Three-Dimensional Nonlinear Seismic Analysis of PileGroups Using FE-CIFECM Coupling in a Hybrid Domain and

HISS Plasticity ModelB. K. Maheshwari, M.ASCE1; and Pavan K. Emani2

Abstract: This paper reports the three-dimensional (3D) nonlinear dynamic analysis of floating pile groups subjected to seismic loading. Forthe nonlinear dynamic analysis of foundations in general, and pile foundations in particular, a perfect accuracy in modeling the frequency de-pendence of the unbounded soil support, as well as the local temporal nonlinearities, was ensured. In the current work, the finite elements (FEs)and boundary FEs were coupled in a hybrid domain of alternating frequency-time domains, to analyze the nonlinear dynamics of pile founda-tions in a 3D space. The frequency dependence of the stiffness and the damping of the unbounded far-field soil were completely accounted forby the consistent infinitesimal finite-element cell method (CIFECM) in the frequency domain, whereas the nonlinearities and transient condi-tions of the near-field were accounted for by the FE in the time domain. The hybrid frequency time domain algorithm is used for coupling thetwo domains. For the evaluation of nonlinear effects, an advanced, plasticity-based, hierarchical single surface (HISS) soil model was used. Theapplicability of this rigorous approach to pile foundations was validated using well-established results in the literature. The transient dynamicresponse of a 23 2 pile group embedded in clayey soils was performed. Using the results, the frequency response of the pile foundations ispresented under elastic and inelastic soil conditions in terms of dynamic impedances and kinematic interaction factors for different pile spacings.The results showed significant reduction, caused by soil plasticity, in both the lateral stiffness and the damping of the pile groups.DOI:10.1061/(ASCE)GM.1943-5622.0000335. © 2014 American Society of Civil Engineers.

Author keywords: Seismic response; Pile groups; Nonlinear soil-pile interaction; Consistent infinitesimal finite-element cell method(CIFECM); Hybrid frequency time domain (HFTD) method; Hierarchical single surface (HISS) soil plasticity.

Introduction

During the last 3 decades, many numerical and analytical methodshave been developed for computing the response of pile foundationsto external dynamic loads and to seismic loads. The initial workreported in the literature was based on a discrete system of mass,spring, and dashpot (Penzien et al. 1964; Matlock et al. 1978).Conventional time domain methods for nonlinear dynamic soil-structure interaction analyses typically adopt Winkler models forthe far-field and finite elements (FEs) for modeling the near-field.Approximate modeling of the soil domain using one-dimensionalWinkler models (e.g., Nogami et al. 1992; Badoni andMakris 1996;Wang et al. 1998; Maheshwari andWatanabe 2006) cannot give thetrue transient response of the three-dimensional (3D) problems. Themethod is only approximate, especially under nonlinear conditions.Other time domain methods, based on FE formulation, are used forpredicting the transient and/or nonlinear responses of 3D systems(e.g., Kuhlemeyer 1979; Mylonakis and Gazetas 1999; Bentley andEl Naggar 2000; Maheshwari et al. 2004b, 2005). However, in thesemethods, the frequency dependence of the unbounded soil can be

modeled only in an approximate manner, especially when theloading is also transient, as in the case of seismic loading. Further,the local/transmitting boundaries adopted in these methods cannotsimulate the radiation boundary conditions for the oblique incidenceof stress waves.

