Analysis Of Pile Foundations Under Seismic Loading

Analysis of Pile Foundations under Seismic Loading Jayram Ramachandran CBE Institute 2005: Final Report

1

Introduction

Pile foundations are commonly used to transfer loads from a structure to the ground in cases where the structural loads are very high or at locations where the soil at shallow depths cannot carry the imposed loads. Under moderate and strong seismic loading, pile foundations undergo large displacements and the behavior of the pile-soil system can be strongly nonlinear. Such pile-soil systems interact with the supported structure; hence their nonlinear behavior is extremely important in evaluating the seismic response of pile-supported structures. In seismic regions, the analysis and design of pile-supported structures requires an accurate prediction of the pile head response and the load resistance to lateral shaking caused by earthquake ground motions. In order to predict the response of pile groups under seismic loading, it is important to understand the interaction between the piles in a group. This research studies the interaction between piles in pile groups accounting for soil nonlinearity and pile-soil gapping effects. Most pile foundations consist of a group of piles rather than a single pile. Thus, an important component of the analysis of pile supported structures is the ability to perform an accurate and efficient analysis of pile groups. However, most pile group analysis techniques are an extension of techniques used to predict the response of single piles. Therefore, Section 2 presents a brief review of single pile analysis techniques. The Beam on Winkler Foundation model is a computationally efficient tool to estimate the response of single piles accounting for pile and soil nonlinearity. A number of researchers have used this technique and hence, this model is briefly described. Section 3 presents an overview of the current techniques available to perform pile group analysis. These include elasticity based methods, wave propagation based methods and the p-multiplier method. Section 4 presents the results of a numerical study using 3D finite element analysis to investigate pile-to-pile interaction in a pile group, accounting for soil nonlinearity and pile-soil gapping effects.

1

2

Single Pile Analysis

Most pile group analysis techniques are an extension of methods employed to compute single pile response. Therefore, it is highly instructive to review techniques used to compute single pile response, including the various numerical and analytical methods that have been developed. Closed form analytical solutions have been obtained by a number of researchers for single piles under axial or lateral loading assuming the soil to be linear viscoelastic and the pile to be a linear elastic beam [15, 17, 18, 24]. Such closed form solutions are hard to extend to reflect the true nonlinear behavior of the pile-soil system. Some researchers include soil nonlinearity in the analysis using an equivalent viscous dashpot [14]. The most direct method of analysis involves 2D/3D modeling of the pile and the soil continuum using the finite element method. In such an analysis, the soil nonlinearity may be considered explicitly by using appropriate material models [12, 37]. The finite element method, although powerful, is computationally very expensive since it requires discretization of the pile and the large volume of surrounding soil. In contrast, the Beam-on-Winkler-Foundation model is a versatile and efficient approach to pile foundation analysis that has been adopted widely. This section reviews the Winkler model.

2.1

Beam on Winkler Foundation Model

The Beam on Winkler Foundation Model model or Winkler model is commonly used to study the response of single piles. The pile is modeled as a beam while the surrounding soil is modeled using continuously distributed springs and dashpots. Pile nonlinearity may be considered in the analysis using an appropriate nonlinear material model. Hysteretic energy loss in the soil is modeled using nonlinear springs whose parameters are determined experimentally from load-deformation curves [21, 22] or analytically by defining stiffness and strength parameters [3, 7, 8, 28]. The dashpots placed in parallel with the nonlinear

2

Figure 1: Beam on Winkler foundation model for a single pile under lateral loading springs account for energy loss due to radiation damping, under dynamic loading conditions. Since radiation damping is a frequency dependent phenomenon, the dashpot coefficients are usually frequency dependent. The Winkler model is easily implemented for analysis of nonlinear pile-soil systems using most standard 1-D finite element programs. Within the finite element framework, the pile is modeled using discrete beam elements and the continuous soil springs/dashpots are replaced by discrete springs and dashpots. Figure 1 shows the discretized form of the Winkler model for the analysis of a single pile under dynamic (including seismic) lateral loading. A similar model may be used for the analysis of single piles under static lateral loads. However, under static loads, the dashpots would be irrelevant. The

3

Winkler model has been shown to provide a good estimate of the pile response by several researchers [3, 32].

