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Dynamic soil-structure interaction: pile foundations and retaining structures Wu, Guoxi 1994-04-15

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DYNAMIC SOIL-STRUCTUREINTERACTION:PILE FOUNDATIONS AND RETAININGSTRUCTURESByGuoxi WUB.Eng. Nanjing Institute of Architectural Engineering,1984M.A.Sc. Tongji University, Shanghai, PRC,1987M.A.Sc. The University of British Columbia,Canada, 1992A THESIS SUBMITTED IN PARTIAL FULFILLMENTOFTHE REQUIREMENTS FOR THE DEGREEOFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISHCOLUMBIAAugust 10, 1994©Guoxi WU, 1994In presenting this thesis inpartial fulfilment of the requirements foran advanceddegree at the University of BritishColumbia, I agree that the Library shallmake itfreely available for reference andstudy. I further agree that permissionfor extensivecopying of this thesis for scholarly purposesmay be granted by the headof mydepartment or by his or herrepresentatives. it is understood thatcopying orpublication of this thesis forfinancial gain shall not be allowed withoutmy writtenpermission.(Signature)Department of C-tIThe University of British ColumbiaVancouver, CanadaDate0cf.7(99DE-6 (2188)ABSTRACTThis thesis deals with two important topics in soil-structureinteraction: seismic earthpressures against rigid walls and the seismic responseof pile foundations. These two disparate problems are linked by a common methodof solution which is an approximationto the response of the half-space, either linearor non-linear.The approximate formulation permits analytical solutions againstrigid walls whenthe backfill is uniform and elastic. The solutionagrees very closely with an existing exact solution. For elastic non-homogeneous backfillsand for non-linear soil response theapproximate formulation is expressed using the finiteelement method.An efficient computer program SPAW has been developedto determine dynamicthrusts and moments against rigid walls for arbitrarynon-homogeneous soil layers. Results of analyses show that the peak dynamic thrustsare larger for a uniform soil profilethan when the shear modulus of the soil varieslinearly or parabolically with depth. Theprogram SPAW also possesses the abilityof modelling the effect of soil non-linearity ondynamic thrusts. Studies showed that an increaseof peak dynamic thrust may be expected due to soil non-linearity, compared with resultsfrom a linear elastic analysis.A quasi-3D finite element method of analysis hasbeen proposed to determine dynamic response ofpile foundations subjected to horizontalloading. A computer programPlUMP has been developed for the analysesof elastic response of pile foundations including the determination of pile impedances asa function of frequency. The analysis11is conducted in the frequency domain.The program can analyze single pilesand pilegroups in arbitrary non-homogeneous soil layers.Another quasi-3D finite element computerprogram PILE3D has been developedforthe analysis of non-linear responseof pile foundations in the time domain.The programis suitable of dynamic analyses of singlepiles and pile groups. The soil non-linearityduring shaking is modelled using a modifiedequivalent linear method. Yieldingof thesoil is taken into account and there isa no-tension option controffing theanalysis.The proposed quasi-3D model has been validatedusing the elastic solutions fromKaynia and Kausel (1982), Novak andNoganii (1977) and Novak (1974), Fanet al.