Soil Dynamics and Earthquake Engineering Volume 24 Issue 4 2004 [Doi...

8/10/2019 Soil Dynamics and Earthquake Engineering Volume 24 Issue 4 2004 [Doi 10.1016%2Fj.soildyn.2004.01.001] B.K.

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Three-dimensional nonlinear analysis for seismicsoilpile-structure interaction

B.K. Maheshwaria,*, K.Z. Trumana, M.H. El Naggarb, P.L. Goulda

aDepartment of Civil Engineering, Washington University, St. Louis, MO 63130, USAbDepartment of Civil Engineering, The University of Western Ontario, London, Ont., Canada N6A 5B9

Received 19 November 2003

Abstract

A three-dimensional method of analysis is presented for the seismic response of structures constructed on pile foundations. An analysis is

formulated in the time domain and the effects of material nonlinearity of soil on the seismic response are investigated. A subsystem model

consisting of a structure subsystem and a pile-foundation subsystem is used. Seismic response of the system is found using a successive-

coupling incremental solution scheme. Both subsystems are assumed to be coupled at each time step. Material nonlinearity is accounted for

by incorporating an advanced plasticity-based soil model, HiSS, in the finite element formulation. Both single piles and pile groups are

considered and the effects of kinematic and inertial interaction on seismic response are investigated while considering harmonic and transient

excitations. It is seen that nonlinearity significantly affects seismic response of pile foundations as well as that of structures. Effects of

nonlinearity on response are dependent on the frequency of excitation with nonlinearity causing an increase in response at low frequencies of

excitation.

q 2004 Elsevier Ltd. All rights reserved.

Keywords:Soilpile-structure interaction; Pile foundation; Plasticity; Seismic response

1. Introduction

The high level of risk associated with nuclear contain-

ment facilities has created the need to undertake rigorous

dynamic analyses, including soilpile-structure interaction

(SPSI), for the design of such complex structures.

Substantial research efforts, e.g. Refs. [15] have been

carried out to investigate the kinematic seismic behavior of

single piles and pile groups. Further, several studies byMakris et al. [6], Mylonakis and Gazetas [7], Guin and

Banerjee[8] have focused on SPSI analyses.

The performance of pile-supported structures (such as

bridges) during recent devastating earthquakes (e.g. Bhuj

Earthquake of 2001, Chi-Chi Earthquake of 1999, and

Kocaeli Earthquake of 1999) has come under scrutiny. The

damage caused to pile foundations during these earthquakes

has emphasized the importance of understanding SPSI.

Moreover, the advances in computer technology justify

the use of rigorous SPSI analysis in many important

practical engineering structures.

The seismic response analysis of piled foundations

should be performed in the time domain to properly

account forthe soil nonlinearity. Nogami and Konagai [9,10]

analyzed the dynamic response of pile foundations in the

time domain using a Winkler approach. Nogami et al. [11]

introduced material and geometrical nonlinearity in the

analysis using discrete systems of mass, spring and

dashpots. El Naggar and Novak [12,13] presented a

nonlinear analysis for pile groups in the time domainwithin the framework of the Winkler hypothesis. How-

ever, proper representation of damping and inertia effects

of continuous soil media is difficult with such discrete

systems.

The inclusion of material nonlinearity caused by soil

plasticity requires that an analysis be performed using the

finite element approach. Wu and Finn [14] presented a

quasi-3D method for the analysis of nonlinear pile response.

Bentley and El Naggar [15] investigated the kinematic

response of single piles considering soil plasticity but did

not consider work hardening. Cai et al. [16] included the

material nonlinearity of soil using a finite element technique

in the time domain. However, in that analysis fixed

boundary conditions were used and damping in

0267-7261/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.soildyn.2004.01.001

Soil Dynamics and Earthquake Engineering 24 (2004) 343356www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: 1-314-935-8220; fax: 1-314-935-4338.E-mail address:[email protected] (B.K. Maheshwari).
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the foundation subsystem was neglected. Moreover the

effects of soil nonlinearity were not discussed.

