10.1139@t00-066

Numerical analysis of kinematic response ofsingle piles

Kevin J. Bentley and M. Hesham El Naggar

Abstract: Recent destructive earthquakes have highlighted the need for increased research into the revamping of designcodes and building regulations to prevent further catastrophic losses in terms of human life and economic assets. Thepresent study investigated the response of single piles to kinematic seismic loading using the three-dimensional finiteelement program ANSYS. The objectives of this study were (i) to develop a finite element model that can accuratelymodel the kinematic soilstructure interaction of piles, accounting for the nonlinear behaviour of the soil, discontinuityconditions at the pilesoil interface, energy dissipation, and wave propagation; and (ii) to use the developed model toevaluate the kinematic interaction effects on the pile response with respect to the input ground motion. The static per-formance of the model was verified against exact available solutions for benchmark problems including piles in elasticand elastoplastic soils. The geostatic stresses were accounted for and radiating boundaries were provided to replicateactual field conditions. Earthquake excitation with a low predominant frequency was applied as an accelerationtimehistory at the base bedrock of the finite element mesh. To evaluate the effects of the kinematic loading, the responsesof both the free-field soil (with no piles) and the pile head were compared. It was found that the effect of the responseof piles in elastic soil was slightly amplified in terms of accelerations and Fourier amplitudes. However, forelastoplastic soil with separation allowed, the pile head response closely resembled the free-field response to thelow-frequency seismic excitation and the range of pile and soil parameters considered in this study.

Key words: numerical modelling, dynamic, lateral, piles, kinematic, seismic.

Rsum : De rcents tremblements de terre destructeurs ont mis en lumire la ncessit daugmenter les efforts derecherche pour la refonte des codes de conception et des rglements pour les btiments de faon prvenir dautrespertes catastrophiques en termes de pertes de vies et dactifs conomiques. Dans la prsente tude, on a examin laraction de pieux simples un chargement cinmatique sismique au moyen dun programme tridimensionnel enlments finis (ANSYS). Les objectifs de cette tude taient: (i) de dvelopper un modle dlments finis qui puissemodliser prcisment linteraction cinmatique solstructure des pieux, prenant en compte le comportement nonlinaire du sol, les conditions de discontinuit de linterface solpieu, la dissipation dnergie et la propagation desondes; et (ii) dutiliser le modle mis au point pour valuer les effets dinteraction cinmatique sur la raction du pieuen fonction du mouvement induit par le terrain. La performance statique du modle a t vrifie avec des solutionsexactes disponibles pour des problmes dvaluation de performance comprenant des pieux dans les sols lastiques etlasto-plastiques. Les contraintes gostatiques ont t prises en compte et des frontires radiales ont t utilises pourreproduire les conditions relles du champ de contraintes. On a appliqu une excitation sismique basse frquenceprdominante comme histoire dacclrationtemps la base du socle rocheux de la grille dlments finis. Pourvaluer les effets du chargement cinmatique, on a compar la rponse du sol sans pieu et galement la rponse de latte du pieu. Lon a trouv que leffet de la raction des pieux dans un sol lastique tait lgrement amplifi enfonction des acclrations et des amplitudes de Fourier. Cependant, pour un sol lasto-plastique avec sparationpermise, la raction de la tte du pieu ressemblait de prs la raction sans pieu pour une faible frquencedexcitation et pour la plage des paramtres de sol considrs dans cette tude.

Mots cls : modlisation numrique, dynamique, latral, pieux, cinmatique, sismique.

[Traduit par la Rdaction] Notes 1382

Introduction

Catastrophic damage that has resulted from recent earth-quakes (e.g., Yugoslavia Earthquake of 1998, Kobe Earth-

quake of 1995, North Ridge Earthquake of 1994, and CairoEarthquake of 1992) has raised concerns about the currentcodes and approaches used for the design of structures andfoundations. In the past, free-field accelerations, velocities,and displacements have been used as input ground motionsfor the seismic design of structures without considering thekinematic interaction of the foundation or the site effectsthat have resulted from the introduction of piles or the soilstratigraphy. Depending on the pile or pile group configura-tion and soil profile, free-field response may underestimate

Can. Geotech. J. 37: 13681382 (2000) 2000 NRC Canada

1368

Received May 12, 1999. Accepted May 9, 2000.Published on the NRC Research Press web site onDecember 18, 2000.