2.2

Model parameters

The Winkler model requires definition of parameters for the nonlinear springs as well as viscous dashpots. This section defines the parameters for the Winkler model under lateral loading. The nonlinear spring may be represented using either the bilinear or Bouc-Wen [35] models. The bilinear model is a very simple representation of the hysteretic behavior of the soil, but its main attraction is that it can be easily implemented in most finite element programs. The Bouc-Wen model is much more versatile as it allows a smooth transition from linear to nonlinear behavior. However, it is not implemented in a number of commonly used analysis programs. Both the bilinear and Bouc-Wen models require definition of the initial spring stiffness and strength at large displacements (assuming a zero post yield stiffness [3]). Based on a number of analytical studies that compare pile head displacements from Winkler model analyses with 3-D finite element analyses, the stiffness of the soil springs per unit length under lateral loading can be approximated realistically by [14, 17]:

kx = 1.2Es (z)

(1)

where kx is stiffness of the continuously distributed spring and Es (z) is the Young’s modulus of the soil as a function of depth z. The value in equation 1 is multiplied by the spacing between two adjacent springs to obtain the discrete spring stiffness. The maximum force in the spring is equal to the ultimate lateral reaction per unit length of pile. For cohesive soils, the lateral soil strength based on theoretical studies [3, 8, 28] using the theory of plasticity

4

under plane-strain conditions is given by:

Fs,max (z) = λSu (z)d

(2)

where Fs,max (z) is the maximum lateral force exerted by the soil on the pile (as a function of depth, z), d is the pile diameter, Su (z) is the variation of shear strength with depth and λ is a dimensionless parameter. Considerable research has been done to quantify λ [8, 19]. λ values between 9 and 12 may be appropriate at depths where plane strain conditions are valid. At shallow depths, plane strain conditions are not valid due to vertical deformation of the soil during lateral motion of the pile. Hence, λ values of 2 or 3 have been suggested. The following expression for the variation of λ with depth is recommended [19]:

λ(z) = 3 +

σx z +J Su d

(3)

where σx is the overburden pressure and J is a co-efficient that is obtained by calibrating the model against known experimental data. In the absence of such data, the recommended value of J = 0.5 may be used. For cohesionless soils, the following expression for the limiting force of the soil on the pile [7] is used: Fs,max (z) = µγs d

1 + sin φ z 1 − sin φ

(4)

where Fs,max (z) is the maximum lateral force exerted by the soil on the pile (as a function of depth, z), γs is the specific weight of the soil, φ is the angle of internal friction and µ is a dimensionless parameter. A value of µ=3 has been suggested [7]. The maximum force (or strength) of the spring is obtained by multiplying the maximum reaction force of the soil in equations 2 and 4 with the spacing between two adjacent springs, s. A number of expressions are available for the dashpot coefficient [14, 34]. The expression

5

used in this study defines a frequency dependent dashpot coefficient as used by [3]:

c(a0 , z) = Qa−0.25 ρs Vs ds 0

(5)

where c is the dashpot coefficient, a0 is a non-dimensional frequency parameter (= ωd/Vs), ω is the angular frequency, ρs is the soil density, Vs is the shear wave velocity in the soil medium, d is the pile diameter, and s is the spacing between two adjacent dashpots. The coefficient Q is given by the expression:

Q=

   3

  2[1 +

if z < 3d, 3.4 ]1.25 ( π4 )0.75 π(1−νs

where νs is the Poisson’s ratio of the soil.

6

if z > 3d.

