinforcement ratio is defined as the ratio of the total longitudinal reinforcement area Aslong to the gross area of the section. The cover of all sections was 1.5 in. (40 mm). The bar diameter was 1.125 in. (28.58 mm). The rest of the properties of Sections 1 to 6 (Fig. 1(a)) are shown in Table 1. The analyses were performed using a moment-curvature analysis program (Montejo and Kowalsky 2007). This program uses the constitutive models developed by Mander et al. (1988) and King et al. (1986) for concrete and the reinforcing steel, respectively. The general stress-strain curves for the concrete and steel are shown in Fig. 2(a) and (b), respectively. In these figures, fcc is the compressive strength of confined concrete, εcc is the confined concrete strain, E and Esec are the tangent and secant modulus of elasticity of the concrete, Fu is the ultimate strength of the steel, and εsh is the hardening strain. From past comparisons with experimental data, these expressions have been shown to be accurate for concrete strengths in the range of 3.6 to 7.2 ksi (25 to 50 MPa), and for steel yield strengths typically encountered for ASTM A706 steel (fy = 58 to 72 ksi [400 to 500 MPa]). A compressive concrete strength fc′ of 4 ksi (28 MPa) and steel yield strength fy of 65 ksi (450 MPa) was used in all analyses. The results in this study should be used with caution when the shapes of the stress-strain curve vary in a significant manner from those shown in Fig. 2(a) and (b) (for example, high-strength or lightweight concrete or high-strength or low ductility steel). The stress-strain models used in this work will be appropriate in the vast majority of cases for building design in the U.S. that utilizes ASTM A706, Grade 60, or, in some cases, ASTM A615, Grade 60, steel. From the moment-curvature analyses, the curvatures were identified at discrete levels of concrete compression and steel tension strain. Concrete compression strain is that measured at the extreme compression fiber, with the neutral axis also measured from the extreme fiber. Steel tension strains are those at the location of the extreme tension bar. The curvatures were normalized to the total section depth and were plotted against material strains to study the impact of the previously described variables. The dimensionless curvatures are represented by K = φH, where H is the depth of the section. The results of the analyses on the sections shown in Fig. 1(a) are shown in Fig. 3. Figure 3(a) represents the curvature as a function of steel tension strain, whereas Fig. 3(b) represents the curvature as a function of concrete compression strain. From this data, it is clear that, in most cases, reinforcement layout has little impact on the relationship between curvature and strain, with the exception of when the reinforcement is concentrated solely in one layer on the tension and compression sides (Section 1). This section has the largest amount of compression steel and the smallest amount of tension steel of any of the sections, thus resulting in the trends shown in Fig. 3, which were expected. Of

Aidcer L. Vidot-Vega is an Assistant Professor in the Department of Engineering Science and Materials at University of Puerto Rico at Mayagüez (UPRM), Puerto Rico, where she received her BS and MS. She received her PhD from North Carolina State University, Raleigh, NC. Mervyn J. Kowalsky is a Professor in the Department of Civil, Construction, and Environmental Engineering at North Carolina State University. He is a member of ACI Committees 213, Lightweight Aggregate and Concrete; 341, Earthquake-Resistant Concrete Bridges; 374, Performance-Based Seismic Design of Concrete Buildings; and Joint ACI-ASCE-TMS Committee 530, Masonry Standards Joint Committee.

Fig. 1—Cross sections: (a) in initial set of analyses (Sections 1 to 6); and (b) with different transverse steel percentage.

Fig. 2—Stress-strain curve for: (a) unconfined and confined concrete (Mander et al. 1988); and (b) reinforcing steel (King et al. 1986).

Fig. 3—Dimensionless curvatures versus material strains for first set of analyses: (a) steel; and (b) concrete.

