He_EM2004_final.pdf

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A THREE-DIMENSIONAL FINITE ELEMENT STUDY TO OBTAIN

P-Y CURVES FOR SAND Liangcai He 1 (Student Member, ASCE), Ahmed Elgamal 2 (Member, ASCE), Zhaohui Yang 3

(Member, ASCE), and Jinchi Lu 4 (Student Member, ASCE)

ABSTRACT

For pile foundations subjected to lateral loads, an analysis method should account for the response ofthe combined soil-pile system. Current practice for pile-soil interaction analysis usually employs a series ofsprings to model the lateral behavior of soil-pile interaction. In this method, the force (p) deformation (y)function, widely known as p-y curve, of the spring characterizes the pile-soil interaction mechanism. Forsand, traditional p-y curves are independent of loading conditions with an initial slope assumed to varylinearly with depth. This paper presents a framework using three-dimensional finite element method (3DFEM) nonlinear analysis to estimate the p-y curves response characteristics for sand. A linear analysis wasfirst conducted as a routine check on laterally loaded piles using the 3D FEM. A nonlinear 3D FEM wasthen carried out to model full-scale field lateral pile response. Different meshes were examined to obtain areliable 3D FEM representation for pile-soil interaction. This procedure was subsequently used to study

p-y curves for sand under other conditions. It is found that at greater depth, p-y curves show some

dependence on loading conditions. Near ground surface, p-y curves show little influence of loadingconditions. In this pilot study, the initial slope of the p-y curves appeared to be about the same at differentdepths.

Keywords: Laterally Loaded Piles, p-y curves, Sand, Three-Dimensional Finite Element Method

INTRODUCTIONThe response of a laterally loaded pile is of critical relevance to foundation engineering under

earthquake loading conditions. The most commonly used approach to design and analyze such pile response scenarios is the p-y method based on the Winkler beam on elastic foundation model

1 Ph.D. Candidate, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093,USA. Email: [email protected] 2 Professor, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093, USA.Email: [email protected] 3 Assistant Research Scientist, Department of Structural Engineering, University of California, San Diego, La Jolla,CA 92093, USA. Email: [email protected] 4 Ph.D. Candidate, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093,USA. Email: [email protected]


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(Fig. 1). In this method, the pile is modeled as a beam and the interaction between pile and soil ismodeled by a series of uncoupled nonlinear springs (e.g., McClelland and Focth 1958; Reese etal. 1974; Bowles 1988). The force-deformation relationship of these nonlinear springs is widelyknown as the p-y curve. Existing p-y curves were empirically back calculated from full-scalefield lateral load tests under specific conditions (e.g., Cox et al. 1974; Reese and Impe 2001). Forinstance, the tests were often carried out in site-specific soil conditions and on free-head pileswith a relatively small pile diameter. These back calculated p-y curves were then extrapolated touse for other conditions (Reese et al. 2000).

Prototype p-y method

FIG. 1. p-y method for laterally loaded pil es (After Reese and Impe 2001).

Increasingly, the 3D FEM is being used as a tool for such soil-foundation-interaction (SFI) problems. The method is quite versatile for simulation of piles under a variety of conditions. This paper presents a pilot study for studying p-y curves characteristics at various conditions using the3D FEM method. The 3D FEM simulation was conducted using the computer program CYCLICdeveloped at the University of California, San Diego.

A linear 3D FEM study on laterally loaded piles was first conducted as a routine check.Preliminary nonlinear 3D analyses on the widely referenced full-scale field lateral load testconducted at the Mustang Island (Cox et al. 1974; Reese and Impe 2001) were then carried out to

study the pile response. P-y curves based on the nonlinear 3D FEM study were subsequentlysynthesized.

MODEL DESCRIPTION

Finite element meshTaking advantage of symmetry, only half of the domain was meshed for the 3D FEM study. A

MH

MH


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number of meshes were employed in the linear study. Figs 2-4 show meshes that were used inconducting both linear and nonlinear studies. Table 1 lists the parameters of these two meshes.

Eight-node solid elements were used to model the soil and beam elements were used to modelthe floating pile. Rigid beam elements with rigidity 1000 times larger than the pile rigidity wereused to connect the pile and the soil in order to model the pile size. No special pile-soil interfaceelements were implemented since this is not required in the linear elastic cases and the employedsoil constitutive model itself plays this role in the nonlinear soil case. The boundary conditionsimposed on the mesh are:

Nodes at the bottom of the mesh are fixed in all directions. Nodes on the plane of symmetry cannot displace out-of-plane. Nodes on the periphery of the mesh are fixed in both horizontal directions, yet remain free

to move vertically.