K.J. Bentley and M.H. El Naggar. Geotechnical ResearchCentre, The University of Western Ontario, London, ONN6A 5B9, Canada.

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or overestimate actual in situ conditions that will result inradically changed design criteria.

Earthquake-induced loading can be separated into two ba-sic loading conditions: kinematic and inertial. The presentstudy is concerned with the response of single piles to kine-matic loading over a range of soil and pile parameters thatcan be used to help model inertial-interaction models.

Fan et al. (1991) performed an extensive parametric studyusing an equivalent linear approach to develop dimension-less graphs for pile head deflections versus the free-field re-sponse for various soil profiles subjected to verticallypropagating harmonic waves. Makris and Gazetas (1992) ap-plied free-field accelerations to a one-dimensional Beam-on-Dynamic-Winkler-Foundation model with frequency-dependent springs and dashpots to analyze the response ofsingle piles and pile groups. The results from both studiesconcluded that interaction effects on kinematic loading arenot significant at low frequencies but are significant for pilehead loading (inertial interaction). These studies were lim-ited to linear (equivalent linear) elastic analysis and one-dimensional (1D) harmonic loading.

A full three-dimensional (3D) transient nonlinear dynamicanalysis was performed in the current study to investigatethe effects of kinematic interaction on the input motion atthe foundation level. This analysis accounted for pilesoilgapping and slippage, soil plasticity, and 3D wave propaga-tion. The finite element program ANSYS (ANSYS Inc.1996) was used in the analysis.

Assumptions and restrictions

The problem to be addressed is shown in Fig. 1. The ac-tual system consisted of a pile foundation supporting a typi-cal bridge pier. The current codes use the free-field motionas the input ground motion at the foundation level. The anal-ysis described herein attempted to evaluate the interaction ofthe pilesoil system and how it alters the free-field motionand modifies the ground motion at the foundation level.

The dynamic loading was applied to the rigid underlyingbedrock (Fig. 1) as 1D horizontal acceleration (X direction

in the finite element model) and only horizontal responsewas measured. Vertical accelerations were ignored becausethe margins of safety against static vertical forces usuallyprovided adequate resistance to dynamic forces induced byvertical accelerations. Wu and Finn (1997), using a 3D elas-tic model, found that deformations in the vertical directionand normal to the direction of shaking are negligible com-pared with those in the direction of horizontal shaking.

Although the finite element analysis used in this study in-cludes important features such as soil nonlinearity and gap-ping at the pilesoil interface, it does not account forbuildup of pore pressure due to cyclic loading. Thus, neitherthe potential for liquefaction nor the dilatational effect ofclays and the compaction of loose sands in the vicinity ofpiles is accounted for, in the current analysis. Furthermore,the inertial interaction between the superstructure and thepilefoundation system is not considered. The analysis islimited to the response of free-headed piles with no externalforces (DAlembert forces) from the superstructure to betterunderstand the kinematic interaction effects in seismicevents.

Three-dimensional finite element model

Model formulationFull 3D geometric models were used to represent the pile

soil systems. Exploiting symmetry, only one half of the ac-tual model was built, thus significantly reducing computingtime and cost. Figure 2b depicts the pilesoil system consid-ered in the analysis and shows an isometric view of the halfof the model used. Figures 2 and 3 show the finite elementmesh (mesh 3) used in the analysis. The pile and soil weremodelled using eight-noded block elements. Each node hadthree translational degrees of freedom, i.e., X, Y, and Z coor-dinates, as shown in Fig. 4a. A 3D point-to-surface contactelement was used at the pilesoil interface to allow for slid-ing and separation in tension, but ensured compatibility incompression. The contact element had five nodes with threedegrees of freedom at each node, i.e., translations in the X, Y,and Z directions as shown in Fig. 4b. Transmitting boundaries

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Notes 1369

Fig. 1. Definition of the problem and terminology. Uff, free-field acceleration; Ug, bedrock acceleration; Up, acceleration due to kine-matic interaction only; Us, actual acceleration.

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were used to allow for wave propagation and to eliminatethe box effect (i.e., the reflection of waves back into themodel at the boundaries) during dynamic loading. The ele-ment used to model the transmitting boundary consisted of aspring and a dashpot arranged in parallel, as illustrated inFig. 4c.