Table 1—Distance of longitudinal steel bars from top of Sections 1 to 6 Dlayers, in. (mm)

292

Section

1

2

3

4

5

6

1

2.14 (54.29)

27.40 (695.71)

—

—

—

—

2

2.14 (54.29)

10.60 (268.20)

18.98 (482.12)

27.40 (695.71)

—

—

3

2.14 (54.29)

6.65 (168.87)

22.89 (581.45)

27.40 (695.71)

—

—

4

2.14 (54.29)

22.59 (573.87)

27.40 (695.71)

—

—

—

5

2.14 (54.29)

7.19 (182.67)

12.25 (311.05)

17.30 (439.43)

22.35 (567.81)

27.40 (695.71)

6

2.14 (54.29)

8.45 (214.76)

14.77 (375.2)

21.09 (535.66)

27.40 (695.71)

—

ACI Structural Journal/May-June 2010

course, this is not surprising; however, it is interesting to note that all other sections that had different layouts had very minor differences between strain and curvature. In the analysis of the sections shown in Fig. 1(b), the primary objective was to study the effects of transverse steel ratio and axial load on the relationship between longitudinal strain and section curvature. These results are shown in Fig. 4. Note that in Fig. 4 (a) and (b), which compare Sections 1 and 1(a), where the only difference is transverse steel amounts (0.3% and 0.6%, respectively), there is little impact on the relationship between strain and curvature. A similar result can be seen in Fig. 4(c) and (d), which compares Sections 2, 2(a), and 2(b) (0.3%, 0.6%, and 1.0%, respectively). It is important to note that this does not imply that all of these sections have the same strain capacity. It is clear that a more highly confined section will have a higher strain capacity. What is shown in these figures is that up to the deformation capacity of the most lightly confined section, the relationship between strain and curvature was not affected by the amount of transverse steel. On the basis of these initial analyses, additional analyses were carried out with reinforcing bars distributed along the entire perimeter because such a configuration was effective at describing all reinforcement bar layouts, with the exception of the concentrated reinforcement bar case. It is also worth noting that sections (both beam and column) with distributed reinforcing steel were heavily favored for seismic design (Priestley et al. 1996), making this a fortuitous outcome. Subsequent analyses also use only one level of transverse steel, as it had little impact on the results. SUBSEQUENT ANALYSIS FOR CURVATURE STRAIN RELATIONSHIPS Following the initial series of analysis, rectangular sections used in the subsequent analysis included sections with the following geometry: 20 x 30, 12 x 20, 16 x 24, 10 x 17, 20 x 40, 20 x 47, 24 x 40, and 28 x 40 in. (500 x 750, 300 x 500, 400 x 600, 255 x 425, 500 x 1000, 500 x 1200, 600 x 1000, and 700 x 1000 mm). The longitudinal reinforcement ratio varied from 0.5 to 6% to cover typical beam and column reinforcement ranges. The longitudinal bar sizes varied from 0.5 to 1.75 in. (12.70 to 44.75 mm) to reach the defined longitudinal reinforcement ratios. The axial load ratio was varied from 0 to 0.5 for compression loads and from 0 to –0.10 for tension loads. As previously noted, the transverse reinforcement ratio was kept constant in all sections because the first series of analysis described in this paper indicated that different levels of transverse steel ratio had little impact on the relationship between curvature and strain. A total of 648 section configurations were analyzed. Section curvatures were obtained for concrete compression strains εc of 0.004, 0.010, 0.014, 0.018, 0.03, and 0.04 and steel tension strains εs of 0.010, 0.015, 0.03, 0.04, 0.06, and 0.09. Curvatures were again normalized to the section depth and were plotted versus axial load ratio for the nine longitudinal steel ratios considered. Examples of these plots are shown in Fig. 5(a) and (b) for a section with dimensions of 20 x 30 in. (500 x 750 mm) and a longitudinal steel ratio of 2% and 4%, respectively. Similar plots can be obtained for the other sections (Vidot-Vega 2008). The solid lines in Fig. 5(a) and (b) represent curvatures φ based on concrete compression strains (at the unconfined extreme compression fiber), whereas the dashed lines represent curvatures defined by extreme steel tension bar strains. It can be noted that the ACI Structural Journal/May-June 2010

Fig. 4—Dimensionless curvatures versus steel and concrete strains for Sections 1 and 1A, 2, 2A, and 2B.