FIG. 2. 3D FEM mesh 1 used in conduct ing both l inear and nonlin ear stu dies.


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FIG. 3. Pile head close up o f mesh 1.

FIG. 4. Mesh 2 wit h a smaller domain size used in the 3D FEM nonli near analys is.


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Table 1 Parameters o f th e two meshes (pile diameter = 0.6 m)

Bottom boundary

from pile tip

Periphery boundaryfrom pile

Number ofsolid

elements

Number of beam elements

for pile

Number of rigid beam elements

for pile size

Mesh 1 in Figs. 2and 3

6 m 60 m 3472 25 234

Mesh 2 in Fig. 4 6 m 15 m 1764 18 171

Soil Constitutive ModelIn the linear study, the soil was modeled as an elastic isotropic material. The material is

characterized by two elastic constants, Youngs modulus E and Poissons ratio .

In the nonlinear study, a constitutive model able to reproduce salient sand responsecharacteristics including shear-induced nonlinearity and dilatancy (Parra 1996; Elgamal et al.2003; He 2004) was employed. This soil constitutive model (Parra 1996; Yang and Elgamal 2002;Elgamal et al. 2003; Yang et al. 2003) was based on the original multi-surface-plasticity theoryfor frictional cohesionless soils (Prevost 1985). In this soil model, a number of similar conicalyield surfaces with different tangent shear moduli are employed to represent shear stress-strainnonlinearity and the confinement dependence of shear stiffness and shear strength (Fig. 5).

FIG. 5. Conical yield surfaces for granular soils in principal stress space anddeviator ic p lane (Prevos t 1985; Lacy 1986; Parra 1996; Yang 2000).

The constitutive equation is written in incremental form as follows (Prevost 1985):

)(: p E (1)


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where is the rate of effective Cauchy stress tensor, the rate of deformation tensor, p

the plastic rate of deformation tensor, and E the isotropic fourth-order tensor of elasticcoefficients. The rate of plastic deformation tensor is defined by: p = P L , where P is a

symmetric second-order tensor defining the direction of plastic deformation in stress space, L the plastic loading function, and the symbol denotes the McCauley's brackets (i.e.,

L =max( L, 0)). The loading function L is defined as: L = Q : / H where H is the plasticmodulus, and Q a unit symmetric second-order tensor defining yield-surface normal at the stress

point (i.e., Q = f f / ), where f is yield function.

The yield function f selected has the following form (Prevost 1985; Lacy 1986; Elgamal et al.2003):

0)())(())((23 2

02

00 p p M p p p p f ss : (2)

in the domain of 0 p . The yield surfaces in principal stress space and deviatoric plane areshown in Fig. 1. In eq. 2, s p is the deviatoric stress tensor, p the mean effectivestress, 0 p a small positive constant (1.0 kPa in this paper) such that the yield surface sizeremains finite at 0 p for numerical convenience (Fig. 5), a second-order kinematicdeviatoric tensor defining the surface coordinates, and M dictates the surface size. In the contextof multi-surface plasticity, a number of similar surfaces with a common apex form the hardeningzone (Fig. 5). Each surface is associated with a constant plastic modulus. Conventionally, thelow-strain (elastic) moduli and plastic moduli are postulated to increase in proportion to the

square root of p (Prevost 1985).

A purely deviatoric kinematic hardening rule (Prevost 1985) is employed in order to generatehysteretic response under cyclic shear loading. This kinematic rule dictates that all yield surfacesmay translate in stress space within the failure envelope (Hill 1950).

The flow rule is chosen so that the deviatoric component of flow P = Q (associative flowrule in the deviatoric plane), and the volumetric component P defines the desired amount ofdilation or contraction in accordance with experimental observations. During shear loading, thesoil contractive/dilative (dilatancy) behavior is handled by a non-associative flow rule (Elgamalet al. 2003) so as to achieve appropriate interaction between shear and volumetric response.

The employed model has been extensively calibrated for clean Nevada Sand at r D 40%(Elgamal et al. 2002). The calibration phase included results of monotonic and cyclic laboratorytests, as well as data from level-ground and mildly inclined infinite-slope dynamiccentrifuge-model simulations.


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RESULTS OF ANALYSIS

Linear 3D FEM AnalysisPile displacements at ground line and maximum moment from the linear 3D FEM analysis

are compared with the widely referenced solutions of Davies and Budhu (1986). Displacement profile and moment distribution along the pile from the linear 3D FEM study were comparedwith the rigorous continuum mechanics solution of Abedzadeh and Pak (2004). Results frommesh 1 were in good agreement with these benchmark solutions (He 2004).