Soil propertiesTo evaluate the effect of soil plasticity on pile response,

the soil was modelled as a homogeneous elastic medium andan elastoplastic material using the Drucker-Prager failurecriteria. For cases involving plasticity, the angle of dilatancywas assumed to be equal to the angle of internal friction (as-sociated flow rule). There was no strain hardening and there-fore no progressive yielding was considered. Since porepressures were not considered in this analysis, effective pa-rameters and drained conditions were assumed. The materialdamping ratio of the soil, x , was assumed to be 5%. Thegoverning equations of the system are given by

[1] [ ]{&&} [ ]{&} [ ]{ } { ( )}M u C u K u F t+ + =where {&&}u , {&}u , and { }u are the acceleration, velocity, and dis-placement vectors, respectively; and [M], [C], and [K] arethe global mass, damping, and stiffness matrices, respec-tively, and F(t) is the forcing function. The damping matrix[C] = b [K], in which the damping coefficient b = 2 x / w o,where w o is the predominant frequency of the loading

(rad/s). Material damping was assumed to be constantthroughout the entire seismic event, although the dampingratio varies with the strain level.

Pile propertiesCylindrical reinforced concrete piles with linear elastic

properties were considered in this study. The piles weremodelled using eight-noded brick elements. The cylindricalgeometry was approximately modelled using wedge-shapedelements (Fig. 2a). No damping was considered within thepiles and relevant parameters are listed in Fig. 5.

Pilesoil interfaceThe modelling of the pilesoil interface is crucial because

of its significant effect on the response of piles to lateralloading (Trochanis et al. 1988). Two cases were consideredin the analysis. First, the pile and soil are perfectly bonded,in which case the perimeter nodes of the piles coincide withthe soil nodes (elastic with no separation). Second, the pileand soil are connected by frictional interface elements thatare described later in the paper. The contact surface (pile) issaid to be in contact with the target surface (soil) when thepile node penetrates the soil surface. A very small tolerancewas assumed to prevent penetration and to achieve instantcontact as pile nodes attempt to penetrate the soil nodes (orvice versa). Coulomb friction was employed between thepile and soil along the entire pile length and the pile tip (forfloating piles). The coefficient of friction relating shearstress to the normal stress was chosen according to recom-mendations of the American Petroleum Institute (API 1991)and was assumed to be 0.7. The contact surface coordinatesand forces were fully updated to accommodate large or

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1370 Can. Geotech. J. Vol. 37, 2000

Fig. 2. (a) Detail of wedge-shaped pile elements surrounded bysoil elements (plan view). (b) Isometric view of complete soiland pile mesh model.

Fig. 3. Finite element mesh (mesh 3) showing boundary condi-tions: (a) plan view; (b) front cross-sectional view with geostaticpressure distribution.



small deflections that may occur. The penalty functionmethod was used to represent contact with a normal contactstiffness (Kn). Kn allowed the interface element to deformelastically before slippage occurred and was chosen to beequal to the shear modulus of the soil. Convergence wasachieved and overpenetration was prevented using Kn =6800 kN/m.

Boundary conditions

Boundary conditions varied depending on the type ofloading. For static loading, the bottom of the mesh (repre-senting the top of the bedrock layer) was always fixed in alldirections. All symmetry faces were fixed against displace-ment normal to the symmetry plane, but were free to moveon the surface of the plane. The nodes along the top surfaceof the mesh were free to move in all directions. The nodes

along the sides of the model were free to move vertically butwere constrained in the horizontal direction by a Kelvin ele-ment to represent a horizontally infinite soil medium duringstatic and dynamic analyses. The constants were calculatedusing the solution due to Novak and Mitwally (1988), givenby

[2] k Gr

S a iS aro

= +[ ( , , ) ( , , )]1 0 2 0n x n x

where kr is the total stiffness; G is the soil shear modulus; rois the distance to the finite element boundary; S1 and S2 arethe dimensionless parameters from closed-form solutions; nis Poissons ratio; i is the imaginary unit = - 1; and ao isthe dimensionless frequency = ro w /Vs, where w is the circu-lar frequency of loading and Vs is the shear wave velocity of

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Notes 1371

Fig. 4. (a) Block element used for soil and pile. (b) Surface contact element used between pile and soil to allow for slippage and sep-aration. (c) Transmitting boundary element consisting of spring (K) and dashpot (C) to allow for radiating boundaries.