(1991), data from full scale vibrationtests of a single pile and a 6-pile group,and datafrom centrifuge tests of a single pile anda 2x2 pile group under strong shaking fromsimulated earthquake. Excellent resultshave been obtained using the proposed method.Time-dependent variations of dynamicimpedances of pile foundations duringshakinghave been evaluated for the modelpile foundations used in the centrifugetests. Theanalyses quantify the reduction in thestiffnesses ofthe pile foundations with theincreasedlevel ofshaking. The translational stiffnessdecreases the most during strong shaking;the rotational stiffness k60 decreases theleast. However, the damping of pile foundationsincreases with the level of shaking.UiTable of ContentsABSTRACTiiList of TablesviiiList of FiguresixAcknowledgementxv1 Introduction1I Dynamic Thrusts onRigid Walls62 Dynamic analyses ofrigid walls72.1 Literature review72.2 Objectives of current research123 Dynamic Thrusts on Rigid Wallswith Uniform Elastic Backfllls143.1 Introduction143.2 Dynamic analysis of rigid wall-soilsystem153.3 Static 1-g solution: Validation of model243.4 Dynamic thrusts under sinusoidalmotions313.5 Dynamic thrusts under earthquake motions353.6 Accuracy of the response spectrum method40iv4 Dynamic Thrusts on Rigid Walls withNon-homogeneous Soil Profiles4.1 Introduction4.2 Finite element formulation and its validation4.3 Linear elastic analyses with non-homogeneoussoil profiles4.4 Equivalent linear simulation of non-linear responseunder earthquake loadsII Dynamic Analyses of Pile Foundations665 Dynamic analyses of pile foundations5.1 Dynamic analyses of single pile response5.2 Dynamic analysis of pile groups5.3 Objectives of this research6 Elastic Response of Single Piles: Theoryand Verification6.1 Introduction6.2 Dynamic analyses of pile foundations: formulation6.3 Pile head impedances6.4 Verification of the proposed model: pile headimpedances . .6.5 Verification of the proposed model: kinematic interaction6.5.1 Kinematic interaction factors6.5.2 Computed kinematic interaction factors6.6 Verification of the proposed model: forced vibration testing6.6.1 Description of site condition and test results6.6.2 Computed results using the quasi-3D model7 Elastic Response of Pile Groups: Theoryand Verification 1107.1 Introduction1104949505561676776788080818689989999100100104V7.2 Rocking impedance of pile group1107.3 Dynamic equation of motions in the vertical direction1127.4 Determination of rocking impedance1147.5 Elastic response of pile group: results and comparisons1167.6 Full-scale vibration test on a 6-pile group1227.6.1 Description of vibration and its testing results. 1227.6.2 Computed results using the proposed model1258 Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction1328.1 Introduction1328.2 Quasi-3D finite element analysis in the time domain1338.3 Solution scheme for dynamic equation1358.4 Non-linear analysis1368.5 Features in dealing with yielding, tension1418.6 Soil parameters required in PILE3D analysis 1428.7 Aspects relative to analysis of pile group1459 Analyses of Centrifuge Tests of Pile Foundations1489.1 Introduction1489.2 Dynamic analysis of centrifuge test of a single pile1489.2.1 Description of centrifuge test on a single pile1489.2.2 Dynamic analysis of the single pile1519.2.3 Non-linear pile impedances1589.2.4 Computational times1699.3 Dynamic analysis of centrifuge test of a pile group 1699.3.1 Description of centrifuge test on a 4-pile group (2x2)1699.3.2 Dynamic analysis of the pile group170vi9.3.3 Non-linear impedances of the 4-pile group177III Summary and Suggestions for Future Work17810 Summary and Suggestions for Future Work17910.1 Dynamic thrusts on rigid walls17910.2 Dynamic analyses of pile foundations183Bibliography188viiList of Tables3.1 Peak dynamic thrusts for walls with L/H=5.0and 11=10 m, )=10% . . .383.2 Peak dynamic thrusts for walls with L/H=1.5and 11=10 m,A=10% . . . 394.3 Patterns offirst natural frequencies forthree types of soil profiles (wa,rad/sec)616.4 Structural properties of pile cap and test pile(after Sy and Siu, 1992) . . 1057.5 Computed resonant frequencies and damping ratios withoutthe effect ofpile cap embedment1287.6 Computed stiffness and damping of the transformer pilefoundation . 