soil dynamics and earthquake engineering volume 24 issue 4 2004 [doi...
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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
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* 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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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)
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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.
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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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