(6)

3

Pile Group Analysis

Most pile foundations consist of a group of piles rather than a single pile. Thus, an important component of the analysis of pile supported structures is the ability to perform an accurate and efficient analysis of pile groups. Just as for single piles, various numerical and analytical methods have been developed for the analysis of pile groups. Currently, the most accurate method of pile group analysis may be the 3D finite element analysis method. Such analyses have been performed under static [11, 38] as well as dynamic (including seismic) [16] loading conditions. However, solving any problem with the 3D finite element method is extremely time consuming and hence numerous attempts have been made to find more computationally efficient solution procedures. A quasi-three-dimensional method is presented in [36], where nonlinear dynamic analysis of single piles and pile groups is performed in the time domain using strain-dependent moduli and damping, and yielding at failure. Since most pile groups are made up of identical piles and the soil profile at the locations of all piles in the group may also be assumed to be identical, one of the most common techniques of pile group analysis involves modifying the response of a single pile by a suitable factor to obtain the response of the group [13, 15, 17, 27]. The response of the single pile may be obtained using any of a wide range of techniques, but, most commonly a Winkler model is used, as discussed in Section 2. The resistance provided by the group under vertical or lateral loading is generally not equal to the sum of the resistances of the individual piles. Most often, the group resistance is less than the sum of the individual pile resistances and is a function of the pile group configuration as well as pile spacing. A method for analysis of pile groups using interaction factors and based on the theory of elasticity was proposed by Poulos and Davis [27]. This method is applicable to pile groups in elastic soils under static loading conditions. Approximate analytical closed form solutions for pile-to-pile interaction factors under dynamic loads, based on frequency domain solutions are also available [13, 15, 17].

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These solution procedures have also been modified to accommodate time domain analysis procedures for nonlinear problems [4]. This section reviews some of the current procedures used to analyze pile groups.

3.1 3.1.1

Use of pile-to-pile interaction factors in pile group analysis Elasticity based solution

Poulos and Davis [27] developed a method of analysis of pile groups using single pile analysis results along with pile-to-pile interaction factors. These solutions are based on elasticity and apply to pile groups under static loading. Separate but similar procedures were developed for the analysis of pile groups under both axial and lateral loading conditions. In this method, the response of each pile in a group is a sum of the response of the pile if it were a single pile and the additional displacements induced in the pile due to its interaction with each of the other piles in the group. In computing the response of the single pile under lateral loads, the pile is assumed to be a thin rectangular vertical strip of constant flexibility and the soil is considered to be a continuum. The pile is divided into a number of elements, each acted upon by a uniform horizontal stress, which is assumed to be constant across the width of the pile. The soil is assumed to be an ideal, homogeneous, isotropic, semi-infinite elastic material with properties unaffected by the presence of the pile. A solution is obtained by imposing compatibility between the displacements of the pile and the soil for each element of the pile. The soil displacement influence factors, relating the soil displacements with the horizontal stress, are evaluated by integration over a rectangular area of the Mindlin equation for the horizontal displacement of a point within a semi-infinite mass caused by a horizontal point load within the mass. This solution is also available for piles in soils with Young’s modulus varying with depth. The soil displacement influence factors are a function of the pile flexibility factor which measures the relative pile to soil stiffness, pile length to diameter ratio

8

and the Poisson’s ratio of the soil. These influence factors have been computed considering the effect of the different parameters [25, 27]. The procedures used in the elasticity based analysis of single piles may be found in great detail in the references cited. A major advantage of the elasticity based method of analysis is that since the soil is treated as a continuum, it is possible to consider the effects of interaction between piles. The method employed to compute the pile group response is an extension of the procedure used to compute the response of single piles [26]. The interaction between two identical piles is considered first and the analysis is then extended to general pile groups. The two identical, equally loaded piles are each divided into a certain number of elements. While elastic conditions prevail, the horizontal displacements of the soil and pile at each element may be equated, and together with the equilibrium equations may be solved for the unknown pressures. In this analysis, the only interaction effect that is considered is the horizontal movement of one pile that results from loading on another pile. The interaction factors are expressed as a ratio of the additional displacement caused by the adjacent pile to the displacement of the pile caused by its own loading. Interaction factors as a function of the spacing between piles, pile flexibility factor, departure angle (for laterally loaded piles – the angle between the horizontal line joining the two piles and the line of action of the load), pile head fixity condition, pile length to diameter ratio and the Poisson’s ratio of the soil are available [27]. The method of analysis of pile groups with two piles is extended to a general pile group using the principle of superposition. The additional horizontal displacement caused in a particular pile in the group, due to the loading on every other pile in the group is computed by considering the interaction between piles in pairs and summing the additional displacements caused by each of the adjacent piles in the group. One of the major disadvantages of this method is that it is hard to extend this method to compute the load-deformation response of pile foundations with significant nonlinear demands. It is also hard to analyze the foundation under complex loading scenarios such as 9