Rigorous analytical methods in frequency domain usingboundary integral methods (Novak 1974; Kaynia and Kausel 1982;Sen et al. 1985; Fan et al. 1991; Maheshwari and Watanabe 2005)are restricted to a steady-state response analysis of linear or at most,equivalent linear, systems. Knappett and Madabhushi (2009) in-vestigated the effect of simple idealizations of soil behavior onamplification through a set of centrifuge and numerical studiesof the axial load. Haldar and Sivakumar Babu (2010) researchedthe failure mechanisms of pile foundations in liquefiable soils.Maheshwari and Sarkar (2011) considered the effect of liquefactionon soil-pile-structure interaction. Sarkar and Maheshwari (2012b)investigated the effect of separation between soil and pile on thedynamic behavior of the soil-pile interaction under liquefyingconditions. Sarkar and Maheshwari (2012a) researched the effectsof plasticity and liquefaction on the dynamic stiffness of pilegroups. Huang and Shi (2013) investigated the application of rowsof piles for attenuating horizontal vibration. Hajialilue-Bonab et al.(2013) used strain wedge parameters to derive the lateral responseof piles.

To obtain the true nonlinear response, the frequency de-pendence of the system must be fully accounted for, while con-sidering the temporal and 3D spatial variation of the system’sresponse. These requirements are fulfilled by coupling the ad-vantages of the FEM and the boundary-element method (BEM),together with the coupling of the frequency domain and timedomain techniques. The coupling of FEM and BEM in a frequency

1Professor, Dept. of Earthquake Engineering, Indian Institute of Tech-nology Roorkee, Roorkee 247667, India (corresponding author). E-mail:bkmahfeq@iitr.ernet.in

2Professor, Dept. of Civil Engineering, Graphic Era Univ., Dehradun248002, India.

Note. This manuscript was submitted on February 7, 2013; approved onJune 18, 2013; published online on April 25, 2014. Discussion period openuntil September 25, 2014; separate discussions must be submitted forindividual papers. This paper is part of the International Journal ofGeomechanics, © ASCE, ISSN 1532-3641/04014055(12)/$25.00.

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domain was carried out for the elastic analysis of two-dimensional(Doherty and Deeks 2006) and 3D (Wegner and Zhang 2001) soil-structure systems. A similar coupling of FE and BE was alsoperformed in a time domain (Bazyar and Song 2006). The authorsdemonstrated the ability of a FE-BE-coupled model to analyze 3Dsoil-pile systems with elastic behaviors (Emani and Maheshwari2009) andwith cyclic nonlinear behaviors (Emani andMaheshwari2008). Syed and Maheshwari (2014) implemented the coupling ofFEM–scaled boundary FEM (FEM-SBFEM) in the time domain.

Recently, many procedures have been presented for nonlinearanalyses in the frequency domain. The method presented here isbased on the hybrid frequency time domain (HFTD) formulation(Wolf 1988). Further, in the current study, to satisfy the radiationconditions, the consistent infinitesimal finite-element cell method(CIFECM) (Wolf and Song 1996) was used, and to account for thematerial nonlinearity of the soil, the hierarchical single surface(HISS) model (Desai et al. 1991) was used. This was implementedthrough a MATLAB code (MATLAB 2008) developed by theauthors because no commercial software was capable of executingthis task.

Physical System and Modeling

The 3D model of the pile-soil system (Fig. 1) consisted of twosubstructures, namely, an unbounded far-field (henceforth referredto as the ground subsystem or the ground) and a bounded near-field(referred to as the structure subsystem or the structure). The dynamicstiffness of the system (which included the effects of inertia,damping, and stiffness) was obtained by coupling the two sub-structures. The structure subsystem was minimally selected to ac-curately represent the near-field effects, such as nonlinearities. Inthe current study, the bounds of the structure were fixed by trialand error, starting from the edge of the pile cap.

It should be noted that the geometrical nonlinearity was notconsidered here, and a perfect bond between soil and pile was as-sumed. Thus, the slip and separation at the soil-pile interface wasnot considered. In this paper, the use of HFTD is demonstrated fornonlinear dynamics in 3D domains, it is believed for the first time.Therefore, only material nonlinearity was studied. At the sametime, the HFTD method is general and therefore can be applied tocases with slip and separation. For example, Wolf (1988) demon-strated the use of the HFTD method for obtaining the seismicresponse of a nuclear containment building with separation at thebase. Further, the HISS model adopted for simulating soil plasticitycan be used for modeling the interface between the soil and the pile(Desai 2001). Therefore, incorporating the geometric nonlinearitycaused by slip and separation in the developed algorithm wasstraightforward.