Fig. 5—Dimensionless curvatures for equal strain lines: (a) 2%; and (b) 4% ρlong. section curvatures at constant concrete compression strains are strongly influenced by the axial load ratio (ALR) and longitudinal steel ratio. For tension or low compression axial loads, the section curvatures varied considerably, especially when the section had a low longitudinal steel ratio. Section curvatures became more linear for both compression and tension strains as the longitudinal steel ratio increased, particularly for low levels of strains. The section curvatures at constant steel tension strains showed very little variation with the longitudinal steel ratio. In addition, the ALR had a minimal impact on section curvature at constant steel strains, especially at low levels of tension strains. The entire data set was analyzed to obtain a relationship for dimensionless curvature that depends on the ALR, longitudinal reinforcement ratio, concrete compression strain εc , and steel tension strain εs. STRAIN-BASED CURVATURE EQUATIONS Equation 1(a) to (c) (concrete compression strain) and Eq. (2) (steel tension strain) represent the proposed equations based on analysis of the data. These equations represent the average values of the fits performed on the eight sections described previously to provide a best estimate of the section curvatures. In Eq. (1) and (2), the longitudinal steel ratio ρlong was expressed by the mechanical reinforcing ratio ωs, which is given by Eq. (3). ALR are expressed (Eq. (4)) as the ratio of the axial load over the product of the compressive concrete strength fc′ and the gross section area Agross. The equations are also a function of the cover ratio cR, given by Eq. (5), which varied from 0.65 to 0.92. In Eq. (4), fy is the 293

Table 2—Statistical parameters for Kcalc /KMC using Eq. (1a) for concrete strain εc

0.004

0.01

0.014 to 0.018

0.02 to 0.022

0.025 to 0.03

0.04

Total

x

0.938

0.978

0.990

1.001

1.019

1.053

0.998 0.122

σ

0.099

0.099

0.114

0.124

0.130

0.135

ρlong

0.01 to 0.015

0.02 to 0.023

0.025 to 0.026

0.03 to 0.033

0.04 to 0.043

0.048 to 0.052

Total

x

0.993

1.017

1.008

1.010

0.996

0.974

0.998

σ

0.152

0.122

0.113

0.109

0.105

0.088

0.122

Table 3—Statistical parameters for Kcalc /KMC using Eq. (2) for steel strain εs

0.01

0.015

0.03

0.04

0.06

0.09

Total

x

1.051

1.048

1.027

1.021

1.013

1.007

1.031

σ

0.081

0.087

0.095

0.093

0.088

0.083

0.090

ρlong

0.01 to 0.015

0.016 to 0.023

0.025 to 0.027

0.028 to 0.033

0.04 to 0.043

0.048 to 0.06

Total

x

1.042

1.029

1.034

1.027

1.020

1.022

1.031

σ

0.085

0.088

0.094

0.095

0.095

0.088

0.090

yield strength of the longitudinal steel. The range of application of the equations is shown. Concrete compression strain based – 0.5

– 0.3 ( – 0.8∗ ALR∗ ω s

K c = ΦH = 2.8ε c ω s

e

)

(1a)

0.009 ≤ ρ long ≤ 0.05, 0.05 ≤ ALR ≤ 0.40, 0.004 ≤ ε c ≤ 0.04 – ( 1.7cR – 1 )

K c = ΦH = ( 2.3cR + 0.8 )ω s

ε c + 0.005

(1b)

0.005 ≤ ρ long ≤ 0.021, ALR = 0, 0.004 ≤ ε c < 0.040, 0.60 ≤ cR < 0.95 K c = ΦH = ( – 1.6∗ ω s + 4 )ε c + 0.005

(1c)

0.022 ≤ ρ long ≤ 0.06, ALR = 0, 0.004 ≤ ε c < 0.040, 0.60 ≤ cR < 0.95 Steel tension strain based – 0.30

0.15 ( 0.75∗ ALR∗ ω s

K c = ΦH = 1.75ε s ω s

e

)

(2)

0.005 ≤ ρ long ≤ 0.06, – 0.01 ≤ A LR ≤ 0.05 0.001 ≤ ε s < 0.090 f ω s = ρ long ----yfc ′

(3)

P ALR = -------------------A gross f c ′

(4)

cR = (d – d′)/H

(5)

