a three-dimensional finite element study to obtain p-y curves for sand.pdf

8/12/2019 a three-dimensional finite element study to obtain p-y curves for sand.pdf

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A THREE-DIMENSIONAL FINITE ELEMENT STUDY TO OBTAIN

P-Y CURVES FOR SAND

Liangcai He1 (Student Member, ASCE), Zhaohui Yang2 (Member, ASCE), Jinchi LU3, and

Ahmed Elgamal4(Member, ASCE)

ABSTRACT

For pile foundations subjected to lateral loads, a realistic design approach and analysis method should

account for the response of both the soil and the pile. Current design practice and pile-soil interaction

analysis usually use a series of springs 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. For sand, p-y curves are independent of loading conditions with an initial

slope assumed to vary linearly with depth. This paper presents a framework using 3D FEM nonlinear

analysis to obtain p-y curves for sand. A linear analysis was first conducted on laterally loaded piles using

three-dimensional finite element method (3D FEM). A nonlinear 3D FEM was then carried out to model a

full-scale field lateral pile test. Different meshes were examined to obtain a reliable 3D FEM procedure for

pile-soil interaction. This procedure is then used to study p-y curves for sand under other conditions. It isfound that at greater depth, p-y curves show somewhat dependence on loading conditions. Near ground

surface, p-y curves show no apparent loading conditions dependence. In addition, the initial slope of the

p-y curves is about the same for different depths, different from commonly used ones

Keywords: Laterally Loaded Piles, p-y curves, Sand.

INTRODUCTION

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

demanding structural, soil and loading conditions. The most commonly used approach to design

1Ph.D. Candidate, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093,

USA. Email: [email protected] Research Scientist, Department of Structural Engineering, University of California, San Diego, La Jolla,

CA 92093, USA. Email: [email protected]. Candidate, Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093,

USA. Email:[email protected], Department of Structural Engineering, University of California, San Diego, La Jolla, CA 92093, USA.

Email: [email protected]


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and analyze such piles is the p-y method based on the Winkler beam on elastic foundation model.

In this approach the pile is modeled with a beam and the interaction between pile and soil is

modeled by a series of uncoupled nonlinear springs. The force-deformation relationship of these

nonlinear springs is widely known as p-y curve. Existing p-y curves were empirically back

calculated from full-scale field lateral load tests under specific conditions due to various

difficulties. For instance, the tests were often carried out in site-specific soil conditions and on

free-head piles with relatively small pile diameter. These p-y curves back calculated from specific

conditions were then extrapolated to use for other conditions. Such an application may not be

appropriate and it is preferable to use p-y curves derived from conditions the same as what the

pile is expected to experience.

3D FEM can model the soil as a continuum as well as the 3D nature of the problem. This

method is quite versatile to simulate piles under a variety of conditions. This paper presents a

framework for studying p-y curves at various conditions using 3D FEM method. The 3D FEMsimulation was conducted using a computer program CYCLIC developed at the University of

California, San Diego.

A linear 3D FEM study on laterally loaded piles was first conducted as routine checks.

Displacement profile and moment distribution of the pile from the linear 3D FEM were

compared with a rigorous continuum mechanics solution to investigate the level of spatial

resolution and domain size of the ground surrounding the pile that is required in order to

satisfactorily represent the involved soil-structure interaction mechanisms.

Nonlinear 3D FEM analyses on the full-scale field test conducted at the Mustung Island (Cox

et al. 1974; Reese and Impe 2001) were then carried out to study the pile response in real site

conditions. P-y curves based on the nonlinear 3D FEM study were subsequently presented.

MODEL DESCRIPTION

Finite element mesh

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

number of meshes were employed for linear study. FIG. 1 shows one of the meshes that used in

both linear and nonlinear studies. FIG. 2 shows another mesh of smaller domain size also used in

both linear and nonlinear study. Table 1 lists the parameters of the two meshes.

Eight-node solid elements were used to model the soil and beam elements were used to model

the 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 interface

elements were implemented since this is not required in the linear elastic cases and the employed

soil constitutive model itself plays this role in the nonlinear soil case. The boundary conditions

imposed on the mesh are:

The nodes at the bottom of the mesh are fixed against displacement in all directions.


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The nodes on the plane of symmetry cannot displace normally to that plane.

The nodes on the periphery of the mesh are fixed against displacement in both horizontaldirections; yet remain free to move vertically.

FIG. 1. 3D FEM mesh used in both linear and nonlinear study.

Employed mesh

Pile head close-up


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FIG. 2. A mesh with a smaller domain size used in 3D FEM nonlinear analysis.