Modeling of Far-Field

The far-field was modeled using the CIFECM (Wolf and Song1996), wherein the dynamic stiffness of the unbounded domain

was evaluated at the interface of the near-field and the far-field fordifferent frequencies. This method exploited the advantage ofsimilarity and thus reduced the model by one dimension. Theassembled dynamic stiffness matrix of the far-field was a fullypopulated complex Hermitian matrix and contained all thestiffness, inertial, and damping (hysteretic) characteristics of thefar-field.

Modeling of Near-Field

The near-field (Fig. 2) consisted of soil and a 23 2 pile group (witha pile cap of width 2B and thickness t). The center-to-center spacingbetween the piles was s, and the length of each pile was L. Thecircular piles were modeled as equivalent square piles. In all threedirections, the size of the near-field, obtained by extending theedges of the pile group by length B, was found to be satisfactory[Fig. 2(a)]. Both the soil and the pileweremodeledwith eight-noded3D solid (finite) elements, with consistent mass matrices. Thedamping in the soil was assumed to be of the hysteretic type. Takingadvantage of symmetry, only one-half of the actual bounded do-main was built, and the corresponding symmetric boundary con-ditions were applied to the respective nodes on the planes ofsymmetry [Fig. 2(b)].

Material Modeling

The piles were assumed to be made of homogeneous and isotropicelastic material. Young’s modulus of the pile material was Ep, andPoisson’s ratio of the pile material was yp. The pile cap was assumedto be massless and rigid. It rested on the pile heads, which wereprojected a little above the soil surface; thus, the pile cap connectedall the pile heads rigidly, without interacting directly with the soil.The soil was considered to be homogeneous, with Young’s modulusEs, Poisson’s ratio ys, mass density rs, and damping ratio z.

The inelastic behavior of the soil elements wasmodeled using theHISS soil model (Desai 2001). In the current work, the dpo version ofthe HISS series was considered. Both the plasticity and the workhardening of the soil were considered in this model with associativeplasticity. Amaterial parameterbwas used to define the shape of the

Fig. 1. Substructuring of the pile-soil system into far-field and near-fieldFig. 2. Physical model and FEM of near-field: (a) full model; (b)half model

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yield surface in the octahedral plane. Assuming b5 0, the dimen-sionless yield surface F could be simplified to

F ¼�J2Dp2a

�þ a

�J1pa

�n

2 g

�J1pa

�2

¼ 0 (1)

where J1 5 first invariant of the stress tensor sij; J2D 5 secondinvariant of the deviatoric stress tensor; pa 5 atmospheric pressure;a 5 hardening function; and g and n 5 material parameters thatinfluence the shape of F in the J1 2

ffiffiffiffiffiffiffiJ2D

pspace. The parameter n is

related to the phase change point, which is defined as the point atwhich the material changes from contractive to dilative behavior(Fig. 3). The hardening function, a, is defined in terms of the plasticstrain trajectory, jv, as

a ¼ a1jh1v

(2)

where a1 and h1 5 material parameters; and jv 5 trajectory of thevolumetric plastic strain. Typical yield surfaces for this model areshown in Fig. 3.

Loading and Boundary Conditions

Harmonic loading was applied either as an external dynamicloading at the centroid of the pile cap or as seismic excitation atthe interface of the near-field and the far-field. In both cases, the

accurate evaluation of the initial transient response required zeropadding of the loading history, beyond the time of interest (Humar1990). This modification leads to a significant change in the Fourieramplitude spectrum compared with a continuous harmonic loadtime history. Because of this, many more frequencies must beanalyzed even for a harmonic loading (Veletsos and Ventura 1984).The near-field was bounded by radiation boundaries (far-field) onthe sides and below, whereas the top surface was a free surface. Forhalf-models, symmetric boundary conditions existed on one of theside faces. The pile cap was (1) restrained against rotation underlateral loading [Fig. 4(a)], (2) restrained against rotation undervertical loading [Fig. 4(b)], and (3) free to rotate under seismicexcitation [Fig. 4(c)].