There are three equations to define the relationship between curvature and concrete compression strain and only 294

one to define the relationship between curvature and steel tension strain. This reflects the higher degree of variation between concrete compression strain and curvature. Equation (1a) attempts to define the entire concrete compression strain versus curvature relationship; however, as will be seen in the following, it results in a higher error than Eq. (1b) and (1c), which subdivide the data set into two parts while eliminating axial load as a variable. Because the primary interest will be for beam members, Eq. (1b) and (1c) were developed. Whereas beams are clearly subjected to axial forces due to column shear forces in a moment frame, the response on a story-by-story basis is best described by the average beam axial load of zero. Also, beam axial loads are generally small, supporting the use of expressions with zero axial load for beams. The ratio between the curvatures calculated Kcalc with the proposed equations and with moment-curvature analyses KMC was obtained and statistical analyses were performed to determine the accuracy of these equations. Tables 2 to 3 present the mean and standard deviation of the curvature ratio (Kcalc/ KMC) according to several values of longitudinal steel ratio and material strains using Eq. (1a) and (2). It can be seen that for Eq. (1a) the mean varies from 0.938 to 1.053 and the standard deviation from 0.08 to 0.152. In general, the standard deviation decreased as the axial load and ρlong increased, and the concrete compression strains decreased. However, the mean obtained from Eq. (1b) and (1c) for axial loads equal to zero was 0.96 considering concrete compression strains. The standard deviation was less than 0.11 for Eq. (1b) and (1c). It can be observed in Table 3 that the standard deviation was less than 0.095 and the mean varied from 1.007 to 1.051 using Eq. (2). Figures 6(a) and (b) and 7(a) and (b) show the curvature ratio Kcalc/ KMC as a function of the longitudinal steel ratio and material strains using Eq. (1a) and (1c) and Eq. (2), respectively. The agreement is much better for the Eq. (1b) and (1c) and Eq. (2) when compared to Eq. (1a) and, in general, it improved for longitudinal steel ratios higher than 2%. DRIFT-STRAIN RELATIONSHIPS FOR RC FRAME BUILDINGS This section investigates the relationships between strain and interstory drift. This was accomplished by using the previous dimensionless curvature Eq. (1) and (2) and the ACI Structural Journal/May-June 2010

Table 4—Beam aspect ratios for cases with n stories and numbers of bays Beam aspect ratios (LB/HB) n Twobay

Threebay

2

4

8

12

16

7.0, 9.0, 9.1, 7.0, 8.0, 9.0, 6.5, 9.0, 7.60, 9.50, 7.0, 9.0, 9.5, 9.25, 9.5, 10.0, 11.0, 9.25, 9.5, 11.25 9.78, 12.0 10.0, 10.59, 11.1, 15.0 12.0 12.0 —

7.0, 8.5, 10.0, 10.0, 13.0. 14.0, 14.5, 15.0

7.0, 10.0, 13.0. 14.0, 14.5, 15.0

7.0, 8.5, 10.0, 11.5, 12.0, 13.0. 14.0, 15.0

7.0, 10.0, 11.5, 13.0. 14.0, 15.0

results from computer analyses. A total of 54 frame buildings having 2, 4, 8, 12, and 16 stories were designed using direct displacement-based design (Priestley et al. 2007) and capacity design principles. The buildings were symmetrical with two and three bays. Beam aspect ratios, defined by the ratio of the beam length to the beam depth (LB/HB), were varied from 6.5 to 15 (Table 4). These frames deformed according to the beam-sway mechanism, in which plastic hinges at the ends of the beams and at the column base of the first floor were formed. As a consequence, the results should not be extrapolated to frames that do not satisfy this failure mechanism (for example, older RC frames with deficient detailing resulting in soft story mechanisms or extensive column hinging due to dynamic amplification of moments in taller buildings). Also, the equations should not be used for beam aspect ratios smaller or higher that the ones considered in this study. Rectangular and square reinforced concrete sections were used for the beams and columns, respectively. The sections were modeled using the fiber element approach. The elements were modeled using a lumped plasticity approach, which in OpenSees (McKenna et al. 2000) program is achieved by the use of the “beam with hinges” element (Scott and Fenves 2006). The confined and unconfined concrete in the fiber sections was modeled using the Kent and Park (1971) concrete model with degraded linear unloading/reloading stiffness (Karsan and Jirsa 1969). The steel was modeled using a reinforcing steel model developed by Moehle and Kunnath (2006). The bond slip (yield penetration) was not directly modeled in the members. The plastic hinge length does account for this indirectly through the length of strain penetration, however, thus addressing the impact of bar slip on the strain versus displacement. The 54 frame models were analyzed in OpenSees (McKenna et al. 2000) program under monotonic loading until 3 to 5% drift. The strains were taken at the extreme steel tension bar and at the extreme concrete fiber (cover). Until yield, the interstory drift θj is assumed to be approximately equal to the beam rotation. This assumption will be verified in the following. After yield, the interstory drift was expressed in terms of the beam rotation and frame yield drift. The beam rotation θbeam is expressed in terms of the curvatures (K/HB) using Eq. (6). The beam yield drift (Eq. (8)) can be found elsewhere (Priestley et al. 2007) and was derived by engineering mechanics principles. In these equations, Lp is the plastic hinge length (Priestley et al. 2007), HB is the depth of the beam, LB is the length of the beam, Fu is the ultimate steel strength, dbl is the longitudinal bar diameter, and εy is the steel yield strain. The dimensionless yield curvature (Priestley and Kowalsky 1998) is given by Eq. (9). The dimensionless curvatures K were obtained by ACI Structural Journal/May-June 2010