Maheshwari et al. [17,18] examined the effects of

plasticity and work hardening of soil on the free field

response as well as on the kinematic response of single piles

and pile groups using the hierarchical single surface (HiSS)

soil model. In this article, the analysis is extended to include

the superstructure in order to evaluate the effects of SPSI for

a fully coupled system. The analysis is performed for both

harmonic and transient excitations and both linear and

nonlinear responses are compared.

2. Modeling the soilpile-structure system

Two soilpile-structure (SPS) systems are considered:

one involves a single pile (Fig. 1a) and the other involves a

2 2 pile group (Fig. 1b). The proposed SPS model is a

three-dimensional nonlinear finite element model that

consists of two subsystems: a structure subsystem and a

pile-foundation subsystem (for brevity, foundation subsys-

tem). As shown inFig. 1, the two subsystems are connected

at the junctions between the pile cap (pile head for a single

pile) and the column base(s) of the structure. The interaction

between the two subsystems is transmitted through the

motions and dynamic forces of the pile cap and column

base(s). The subsystem approach facilitates the compu-

tational efficiency of the model as well as the investigationinto the effects of kinematic and inertial interaction.

Full three-dimensional geometric models are used to

represent the SPS systems. Taking advantage of symmetry

and anti-symmetry, only one fourth of the actual model was

built, which dramatically improves the efficiency of

computation. Finite element models of the foundations of

SPS systems considered are shown inFig. 2. The piles have

square cross-sections, are fully embedded in the soil and are

socketed in the bedrock. For pile groups, the pile spacing

(center-to-center) ratio, s=d 5: The superstructure con-sidered for the single pile case consists of a massive column

with a rectangular cross-section and is directly attached to

the pile head (Fig. 1a). For the pile group, a rigid massless

cap connects all the pile-heads and the superstructure

consists of four massive columns (each is similar to that

used in the single pile case). The superstructure is placed

such that its center of gravity coincides with the center of

the foundation (pile cap), thus maintaining full symmetry

(Fig. 1b).

The soil and piles are modeled using eight-nodehexahedral elements. Each node has three translational

degrees of freedom along the X; Y and Zcoordinates as

shown inFig. 3a. These elements are selected because in the

present study response is dominated by shear deformations.

However, for accurate analyses 20-node solid elements can

be used. Kelvin elements (spring and dashpot as shown in

Fig. 3b) are attached in all three directions (i.e. X;YandZ)

along the mesh boundaries (Fig. 4a) in order to model the far

field conditions and allow for wave propagation. The

coefficients of the springs and dashpots are derived

separately for the horizontal and vertical directions. The

structure is modeled using simple two-node beam elementswith six degrees of freedom (three translations and three

rotations) at each node as shown in Fig. 3c. Boundary

conditions at the axes of symmetry and anti-symmetry are

discussed later.

Fig. 1. Soilpile-structure systems considered in the analyses: (a) a single pile system, (b) a 2 2 pile group system.

B.K. Maheshwari et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 343356344



It is assumed that the soil and pile are perfectly bonded.

However, separation between the soil and pile can be

considered using no tension elements as shown by

Maheshwari et al. [18]. The piles are assumed to behave

linearly but their nonlinear behavior can also be modeled

using an appropriate constitutive relation. For the nonlinear

soil model (HiSS), the initial stress condition in the soil is

governed by the confining pressure of the soil and isproportional to the depth (Fig. 4b). The seismic excitation

is assumed to act on the fixed base nodes and consist of

vertically propagating shear waves.