Methodology

In the current work, the HFTD method was implemented to eval-uate the inelastic dynamic response of the pile foundations. Thetwo major steps in this method were (1) evaluation of the transientelastic response and (2) iterative correction of the response byapplying pseudoloads arising out of nonlinear stiffness. The pre-sented nonlinear analysis is an extension of the linear analysispresented by Emani and Maheshwari (2009), wherein the appli-cability of the coupling of FEM and CIFECM to 3D soil-structure systems was validated using elastic analysis of pilegroups. Therefore, the coupling between the near-field (FEM)

Fig. 3. Shape of yield surfaces in J1 2ffiffiffiffiffiffiffiJ2D

pspace for HISS soil model

Fig. 4. Pile cap restraints under (a) horizontal, (b) vertical, and (c) seismic excitations

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and the far-field (CIFECM) is not discussed in detail here. In theHFTD algorithm, pseudoforces are evaluated in the time domain andare used to correct a linear estimate of the system’s response in aniterative manner. In each iteration, the response approximated in theprevious iteration is considered, and the difference between thetrue nonlinear internal forces and the linear internal forces at the cor-responding strain level is evaluated for each time step. The HISSsoil plasticity model was used for this purpose. This gave the timehistory of the pseudoforces, which, when solved in the frequencydomain, gave an improved estimate of the response for the nextiteration.

Validation

The accuracy of the proposed FE-CIFECM coupling method inevaluating the frequency response of the soil-structure systemswas already validated using elastic analysis results (Emani andMaheshwari 2009). In the current work, the validation of the HFTDalgorithm and its implementation in 3D FE problems were per-formed through the elastoplastic dynamic analysis of a 3D cantileverbeam (Soares and Mansur 2005) subjected to a suddenly applied,uniformly distributed, load. In addition, the validation of the seismicanalysis of pile groups (Kaynia and Kausel 1982) is presented.

Nonlinear Transient Dynamic Analysis ofCantilever Beam

Fig. 5 shows the geometry, boundary conditions, and FEmesh used,along with the spatial variation of loading on the cantilever beam.The beam was assumed to be made of a perfect plastic materialobeying the von Mises yield criterion. The material properties arealso shown in Fig. 5. The damping matrix was assumed to beproportional to the mass matrix (Soares and Mansur 2005). Thenonlinear dynamic response histories are shown in Fig. 6. It can beobserved that the results using the current algorithm are in perfectagreement with the published results (Soares and Mansur 2005),which validates the HFTD algorithm.

Kinematic Interaction Factors for 2 3 2 SquarePile Group

Using the proposed method, a frequency response analysis wascarried out for elastic soil behavior in a frequency domain to obtainthe steady-state kinematic response of a single pile and 23 2 pilegroups subjected to harmonic seismic excitations [vertically prop-agating shear (SH)waves] at the interface. The kinematic interactionfactor (KIF) for translation was the ratio of the steady-state values ofthe pile cap displacement (Up) to the amplitude of the free-fielddisplacement (Uf ) at the same level. Similarly, the KIF for rotationwas defined as the ratio of product of the pile diameter (d) and theamplitude of the steady-state rotation (fp) of the pile cap to theamplitude of the free-field displacement. The KIFs obtained fromthe current method and those obtained from using a rigorous bound-ary integral formulation (Kaynia and Kausel 1982) are compared inFig. 7. In this figure, the dimensionless frequency a0 5vd=Vs,where v is the frequency of excitation, d is the diameter of the pile,and Vs is the shear wave velocity of the soil. Fig. 7 shows a rea-sonably good agreement between the current results and thosepresented by Kaynia and Kausel (1982). The differences in the KIFs,especially in rotations, were attributed to the error introduced by FEdiscretization in the wave-propagation analysis. This validated thedeveloped algorithm for seismic excitation.