1297.7 Measured and computed resonant frequencies and dampingratios including the effect of pile cap embedment1318.8 Relationship between Hardin and Drnevich constant k and plasticityindexPT (after Hardin and Drnevich, 1972)1449.9 Parameters of dynamic impedances of single pile167vi”List of Figures2.1 Wall-soil system used in Wood’s study(after Wood, 1973)93.1 Definition of rigid-wall problem (a) originalproblem (b) equivalent problem by using antisymmetric condition163.2 Comparison ofthe accuracy ofapproximate solutionsfor rigid-wall systems(a) L/H=5.0 (b) L/H=1.5273.3 Normalized thrust ratios for 1-g static solution(a)Wood’s solution (b)author’s solution293.4 Heights of thrusts due to 1-g static horizontal force(wall height H=lOm) 303.5 Accuracy of solutions versus number of modes used323.6 Normalized thrust ratios for sinusoidal motions (a) L/H=5.0 (b)L/H =1.5 343.7 Time histories of dynamic thrusts using the El Centroinput (a) L/H=5.0(b) L/H=1.5363.8 Time histories ofdynamic thrusts using the Loma Prieta input (a)L/H=5.0(b) L/H=1.5373.9 A time history of the height of dynamic thrust,L/H=5.0 393.10 Normalized thrust ratios versusfRifor earthquake motions (a) L/H=5.0(b) L/H=1.5413.11 Normalized thrust ratios versusfR2for earthquake motions (a) L/H=5.0(b) L/H=1.5423.12 Pseudo-spectral velocities of(a) the El Centro input and (b) the LomaPrieta input44ix3.13 Variations of thrust factorCp versus frequency ratiofRi(RSS method)(A) L/Hz=5.0, (B) L/H=1.5463.14 Variations of thrust factorCp versus frequency ratiofRi(ABS method)(A) L/H=5.0, (B) L/H=1.5474.1 A composition of non-homogenoussoil profile504.2 A composition of the finite elementused in SPAW514.3 A finite element mesh used for dynamicanalyses534.4 Comparisons of dynamic thrustsbetween the F.E. method and theclose-form solution for uniform soils(a)L/H=5 (b) L/11=1.5544.5 Relationships between thrust ratioand frequency ratiofR2for linear soilprofiles (a) sinusoidal motions (b)the El Centro input574.6 Relationships between thrust ratioand frequency ratiofR2for parabolicsoil profiles (a) sinusoidal motions(b) the El Centro input584.7 Comparison ofdynamic thrust ratiosfor parabolic soil profiles and uniformsoil profiles under sinusoidal motions (L/H=5)594.8 Typical time histories of heightsof dynamic thrusts for three types ofsoilprofiles (H=lOm)604.9 Dynamic responses ofa stiff site due to non-linear effect,G0=132,000kPa 634.10 Effect of level of shaking on thedynamic thrust,G0=132,000 kPa . . .644.11 Dynamic responses of a soft sitedue to non-linear effects,Go=66,000 kPa645.12 Variation of pile horizontal stiffness,k, with force and frequency duetosoil non-linearity (after Angelidesand Roesset, 1981)736.1 The principle of quasi-3D dynamicpile-soil interaction in the horizontaldirection82x6.2 Finite element compositions formodelling horizontal motions836.3 Pile head impedances866.4 A pile-soil system used for computing impedancesof single piles906.5 Finite element modelling of single pile for computingimpedances916.6 Normalized stiffness k,,, and dampingC,,,, versusa0for single piles (E/E31000, v =0.4, A=5%)926.7 Normalized stiffness k,,8and dampingC,, versusa0for single piles (E/E3=1000, ii =0.4, )=5%)936.8 Normalized stiffnessk88 and dampingC89 versusa0for single piles (E/E8=1000, v =0.4, A=5%)946.9 Comparison of stiffness k,,,, and dampingC with solutions by Novak andNogami (1977), Novak (1974)966.10 Comparison of stiffnessk,,,, for different mesh size976.11 Pile foundation for analysis of kinematic response986.12 Kinematic interaction factors versusa0 for E/E8= 1, 0001016.13 Kinematic interaction factors versusa0 for E/E3= 10, 0001026.14 The in-situ measured geotechnicaldata (after Sy and Siu, 1992)1036.15 The layout of the full-scale vibrationtest on a single pile (afterSy andSiu, 1992)1046.16 The soil parameters used in the analysis(afterSy and Siu, 1992) . . . 