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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams ASCE 7-05 reference section(s) 12.13, 14.2.3, 14.3.2

2006 IBC (2007 CBC) reference section(s) 1808, 1809, 1810, 1811, 1908.1.10

Other standard reference section(s) ACI 318-05, 21.10

Design Overview An overview of the design of piles, pile caps, and grade beams under ASCE 7-05 and the 2006 IBC (2007 CBC) is provided in Article 7.01.001, Foundation Design Overview. That article explains that the 2006 IBC is based upon provisions initiated by the 2000 NEHRP/2000 IBC provisions. ASCE 7-05 also incorporates substantial changes from Chapter 7 of the 2003 NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures. The 1999 Blue Book provisions (SEAOC 1999) do not cover detailing provisions in as much depth as ASCE 7-05 does. However, the 1999 Blue Book provisions are still relevant to the design of piles, pile caps, and grade beams, particularly where a capacity design approach is taken. A performance-based design approach for pile foundations has not yet been addressed by any of the above codes, standards, or guidelines. However, researchers at the University of California, San Diego did propose performancebased design criteria for specific pile types (Silva, Seible, Priestley, 1997). This work can be used as a basis for the performance-based design of other piles and for pile connections where progressive cyclic testing has been done to establish various damage levels versus deformation. The design force level for the vertical system that resists lateral forces and for the foundation (except the superstructure-to-footing connection) are the same. This has been and still is a recognized area requiring research and code changes (Harden, Hutchinson, and Moore 2006), and it is therefore uncertain as to where the majority of the inelastic deformation occurs. To address this uncertainty and to prevent excessive concentration of inelastic deformation in the foundation system, a BSSC Issue Study (BSSC 2007) has recommended that the foundation design force be higher than the code level design force where there is a high system ductility (high R value). This does not preclude nonlinear behavior in the foundation system, it simply endeavors to limit the ductility demand level imposed on the foundation system. This article recommends, in addition to the philosophical approach of limiting the ductility demand on the foundation system, that a capacity design approach be applied to preclude the formation of any non-ductile mechanisms. The following outlines an approach for the design of the pile cap and reinforced concrete piles. The approach can be extended to other forms of deep foundations and to shallow foundations. Pile Cap Design The above references have a limited number of seismic design and detailing provisions for pile caps. If the pile cap should yield or fail in shear before developing the strength of the piles, then the displacement capacity of a yielding foundation system may be compromised. Such a pile cap mechanism can occur due to joint shear stresses from column or wall pier bending moments or from under-reinforcement for bending moments necessary to transfer the overturning forces to the piles. Joint shear failure in footings has occurred prematurely under large-scale testing of column to footing connections (Xiao, Priestley, and Seible, 1996). This type of mechanism lacks the ductility capacity to achieve the performance intended by the code. The exception to this is where flexural yielding of a grade beam or pile cap occurs first. The component in this scenario should be treated as a flexural element and either meet the detailing requirements of ACI 318-05 Section 21.3 or be demonstrated to afford sufficient ductility. Below are suggested strength and ductile detailing provisions for the pile cap as a flexural member and an evaluation approach to prevent pile cap joint shear failure before flexural yielding of the pile cap.

Article 7.02.020

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March 2008

SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams Pile Cap Seismic Design Provisions Unless explicitly detailed for ductility, the pile cap should be sized and reinforced to develop the capacity of either the vertically oriented system that resists lateral forces or the pile capacities. This is applicable to both overturning and lateral forces. The pile cap should be designed to transfer the forces of the vertical lateral force-resisting system to the piles, in particular the joint formed between structural components. The joint(s) should be reinforced and detailed to resist the diagonal tension stress induced by the pile, column, or structural wall using the following provisions. 1.

2.

3.

The pile caps shall have continuous top and bottom longitudinal reinforcing using a steel ratio not less than 0.0020 for each layer and terminated with seismic hooks in accordance with ACI 318 Section 21.1 at the edges. For pile caps not containing shear reinforcement, the top and bottom reinforcing layers of the pile cap shall be connected by not less than the equivalent of #5 vertical bars at 24 inches on-center each way with 90 degree hooks on one end and seismic hooks on the other end alternated. For piles located at pile cap edges, the distance of pile centerline away from the cap or grade beam edges shall be the greater of the diameter or least dimension of the pile, but not less than 20 inches unless transverse reinforcing consisting of spirals, with a volumetric steel ratio of not less than 0.010, confines the pile or pile longitudinal reinforcing throughout its development length into the pile cap. The diagonal tension stress, ft, within the pile cap induced by the plastic moment demand from the column, structural wall or pile shall not exceed 3.5 ϕ¥f’c, where ϕ = 1.0. The axial compression force in the column, structural wall or pile may be used to reduce the diagonal tension stress using the relationship 2

fa § fa · ft = − ¨ ¸ + vjv 2 © 2¹ 2 where vjv is the average shear stress on the joint area, fa the average effective axial compressive stress at the mid-depth of the pile cap and ft is negative for tension.