3. Successive-coupling incremental solution scheme

A successive-coupling incremental solution scheme in

the time domain is used to solve the seismic response of the

SPS system. In this methodology, the motions from the pile

cap and forces from the column bases are transmitted from

one subsystem to another while moving to a forward time

step (Fig. 5). The time history of seismic excitation isdivided into small time steps, each equal to Dt:

At the first time stept Dt; the pile head dynamicforces are zero and the bedrock acceleration input is VbDt:

Fig. 3. (a) Block element used for soil and pile, (b) boundary element, (c) space frame element used for the structure.

Fig. 2. Three-dimensional finite element quarter models used for the foundation subsystems: (a) a single pile, (b) a 2 2 pile group.

B.K. Maheshwari et al. / Soil Dynamics and Earthquake Engineering 24 (2004) 343356 345



The pile head acceleration VpDt is obtained by solving theresponse of the foundation subsystem due to the input

bedrock motion VbDt:The pile head acceleration, VpDt;is calculated and is used as the support excitation for the

structure subsystem. The dynamic forces acting on the

column bases of the structure, which are equivalent to pile

head forces FpDt; can then be obtained.At the second time step t 2Dt; the foundation

subsystem is subjected to both bedrock acceleration input,Vb2Dt;and the pile head dynamic forces, FpDt:The pilehead acceleration, Vp2Dt; is evaluated by solving theresponse of the foundation subsystem. The pile head

forces, Fp2Dt; are equivalent to the column base forceson the structure at the second time step, and are then

obtained by solving the response of the structure subsystem

for the corresponding support excitations. This successive-

coupling incremental procedure is repeated until the entire

response history is determined. It is seen that the

continuous response history is well approximated by the

discrete step approach using a sufficiently small time step(i.e. Dt T=80; where Tis the period of the excitation).

The substructuring technique coupled with a successive-

coupling scheme is though approximate for a nonlinear

analysis but quite effective. The effect of this approximation

on the accuracy of the results depends on the size of the time

step considered. For a very small time step

T=80

used in

the present analysis, the results converged and noaccumulation of errors was noticed. Further, a fully coupled

system would require enormous computation. Thus, the

additional accuracy, which may not be significant, obtained

at the expense of large computational cost, may not be

justified.

4. Structure subsystem

The structure subsystem may be subjected to non-

uniform foundation motion (support excitation) that is

equal to the pile heads motion. The non-uniform foundation

motion is obtained from the coupling kinematicinertialinteraction of the SPS system (Fig. 5).

Fig. 4. Finite element mesh for the pile group system: (a) top plan, (b) front

elevation with initial pressure distribution.

Fig. 5. Schematic of successive-coupling scheme.




The incremental equations of motion for a structure

subjected to uneven support excitations are derived assum-

ing the principle of superposition to be valid within each

incremental time step, provided that the time step is

sufficiently small. For a structure subsystem with n active

DOF andm support DOF, the multiple-support excitation is

obtained by the superposition of the dynamic responses of

the subsystem due to each independent support input. The

total dynamic equilibrium equations of the structure

subsystem with all m support DOF having individual

motions can be written as [19]:

Ms U Cs _U KsU 2MsRs Ug 1

where Ms; Cs; and Ks are n n matrices of the mass,

damping and stiffness of the structure subsystem, respect-

ively; U; _U and U are n 1 vectors of displacement,velocity and acceleration of the subsystem, respectively; Rsis an n m matrix that contains the pseudo-static response

influence coefficients and may be updated at each time step

for a nonlinear structure; Ugis anm 1 vector that contains

the non-uniform support motion and is equivalent to the pile

head motion, Vp:

Eq. (1) is constructed in an incremental form using the

Newmark time integration scheme (constant average

acceleration method [20]). The incremental equations of

motion of the subsystem at time t Dtcan be written as:4Ms

Dt2

2Cs

DtKs DU

2MsRstDt Ug2KstUMs4t _U

Dt t U

!Cst _U 2

where Dtis the time step; superscript t indicates the time;

and DU are increments of the dynamic response vector U;

such that tDtU tUDU:For the nonlinear response of the structure subsystem,

Eq. (2) is formulated using the modified Newton Raphson

iteration scheme. The stiffness and damping matrices, Ksand Cs; are replaced by the corresponding tangent stiffness

and damping matrices, tKs and tCs; which are updated at

each time step t: The stiffness and mass matrices of thestructure subsystem are constructed using normal finite

element method procedures. The damping is assumed to be

proportional to the stiffness[8].