seismic or other dynamic loads. 3.1.2

Wave propagation based solution

Wave propagation based methods use wave propagation theory to compute the interaction between piles in groups under dynamic loads. These interaction factors are combined with the single pile response computed using any procedure (usually Winkler model type procedure) to compute the response of pile groups. These procedures were originally developed in the frequency domain and applied to foundation systems in which the pile response may be considered elastic and the soil is modeled as a linear viscoelastic medium. Dobry and Gazetas developed a simple analytical solution for the dynamic stiffness and damping of floating pile groups [13] subjected to harmonic loading at the pile head. The vibration of each pile causes the generation of waves that propagate in all directions, causing additional displacements in the adjacent piles. The vibrating pile is referred to as the active pile, while the adjacent pile is referred to as the passive pile. Dobry and Gazetas introduced some simplifying assumptions on the nature of the generated wave field and the consequent displacement of adjacent piles. They used closed form expressions for the generated wave field (in frequency domain) and assumed that the additional displacement caused in the passive pile is equal to this dynamic displacement field. For piles under axial loading, the interaction factor is derived from the asymptotic cylindrical wave expression [23] that was assumed for the generated wave field around a vibrating pile:

αv (S) ≈ (

−βωS −iωS d 0.5 ) exp( ) exp( ) 2S Vs Vs

(7)

where αv (S) is the pile interaction factor, S is the pile spacing, d is the pile diameter, β is the soil damping, ω is the angular frequency of the input loading, and Vs is the shear wave velocity in the soil medium. For laterally loaded piles, the interaction factor is a function of

10

the angle θ between the line of the two piles and the direction of the horizontal force applied. However, it is sufficient to compute the interaction factor for θ = 0◦ and 90◦ , and then use the following expression to compute the interaction factor for any angle θ [26]: αh (S, θ) ≈ αh (S, 0◦ ) cos2 θ + αh (S, 90◦ ) sin2 θ

(8)

To compute the interaction factors for θ = 0◦ and 90◦ , it was assumed that the 90◦ pile is affected only by S-waves which emanate from the active pile with a velocity Vs , while the 0◦ pile is affected only by compression-extension waves that emanate from the active pile and travel with a velocity VLa . VLa is the so called Lysmer’s analog velocity, VLa = 3.4Vs /[π(1 − ν)]. Thus, the following expressions were proposed: −βωS d 0.5 −iωS ) exp( ) exp( ) 2S Vs Vs −iωS d −βωS αh (S, 0◦ ) ≈ ( )0.5 exp( ) exp( ) 2S VLa VLa

αh (S, 90◦ ) ≈ (

(9) (10)

The work by Dobry and Gazetas was carried forward by Gazetas and Makris [15, 17]. They retained the expressions for the wave field generated by the vibration of the active pile. However, they assumed that the adjacent pile modifies the arriving wave field (us (ω, z)) depending upon its relative flexural rigidity and the vertical fluctuations of the arriving wave field. The additional displacements in the passive pile are approximately equal to the arriving wave field for long flexible piles and smoothly varying us (ω, z). The additional displacements are nearly zero for rigid piles and rapidly varying us (ω, z). However, in general, the additional response will be something in between these two extremes. These additional displacements may be computed by applying us (ω, z) as a support excitation to the passive pile. This may be accomplished using the Winkler model [3, 17]. The use of this technique as suggested above is restricted to linear systems, where nonlinear

11

effects may be included indirectly in the form of equivalent viscous damping to account for energy dissipation. However, the wave propagation based technique has been modified to obtain time domain expressions for the wave field generated by the active pile. This allows the use of time domain procedures in computing pile groups response, thereby allowing nonlinear effects to be included directly using nonlinear material models [3, 4]. Although the time domain procedures allow the solution of a larger class of problems, they still have some limitations. The consideration of nonlinear effects is restricted to computing the single pile response. The pile-to-pile interaction factors are based on Fourier transforms of the closed form expressions obtained in frequency domain. Thus, the effect of soil nonlinearity on pile-to-pile interaction is not considered.