Fig. 5. Two-dimensional representation of floating and socketed pile in either a homogeneous (used for verification) or layered soilprofile.



the soil. The real and imaginary parts of eq. [2] represent thestiffness and damping, respectively, i.e.,

[3] K GSr

CGS

r= =

1 2

o o

andw

To determine the stiffness and damping of the Kelvin ele-ments, the constants given by eq. [3] were multiplied by thearea of the element face (normal to the direction of loading)because they assume constant unit area of contact. For staticloading, i.e., zero frequency, the damping term vanishes andthe element reduces to a spring only.

For dynamic loading, w was taken as the predominant fre-quency of the earthquake load and was determined from adiscrete Fourier transform of the time history of the inputmotion. Figure 6 shows the Fourier amplitude (cn) versusfrequency (w n) content for the strong-motion record used inthe study. It is evident that a narrow spectrum exists at adominant frequency of approximately 2 Hz.

Time-dependent displacements were applied to the stra-tum base to simulate seismic loading. All other boundaryconditions remained unchanged and are graphically por-trayed in Fig. 3.

Loading conditions

Initial loadingThe state of stress in the pilesoil system in actual in situ

conditions was replicated as an initial loading conditionprior to any additional dynamic or static external load. Thatis, geostatic stresses were modelled by applying a globalgravitational acceleration, g, to replicate vertically increasing

stress with an increase in depth. A linearly increasing pres-sure with an increase in depth was applied to the peripheryof the soil block to replicate horizontal stresses as shown inFig. 3b. A coefficient of lateral earth pressure, K0 = 0.65,typical of many geological conditions, was used. Due to thedifference in density and stiffness for the pile and soil, thesoil tended to settle more than the pile in the vertical direc-tion, resulting in premature slippage at the pilesoil inter-face. To eliminate this false representation of initialconditions, the difference between the relative displacementbetween the soil and the pile was accounted for by adding acorresponding body load to the pile. The resulting mesh rep-resented in situ conditions, especially for drilled caissons.

Static loadingAll static loads were applied as distributed loads along the

perimeter of the pile head that was level with the ground sur-face. Only one half of the total load was applied to the pilein the finite element analysis due to the symmetric geometryof a full circular pile.

Dynamic loadingStrong-motion records from the Loma Prieta Earthquake

in California (ML = 7.1) in 1989 were used in the finite ele-ment study. The accelerogram and displacement data usedwere from the Yerba Buena Island rock outcrop station inthe Santa Cruz Mountains (National Center for EarthquakeEngineering Research 1998). The measured displacementswere applied to the top of the rigid bedrock layer at 0.02 sintervals. Considering that the maximum acceleration of themeasured 1D motion was 0.03g, the data were multiplied bya factor of seven to simulate a peak horizontal acceleration(PHA) of approximately 0.2g for the bedrock input motion.It is important to note that the acceleration data were forbedrock motions and not free-field motions that can eitherincrease or decrease in terms of PHA due to the site effects.Motions of 20 s duration were modelled to include all of theimportant features of the earthquake. The predominant fre-quency was approximately 2 Hz, which is typical of destruc-tive earthquakes (Kramer 1996).

Verification of the finite element model

The verification process followed incremental steps to en-sure that pile, soil, and boundary conditions were separatelyaccounted for to minimize error accumulation. The size ofthe mesh was mainly dependent on the loading conditions(static or dynamic) and geometry of the piles. The mesh wasrefined near the pile to account for the severe stress gradi-ents and plasticity encountered in the soil, with a gradualtransition to a coarser mesh away from the pile in the hori-zontal X and Y directions. The vertical Z direction subdivi-sions were kept constant to allow for an even distribution ofvertically propagating SH waves. The maximum elementsize, Emax, was less than one-fifth to one-eighth the shortestwavelength ( l ) to ensure accuracy (Kramer 1996), i.e.,[4] Emax ( / / )< -1 5 1 8 lwhere l = Vs /f, in which Vs is the soil shear wave velocity,and f is the excitation frequency in Hz. The minimum Vs was

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Fig. 6. Fourier amplitude spectrum for earthquake loading at thebedrock level.