For a structure subjected to non-uniform support

motions, the support reactions of the structure depend on

the total displacements of the active DOF Ut as well as therelative displacements of the support DOF Ug:Therefore,the support forces for non-uniform support motion cases are

different from those for rigid ground motion cases and

should be calculated as shown by[19]:

Fg KgsUt KggUg KgsURsUg KggUg 3

whereFgis an m 1 support force vector that is equivalentto the dynamic pile head force vector Fp; Kgs is an m n

matrix that represents the coupling of the support forces and

the motions of the active DOFs; andUt represents the total

displacement vector.Kggis anm mmatrix that represents

the coupling of the support forces and the motions of the

support DOFs. For each time step, the support forces Fgare

calculated and used as the pile head inertial force input for

the pile foundation subsystem. For nonlinear seismic

response of the structure subsystem, the support force-

active DOFs motion coupling matrix Kgs and the support

force-support DOF motion coupling matrixKggare updated

at the beginning of each time step t:

Although the whole structure subsystem is considered,

only a quarter of the foundation subsystem is used.

Therefore, only one fourth of the column base force (for

the single pile system) or one fourth of the sum of all four

column base forces (for the pile group system) is transferredfrom the structure to the foundation, i.e. Fp Fg=4:

5. Foundation subsystem

5.1. Governing equation and solution

The foundation subsystem is composed of piles and the

surrounding soil. The equation of motion for this subsystem

is written as:

MFV CF_V KFV 2MFRFVb Fp 4

where MF; CF; and KF are matrices of the mass, dampingand stiffness of the foundation subsystem, respectively;V; _V

and V are the vectors of displacements, velocity and

acceleration of the subsystem, respectively; RF is the

pseudo-static response influence coefficients matrix that isupdated at each time step; Vb is the vector of bedrock

acceleration due to seismic excitations and is assumed to

consist of vertically propagating shear waves; and Fp is the

pile head force vector that is calculated from the support

force vectorFg (Eq. (3)). Eq. (4) is formulated in the same

incremental form as for the structure subsystem (Eq. (2)).

For the nonlinear soil model, the incremental dynamic

equilibrium of the foundation subsystem for theith iterationat time t Dtcan be written as:

4MF

Dt2 2

tCF

Dt tKF

DV

i

2MFRFtDt Vb tFp 2 tDtFi21

2tCF

2

DttDtVi21 2 tV2 t _V

2MF4

Dt2tDtVi21 2 tV2 4

Dt

t _V2t V

5a

where superscripts t and i indicate the time t and ith

iteration, respectively; tDtFi21 is the vector of nodalforces in thei 2 1th iteration of the current time step.




The stiffness and damping matrices,KFand CF;are replaced

by the corresponding tangent matrices, tKF and tCF; which

are updated at each time step t: DVi are increments of thedynamic response vector Vat theith iteration, such that

tDtVi tDtVi21 DVi 5b

For the foundation subsystem, the mass matrix MF is

diagonal because all masses are lumped at the nodal points.

The global damping matrix, tCF;includes the contributions

of both material damping and radiation damping (including

dashpots along the boundary). The stiffness matrix, tKF; is

symmetric and is determined assuming full coupling in all

three directions of motion and includes the stiffness of

springs at the boundary nodes.