fa =

P Aeff

where the effective Aeff, over which the total axial load P at the column, wall or pile is distributed, is found from a 45 degree spread of the zone of influence for a rectangular section Aeff = (Bc + df)(Dc + df) and for a circular section Aeff = ʌ (Dc + df)2/4 where Dc is the overall section depth of the rectangular section or the diameter of the circular section and Bc is the width of the rectangular section and df is the effective depth of the footing. The effective joint width and depth shall not exceed the pile cap dimensions. The vertical joint shear Vjv may be taken as the plastic moment of the column, wall, or pile divided by its effective moment arm force couple. Where a pile and column align, the joint shear shall be based on the summation of the internal actions of each component. The effective joint shear stress is taken as

Article 7.02.020

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams

vjv =

Vjv bjeffdf

where the effective joint width, bjeff is taken for rectangular sections as bjeff = Bc + Dc and for circular sections as bjeff = ¥2 Dc Exception: Where grade beams, separate from the pile cap, are designed to develop the moment strength of the column or wall above, the pile cap need not be designed and detailed to resist the plastic moment capacity of the column or wall. Limit State analysis of Concrete Piles for Lateral Seismic Forces A simplified limit state analysis of fixed-head concrete piles for seismic lateral loading has been developed by Song, Chai, and Hale (2005), which can be used for a capacity design approach. This analysis has been rearranged and a condensed version reproduced in this article. The method is based on characterizing the response of the pile for a selected number of limit states and applies to a fixed-head pile subject to lateral inertial loading at the pile head. The model assumes sequential yielding of the pile under progressive lateral deformation until a plastic mechanism is fully developed. Idealized ultimate soil pressure distributions are assumed in the ultimate state. It is assumed in the methodology that the pile shear capacity is sufficient to develop the plastic moment capacity of the section. Similar procedures can be used to develop a simplified limit state analysis for pinned-head piles (Chai 2002) and steel piles. Figure 1 shows the progressive limit states of the pile, where uniform flexural strength is assumed along the length of the pile. The first-yield limit state occurs when the flexural strength of the pile equals the moment demand

Figure 1. Limit states of Fixed-head Concrete Piles under Lateral Seismic Loads (from Song, Chai, and Hale, 2004)

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams occurring at the pile head. Further displacement beyond the first-yield limit state results in the formation of the first plastic hinge at the pile cap. Upon even further displacement, the bending moment redistributes to a lower level in the pile forming a second plastic hinge at the depth Lm. Inelastic rotation in both hinges occurs upon further displacement until the pile reaches its ultimate limit state. The ultimate limit state is taken to occur when the pile section curvature capacity is exhausted in the first plastic hinge of the pile, or where the pile section curvature capacity is something less than that of the second plastic hinge where it may control the ultimate limit state. In other words, the second plastic hinge curvature capacity may become exhausted before that of the first plastic hinge if lesser confinement of concrete is provided. In this method, the lateral soil-pile stiffness is based on the elastic response of the soil-pile system. For a cohesive soil, the soil reaction is proportional to the lateral deflection of the pile, or kh y, where kh is the constant for horizontal subgrade reaction and y is the lateral deflection. The initial lateral stiffness can be derived from the applicable differential equation (Poulos and Davis 1980) to produce

K1 ≡

V = Δ

2

EI e 3 Rc

where R c ≡ 4 EI e / k h is the characteristic length of the pile in cohesive soil and EIe the effective flexural rigidity of the pile. The force-deformation curve for the pile-soil system, assumed to be tri-linear in characterization, is shown in Figure 2. The reduced lateral stiffness beyond the first yield limit state is derived as K2 ≡

V − Vy Δ − Δ y1

and the rotation of the first plastic hinge θ = (Δ − Δ y1 )

=

EI e 2 Rc

3

2 Rc . The pile-head displacement and lateral force at the

first-yield limit state is taken as Δ y1 = M u Rc2 EI e and V y = K 1 Δ y1 = 2 M u Rc . The term Mu is the expected ultimate moment capacity of the section. For lateral strength estimation, the ultimate soil pressure distribution recommended by Reese and Van Impe (2001) is assumed. Using the assumed ultimate soil pressure for cohesive soils, the normalized depth of the second plastic hinge may be determined by solving the equation for L*m and for the actual depth Lm = L*m D . 3