Table 1 Parameters of the two meshes

Bottom

boundary

from pile tip

Periphery

boundary

from pile

Number

of solid

elements

Number of

beam elements

for pile

Number of rigid

beam elements

for pile size

Mesh in FIG. 1 10 100 3472 25 234

Mesh in FIG. 2 10 25 1764 18 171

Soil Constitutive Model

In linear study, the soil was modeled by elastic isotropic material. The material is

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

In nonlinear study, a constitutive model able to reproduce salient sand response characteristics

including shear-induced nonlinearity and dilatancy (Parra 1996; Yang 2000; Elgamal et al. 2003)

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 theory for frictional

cohesionless soils (Prevost 1985). In this soil model, a number of similar conical yield surfaces

with different tangent shear moduli are employed to represent shear stress-strain nonlinearity and


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the confinement dependence of shear stiffness and shear strength (FIG. 3).

FIG. 3. Conical yield surfaces for granular soils in principal stress space and

deviatoric plane (Prevost 1985; Lacy 1986; Parra et al. 1996; Yang 2000).

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

)(: pE &&& = (1)

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 elastic

coefficients. 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,Lthe

plastic loading function, and the symbol denotes the McCauley's brackets (i.e.,

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

point (i.e., Q= ff / ), wherefis yield function.

The yield functionfselected has the following form (Elgamal et al. 2003):

0)())(())((23 20

200 =ppMppppf ss : (2)

in the domain of 0p . The yield surfaces in principal stress space and deviatoric plane are

shown in Fig. 1. In eq. 2, s p is the deviatoric stress tensor, p the mean effectivestress, 0p a small positive constant (1.0 kPa in this paper) such that the yield surface sizeremains finite at 0p for numerical convenience (FIG. 3), a second-order kinematic


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deviatoric tensor defining the surface coordinates, and Mdictates the surface size. In the context

of multi-surface plasticity, a number of similar surfaces with a common apex form the hardening

zone (FIG. 3). Each surface is associated with a constant plastic modulus. Conventionally, the

low-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 generate

hysteretic response under cyclic shear loading. This kinematic rule dictates that all yield surfaces

may 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, the

soil contractive/dilative (dilatancy) behavior is handled by a non-associative flow rule (Elgamal

et al. 2003) so as to achieve appropriate interaction between shear and volumetric response. In

particular, nonassociativity is restricted to the volumetric component of the plastic flow tensor

(outer normal to the plastic potential surface in stress-space). Therefore, depending on the

relative location of the stress state (FIG. 3) with respect to the phase transformation (PT) surface

(Ishihara et al. 1975), distinct contractive/dilative behaviors are reproduced by specifying

appropriate expressions for the nonassociativity (Elgamal et al. 2003). Consequently, P defines the degree of non-associativity of the flow rule and is given by (Parra 1996):

P1)/(

1)/(2

2

+

=

The employed model has been extensively calibrated for clean Nevada Sand at rD 40%(Elgamal et al. 2002). The calibration phase included results of monotonic and cyclic laboratory

tests, as well as data from level-ground and mildly inclined infinite-slope dynamic

centrifuge-model simulations.

RESULTS OF ANALYSIS

Linear 3D FEM Analysis

Abedzadeh and Pak (2004) presented a benchmark solution of laterally loaded piles. This

solution is a rigorous mathematical formulation for a flexible tubular elastic pile of finite length

embedded in a semi-infinite homogenous elastic soil medium under lateral loading in theframework of three-dimensional elastostatics and the classical Bernoulli-Euler beam theory. A

series of 3D FEM analyses were conducted with various domain sizes for the case of a pure

pile-head horizontal load H using the following pile and soil properties:

Homogenous elastic soil: Youngs modulus Es= 50 MPa, Poisson ratio s= 0.4, and thus

shear modulus Gs = 17.857 Mpa


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10

0

10

20

30

40

50

60

70

80

0 1 2 3 4

Pile Diameter (m)

PileDisplacementatGroundLine(mm)

Free-head by Davies and Budhu (1986)

Free-head by 3D FEM

Fixed-head by Davies and Budhu (1986)

Fixed-head by 3D FEM

0

500

1000

1500

2000

2500

3000

0 1 2 3 4

Pile Diameter (m)

MaximumM

omentin

Free-HeadPileorMomentinFixedPile-Head

(kN*m)

Free-head by Davies and Budhu (1986)

Free-Head by 3D FEM

Fixed-head by Davies and Budhu (1986)

Fixed-head by 3D FEM

FIG. 6. 3D FEM results compared with Davies and Budhu (1986).

Nonlinear 3D FEM Analysis

An effort to model a full-scale field lateral-load pile test was attempted in this study. The test

modeled herein was conducted at Mustung Island near Corpus Christ, Texas (Cox et al. 1974;

Reese and Impe 2001). 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: moment of inertial Ip = 8.084510-4 m4;

bending stiffness EpIp= 163,000 kN-m2; yield moment = 640 kN-m; and ultimate moment Mult=

828 kN-m. The soil at the site was uniformly graded, fine sand with a friction angle of 39 degrees.