Nonlinear 3D FEM AnalysisA pilot nonlinear 3D FEM modeling of the well-known Mustang (near Corpus Christ, Texas)

full-scale field lateral-load pile test was conducted in this study (Cox et al. 1974; Reese and Impe2001). The pile was a steel-pipe pile with a 0.61m outside diameter and a 0.095m wall thickness.It was driven open-ended into the ground leading to an embedded length of 21m. The mechanical

properties of the pile were (Reese and Impe 2001): moment of inertial I p = 8.084510-4 m4;

bending stiffness E pI p = 163,000 kN-m 2; yield moment = 640 kN-m; and ultimate moment M ult =828 kN-m. Soil at the site was uniformly graded, fine sand with a friction angle of 39 degrees.The submerged unit weight was 10.4 kN/m 3. Water table was maintained at 0.15 m or so abovethe ground line throughout the tests. Lateral load was applied at 0.305 m above the ground line(Cox et al. 1974; Reese and Impe 2001).

In the 3D mesh, the pile was modeled as a linear elastic beam with the above mechanical properties. Lateral load at increments of 1 kN was applied at 0.305 m above the ground line. Thefinal lateral load was 280 kN, below which the pile behaved linearly (Cox et al. 1974; Reese and

Impe 2001). Table 2 lists the main soil constitutive parameters in addition to above soil propertiesused in nonlinear modeling.

Table 2 Soil Cons titutiv e Parameters fo r Mustang Island Lateral Pile Test An alysis

Parameter Value

Submerged unit weight 10.4 kN/m 3 Reference mean pressure p 0 80 kPaShear modulus G 0 at p 0 90,000 kPaPoissons ratio 0.4Pressure dependence coefficient n p 0.5Friction angle 39

Number of yield surfaces 18

Fig. 6 shows the deformed mesh 2 at lateral load 200 kN exaggerated 50 times. Soil heave infront of the pile and settlement behind the pile are observed, which is consistent with fieldobservation. It is also noted that the soil very close to the pile, about 3 pile diameters in the lateraldirection and 5 pile diameters in depth, has undergone significant deformation.


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FIG. 6. Deformed mesh 2 (exaggerated 50 times) at lateral load 200 kN.Fig. 7 shows the comparisons of experimental and computed pile deflection at ground line. Goodagreement between experimental and computed response is observed. Fig. 7 shows that mesh 1 issomewhat softer than the mesh 2 as expected, since mesh 1 has a larger lateral domain size andalso has more beam elements representing the pile. Fig. 8 shows the lateral pile responsecomputed using meshes 1 and 2, where the soil pressure is determined by differentiating theshear force with respect to depth. Figures 7 and 8 show that the results from mesh 1 and 2 arequite close. Mesh 2 was used afterward to study p-y curves in order to save computing time.


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FIG. 7. Comparis on of experimental and comput ed pile deflection at ground line.


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(a) Mesh 1

(b) Mesh 2

FIG. 8. Comp uted respons e at lateral load 20, 60, 100, 140, 180, 220, and 260 kN.


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P-y curve from Nonlinear 3D FEM Analysis

As mentioned earlier, the p-y curve is the force-deformation relationship of the springsrepresenting soil-pile interaction. In this relationship, p is lateral soil pressure per unit pile lengthand y is pile displacement. The lateral soil pressure p can be determined by differentiating theshear forces obtained from 3D FEM with respect to depth. The associated pile displacement can

be directly obtained from 3D FEM. Fig. 8 shows the 3D FEM nonlinear analysis obtainedrelationships. The corresponding p-y curves are easily synthesized (denoted as L1 in Fig. 9).Using the same 3D FEM procedure, another set of p-y curves is also obtained for the case wherethe lateral load is applied right at ground line. This set of p-y curves is denoted as L2 in Fig. 9.

Fig. 9 shows that at greater depth, p-y curves show some dependence on loading conditions.The p-y curves are softer when lateral load was applied above the ground line. Commonly used

p-y curves do not distinguish loading conditions (e.g, Reese and Impe 2001). Near ground

surface, p-y curves show little influence of loading conditions, similar to the traditional p-ycurves. Fig. 9 also shows that the initial slope of the p-y curves is about the same at differentdepths, unlike the traditional curves (e.g, Reese and Impe 2001). The p-y curve yields at smaller

pile displacement near ground surface. In the two cases, L1 and L2, the p-y curves above half pile diameter depth yielded completely. The p-y curves at greater depth did not fully yield at thestudied load levels.