Fig. 6—Curvature ratio Kcalc/KMC versus: (a) longitudinal steel ratio; and (b) concrete strain for Eq. (1a) to (1c).

Fig. 7—Curvature ratio Kcalc/KMC versus: (a) longitudinal steel ratio; and (b) steel strain for Eq. (2).

Fig. 8—Rotations versus steel strains for: (a) four stories, two-bays; and (b) 12 stories, two-bays. using Eq. (1) and (2) for concrete compression and steel tension strains, respectively. L θ beam = θ by + ( K – K Y ) ⎛ ------p-⎞ ⎝ H B⎠

(6)

⎧ kL + 0.022F y d bl ⎫ ⎛ Fu ⎞ L p = max ⎨ ⎬; k = 0.2 ⎝ ------ – 1⎠ ≤ 0.08; (7) FY 0.044Fy d bl ⎩ ⎭ L = L B ⁄ 2 (double bending) LB ⎞ θ by = K Y ⎛ --------⎝ 6H B⎠

(8)

K Y = φ Y H B = 2.1ε y

(9)

Figure 8 (a) and (b) show the beam rotations and interstory drift as a function of steel tension strains for two cases. Similar results were obtained for all 54 cases (Vidot-Vega 2008). It can be seen that the interstory drift and beam rotations are very similar. After yield, the beam rotations were slightly larger than the interstory drift. This is likely due to the shortened length between beam plastic hinges, thus resulting in higher beam rotations when compared to the interstory drift. As the difference is small, however, the interstory drift can be approximated by the use of the beam rotations. This implies that the interstory drift (or joint rotations) can be estimated 295

using the beam rotation expressions developed in the previous section with some minor changes. STRAIN-BASED DRIFT AND DISPLACEMENT DUCTILITY EQUATIONS To develop the strain-based drift equations, the beam rotation expressions (Eq. (6)) shown previously were used with some modifications. To be useful for design, the axial load in the beams was assumed to be equal to zero because under the action of seismic loads in both directions, there is the possibility for tension and compression loads at opposite ends of the beams; and their average will be approximately equal to zero. Also, to simplify the drift expressions, only Eq. (1c) was used for concrete compression strains. The beam yield drift was modified to account for shear deformation components. The yield drift (Eq. (11)) includes the contribution of several components according to Priestley (1998) such as 1) the joint rotation due to shear and flexure; and 2) column deformation due to shear and flexure. The general form of the drift θj expressions is shown in Eq. (10). A factor (β) was added to the equation to account for the slightly larger beam rotations (compared to the joint rotation) as the inelastic deformations increases (Fig. 8). A power fitting was performed to the data from moment-curvature analyses to find the factor (β) to modify the beam rotation expressions previously shown. L θ j = θ y + ( K – K Y ) ⎛ ------p-⎞ β ⎝ H B⎠

(10)

L θ y = 0.5ε y ⎛ ------B-⎞ ⎝ H B⎠

(11)