5.2. Boundary conditions

The boundary conditions are shown in Fig. 4a,b. The

presence of the springs provides stiffness, giving this

boundary a distinct advantage over the standard viscous

boundary[21,22].The constants of the Kelvin elements in

the two horizontal directionsare calculated usingthe solution

developed by Novak and Mitwally[22],and the constants of

the vertical Kelvin elements are calculated using the solution

developed by Novak et al. [23]. These constants are

frequency dependent. For transient excitation, the constants

are determined based on the predominant frequency of the

excitation. The stiffness and damping of the Kelvin elementsareevaluated using thearea of theelement face (normal to the

direction of loading). Further details on the evaluation of

these constants are described in Maheshwari et al. [17,18].

All the nodes along the base are fixed in all three

directions. The nodes on the axis of symmetry are free to

move in the vertical direction and along the direction of the

axis of symmetry, and are fixed in the perpendicularhorizontal direction (Fig. 4a). The nodes on the axis of anti-

symmetry are constrained in the vertical direction and along

the direction of this axis and are free to move in the

perpendicular horizontal direction (Fig. 4a,b).

It should be noted that the boundary conditions at the axis

of symmetry and anti-symmetry are developed with dueconsideration of waves and loading patterns and thus they

reflect mirror images. This means they should reproduceexactly the same effects of the missing part (including other

piles). The effects of pilesoilpile interaction would be

reproduced through these boundaries. For the foundation

subsystem only (without the superstructure), the results of

the quarter model were compared with that obtained using

the full model for (2 2) and (3 3) pile groups. The

results from both cases were exactly the same.

5.3. Damping matrixCF

To adequately represent damping in the foundationsubsystem, both radiation dampingCrand material damping

Cm are considered. Thus, damping matrix CF is:

CF Cr Cm 6aRadiation damping Cr is a diagonal matrix and has

nonzero terms only at the nodes on the boundary where

Kelvin elements are attached. Material damping Cmis taken

as proportional to stiffness and is given by:

Cm aK where a 2D=v0 6b

where D is the material damping ratio and v0 is the

predominant circular frequency of loading.

5.4. Pile cap

The piles are assumed to be connected at their heads to arigid massless cap. Therefore, deformations of all pile heads

will be the same as that of the cap and the motion

transmitted to all column base(s) from the foundation

subsystem will be equal. Also, the force equilibrium is

satisfied between the pile heads and the cap.

5.5. Rocking of the pile group

For the finite element formulation considered here, eight

noded brick elements with three degrees of freedom at each

node were used in the analysis. Rotations in the analysis can

be represented by three independent displacements (DOFs).

Once these displacements are known, three components ofnormal strains as well as three components of shear strains

(rotations) can be found. In the present analysis, the pile cap

is assumed rigid in all three directionsx;y;z: Rocking ofthe pile cap can be calculated through the evaluation of

rotation of the pile connected to it (rigid body movement for

the cap). This is a reasonable assumption, as pile caps with

dimensions used in practice tend to behave as a rigid body,

especially in the lateral direction.

5.6. HiSS soil model

The dp0 version of the HiSS model [24] is used to

introduce the effect of soil plasticity and work hardening.The model is based on an incremental stressstrain

relationship and assumes associative plasticity. In this

model, the dimensionless yield surface Fis simplified as:

F J2Dp2a

aps

J1

pa

h2g

J1

pa

2 0 7a

whereJ1is the first invariant of the stress tensorsij;J2Dis the

second invariant of the deviatoric stress tensor; pa is the

atmospheric pressure; gand hare material parameters that

influence the shape ofFinJ1 2p

J2Dspace; the parameterh

is related to the phase change point that is defined as the point

where material changes from contractive to dilative behavior(Fig. 6).




The hardening function,aps;is defined in terms of plastic

strain trajectory jv;as:

apsh1

jh2v

7b

where h1 and h2 are material parameters and jv denotes

the trajectory of the volumetric plastic strain. The soil

parameters used in the model are for a marine clay

found near Sabine Pass, Texas, and were determined

from laboratory tests (Katti [25]) and verified with

available data of field tests from Pile Segment [26].Typical yield surfaces for this model are shown in

Fig. 6.