Fig. 5. FEMof cantilever used for validating the 3D nonlinear dynamicanalysis

Fig. 6. Elastic and elastoplastic response history of the tip node A of the cantilever beam

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Results: Input Data Used

The validated analytical methodwas applied to analyze the responseof the pile groups. This method can account for inelastic behaviorwhile simultaneously accounting for the pile-soil-pile interaction(PSPI), under a variety of near-field irregularities (such as inclined

piles, pile caps with ground contact, etc.). Four FEmodels, shown inFig. 8, were analyzed. The size of the pile, the spacing of the piles,and the size of the model can be evaluated from the figure. In all themodels, the individual circular piles with 0.6-m diameter weremodeled as square piles,with equivalent side dimensions of 0.527m.All the pileswere 9.0m in length; thereforeL=d5 15 for all the piles.

Fig. 7. Kinematic interaction factors in translation and rotation: (a) displacement (single pile); (b) displacement (s=d5 2); (c) rotation (s=d5 2);(d) displacement (s=d5 5); (e) rotation (s=d5 5); (f) displacement (s=d5 10); (g) rotation (s=d5 10)

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The following material properties for the soil, pile, and pile capwere used in the computation:

Es ¼ 25 MPa, ns ¼ 0:40, rs ¼ 1,750 kg=m3

Ep ¼ 25 GPa, np ¼ 0:25, rp ¼ 2,500 kg=m3

Ec ¼ 1� 1015 Pa, nc ¼ 0:25, rc ¼ 0 kg=m3

for a rigid massless pile cap

g ¼ 0:047, b ¼ 0, n ¼ 2:4, a1 ¼ 0:0034, h1 ¼ 0:78

for the nonlinear HISS model of the soil.The evaluation of the HISS model parameters and the validation

of the prediction of the triaxial test results were reported inWathugala and Desai (1993) and Desai et al. (1993). The analysiswas performed for both harmonic and transient excitations. Forharmonic loading, the effect of nonlinearity on the dynamic stiffnessand seismic response at different frequencies of excitation wasinvestigated. However, for transient excitation, the seismic responsefor real earthquake excitation under inelastic behavior was analyzed.The results are discussed in the following sections.

Effect of Nonlinearity on Dynamic Stiffness

The results are presented for dynamic impedances of a single pile and23 2 floating pile groups embedded in an elastoplastic half-space.

For evaluating the horizontal (or vertical) impedances of a single pileand pile groups, the corresponding model was subjected to a har-monic horizontal (or vertical) force at the top of the pile cap. This loadwas applied symmetrically at the centroid of the pile cap, whilerestraining the degrees of freedom of the loading nodes, in directionsother than the loading direction, as shown in Figs. 4(a and b). Ineffect, this is equivalent to a fixed-head pile.

For harmonic loading, the effect of frequency was investigated bycomputing the dynamic stiffness at different frequencies of excitation inthe range of about 0.5 to 20 Hz. For this range, the dimensionless fre-quency a0 was in the range of approximately 0.025 to 1, typically usedfor soil-structure interaction analyses in the literature. The effect ofnonlinearity is verymuch influencedby the amplitudeofdynamic force;therefore, itmust be indicated. For computing dynamic impedances, theamplitude of harmonic force (P0) for 23 2 pile groups was 300 kN(applied at the centroid of the pile cap). For the single pile model, onlyone-fourth of this load (i.e., 75 kN) was applied at the pile head. Thislevel of loading can also be written in nondimensional form (e.g.,assuming the shear strength of soil stratum tf to be 80 kN=m2) andgiven that pile length L5 9 m, the ratio Pmax=ðtf L2Þ5 0:012 can beused to characterize the level of loading (Nogami and Konagai 1987).