Substituting Eq. (2) for ALR = 0 and Eq. (11) into the drift equation based on steel tension strain results in Eq. (12). Using steel tension strain L L 0.15 – 0.080 (12) θ j = 0.5ε Y ⎛ ------B-⎞ + ( 1.75ε s ω s – K y ) ⎛ ------p-⎞ 0.70ε s ⎝ H B⎠ ⎝ H B⎠ Now, substituting the yield drift (Eq. (11)) and the curvatures (Eq. (1b) and (1c)) into Eq. (10), the drift equation for concrete compression strain can be shown as Eq. (13). Using concrete compression strain L θ j = 0.5ε Y ⎛ ------B-⎞ + ⎝ H B⎠ L – 0.065 ( ( – 1.6∗ ω s + 4 )ε c + 0.004 – K y ) ⎛ ------p-⎞ 0.85ε c ⎝ H B⎠

(13)

Displacement ductility expressions based on concrete compression and steel tension strains were obtained by dividing the (Eq. (12) and (13)) by the yield drift and are given by Eq. (14) and (15). Using steel tension strain 0.15 – 0.080 2 (14) μ Δs = 1 + ⎛ ----⎞ ( 1.75ε s ω s – K y ) ( L p ⁄ L B ) 0.70ε s ⎝ ε y⎠

Using concrete compression strain 296

Table 5—Statistical parameters for all cases θOpenSees/θcalc Strain Concrete strain

Steel strain

No. of stories

Statistical parameter

2

4

8

12

16

Total

Mean

0.869

0.90

0.96

1.06

1.00

0.95

Standard deviation

0.062

0.059

0.069

0.077

0.091 0.072

Mean

1.06

1.01

0.98

0.97

1.00

Standard deviation

0.037

0.039

0.041

0.029

0.042 0.038

1.00

– 0.065 (15) 2 μ ΔC = 1 + ⎛ ----⎞ ( ( – 1.6∗ ω s + 4 )ε c + 0.004 – K y ) ( L p ⁄ L B ) 0.85ε c ⎝ ε y⎠

In the aforementioned, Ky and Lp are obtained from Eq. (9) and (7), respectively. The accuracy of the proposed equations is explored in the following. ACCURACY OF STRAIN-BASED DRIFT EQUATIONS The ratio from the drift obtained from the computer (θOpenSees) analysis and the proposed equations (θcalc) was tabulated for all cases as a function of steel tension and concrete compression strains to determine the accuracy of these equations. The ratio calculated here uses Eq. (12) and (13) without the inclusion of the shear deformation components in the yield drift equation. This was done for comparison purposes only because the computer analyses did not include the effects of member shear deformations on joint rotation. Thus, the yield drift was calculated only with beam flexural components (Eq. (6)). For design, it is recommended to include all the components in the yield drift equation as presented in Eq. (12) and (13). These results are shown in Fig. 9 (a) to (d) and 10 (a) to (d) for steel tension and concrete compression strain, respectively. The list of number of bays and beam aspect ratios appears in the legend of each plot. The plots were divided according to the number of stories. It can be seen that the difference between the drifts from the analysis and proposed equations for steel tension strains is less than 12% for all cases. For concrete compression strain, the differences between the calculated drift and those obtained from computer simulations are as high as 22% for some cases. In general, errors for concrete compression strain equations are higher for beam aspect ratios less than 7.6 or greater than 14.5 as well as for taller buildings (12and 16-story) with three bays. This can be attributed to changes in the inflection points of the deflected shape for the beams and columns that do not exactly represent the middle points of each element. The results obtained from the proposed equations, however, are quite good for all ranges of interstory drift and strains considered in this study. The mean and standard deviations were calculated for all cases (Table 5). The mean for the ratio θOpenSees/θcalc ranged between 0.87 to 1.00 for concrete compression strain and 0.97 to 1.06 for steel tension strain. The standard deviations varied from 0.059 to 0.091 and 0.029 to 0.042 for concrete and steel strains, respectively. DRIFT AND DUCTILITY EXPRESSIONS VERSUS CURRENT CODE LIMITS The final objective of this paper is to discuss the implications of the strain-based drift and ductility equations proposed in ACI Structural Journal/May-June 2010

Fig. 9—Drift ratio comparison as function of steel tension strain for: (a) two- and fourstory; (b) eight-story; (c) 12-story; and (d) 16-story frame cases with several beam aspect ratios and number of bays at first floor.