6. Computerization

The proposed methodology employs the subsystemconcept to reduce the storage space needed for property

matrices. In addition, some effective computer storage

allocation and matrices storage schemes [27] are used to

further reduce the space needed.

A FORTRAN code named 3dNDPILE was developed to

perform the analysis. For nonlinear analysis, three types ofcriteria are used simultaneously to examine the solution

convergence, namely the displacement criterion, the out ofbalance load criterion and the internal energy criterion[20].

Special procedures are used to ensure the robustness of the

HiSS iterative solution[28]and are further enhanced to deal

with the case when the plasticity parameter l (a constant of

proportionality used to define the flow rule of plasticity

[29]) becomes negative.

The dimensionless yield surface F is assumed toconverge when its absolute value becomes fairly small, i.e.

ABSF , 10210: For harmonic excitation, the size of thetime step is taken to be T=80 where T is the period of

excitation. The algorithm developed is quite efficient andeconomical.

7. Properties of SPS system

7.1. Properties of foundation subsystem

The soil is assumed to be clay at Sabine Pass, Texas[30].

Its properties are as follows: Youngs modulusEF 11:78MPa; mass density rF 1610 kg=m3; Poissons ratio nF0:42 and material damping ratio DF 5%: The materialparameters for the HiSS model are: b 0;g 0:047;h2:4; h1 0:0034 and h2 0:78:

The piles are made of concrete, 10 m long and have

square cross-sections (0.5 m 0.5 m) with the pile slender-ness ratio, l=d 20: Youngs modulus, mass density andPoissons ratio for the pile are, respectively:

Ep 25 GPa; rp 2400 kg=m3; np 0:25

7.2. Properties of structure subsystem

The superstructure considered for the single pile case is a

rectangular column (0.75 m 0.5 m) and is 6 m high with

the following properties: Youngs modulus Es 25 GPa;and material damping ratio,Ds

5%:The total mass of the

column is 54 Mg and the fundamental natural frequency ofthe structure for the fixed base case is 3.43 Hz. For the pile

group case, the superstructure consists of four identicalcolumns (similar to that considered in the single pile case)

placed symmetrically on top of the pile cap so that the C.G.

of the superstructure coincides with that of the foundation.

7.3. Dynamic loading

The seismic loading is applied as either a harmonic or a

transient bedrock motion. The harmonic excitation consists

of sinusoidal waves of unit amplitude and varying

frequency. The transient motion is the NS component of

the El Centro 1940 Earthquake with a PGA equal to 0.32 g[31]. The predominant frequency of the excitation is

Fig. 6. Shape of yield surfaces in J1 2p

J2D space.




approximately 1.83 Hz. The responses are calculated at the

pile cap (or pile head) and at the top of the structure.

8. Verification of the model and algorithm

Maheshwari et al.[17,18]verify the model and algorithm

for the foundation subsystems. They have performed elastic

and elasto-plastic analyses and have compared the results

with published results for a single pile as well as for pile

groups. In Section 9, the effects of kinematic and inertial

interactions are discussed for an elastic soil model.

9. Effects of kinematic and inertial interactions

The harmonic seismic excitation is applied at the base of

the bedrock for the single pile model (Fig. 1a), and the

response is calculated at the pile head and at the top of the

structure. The analysis is performed for two different

situations, the first situation being a fully coupled analysis

that accounts for the inertial interaction of the structure as

well as the kinematic interaction. In this case, the displace-

ments at the pile head and the forces at the column bases are

transferred from one subsystem to another using the

successive-coupling scheme. The second situation focuses

on thekinematic seismic interaction of thepile foundation. In

this case, the pile head response is calculated from the

kinematic interaction of the pile of the seismic wave through

the soil. This pile head movement is then used as the input

excitation (support displacements) to calculate the response

of thestructure. This procedureignores thefeedbackfrom the

structure (the inertial interaction). The analysis is performed

for excitation with different frequencies.