60 m/s, and the dynamic loading had a cutoff frequencyequal to 20 Hz. Thus, a maximum element length of 0.5 mwas adopted. The proposed element division was verifiedusing results from a sensitivity study focusing on verticalpile shaft discretization by El-Sharnouby and Novak (1985),who found that using 1220 elements gave accurate resultswith a minimum of computational effort. Thus, this rangewas adopted for this study.

The pile mesh was first verified by considering the pile asa fixed cantilever in air (no soil). Lateral deflections result-ing from a static load for three different pile mesh sizes werecompared with those from 1D beam flexure theory as shownin Table 1. As noted in Table 1, the maximum differencewas 8.1%. The results were very close; however, the smalldifferences can be explained because beam theory is not ex-

act (ignoring shear deformations) and the finite model wasnot a perfect cylinder. Since the maximum number of ele-ments (6000) and nodes (11 000) available was limited, 180elements were used to model the pile (accuracy within 8%of theoretical solutions). When soil and boundary elementswere added, the total number of elements was close to thelimit.

The soil was added to the model assuming a homoge-neous soil stratum (Fig. 5). The elastic responses of socketedand floating single piles in the homogeneous soil stratumwere compared with the results from two different analyses:(i) the results from Poulos and Davis (1980) using Mindlinsequations and enforcing pilesoil compatibility; and (ii) theresults presented by Trochanis et al. (1988) using a 3D finiteelement analysis (FEA), although their pile had a square

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Notes 1373

Fig. 7. Response of single socketed pile: (a) elastic, (b) elasticgapping, and (c) plasticgapping.



cross section but the same flexural rigidity. Three differentsoil meshes were built with increasing refinement to deter-mine an acceptable level of accuracy while maintaining acomputationally efficient model. Mesh 1 consisted of 1080elements, mesh 2 consisted of 2640 elements, and mesh 3consisted of 3280 elements. Other meshes with a total num-ber of elements equal to 6000 were also tested but were notused due to unreasonable computer processing time.

The results for the linear elastic response under lateralloading at the pile head are shown in Fig. 7a. The mesh thatyielded the closest match (mesh 3, depicted in Figs. 2 and 3)was used in the analysis. The deflections obtained in thisstudy were slightly greater than those from Poulos and Davis(1980). However, they pointed out that their solution mightunderestimate the response of long piles in soft soils. Fig-ures 7b and 7c show pile head deflections considering sepa-ration at the pilesoil interface and soil plasticity,respectively, and good agreement exists with the results fromTrochanis et al. (1988). The differences in the plastic soilcase may be attributed to the use of a different model forsoil plasticity (modified Drucker-Prager model). Figures 8aand 8b show the elastic soil surface displacements awayfrom the pile compared with the results from elastic theory(Poulos and Davis 1980; other FEA by Trochanis et al.1988). Figures 8a and 8b show that the results obtained us-ing mesh 3 agree well with both solutions, especially closeto the pile. The pressure distribution in the soil agreedequally well.

The final step in the verification process was accom-plished by solving the ground response to an earthquake sig-nal using the finite element model and comparing the elasticfree-field response to that obtained using the programSHAKE91 (Idriss and Sun 1992). Considering thatSHAKE91 is a 1D analysis, constraints were applied to thefinite model to allow only displacements in the direction ofshaking (one degree of freedom per node) to replicate 1D re-sults. The results from the finite element analysis andSHAKE91 are plotted in Fig. 9 for elastic response using thesame parameters. A constant shear modulus and materialdamping ratio were used in both the SHAKE91 and FEAmodels. Figure 9 shows that the agreement is quite goodalong the entire time period considered. The maximum free-field accelerations for FEA and SHAKE91 were both ampli-fied to approximately 0.6g from 0.2g (bedrock input motion)and are compared with bedrock accelerations in Fig. 10. Thesame FEA model was modified to allow for 3D response,and the free-field response is plotted against the 1D resultsin Fig. 11. The maximum free-field acceleration obtainedfrom 3D analysis was only 0.35g (Fig. 11). The accelera-tions calculated from the 3D analysis are closer to those ob-served during actual seismic events. Hence, it was concludedthat the 3D analysis resulted in realistic acceleration magni-tudes. Therefore, all further models discussed herein as-sumed full 3D capability.