The submerged unit weight was 10.4 kN/m3. The water table was maintained at 0.15 m or so

above the 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 FEM modeling, the pile was modeled as a linear elastic beam with above

mechanical properties. Lateral load at increment of 1 kN was applied at 0.305 m above the

ground line. The final lateral load was 280 kN, below which the pile behaved linear (Cox et al.

1974; Reese and Impe 2001). Table 2 lists the soil constitutive parameters in addition to above

soil properties used in the nonlinear modeling of the soil.


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FIG. 7. Deformed mesh 2 at lateral load 200 kN.


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(a) Mesh 1

(b) Mesh 2

FIG. 10. Computed response 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, 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 length

and y is pile displacement. The lateral soil pressure p can be determined by differentiating the

shear forces obtained from 3D FEM with respect to depth. The associated pile displacement can

be directly obtained from 3D FEM. As a matter of fact, p and y obtained from 3D FEM nonlinear

analysis have been shown in FIG. 10 for the Mustung Island lateral load test. The corresponding

p-y curves are easily obtained (FIG. 11 denoted as L1).

Using the same 3D FEM procedure, another set of p-y curves is also obtained for the case

where the lateral load is applied right at ground line. This set of p-y curves is denoted as L2 in

FIG. 11.

FIG. 11 shows that at greater depth, p-y curves show somewhat dependence on loadingconditions. The p-y curves are softer when lateral load was applied above ground line than

applied at ground line, while commonly used p-y curves do not distinguish loading conditions

(e.g, Reese and Impe 2001). Near ground surface, p-y curves show no apparent difference from

loading conditions, in agreement with traditional p-y curves. FIG. 11 also shows that the initial

slope of the p-y curves is about the same for different depths, different from commonly used ones

(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. From half to two pile diameter depth, the p-y curves yielded first and displayed

dilatance when associated displacement reached 0.015 m. The p-y curves at greater depth did not

fully yield at studied load levels.


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FIG. 11. p-y curves derived from 3D FEM analysis.

CONCLUTIONSThis study used 3D FEM to laterally loaded piles. A framework is presented using 3D FEM

nonlinear analysis to obtain p-y curves for sand. The following conclusions can be drawn:

1. 3D FEM linear elastic lateral pile analysis is sensitive to lateral domain size; a

boundary 100 times pile diameter from the pile is needed. The bottom boundary is not

critical; 10 times pile diameter from the pile tip is appropriate.

2. When conduction 3D FEM nonlinear analysis, the domain size can be smaller than

linear case; a boundary 25 times pile diameter from the pile is appropriate.

3. At greater depth, p-y curves show somewhat dependence on loading conditions. Near

ground surface, p-y curves show no apparent loading conditions dependence.

4. 3D FEM nonlinear analysis shows that the initial slope of the p-y curves is about the

same for different depths, different from commonly used ones (e.g, Reese and Impe2001).

ACKNOWLEDGMENTS

This research was supported by the Pacific Earthquake Engineering Research (PEER) Center,

under the National Science Foundation Award Number EEC-9701568.


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REFERENCES

Abedzadeh, F., and Pak, R. Y. S. (2004). "Continuum Mechanics of Lateral soil-pile interaction."ASCE J Engrg Mech.

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 in

Saturated Cohesionless Soils."Int. J. Plasticity, 19(6), 883-905.

Hill, R. (1950). The Mathematical Theory of Plasticity, Oxford University Press, London.

Ishihara, K., Tatsuoka, F., and Yasuda, S. (1975). "Undrained Deformation and Liquefaction ofSand under Cyclic Stresses." Soils and Foundations, 15(1), 29-44.

Lacy, S. (1986). "Numerical Procedures for Nonlinear Transient Analysis of Two-phase Soil

System," Ph.D. Thesis, Princeton University, NJ.

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.

Parra, E., Adalier, K., Elgamal, A.-W., Zeghal, M., and Ragheb, A. "Analyses and Modeling ofSite Liquefaction Using Centrifuge Tests."Eleventh World Conference on Earthquake

Engineering, Acapulco, Mexico.

Pak, R. Y. S. (2004). personal communication).

Prevost, J. H. (1985). "A Simple Plasticity Theory for Frictional Cohesionless Soils." Soil

Dynamics and Earthquake Engineering, 4(1), 9-17.Reese, L. C., and Impe, W. F. V. (2001). Single piles and pile groups under lateral loading, A. A.

Balkema Publishers, Brookfield, USA.

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 Shear

Deformation."J. Engineering Mechanics, 128(7), 720-729.

Yang, Z., Elgamal, A., and Parra, E. (2003). "A Computational Model for Cyclic Mobility and

Associated Shear Deformation."J. Geotechnical and Geoenvironmental Engineering,

129(12), 1119-1127.

a three-dimensional finite element study to obtain p-y curves for sand.pdf

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