1066.17 Finite element modelling of the expandedbase pile1066.18 An uncoupled system modeffingthe horizontal motions of structure-pilecap system1076.19 Amplitudes of horizontaldisplacement at the centre of gravityof the pilecap versus the excitation frequency10837.1 The mechanism of rocking in a pilegroup1117.2 The quasi-3D model in the verticaldirection, Z1137.3 A pile-soil system used for computing impedancesof pile groups 1167.4 Comparison ofdynamic interactionfactorawith solution by Kaynia andKausel for 2x2 pile groups (E/E8=1000,s/d = 5.0)1197.5 Dynamic interaction factors a,,aye, aes versus a0 for 2x2 pile groups(E/E8= 1000,s/d = 5.0)1207.6 Comparison of normalized total rotationalimpedance K7/A with solution by Kaynia and Kausel for 2x2pile groups (E/E8= 1000, s/d=5, A=N*Y2rk°2)1217.7 Idealized soil profile at Duwamish Substation(after Crouse and Cheang,1987)1237.8 Setup of a full-scale free vibrationtest on a 6-pile group (after Crouse andCheang, 1987)1247.9 3-D finite element models of the 6-pile foundation(a) NS direction, (b)EW direction1277.10 Response curves ofthe oftransformer-pilecap system (a) NS direction (b)EW direction1308.1 Hysteretic stress - strain relationshipsat different strain amplitudes . .1378.2 Relationships between shear moduli,damping ratios and shear strains(after Seed and Idriss, 1970& Seed et aL,1986)1388.3 The principle of modified equivalent linearmethod1408.4 Simulations of shear yielding andtension cut-off1418.5 Comparison of damping ratios for sands andgravelly soils (after Seed etal., 1986)145xii8.6 A diagram showing the representation of pilegroup supporting structure1469.1 The layout of the centrifuge test fora single pile1509.2 The prototype model of the singlepile test1509.3 Computed Fourier amplitude ratios (a) pileamplitude ratio (APH/AFF)(b) free field amplitude ratio (AFF/AB)(after Gohl, 1991)1529.4 The finite element modeffing of centrifugetest1539.5 The relationships between shear modulus,damping and the shear strainfor the loose sand1549.6 The computed versus measured accelerationresponse at the free field .1559.7 The computed versus measured accelerationresponse at the pile head .1569.8 The computed versus measured displacementresponse at the top of thestructure1569.9 The computed versus measured moment responseat the soil surface .1579.10 The computed versus measured moment responseat depth D=3m . . . 1579.11 The computed versus measured momentdistribution of the pile at peakpile deflection1589.12 3-D plots of the distribution of shearmoduli at t=12.58 sec1599.13 3-D plots of the distribution of shearmoduli at t=17.11 sec1609.14 Variation of stiffnesses k,k9,k98 of the single pile at f=1.91 Hz . . . 1639.15 Variation of stiffnesses kand k98 with time under different excitationfrequency1639.16 Variation of translational dampingC versus time under different frequency1649.17 Variation of hysteretic dampingsC,,C6andC of the single pile . .. 1659.18 Variation of radiation damping constantat different frequencies . . 166xlii9.19 Comparison of dynamic stiffnesses ofpile foundations with full structuralmass and without structural mass1689.20 The layout of centrifuge test for 4-pile group(after Gohi, 1991)1719.21 Finite element modelling of the 2x2pile group1739.22 The relationships between shearmodulus, damping and the shear strainfor the dense sand1749.23 The computed versus measuredacceleration responses at pile cap .. 1759.24 The computed versus measured displacementat top of structural mass 1759.25 The computed versus measuredmoment at depth D=2.63 m1769.26 Distribution of moments at peakpile cap displacement1769.27 Variation of stiffnessesk8 of the 4-pile group at f=1.91 Hz 177xivAcknowledgementThe author sincerely thanks his research supervisor, ProfessorW.D. Liam Finn, for hisguidance, suggestions and encouragement during thecourse of research and preparationof this thesis.The author also wishes to thank Professors P.M. Byrne, Y.P.Vaid, Dr. R. J. Fanninof UBC and Dr. M. K. Lee of B. C. Hydro for serving as membersof the supervisorycommittee and for reviewing the manuscript. Appreciationis also extended to otherfaculty members in the Dept. of Civil Engineeringfor offering many excellent courseswhich established the cornerstone of this research.The postgraduate fellowship awarded by the Canadian National Scienceand Engineering Research Council and the research assistantship providedby the University ofBritish Columbia are gratefully acknowledged.Finally, the author