For each frequency of excitation, the displacement time historywas plotted until a steady-state response was obtained. From thistime history, the amplitude of displacement (U0) and its phase lag(f) with applied load was noted. The results are presented in di-mensionless form, i.e., in terms of real part (Re) (k), and imaginarypart (Im) (c) of the normalized stiffness defined as

k ¼ P0=U0

NKscosðwÞ (3a)

c ¼ P0=U0

NKsa0sinðwÞ (3b)

whereN 5 number of piles in the group; and Ks 5 static stiffness ofa single pile. This procedure was repeated for all frequencies and forall pile spacings (and the single pile). The results are presentedseparately for horizontal and vertical modes of vibration.

In this section, the effect of nonlinearity on the dynamic stiffness,with two key parameters (i.e., frequency of excitation and spacingratios), is studied. Further parametric study, such as the effect of theslenderness ratio of the piles and the relative rigidity of the soil and pile,can also be considered, as shown by Emani (2008) for a linear case.

Normalized Horizontal Impedances

Fig. 9 shows the normalized horizontal impedances of a single pileand pile groups under elastic and HISS conditions. The HISS(i.e., elastoplastic) conditions are characterized by the soil param-eters given in Fig. 3.

For a single pile, Figs. 9(a and b) indicate that there was a cleardecrease in the stiffness and damping values caused by plasticityover the whole range of frequencies. The effect was significant athigher frequencies. For a 23 2 pile group with close spacing(s=d5 2), Fig. 9(c) suggests that the effect of nonlinearity onstiffness was similar to that on a single pile. However, the effect ondamping was not significant at higher frequencies [Fig. 9(d)].

Further, the effect of plasticity was much more significant on thePSPI, as evident in the cases of pile groupswith s=d5 5 and s=d5 10.In both these cases [Figs. 9(e and f) and 9(g and h)], the PSPI peakswere reduced inmagnitude, accompanied by a shift in the frequency ofoccurrence. This shift in frequency was small for s=d 5 10. Thereduction in peak values of stiffness (caused by the group interfer-ence effect) was very significant (i.e., as much as a reduction of

Fig. 8. Details of meshing of the FE half-models: (a) single pile;(b) 23 2 pile group with s=d5 2; (c) 23 2 pile group with s=d5 5;(d) 23 2 pile group with s=d5 10

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Fig. 9. Effect of plasticity on normalized horizontal impedances: (a) Re and (b) Im for single pile; (c) Re and (d) Im for s=d5 2; (e) Re and (f) Im fors=d5 5; (g) Re and (h) Im for s=d5 10

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approximately 30% for s=d5 5 [Fig. 9(e)] and approximately 50% fors=d5 10 [Fig. 9(g)]). In addition, the damping provided by the pilegroups with s=d5 5 and 10 were significantly smaller in the HISSsoils.As the frequency increased, therewas some recoveryof dampingfor s=d5 10. Very similar results were reported by Maheshwari et al.(2004a) for the elastic and inelastic (HISS) responses of 23 2 pilegroups under harmonic loading, using a conventional time-steppingalgorithm considering Kelvin elements at the boundary.

In all the results presented here, the hysteretic or frequency-independent material damping model was used. Therefore, thecontribution of the material damping to the overall damping wasconstant at all frequencies and was not significant compared withradiation damping.

From Fig. 9, it can be observed that the effect of nonlinearity onhorizontal dynamic stiffnesswas verymuch dependent on the spacingbetween piles, and it is difficult to generalize the results. Nevertheless,in general, the effect on stiffness was significant at moderate and highfrequencies but not as significant at low frequencies. However, theeffect of nonlinearity on damping was significant at low frequenciesbut not as significant at higher frequencies.