3.2

Pile group analysis using p-y multipliers

Another class of pile group analysis techniques involves the use of p-y multipliers. These procedures are an extension of the Winkler model technique used for single piles (Section 2). In the Winkler model, the soil is replaced by discrete springs (Figure 1) and soil nonlinearity is represented using a nonlinear load-deformation curve referred to as the p-y curve for lateral load response. These p-y curves may be generated using analytical expressions for stiffness and strength of the p-y curve along with standard hysteretic models such as the bilinear and Bouc-Wen models, as suggested in Section 2. However, p-y curves most commonly are obtained from back calculations based on experimental or 3D finite element response data [12, 19, 21, 30, 31, 37]. The Winkler model for single piles may be extended for analysis of pile groups. In such a model, each pile is modeled as a beam supported on horizontal springs that account for soil response. The nonlinear response of each soil spring is represented by a p-y curve. The p-y curves used in pile group analysis are obtained by modifying p-y curves for single pile analysis using p-y multipliers. The effect of pile-to-pile interaction on soil p-y curves has been studied by a number of researchers [9, 10, 11, 38]. This technique of pile 12

group analysis is widely used and is also incorporated into software programs. However, the major drawback of this pile foundation model is that though it is computationally much more efficient than 3D finite element models, it still requires extensive modeling and may take considerable analysis time, particularly for large pile groups under dynamic loads. Thus, the use of such models may not be very efficient for certain applications such as vulnerability analysis.

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4

Pile-to-Pile Interaction Analysis

Some of the currently available techniques of pile group analysis were reviewed in the previous section. Pile group analysis using pile interaction factors is common. These procedures are applicable to elastic systems as they use the frequency domain. Although these procedures were extended to the time domain in order to incorporate pile and soil nonlinearity effects, they only consider nonlinearity effects in the computation of the single pile response. Currently available techniques for computing pile interaction factors do not account for nonlinearity. Since soil and pile nonlinearity effects do have a significant effect on pile foundation response, it is necessary to understand the effect of these nonlinearities on pile-to-pile interaction, particularly in groups with closely spaced piles. Pile-to-pile interaction studies were conducted using 3-D finite element analysis. A single pile model was first calibrated against experimental response data. The parameters used in this model were then used to perform pile group analysis. Soil nonlinearity and pile-soil gapping effects were considered in the analysis.

4.1

3D Finite Element Model for Pile Foundation Analysis

Two sets of experimental tests performed at the same site were used to calibrate the finite element model. Lateral load tests were performed on a group of nine steel pipe piles and also on an isolated single pile [10]. The location of the tests was an overconsolidated clay site in Houston, TX. In another experimental program at the same site, dynamic lateral load tests were conducted on a cantilevered mass supported by a single steel pipe pile embedded in stiff, overconsolidated clay [6]. 3D models were created using the program ABAQUS [1]. Due to the symmetry of the problem, only half the mesh was generated (Figure 2). The pile was assumed to respond elastically. The steel pipe pile was modeled as an equivalent cylinder. Thus, the moment of inertia and the mass density were modified so that flexural