Fig. 10—Drift ratio comparison as function of concrete compression strain for: (a) fourstory; (b) eight-story; (c) 12-story; and (d) 16-story frame cases with several beam aspect ratios and number of bays at first floor. this paper when compared to ASCE 7-05 code guidelines. Interstory drift ratios for concrete compression (Eq. (13)) and steel tension strains (Eq. (12)) were obtained for different values of ρlong including shear deformation components. These drifts are shown in Fig. 11 as a function of the beam aspect ratio and for a section with ρlong = 2.5%. Different values of concrete compression (Fig. 11(a)) and steel tension strains (Fig. 11(b)) were considered. Each figure also includes the drift limits defined in ASCE 7-05 for RC frames of 0.015 (occupancy Category IV) and 0.025 (occupancy Categories I and II). Whereas drift limits are not necessarily linked to structural damage, such limits exist; and it is interesting to study the implied levels of structural damage as defined by material ACI Structural Journal/May-June 2010

strain for a prescribed drift limit. As noted in Fig. 11, the strains in beam plastic hinges vary considerably depending on the beam aspect ratio. As the beam aspect ratio becomes larger, the code drift limits can be nearly accommodated only with elastic response. This result implies that at the code-based drift limit, varying levels of structural damage will be evident, depending on building configuration. Of course, drift ratio limits have as their primary purpose to control nonstructural damage, rather than structural damage, in which case the code limits are reasonable. It is the interplay between the implied level of structural performance based on code-drift limits and force-reduction factors that is perhaps most interesting to explore, however. 297

Fig. 11—(a) Interstory drift for equal concrete compression strains; and (b) tension steel strain lines.

Fig. 12—(a) Displacement ductility for equal concrete compression strains; and (b) tension steel strain lines. Shown in Fig. 12 are the displacement ductility values computed with Eq. (14) and (15) as a function of beam aspect ratio for a section with a ρlong of 0.025, the implied ductility level based on the ASCE 7-05 Cd of 5.5 for special RC moment frames, and the displacement ductilities achieved using ASCE 7-05 drift limits. Displacement ductilities decreased as the ρlong increased in the section for concrete compression strains and vice versa for steel tension strains. For the implied ductility level of 5.5 based on Cd, the damage level, as defined by strains in beam plastic hinges, will vary greatly. This indicates that the use of a constant force reduction factor does not imply a constant level of damage across a variety of building geometries for a given structure type. Also from Fig. 12, it can be noted that the implied level of ductility based on the code drift limits was rarely in agreement with the displacement amplification (and hence force reduction factor). As a result, in many cases, the code drift limit will govern the design with ductility levels far smaller than that implied by the force reduction factor given by the same code. Whereas such an outcome is acceptable, the problem arises with regard to the method in which drifts are typically evaluated in force-based design. Consider an example of a moment frame with an aspect ratio of 8 that is designed using ASCE 7-05. From Fig. 12, the drift limit for this aspect ratio will govern design; and if the limit of 2.5% applies, then the actual displacement ductility level will be approximately 3. The force reduction factor and displacement amplification factor for a concrete moment frame, however, are 8 and 5.5, respectively. Then, the engineer designing this structure would divide the elastic base shear by 8, magnify the elastic displacements by 5.5, and compare the resulting drifts with the code limit of 2.5% where the actual displacement ductility is 3. If the drifts are greater than 2.5% in any story, the engineer would follow by strengthening (and as a result, stiffening) the structure until the evaluated drifts are less than 2.5%. This approach would be acceptable if the displacement magnification factor equals the actual ductility level at the drift limit; however, that is rarely the 298