Computational time and method

The ANSYS finite element program can solve the dy-namic response of structural systems in either the frequencyor time domain. Because nonlinearity and gapping weredeemed to be important in seismic response, and hence wereintroduced into the model, the time domain was chosen forthe analysis. The response was calculated at intervals of ap-proximately 0.02 s. The Newmark integration method wasused with a = 1/2 and d = 1/4 (where a and d are integrationconstants) to obtain an unconditionally stable scheme. TheModified Newton-Raphson iteration technique was used andconvergence criteria were force and displacement dependent.Static solution processing time averaged between 5 and45 min, whereas the dynamic solutions were extremely timedemanding. For a 20 s earthquake with 0.02 s time intervals,the nonlinear solution with gapping lasted approximately 10days on a Pentium 233 MHz personal computer. FivePentium computers (three 233 MHz and two 266 MHz with64 and 128 Mb RAM, respectively) were used simulta-neously at The University of Western Ontario to optimizetime efficiency.

Numerical study for kinematic interaction

The kinematic effects of piles in a homogeneous soil me-dium were evaluated by comparing acceleration time histo-ries and Fourier spectra of the pile head and the free-field.The same dynamic loading was applied in all cases (i.e.,Loma Prieta data) to the underlying bedrock for a homoge-neous soil profile. Results from seven different pilesoil con-figurations were obtained and are referred to in Figs. 1218.The following notation is used throughout the graphs andliterature to identify each test case (see Fig. 6 for soil and

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Model P (kN) d (m) Error (%)Beam flexure theory 50 0.115 0

200 0.458 0Pile mesh 1 50 0.109 5.2

200 0.435 5.0Pile mesh 2 50 0.108 6.1

200 0.428 6.6Pile mesh 3 50 0.107 7.0

200 0.421 8.1

Note: Deflection d of the pile head is determined according to beam flexuretheory as PL3/3Ep I, where P is the applied static lateral load, L is the pilelength, Ep is the pile Youngs modulus, and I is the pile cross-sectionalmoment of inertia (see Fig. 5). All models assume L = 7.5 m, D = 0.5 m,Ep = 2 107 kPa, and I = p (D/2)4/4. Mesh 1 had 15 vertical divisions and 54elements per division, equaling a total of 810 elements; mesh 2 had 30 verticaldivisions and 12 elements per division, equaling a total of 360 elements; andmesh 1 had 15 vertical divisions and 12 elements per division, equaling a totalof 180 elements.

Table 1. Verification of pile head response as cantilever beam.



pile parameters): (i) EFH refers to the free-field response us-ing linear isotropic viscoelastic constitutive relations (elas-tic, free-field, homogeneous); (ii) PFH refers to the free-field response using a perfect elastic plastic soil model,Druker-Prager criteria (plastic, free-field, homogeneous);(iii) ESNFH refers to the floating single pile head responseusing a linear isotropic viscoelastic soil with no separation atthe pilesoil interface, ratio of pile length to pile depthL/D = 15, ratio of pile Youngs modulus to soil Youngsmodulus Ep /Es = 1000 (elastic single pile, no separation,floating, homogeneous); (iv) ESNSH refers to the socketedsingle pile head response using a linear isotropic viscoelasticsoil with no separation at the pilesoil interface, L/D = 20,Ep /Es = 1000 (elastic, single pile, no separation, socketed,homogeneous); (v) ESSFH is the same as ESNFH, but al-lows for separation at the pilesoil interface (elastic, single

pile, separation, floating, homogeneous); (vi) PSSFH refersto the floating single pile head response using a perfect plas-tic elastic soil model with separation allowed at the pilesoilinterface, L/D = 15, Ep /Es = 1000, effective cohesion inter-cept c = 34 kPa, friction angle f = dilatancy angle y = 16.5(plastic, single pile, separation, floating, homogeneous); and(vii) PSSSH refers to the socketed single pile head responseusing a perfect plastic elastic soil model with separation al-lowed at the pilesoil interface, L/D = 20, Ep /Es = 1000, c =34 kPa, f = y = 16.5 (plastic, single pile, separation, sock-eted, homogeneous).

Results

Figure 12a compares the free-field response for the elasticand plastic soil cases. The difference between the two cases

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Notes 1375

Fig. 8. Comparison of soil displacements (a) along line of loading, and (b) normal to direction of loading.