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Property Value Pile properties Pile diameter 0.273m Pile length 13.1m Flexural stiffness 1.34X104 kN − m2 Lineal pile density 60.3kg/m Soil properties Young’s modulus 100M P a Density 2100kg/m3 Poisson’s ratio 0.49 Undrained shear strength 47kP a Table 1: Pile and soil properties used in the 3D finite element model stiffness (EI) and total mass of the cylinder were equal to the flexural stiffness and total mass respectively of the steel pipe. The soil was assumed to be homogeneous. The model was meshed using 3D brick (denoted C3D8 in ABAQUS) elements. A number of laboratory and in-situ field tests were performed during both the mentioned experimental programs, to determine soil properties. More details may be found in the cited references. However, the tests showed considerable scatter in the results. The value of Young’s modulus for the soil was back calculated so that the stiffness of the load deformation curve in an elastic analysis matched the initial stiffness of the experimental curve. The obtained value was found to fall within the range of experimentally determined values. A constant value of undrained shear strength was assumed based on experimental evidence. Soil and pile properties used in the model are listed in Table 1. Checks were performed to verify the finite element mesh, assuming the soil to be linear elastic. The soil-pile model was fixed at the base (and free on all sides) and subjected to a lateral load at the ground level. The obtained response was compared with the deflection of a cantilever beam subjected to a tip load, as obtained from beam theory. The difference between the two results was about 8% (Table 2), which is acceptable since beam theory is not exact in this case due to the presence of shear deformations. The second check performed,

15

Figure 2: 3D finite element mesh for single pile analysis 16

Analysis procedure Beam theory Beam theory 3D FEA 3D FEA

Applied load (kN) Pile head displacement (m) Error (%) 50 0.0144 0.00 300 0.0863 0.00 50 0.0155 7.73 300 0.0933 8.08

Table 2: 3D FE model check: verification of pile head response as a cantilever beam compared with beam theory Analysis procedure 3D model Analytical solution using Winkler model [29]

Pile head response (m) 0.00128 0.00204

Table 3: 3D FE model check: comparison of elastic response with analytical solution compared the response of the 3D model under a static lateral pile head load of 100kN, with solutions predicted by a beam-on-elastic-foundation (Winkler model) approach (Table 3), obtained analytically from the solution of the differential equation [29]. The observed difference between the 3D model and Winkler model solution is due to the fact the Winkler model assumes that the modulus of subgrade reaction provided by the soil is k = 1.2Es (equation 1) while the 3D model explicitly considers the Young’s modulus of the soil. 4.1.1

Modeling of soil nonlinearity

Since the aim of this study is to understand the effect of nonlinearity on interaction factors, nonlinear soil models were used in the 3D finite element analyses. Abaqus provides a large number of plasticity models to incorporate soil nonlinearity in the analysis. These include the Extended and Modified Drucker-Prager models, the Mohr-Coulomb plasticity model and the Critical state (clay) plasticity model [1]. These are sophisticated plasticity models that require calibration based on experimental data. In this study, the Mohr-Coulomb model is used since it can be calibrated based on the standard Mohr-Coulomb parameters. It uses

17

60

0.8

50 Shear stress, τ (kPa)

1

G/Gmax

0.6

0.4

40

30

20

0.2 10 0 −3 10

−2

10

−1

Strain, γ (%)

0

10

10

0 0

(a) Modulus reduction curve (PI=30) [33]

0.2

0.4 0.6 Shear strain, γ (%)

0.8

1

(b) Shear stress-strain curve

Figure 3: Nonlinear soil model for cohesive soil used in the 3D FEM the classical Mohr-Coulomb yield criterion:

τ = c − σ tan φ

(11)

where τ is the shear stress, σ is the normal stress (negative in compression), c and φ are the Mohr-Coulomb parameters. The definition of material behavior using the Mohr-Coulomb model in Abaqus includes the elastic properties (Young’s modulus and Poisson’s ratio), density (for dynamic analysis), angle of friction, angle of dilation (which governs the flow potential), and cohesive yield stress vs. plastic strain. Since the soil being modeled in the presented set of analyses was cohesive, the angle of friction (φ) was assumed to be zero. The dilation angle (ψ) was also assumed to be zero. The cohesive stress-strain response (Figure 3(b)) was derived from the modulus reduction curve for cohesive soils with plasticity index, P I = 30 [33] (Figure 3(a)). The parameters eccentricity () and deviatoric eccentricity (e), which determine the shape of the flow potential function, assumed default values:  = 1 and e = (3 − sin φ)/(3 + sin φ).