Normalized Vertical Impedances

Fig. 10 shows the normalized vertical impedances of a single pile andpile groups under elastic and HISS conditions. At the consideredload level, because of plasticity, little effect on impedances wasobserved. There was no significant effect of plasticity on the dy-namic stiffness, particularly for a single pile and a pile group withs=d5 2. For the pile groups with s=d5 5 and 10, there was somealteration in stiffness. For the s=d5 5 case, the frequency shift inPSPI was more significant, whereas for the case of s=d5 10, thedecrease in vertical stiffness was not significant. For this case, therewas a slight reduction in damping at moderate frequencies.

Apart from this, there was not much change in the impedancevalues. Thus, in general, the effect of plasticity on vertical impe-dances was not as significant as that observed for the horizontal case.The effect was visible for pile groups with spacing (s=d5 5). In thiscase, the stiffness decreased significantly at higher frequencies. Theeffect on damping was not significant.

Effect of Nonlinearity on Seismic Response

The results are presented for KIFs of a single pile and 23 2 floatingpile groups embedded in an elastoplastic half-space. For evaluatingKIFs, the pile cap was unrestrained, whereas the harmonic seismicexcitation was applied in the form of an equivalent interaction forcehistory at the interface of the near-field and the far-field.

Fig. 11 shows the effect of plasticity on theKIFs of a single pile andpile groups. For all models (single and s=d5 2, 5, and 10), the nor-malized pile head displacements increased slightly at all frequenciesbecause of the elastoplastic (HISS) behavior of the surrounding soil.Similar results were reported by the National Cooperative HighwayResearch Program (2001), although in that study, a Drucker-Pragersoil model was used to deal with the nonlinearity. On the other hand,the pile head rotation increased significantly at frequencies higher than3.5 Hz (a0 5 0:2) for all models of pile foundation.

Nonlinear Seismic Response forEarthquake Excitations

The proposed methodology has already been described and dem-onstrated for the harmonic loading case, but the actual usefulness of

the method lies in the case of transient excitations, which may notnecessarily have a predominant frequency. In such cases, the vari-ation of dynamic properties with frequency becomes very important.To study the response under such cases, an earthquake excitationwas applied as a lateral inputmotion to a 23 2 pile groupmodel withs=d5 5. The response of the pile group to this real earthquakeexcitation was studied under both elastic and inelastic soil con-ditions. Real earthquake excitation considered is acceleration timehistory of 1995 Kobe earthquake (Takarazuka station minus trans-verse component) with peak ground acceleration (PGA) 0.694g.The time histories and Fourier spectra of the input motion are shownin Fig. 12.

Under transient excitations, it was observed (but not presentedin this paper) that the difference between the response for elasticand HISS soil conditions was imperceptibly small. This was to beexpected from the harmonic kinematic interaction analysis resultspresented in the previous section [Figs. 11(d and e)]. For example, atthe predominant frequency of 2.0 Hz (a0 5 0:104) for the Kobeearthquake, there was hardly any difference between the elastic andthe HISS soil conditions. This means that the nonlinearity of thesurrounding soil was not activated to a significant degree for theconsidered soil-pile properties and excitation level.

Further, the kinematic interaction effects in the transient con-ditions were less severe than in the steady-state condition at thecorresponding predominant frequency (Nikolaou et al. 1995).Hence, another case of soil-pile system was chosen such that thenonlinearity in the soil was activated. The considered system wasa pile group with a flexible pile cap having a high density (100 timesmore than the pile material) but of the same dimensions as in theprevious case. In this case also, the bottom of the pile cap wasprojected slightly above the ground surface. Unlike the previouscase, in which only the kinematic interaction effects of the pilegroup were present, this case also included the effect of inertialinteractions.