The results are shown in F ig. 7 i n terms of

amplifications of response with respect to the input

bedrock motion. The free-field response is also shown in

Fig. 7. It can be observed from Fig. 7 that the kinematic

seismic response of the pile head (without inertial

interaction) follows the free-field ground motion. The

response of the pile foundation increases slightly due to

the inertial interaction but the natural period of the

foundation subsystem increases from about 0.22 to 0.4 s.

The increase in the natural period may be attributed to

increased soil nonlinearity due to the inertial interaction.It is also noted that the kinematic interaction amplifies

the input motion at the base of the structure for short

period T, 0:3 s excitations. On the other hand, thekinematic interaction has no effect on the structure

input motion for long period excitationsT. 0:4 s:Fig. 7 also shows that the fundamental period of the

structure increases and the structure peak response decreases

significantly due to the inertial interaction. For the range of

parameters considered in this study, the SPSI elongates the

periodof thestructure andtends to decrease thepeakresponse.

Guin and Banerjee [8] and Veletsos [32] made similar

observations using frequency domain analyses. Therefore,

these results further verify the model and algorithm.

Fig. 7. Effects of kinematic and inertial interactions on the response of the foundation and of the structure for an elastic soil model.




10. Effects of nonlinearity on SPS response

The effects of soil nonlinearity on the seismic response of

the SPS system are examined. The responses at the pile cap

(or pile head) and at the top of the structure are calculated

for both the linear (elastic) and nonlinear (HiSS) soil models

and the results are compared. The analyses are performed

for both harmonic and transient excitations, and for a single

pile and a pile group case.

11. Analyses for harmonic excitations

Harmonic excitations with different frequencies

are applied to the SPS system and the amplitude of

Fig. 8. Effect of nonlinearity on the response for a single pile model at different frequencies.

Fig. 9. Effect of nonlinearity on the response for the pile group model at different frequencies.




the steady-state response is noted in each case. The response

at the pile cap (or pile head) and at the top of the structure are

plotted as a ratio of the input bedrock motion (i.e.

amplification factor) versus the dimensionless frequency,

a0 vd=Vswhere vis the circular frequency of excitation; dis the dimension of the square pile in cross-section andVsis

the shear wave velocity of the soil.

11.1. Single pile model

Fig. 8shows the responses at the pile head and at the top

of the structure. For low frequencies a0 , 0:2 or fv=2p, 3 Hz; the soil nonlinearity increases the pile headresponse significantly. For higher frequencies, however,

the pile head response decreases slightly due to soil

Fig. 10. (a) Linear and nonlinear time histories of pile head response for the single pile model, (b) linear and nonlinear time histories of response of the structure

for the single pile model, (c) linear and nonlinear Fourier spectra for the single pile model.




nonlinearity. It is also noted fromFig. 8that the structural

response increases slightly due to soil nonlinearity at low

frequencies a0 , 0:15 but decreases significantly formoderate frequencies0:15 , a0 , 0:3:

11.2. Pile group model

The response of the pile group model is shown inFig. 9.

Comparing this with Fig. 8, it can be noted that effect of

group interaction decreases the value of peak response for

the structure. Guin and Banerjee [8] made a similar

observation for an elastic soil model.

As far the effect of soil nonlinearity, overall trend of its

effect on the pile and structural response is similar to that

observed for the single pile model. The soil nonlinearity

increases the pile head response significantly and decreasesthe structural response. The comparison between Figs. 8

and 9shows that the effects of soil nonlinearity on the piles

and structural responses for the pile group case are less

significant when compared to the single pile case. This may

be attributed to the fact that the interaction betweenpiles (the

group effect) reduces the effects of soil nonlinearity.

12. Analyses for transient excitations

The SPS system is subjected to the El Centro excitation

and the responses of the pile cap and the structure are

calculated. The effect of soil nonlinearity on the SPS systemresponse is investigated.