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Fig. 10. Response of underlying bedrock and free-field for homogeneous soil (using 1D FEA).

Fig. 9. One-dimensional verification of finite element analysis (FEA using ANSYS) with SHAKE91.

Fig. 11. Elastic free-field response for homogeneous soil (EFH) for 1D and 3D analysis.



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Notes 1377

is not evident over the 20 s duration, but a more detailedevaluation is presented in Fig. 15 for the 210 s interval.The acceleration response is slightly amplified using a plas-tic soil model that can be attributed to the limiting ultimateeffective stress and limiting shear strength. Figure 12b com-pares the Fourier spectra for elastic and plastic soil profilesagainst the input bedrock spectrum using a cutoff frequencyof 20 Hz. It is evident from Fig. 12b that there is anamplification of the Fourier amplitudes for the free-field re-sponse compared with that for the bedrock. There is a nota-ble increase in amplitude for the plastic soil model over theelastic soil model, suggesting that the reduction in soil stiff-ness reduces the natural frequency of the homogeneouslayer. The increase in acceleration and amplitude may be at-

tributed to the fact that the first natural frequency of theelastic homogeneous layer is slightly greater than 2 Hz,whereas the natural frequency of the plastic soil layer isslightly decreased and becomes closer to the predominantfrequency at the free-field (approximately 1.5 Hz). Forhigher frequencies (Fig. 12b), the small-amplitude peaksseen at the bedrock level diminish as the seismic wavespropagate throughout the soil until they reach the free-field.Both the elastic and plastic free-field amplitudes diminish atfrequencies higher than 10 Hz, above which little response isinduced in most structures.

Similar results for the floating and socketed pile head re-sponse are plotted in Figs. 13 and 14. Figures 13a and 13brepresent the corresponding acceleration and Fourier spectrum,

Fig. 12. (a) Comparison between calculated accelerations for elastic free-field (EFH) and plastic free-field (PFH) using the Druker-Prager criteria for a homogeneous soil profile. (b) Fourier spectrum for the response at the bedrock level, elastic soil free-field, andplastic soil free-field.



respectively, for ESNFH compared with EFH. Both dia-grams are almost identical, except the Fourier amplitudes areslightly greater for the floating pile (especially for a fre-quency above 5 Hz). Figure 14 shows the response of theESNSH socketed pile case. Again, the overall accelerationof the pile head is similar to that of the elastic free-field. TheFourier amplitudes of the socketed pile (no separation) showboth a decrease and an increase in magnitude over the elas-tic free-field, depending on the frequency range. At the pre-dominant frequency amplitude (2 Hz), ESNSH seems toslightly decrease compared with EFH, and at frequenciesabove and below the predominant frequency the amplitudesare increased. The increased stiffness of the system due tothe socketed pile may be responsible for the increased am-

plitude at higher frequency ranges compared with that of thefree-field.

Figure 16 introduces the effects of separation between thepile and soil for the floating pile case. Only the 210 s timeinterval is shown to provide a more detailed analysis. Theoverall response is very similar for both cases shown inFig. 16. The floating pile with gapping seems to eliminatesmall fluctuations of acceleration seen when no gapping wasallowed.

Figure 17 introduces the effects of the soil plasticity in ad-dition to separation for the floating pile. PSSFH is comparedwith the elastic model (ESSFH) and the results are verysimilar. The random scatter shown by introducing plasticitymay be attributed to the solution procedure used in the finite

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Fig. 13. (a) Comparison between calculated accelerations for elastic free-field (EFH) and floating pile head (ESNFH) for a homoge-neous elastic soil profile. (b) Fourier spectrum for the response of the elastic soil free-field and floating pile head.



element program. For convergence reasons, smaller timesteps had to be used for the plastic soil model which led tonumerical instabilities. Figure 18 compares the floating andsocketed pile head response including both separation andsoil nonlinearity. The floating pile showed slightly higherpeaks over the socketed pile, but the response remained al-most identical.