2

0.5L*m + 2

1.5L*m = M u* Ψr 2

2.75L*m − 0.75ψ r = M u*

for for

L*m ≤ Ψr L*m > Ψr

where M u* ≡ M u / ( s u D 3 ) , Ψr is the critical depth coefficient which is equal to 9 su / (γ ' D + 2 2 s u ) , D is the diameter of the pile, su is the undrained shear strength of the soil and γ’ the effective unit weight of the soil. The normalized lateral strength at which the second plastic hinge is formed can be found using the equations

Article 7.02.020

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams 2

Vu* = 2 L*m +

4.5L*m Ψr

2

for

2

Vu* = 11L*m − 4.5ψ r

for

L*m ≤ Ψr L*m > Ψr

where the actual lateral strength of the soil-pile system is Vu ≡ Vu* s u D 2 . For a cohesionless soil, the soil reaction can be assumed to be proportional to a modulus of horizontal subgrade reaction which increases with depth, i.e. nh x y, where nh is the constant rate of increase in the modulus of horizontal subgrade reaction and is defined as kh/x. The initial lateral stiffness can be derived from the applicable differential equation (Matlock and Reese 1960) to produce

K1 ≡

EI V = 1.08 3e Δ Rn

where Rn ≡ 5 EI e / n h is the characteristic length of the pile in cohesionless soil and EIe the effective flexural rigidity of the pile. The force-deformation curve for the pile-soil system, assumed to be tri-linear in characterization, is shown in Figure 2. The reduced lateral stiffness beyond the first yield limit state is derived as

K2 ≡

V − Vy Δ − Δ y1

= 0.41

EI e Rn 3

and the rotation of the first plastic hinge is θ = (Δ − Δ y1 ) 1.5 Rn . The pile-head displacement and the corresponding lateral force at the first-yield limit state are taken as Δ y1 = M u Rn2 EI e and

Lateral Force V

V y = K 1 Δ y1 = 1.08 M u Rn respectively. The term Vu

Elasto-Plastic Response

Actual Response

Ultimate Tri-Linear Response

K2

Vy

Limit State of 2nd Hinge

V

P 1st Hinge

Lm

Limit State of 1st Hinge K1

Δ*p

2nd Hinge

μΔ=

Δ** p

Δy1 Δy Δy2

Δu Δy

Δu

Lateral Displacement Δ

(from Song, Chai, and Hale, 2004)

Article 7.02.020

Using the assumed ultimate soil pressure pattern for cohesionless soils, the normalized depth of the second plastic hinge may be determined using the equation L*m = 3 2 M u* and the actual depth

Lm = L*m D , where M u* ≡ M u / ( K p γ ′ D 4 ) , Kp is the coefficient of passive soil pressure, which depends on the friction angle of the soil, γ’ the effective unit weight of the soil and D is the diameter of the pile. The normalized lateral strength associated with the second plastic hinge may then be 2

Figure 2. Idealized Force-Displacement Curve for

Fixed Head Piles

Mu is the expected ultimate moment capacity of the section, same as before.

taken as Vu* = 1.5 L*m or the actual strength of

Vu ≡ Vu* K p γ ' D 3 . The first-yield limit state occurs at a displacement of

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams Δy1 (as given above) and the second yield limit state at Δy2. A lateral displacement beyond Δy2 is characterized by a perfectly plastic force-deformation curve. An equivalent idealized elasto-plastic yield deformation is considered to be related to Vu at a yield displacement of

Vu K1

Δy =

The lateral displacement at the second yield limit state is taken as

Δ y2 =

Vy K1

+

Vu − V y K2

where all terms are defined as above. The lateral displacement of the pile head from the first plastic hinge formation to the second plastic hinge formation corresponds to Δp* and associated plastic rotation θp* as

Δ*p = Δ y 2 − Δ y1 = θ*p =

Vu − V y K2

* p

Δ

η Lm

where η = 2 Rc / Lm for cohesive soils and η ≡ 1.5 Rn / Lm for cohesionless soils, and the other terms are as defined above. The lateral displacement of the pile head from Δy2 to Δu (the ultimate displacement based on the ductility of the hinges) corresponds to Δp** and a plastic rotation of θp** for both hinges as shown in Figure 1. Based on the first plastic hinge capacity, Δp** and θp** can be found by

Δ*p* = (φ u1 − φ i ) L p1 Lm .