case (Fig. 12). It should be noted that this does not suggest that structures designed by such an approach are unsafe. To the contrary, they are likely to have greater than required strength and required ductility capacity, both of which are fine for life safety considerations. Such an approach, however, is of limited value if one desires to implement some form of performance-based seismic design. A much more logical approach would be to determine design displacement levels based on both material strains in plastic hinges as well as nonstructural drift limits, with the lower of the two governing. Then, that displacement and its corresponding ductility level are used for design, thus eliminating the need for an arbitrary choice of the force reduction and displacement amplification factor. CONCLUSIONS This study had three main objectives: 1. Develop dimensionless curvature relationships as a function of material strains for rectangular RC sections typically used for RC building frames. 2. Study the relationship between interstory drift and material strains in beam plastic hinges. 3. Investigate the impacts of the proposed equations on current code-based drift and ductility limits. The proposed expressions depend on the ALR, mechanical steel ratio, material strains (steel and concrete), beam aspect ratio, and cover ratio. These expressions are intended for sections with bars distributed around the perimeter. The research indicated the following: 1) curvatures at concrete compression strains are strongly influenced by the ALR and ρlong, 2) constant steel tension strains show little variation with the ρlong and ALR, and 3) constant values of drift ratios and ductility demands imposed in building codes are generally not in agreement with each other, and that in neither case do they imply uniform levels of damage. The proposed expressions based on steel tension and concrete compression strains are accurate to within less than 12% and 22%, respectively, which is perhaps sufficient when one considers the variations found in seismic design. These expressions are valid for RC moment frames that have been designed to develop beam swing mechanisms and for the range of variables considered in this study. Also, the accuracy of these equations is impacted by the models used for the concrete and steel materials in the analyses. Further studies in this area include the study of the impact of load history on the proposed relationships. NOTATION Agross Aslong cR dbl E and Esec Fu fc ′ fcc fy H HB K Ky L LB Lp β εc and εs εsh εy

= = = = = = = = = = = = = = = = = = = =

gross section area longitudinal steel area cover ratio longitudinal bar diameter concrete tangent and secant modulus of elasticity ultimate steel strength compressive concrete strength confined concrete strength yield strength of longitudinal steel section depth depth of beam dimensionless curvature yield dimensionless curvature member length length of beam plastic hinge length factor to modify drift concrete compression and steel tension strains steel hardening strain steel yield strain

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Φ μΔ μΔC μΔS θbeam θby θj θy ρlong ωs

= = = = = = = = = =

curvature displacement ductility displacement ductility based on concrete compression strain displacement ductility based on steel tension strain beam rotation yield beam rotation interstory drift frame yield drift longitudinal reinforcement ratio mechanical reinforcing ratio

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Mohle, J., and Kunnath, S., “Reinforcing Steel: OpenSees User’s Manual,” 2006, http://opensees.berkeley.edu. Montejo, L. A., and Kowalsky, M. J., “CUMBIA—Sets of Codes for the Analysis of Reinforced Concrete Members,” Report No. IS-07-01, CFL, NCSU, Raleigh, NC, 2007, 41 pp. Priestley, M. J. N., “Brief Comments on Elastic Flexibility of RC Frames, and Significance to Seismic Design,” Bulletin of the New Zealand National Society for Earthquake Engineering, V. 31, No. 4. Dec. 1998, pp. 246-259. Priestley, M. J. N.; Seible, F.; and Calvi, S. M., Seismic Design and Retrofit of Bridges, John Wiley and Sons Inc., 1996, 679 pp. Priestley, M. J. N., and Kowalsky, M. J., “Aspects of Drift and Ductility Capacity of Cantilever Structural Walls,” Bulletin of the New Zealand National Society for Earthquake Engineering, V. 31, No. 2, July 1998, pp. 73-85. Priestley, M. J. N., and Kowalsky, M. J., “Direct Displacement-Based Design of Concrete Buildings,” Bulletin of the New Zealand National Society for Earthquake Engineering, V. 33, No. 4, Dec. 2000, pp. 421-444. Priestley, M. J. N.; Calvi, G. M.; and Kowalsky, M. J., “DisplacementBased Seismic Design of Structures,” IUSS Press, Italy, 2007, 721 pp. Scott, M., and Fenves, G., “Plastic Hinge Integration Methods for ForceBased Beam-Column Elements,” Journal of Structural Engineering, ASCE, V. 132, No. 2, Feb. 2006, pp. 244-252. Structural Engineers Association of California, SEAOC Vision 2000, Sacramento, CA, 1995. Vidot-Vega, A. L., “The Impact of Load History on Deformation Limit States for the Displacement-Based Seismic Design of RC Moment Frame Buildings,” PhD dissertation, Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC, 2008.

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