Summary and conclusions

A 3D finite element analysis was performed to investi-gate site effects and pile kinematic interaction effectsfrom seismic loading. The analysis considered floating

and socketed piles, including nonlinear soil properties,slippage and gapping at the pilesoil interface, and dissi-pation of energy through damping. Based on the resultsfrom the kinematic interaction study, it was concludedthat the pile (floating and socketed) head response closelyresembled the free-field response for the low predominantfrequency seismic loading. Fan et al. (1991) made asimilar conclusion from their parametric study using aboundary element solution.

The following conclusions are drawn:(1) The effect of a soil layer overlaying the bedrock was

to amplify the bedrock motion, resulting in a higher free-field motion for the soil parameters used in the analysis.

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Notes 1379

Fig. 14. (a) Comparison between calculated accelerations for elastic free-field (EFH) and socketed pile head (ESNSH) for a homoge-neous soil profile. (b) Fourier spectrum for the response at the plastic soil free-field and socketed pile head.



(2) The effect of allowing a 3D behaviour as opposed to a1D behaviour, with seismic loading applied in one dimen-sion, was to decrease the acceleration amplitudes by a factorof 1.6 for the soil profiles considered.

(3) The effect of soil plasticity was to increase the Fourieramplitudes at the predominant frequency but to slightly de-crease the maximum acceleration amplitudes.

(4) The elastic kinematic interaction of single piles (bothfloating and socketed) has slightly amplified the bedrockmotion when compared with the free-field response andslightly decreased the Fourier amplitudes of all frequenciesconsidered (020 Hz).

(5) Overall, the kinematic interaction response includingsoil plasticity, slippage and gapping at the pilesoil interface,

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Fig. 16. Pile head response for floating pile (elastic, elastic with gapping).

Fig. 15. EFH and PFH response (elastic and plastic free-field, Es = 20 000 kPa).



and damping is equivalent to the free-field response. How-ever, the conclusions are limited to the pile and soil parame-ters, and to the earthquake loading used in the analysis.

References

ANSYS Inc. 1996. General finite element analysis program. Ver-sion 5.4. ANSYS, Inc., Canonsburg, Pa.

API. 1991. Recommended practice for planning, designing andconstructing fixed offshore platforms. API RecommendedPractice 2A (RP 2A). 19th ed. American Petroleum Institute,Washington, D.C., pp. 4755.

El-Sharnouby, B., and Novak, M. 1985. Static and low frequencyresponse of pile groups. Canadian Geotechnical Journal, 22: 7984.

Fan, K., Gazetas, G., Kaynia, A., Kausel, E., and Ahmad, S. 1991.Kinematic seismic response of single piles and pile groups.

2000 NRC Canada

Notes 1381

Fig. 18. Pile head response for floating and socketed pile (plastic gapping).

Fig. 17. Pile head response for floating pile (elastic gapping, plastic gapping).



2000 NRC Canada


Journal of Geotechnical Engineering, ASCE, 117(12): 18601879.

Idriss, I.M., and Sun, J.I. 1992. Modifications to SHAKE programpublished in Dec. 1972 by Schnabel, Lysmer & Seed. Usersmanual for SHAKE91 (with accompanying program). Center forGeotechnical Modeling, University of California, Berkeley.

Kramer, S.L. 1996. Geotechnical earthquake engineering. Prentice-Hall Inc., Englewood Cliffs, N.J.

Makris, N., and Gazetas, G. 1992. Dynamic pilesoilpile interac-tion. Part II: Lateral and seismic response. Earthquake Engi-neering and Structural Dynamics, 21: 145162.

National Center for Earthquake Engineering Research. 1998. Infor-mation Service, State University of New York at Buffalo, website: http://nceer.eng.buffalo.edu, January 1998.

Novak, M., and Mitwally, H. 1988. Transmitting boundary foraxisymmetrical dilation problems. Journal of Engineering Me-chanics, ASCE, 114(1): 181187.

Poulos, H.G., and Davis, E.H. 1980. Pile foundation analysis anddesign. John Wiley & Sons, New York.

Trochanis, A.M., Bielak, J., and Christiano, P. 1988. A three-dimensional nonlinear study of piles leading to the developmentof a simplified model. Technical Report of Research Sponsoredby the National Science Foundation, Grant ECE-86/1060, Car-negie Mellon University, Washington, D.C.

Wu, G., and Finn, W.D.L. 1997. Dynamic nonlinear analysis ofpile foundations using finite element method in the time domain.Canadian Geotechnical Journal, 34: 4452.



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