θ*p* = (φ u1 − φ i ) L p1

for

φu1 ≥ φi ≥ φ y

where Lp1 is the equivalent plastic hinge length for the first hinge, φ u1 is the ultimate curvature of the first plastic hinge which may be obtained from a moment-curvature analysis of the section, and φ i is the curvature in the first plastic hinge at the lateral displacement of Δy2. The term φ i may be found from

ª K β Lm º (1 − α )» φi = φ y «1 + 1 ¬« K 2 η L p1 ¼»

Article 7.02.020

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams

(

where α = V y / Vu = Δ y1 Δ y , β = Δ y / φ y Lm

2

) and η =

2 Rc / Lm cohesive soils and η ≡ 1.5 Rn / Lm for

cohesionless soils. φ y is the idealized yield curvature taken from a moment-curvature section analysis. Δp** and θp** are related by θ *p* = Δ*p* Lm . Based on the second plastic hinge capacity, Δp** and θp** can be found by

(

)

Δ*p* = φu 2 − φ y L p 2 Lm θ *p* = (φ u 2 − φ y ) L p 2

for φ u 2 ≥ φ y

where Lp2 is the equivalent plastic hinge length for the second hinge. Based on prior research, Lp2 may be taken as equal to D. The term φ u 2 is the ultimate curvature of the second plastic hinge, and φ y is the idealized yield curvature, both of which may be determined from a moment-curvature section analysis. The plastic displacement and plastic rotation, Δp** and θp**, are related by θ *p* = Δ*p* Lm . Song, Chai and Hale (2004) have also developed equations to determine the curvature ductility demand under a certain design displacement ductility, which are not included here but may be useful to the reader. References Broms, B. B. (1964). “Lateral resistance of piles in cohesionless soils.” Journal of Soil Mechanics and Foundations Division, ASCE, 90(SM3), 123-156. BSSC (2007). “Appropriate seismic load combinations for base plates, anchorage and foundations,” Issue Team #3 Draft White Paper, July 2007. Chai, Y. H. (2002). “Flexural strength and ductility of extended pile-shafts. I: Analytical model.” Journal of Structural Engineering, ASCE, 128(5) 586-594. Harden, C.W., Hutchinson, T.C., and Moore, M. (2006). “Investigation into the effects of foundation uplift on simplified design procedures,” Earthquake Spectra, 22(3), 663-692. Matlock, H. and Reese, L. C. (1960) “Generalized solutions for laterally loaded piles,” Journal of Soil Mechanics and Foundations Division, ASCE, 86(SM5), 63-91. Poulos, H. G. and Davis, E. H. (1980). Pile foundation analysis and design, Wiley-Interscience, New York, NY. Reese, L. C. and Van Impe, W. F. (2001). Single piles and pile groups under lateral loading, Balkema, Rotterdam, Netherlands. SEAOC (1999). Recommended lateral force requirements and commentary, Chapter 3, Structural Engineers Association of California, Seismology Committee, Sacramento, CA. Silva, P., Seible, F. and Priestley, M.J.N, (1997). Response of standard Caltrans pile-to-pile cap connections under simulated seismic loads, Report no. SSRP – 97/09, November 1997, University of California-San Diego, La Jolla, CA.

Article 7.02.020

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SEAOC Blue Book – Seismic Design Recommendations

Limit State Design of Piles, Pile Caps, and Grade Beams

Song, S. T., Chai, Y. H., and Hale, T. H. (2004). “Limit state analysis of fixed-head concrete piles under lateral loads.” Proc. 13th World Conference on Earthquake Engineering, Vancouver, Canada. Song, S. T., Chai, Y. H., and Hale, T. H. (2005). “Analytical model for ductility assessment of fixed-head concrete piles.” Journal of Structural Engineering, ASCE, 131 (7), July 2005, p. 1051. Xiao, Y., Priestley, M.J.N. and Seible, F. (1996). “Seismic assessment and retrofit of bridge column footings,” ACI Structural Journal, January-February 1996. Keywords piles, pile caps, grade beams, limit state design

How to Cite This Publication In the writer’s text, the article should be cited as: (SEAOC Seismology Committee 2008) In the writer’s list of cited references, the reference should be listed as: SEAOC Seismology Committee (2008). “Limit State Design of Piles, Pile Caps, and Grade Beams,” March, 2008, The SEAOC Blue Book: Seismic design recommendations, Structural Engineers Association of California, Sacramento, CA; accessible via the world wide web at: http://www.seaoc.org/bluebook/index.html.

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