FIG. 9. p-y curves at dif ferent depth derived from 3D FEM analysi s.


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CONCLUTIONS

This pilot study used a 3D FEM to analyze laterally loaded piles. A framework is presentedusing 3D FEM nonlinear analysis to obtain p-y curves for sand. The following conclusions can be drawn:

1. At greater depth, p-y curves show some dependence on loading conditions. Nearground surface, p-y curves were similar.

2. In this preliminary study, 3D FEM nonlinear analysis shows that the initial slope of the p-y curves is about the same at different depths. Additional research is underway tofurther clarify this mechanism.

ACKNOWLEDGMENTSThe authors would like to thank Professor Ronald Y. S. Pak (University of Colorado, Boulder),

for his rigorous analytical solutions and valuable comments on the 3D FEM linear analysis. Thisresearch was supported by the Pacific Earthquake Engineering Research Center (PEER), underthe Earthquake Engineering Research Centers Program of the National Science Foundation(award number EEC-9701568), and by the National Science Foundation (awards numberCMS-0084616, and 0200510).

REFERENCESAbedzadeh, F., and Pak, R. Y. S. (2004). "Continuum Mechanics of Lateral Soil-Pile Interaction."

Journal of Engineering Mechanics, ASCE , (in review).Bowles, J. E. (1988). Foundation Analysis and Design, 4th Ed., McGraw-Hill.Cox, W. R., Reese, L. C., and Grubbs, B. R. (1974). "Field testing of laterally loaded piles in

sand." Proc. 6th Offshore Technology Conference , Paper 2079, Houston, Texas, 459-472.Davies, T. G., and Budhu, M. (1986). " Non-linear analysis of laterally loaded piles in heavily

overconsolidated clays." Geotechnique , 36(4), 527-538.Elgamal, A., Yang, Z., and Parra, E. (2002). "Computational Modeling of Cyclic Mobility and

Post-Liquefaction Site Response." Soil Dynamics and Earthquake Engineering , 22(4),259-271.

Elgamal, A., Yang, Z., Parra, E., and Ragheb, A. (2003). "Modeling of Cyclic Mobility inSaturated Cohesionless Soils." Int. J. Plasticity , 19(6), 883-905.

He, L. (2004). "Experimental and Numerical Modeling of Laterally Loaded Piles," Ph.D. Thesisin Progress, University of California, San Diego, La Jolla, California.

Hill, R. (1950). The Mathematical Theory of Plasticity, Oxford University Press, London.Lacy, S. (1986). "Numerical Procedures for Nonlinear Transient Analysis of Two-phase Soil

System," Ph.D. Thesis, Princeton University, NJ.McClelland, B., and Focth, J. A. J. (1958). "Soil modulus for laterally loaded piles." Transactions,

ASCE , 123(1049-1063).Parra, E. (1996). "Numerical Modeling of Liquefaction and Lateral Ground Deformation

Including Cyclic Mobility and Dilation Response in Soil Systems," Ph.D. Thesis,Rensselaer Polytechnic Institute, Troy, NY.

Prevost, J. H. (1985). "A Simple Plasticity Theory for Frictional Cohesionless Soils." Soil Dynamics and Earthquake Engineering , 4(1), 9-17.


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Reese, L. C., Cox, W. R., and Koop, F. D. (1974). "Analysis of laterally loaded piles in sand."Proc., 6th Offshore Technology Conf., Houston, Texas (Paper No. 2080), 473-483.

Reese, L. C., and Impe, W. F. V. (2001). Single piles and pile groups under lateral loading, A. A.Balkema Publishers, Brookfield, USA.

Reese, L. C., Wang, S. T., Isenhower, W. M., and Arrellaga, J. A. (2000). "Computer ProgramLPILE Plus Version 4.0 Technical Manual." Ensoft, Inc., Austin, Texas.

Yang, Z. (2000). "Numerical Modeling of Earthquake Site Response Including Dilation andLiquefaction," Ph.D. Thesis, Columbia University, New York, NY.

Yang, Z., and Elgamal, A. (2002). "Influence of Permeability on Liquefaction-Induced ShearDeformation." J. Engineering Mechanics , 128(7), 720-729.

Yang, Z., Elgamal, A., and Parra, E. (2003). "A Computational Model for Cyclic Mobility andAssociated Shear Deformation." J. Geotechnical and Geoenvironmental Engineering ,129(12), 1119-1127.

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