12.1. Single pile model

Fig. 10ashows a comparison of the pile head responsefor the linear and nonlinear soil models (only the first 10 s of

the record are shown). It is seen that the response of the pile,

in general, increases due to soil nonlinearity. The peak

values of linear and nonlinear accelerations are 0.43 and

0.54 g, respectively.

Fig. 10b shows the effect of soil nonlinearity on the

response of the structure. It increases the structural

response, and the peak acceleration increases from 1.04 to

1.2 g. These observations are consistent with the results

obtained using the harmonic excitations.Fig. 8shows that at

the predominant frequency of the transient excitationf1:83 Hz or a0 0:11; both the pile head and structuralresponses increase due to soil nonlinearity.

Fig. 10c shows the smoothed Fourier spectra for the

responses. It is noted that the soil nonlinearity increases the

pile response slightly for most of the frequency content of

the excitation. Also, the structural response increases

substantially due to soil nonlinearity for the entire frequency

content of the excitation, and especially for the frequency

range,f , 7 Hz:

The effect of soil nonlinearity on the structural response

in this case is different from that observed for the harmonic

excitations (Fig. 8). This may be attributed to the fact that

the amplitude of the harmonic excitation is 0.1 g while

the peak amplitude of the transient excitation is 0.32 g. Thehigher amplitude increases the level of nonlinearity

Fig. 10 (continued)




and results in a nonlinear structural response significantly

higher than the linear response.

12.2. Pile group model

Fig. 11a,b show the responses of the pile cap and

structure for the pile group model, respectively. The trends

of the results are similar to that observed for the single pile

model. The soil nonlinearity increases the peak pile cap

acceleration from 0.44 to 0.62 g and increases the peak

structure acceleration from 0.94 to 1.02 g. Also it can be

seen that the peak values of response is decreased as

compared to that observed for the single pile model.

Fig. 11cshows smoothed Fourier spectra for the pile cap

and structural responses. Similar to the single pile case, the

soil nonlinearity increases the piles response slightly for

Fig. 11. (a) Linear and nonlinear time histories of pile cap response for the pile group model, (b) linear and nonlinear time histories of response of the structure

for the pile group model, (c) linear and nonlinear Fourier spectra for the pile group model.




most of the frequency content of the excitation. For the

structural response, it appears that the group effect reduces

the difference between the linear and nonlinear response.

13. Conclusions

The effects of soil plasticity on the seismic response of

SPS systems are investigated using three-dimensional finite

element analyses in the time domain. The analyses involve a

subsystem model and a successive-coupling incremental

scheme. Analyses are performed for both single piles and

pile groups. The following conclusions can be drawn:

1. The effect of inertial interaction (for the range of

foundations and structure parameters considered) is, ingeneral, to increase the pile head response but to

significantly decrease the response of the structure.

2. For a harmonic excitation, the soil nonlinearity increases

the pile head and structural responses at low frequencies.

At high frequencies, both the pile head and the structural

responses are slightly affected by the soil nonlinearity.

3. For the transient excitation, soil nonlinearity increases

both the pile head and the structural responses. Smoothed

Fourier spectra show that in general, nonlinearity

increases the responses at low and moderate frequencies

but its effect is negligible at high frequencies.

4. The pile group effect decreases the peak values of the

response for both the harmonic and transient excitations(i.e. reduces the effect of soil nonlinearity).

Based on the range of parameters considered in this

study, soil nonlinearity increases the response at low

frequencies

a0 , 0:2

;which represent the range of interest

for earthquake loading. However, generalization of theseresults may require further analyses with different soil and

pile parameters.

Acknowledgements

The research presented here was partially supported by

the Mid-America Earthquake Center under National

Science Foundation Grant EEC-9701785 and the US

Army Corps of Engineers. This support is gratefully

acknowledged. The authors are thankful to Dr Y.X. Cai

for his cooperation.

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