Fig. 13 shows the lateral displacement response at the centroidof the inertial pile cap under the excitation of the Kobe earthquakeground motion. It can be seen that there was considerable changebetween the elastic and the HISS responses. It is clear that the in-elastic response was significantly higher than the elastic response atall times. This was to be expected because the frequency content ofthe ground motion input [Fig. 12(a)] was below 4 Hz, and in thisrange the dynamic stiffness of the inelastic (HISS) soil-pile systemwas slightly lower than that of the elastic system [Fig. 9(e)]. Further,the response was increased because of inertial interaction.

InFig. 13, the coincidence of the response peaks indicates that thesoil softening was not enough to change the overall stiffness of thesoil-pile system. This was mainly because of the localized nature ofthe nonlinearity or plasticity. Under the condition of inertial loadingcoming from the top of the pile cap, most of the nonlinearity waslocalized in the top portion of the soil, so the overall stiffnessremained almost the same. Therefore, the natural frequency of thesoil-pile system was unaltered, although the displacement timehistory at a specific location (the top of the pile cap) showed sub-stantial softening (i.e., reduction in stiffness and subsequent increasein displacement response).

Summary and Conclusions

A new hybrid method was developed and implemented for inelasticdynamic analysis of 3D soil-structure systems with frequency-dependent stiffness properties. In the current work, an advancedplasticity-based HISS soil model was used to deal with the soilnonlinearity. The elastic response obtained from the frequency

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domain formulation was corrected using the pseudoforce approachof the HFTD algorithm. The method showed excellent agreementwith the published results, under both external dynamic loading andseismic loading.

The developed method was used to evaluate the inelastic dynamicimpedances of a single pile and a 23 2 floating pile group square in

plan. Both horizontal and vertical impedances were evaluated underinelastic conditions. The results showed significant reduction, causedby soil plasticity, in both lateral stiffness and damping of the pilegroups. The reduction in the stiffness, especially at the PSPI resonancefrequencies, could be as high as 50%, at the considered load level. Thestiffness reduction was not very pronounced in the vertical direction,

Fig. 10. Effect of plasticity on normalized vertical impedances: (a) Re and (b) Im for single pile; (c) Re and (d) Im for s=d5 2; (e) Re and (f) Im fors=d5 5; (g) Re and (h) Im for s=d5 10

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Fig. 11. Effect of plasticity on the kinematic interaction factors: (a) displacement (single pile); (b) displacement (s=d5 2); (c) rotation (s=d5 2);(d) displacement (s=d5 5); (e) rotation (s=d5 5); (f) displacement (s=d5 10); (g) rotation (s=d5 10)

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partly because the slip between the soil and pile was not modeled. Atcertain frequencies around the PSPI resonance, there was a marginalincrease in the resonance peaks in stiffness under inelastic soilconditions.

The inelastic seismic analysis, in terms of KIFs, indicated that theeffect of soil plasticity was not significant in the kinematic in-teraction effects of the considered pile group model. The same wasconcluded from both the harmonic (all pile group models) andtransient (only the pile group with s=d5 5) excitation analyses. Forthe same model (with s=d5 5), inertial interaction effects caused bythe presence of a heavy pile cap showed a significant increase in theresponse under inelastic conditions. To make general comments,more comprehensive parametric studies must be performed underdifferent load levels and soil conditions.

The model used in the current research work has many merits. Inthe model, the FE and boundary FE are coupled. The latter helps inprecisely satisfying the radiation conditions at the boundary and thusaccurately model the unbounded domain of the soil medium usingCIFECM, and the FEs used for the near-field facilitate modeling forthe nonlinear effects.

The computation was carried out in the hybrid domain, where theadvantage of both the frequency domain and the time domainalgorithms was exploited. This was because the dynamic stiffness ofthe soil-pile system, which depends on the frequency, could best bedealt with in the frequency domain, whereas the nonlinearity of thesoil could best be dealt with in the time domain. Thus, the model andalgorithm developed are unique.

Acknowledgments

The research work presented here was supported by an InstituteFellowship to the second author from the Ministry of Human Re-source Development, Government of India. This support is grate-fully acknowledged.

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