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4.1.2

Modeling of pile-soil interface

Modeling of the pile-soil interface is critical under strong loads that cause separation between the pile and the soil. Abaqus provides a number of advanced models for contact behavior. Surface-based contact, which allows modeling of contact between two deformable bodies that undergo small or finite sliding, was used in the analyses presented here. Abaqus uses the concept of contact pairs - a master surface and a slave surface. The pile surface was chosen as the master and the surrounding soil surface was chosen as the slave. The mechanical contact simulation of the interaction between two bodies includes a constitutive model for the contact pressure-overclosure relationship (i.e. behavior normal to the contact surfaces) and a friction model that defines the force resisting relative tangential motion of the surfaces. The most common contact pressure-overclosure relationship is “hard” contact. When surfaces are in contact, any contact pressure can be transmitted between them. The surfaces separate if the contact pressure reduces to zero. Separated surfaces come into contact when the separation between them reduces to zero. In addition to the pressure-overclosure relationship, frictional behavior may or may not be included to model the shear stress transmitted across the interface. Abaqus has a wide range of friction models to choose from. The classical isotropic Coulomb friction model, which defines a friction coefficient relating shear stress to the contact pressure, was used. A value of 0.7 was assumed for the coefficient of friction, as used by Bentley and Naggar [5], based on the recommendations of the American Petroleum Institute [2]. The 3D finite element model of the single pile was compared with the experimental response with and without the friction model. 4.1.3

Response of single pile using 3D FE model

The experimental response of the single pile [10] under a lateral pile head load was compared with the static response of the 3D finite element model (Figure 4) assuming linear elastic pile response and nonlinear soil response using the Mohr-Coulomb model. Three different 19

100

Pile head load (kN)

80

60

40

Experimental data FEM (perfect soil−pile contact) FEM (hard contact, no friction) FEM (hard contact, friction)

20

0 0

0.01 0.02 0.03 Pile head displacement (m)

0.04

Figure 4: Comparison of single pile response using 3D FEM with experimental response pile-soil contact models were considered: (i)perfect contact (ii)“hard” contact with friction (iii)“hard” contact without friction. The analyses showed that the use of the Mohr-Coulomb model for soil nonlinearity along with the “hard” contact model with no friction gives the best match with experimental data. Therefore, it was decided to use this model in the pile group analysis. The 1D FE model (or Winkler model) for the single pile was also compared with the 3D model and experimental response. Figure 5 shows that by using the p-y material for soft clay [19] to model the nonlinear spring in the Winkler model, it is possible to obtain a good match with the experimental response. This model has been implemented in the current version of Opensees [20]. However, using the bilinear and Bouc-Wen model for the nonlinear springs does not compare quite as well with the experimental response.

4.2

Development of Improved Pile-to-Pile Interaction Factors

To study the effect of soil nonlinearity and soil-pile gapping on the pile interaction factors, 3D finite element analyses were performed on two-pile groups. The Mohr-Coulomb model used in the single pile analysis was used in the pile group model, along with the “hard” contact

20

100

Pile head load (kN)

80

60

40 Experimental data 3D FE model Winkler (soft clay p−y model) Winkler (bilinear) Winkler (Bouc−Wen)

20

0 0

0.01 0.02 0.03 Pile head displacement (m)

0.04

Figure 5: Comparison of single pile response using 1D and 3D FEM with experimental response (no friction) model used for pile-soil contact. The soil and pile properties are listed in Table 1. The factors affecting pile-to-pile interaction include the pile to soil relative stiffness, departure angle and pile length to diameter ratio [26]. However, the most important factor is the pile spacing. Two finite element models were created with pile spacing, S = 2d and S = 5d, where d is the pile diameter. The finite element meshes for the pile groups are shown in Figures 6 and 7. The pile groups were subjected to static lateral pile head loads and the interaction factors at the pile head level were computed by comparing the pile group response with the single pile response. The single pile model was similar to what was used in section 4.1, with the only difference that the pile head was assumed to be fixed. This was achieved by restraining the vertical degrees of freedom on the top surface of the single pile. The pile-to-pile interaction factors were computed as:

αh =

∆uplgrp usglpl

21

(12)

Figure 6: 3D finite element mesh for pile group analysis (S = 2d)

22

Figure 7: 3D finite element mesh for pile group analysis (S = 5d)

23

2 1.8

Interaction factor, αh

1.6 1.4 1.2

Elastic soil Nonlinear soil w/o pile−soil gapping Nonlinear soil with pile−soil gapping

1 0.8 0.6 0.4 0.2 0

50

100 Pile group load (kN)

150

200

Figure 8: Effect of soil nonlinearity and pile-soil gapping on interaction factors where αh is the interaction factor, ∆uplgrp is the additional displacement caused in one pile due to the loading on the other pile, and usglpl is the single pile displacement. Since the system is nonlinear, αh is a function of the applied load. To observe the effect of pile nonlinearity and the pile-soil contact model on the interaction factors, for the case of S = 2d, αh was computed for three different cases: (i)assuming the soil to be linear elastic and soilpile contact to be perfect, (ii)modeling soil nonlinearity using the Mohr-Coulomb model with no soil-pile gapping (iii)using the Mohr-Coulomb model for soil nonlinearity and the “hard” contact (no friction) model for the pile-soil interface. The results of the analysis are shown in Figure 8. Figure 9 shows the effect of pile spacing on the interaction factors. The figure clearly shows that as the spacing increases, the interaction between piles decreases. If the piles in a group are very closely spaced, then interaction between the piles is larger. This implies that the induced additional displacements are also larger. Thus, as the pile spacing is decreased, the pile group undergoes larger deformations for the same load level. At very large spacings, there would be no interaction between piles, in which case, the pile group would carry twice the load of a single pile. Figure 10 shows the effect of pile spacing on the

24

2 1.8

S/d = 2 S/d = 5

Interaction factor, αh

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

50

100 Pile group load (kN)

150

200

Figure 9: Effect of pile spacing on interaction factors

200 180 160

Pile head load

140 120 100 80 60 40

2Xsingle pile response Pile group response: S/d = 2 Pile group response: S/d = 5

20 0 0

0.005 0.01 0.015 Pile head displacement (m)

0.02

Figure 10: Lateral load response of a two-pile group

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lateral load-deformation response of pile groups. As the pile spacing decreases, the efficiency of each pile in the group decreases. Further analyses need to be conducted to study the effect of other factors on the interaction factors. These include the effect of departure angle, relative pile-to-soil stiffness, soil strength and dynamic loading.

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5

Conclusions

A number of techniques currently used to analyze pile foundations were reviewed. The Beam on Winkler Foundation model, commonly used in the analysis of single piles, was reviewed. Two major classes of pile group analysis techniques—the pile-to-pile interaction factor method and the p-multiplier method were also reviewed. The p-multiplier method while highly efficient (when compared to 3D finite element analysis) and widely used may still be very time consuming for large pile-supported structures which have a number of large pile groups. The pile-to-pile interaction factor method is based on modifying the response of a single pile to account for interactions among different piles in a group. The available techniques to evaluate these interactions are generally applicable to linear elastic systems and do not incorporate the effect of soil and pile nonlinearity. 3D finite element analyses were performed to compute pile-to-pile interaction factors accounting for soil nonlinearity and pile-soil gapping effects. Interaction factors were computed for two-pile groups under lateral static loading. The analyses showed that both nonlinearity in the soil and gapping between the pile and the soil have a very significant effect on the interaction. The pile spacing is the most important parameter influencing the interaction factors. The interaction factors were computed for two different pile spacings. A number of other factors may have a significant effect on the interaction—departure angle, relative pile-to-soil stiffness, soil strength, pile nonlinearity, and load amplitude and frequency (under dynamic loading). A very extensive set of finite element analyses need to be conducted to understand the full effect of all these factors.

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Acknowledgment This work was supported by the National Science Foundation through the Mid-America Earthquake Center under award number EEC-9701785. This work was partially supported by National Computational Science Alliance under BCS040006N and utilized the IBM P690 at the National Center for Supercomputing Applications.

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