would like to thank his wife Liffian forher love and faithful supportover the course ofpreparation of this thesis. This thesis is dedicatedto her and our sons,Galen and Allan.xvChapter 1IntroductionThis thesis deals with two important topics in soil-structureinteraction: seismic earthpressures against rigid walls and the seismic responseof pile foundations. These twodisparate problems are linked by a common methodof solution based on an approximation to the response of the half-space, either linearor non-linear. In the case of the pilefoundations, the piles are modelled as linear beamelement inclusions in the half space.The rigid wall solution has important applications: seismicearth pressures againstdeep basement walls, buried containment structuresand the wingwalls of dams. Theseismic analysis of pile foundations remainsa challenging problem both in engineeringpractice and in research. The action of pile foundationsis a key element in evaluating properly the response of pile-supported buildings,bridges and offshore platforms toearthquake loading. Characterization of the stiffnessand damping of pile foundations isa complex major task for large bridges with multiplepoints ofseismic input. The methoddeveloped in this thesis for solving these twomajor soil-structure interaction problems iscapable of simulating important featuresof seismic interaction.The methodology used for solving the rigid-wallproblem is essentially a 2-D planestrain application of the quasi-3D theory developedfor dealing with more general 3-Dsoil-structure systems subjected to horizontal shaking.The basic idea of the proposedquasi-3D theory is that the dynamic motions excitedin a 3-D half-space by shear waves1Chapter 1. Introduction2propagating vertically is governed primarily by compression wavespropagating in theprincipal shaking direction and shear waves propagatingin the two other directions.Other types of waves in the 3-D half-space are ignoredin the analysis because they areassumed to be less significant. This assumption willbe validated later using elastic solution based on a full 3-D formulation.In the seismic analysis and design of rigid-wall systems, thebasic challenge is to evaluate the magnitude and distribution of dynamicsoil pressures against the walls inducedby ground shaking. In addition to the evaluationof dynamic soil pressures against rigidwalls with elastic homogeneous soil backfills, thereare two important issues relative torigid walls. The first issue is the accurate evaluationof dynamic soil pressures againstrigid walls with arbitrary non-homogeneous backfills.The second issue is the appropriatemodelling of soil non-linearity under relatively strongshaking. Therefore, a portion ofthis thesis is devoted to developing a cost-effectivemethod for dealing with arbitrarynon-homogeneous soil proffle and soil non-linearity.In the first part ofthis thesis, the 2-D plane strain formulationofthe quasi-3D theoryis used to obtain solutions of dynamic soil pressures againstrigid walls with uniformbackfills. Then the theory is implemented into aneffective finite element program. Thefinite element method of analysis is usedto explore the effect of soil non-homogeneity ondynamic pressures on rigid walls. Effect of soil non-linearity under strongshaking is alsoinvestigated.The second part of this thesis is devoted to dynamic responseof pile foundationssubjected to horizontal shaking. Pile foundations arewidely used in civil engineeringChapter 1. Introduction3works. Pile-group foundations are used to supportimportant structures such as high-rise buildings, bridges, and large power transmissiontowers. When these pile-supportedstructures are located in a seismic active zone, concernsarise on how piles, either individually or in groups, respond to earthquake loading.Many studies have been conducted onthe dynamic response of pile foundations, most of whichare restricted to elastic response.This thesis describes the development of a cost-effective andreliable numerical procedure which can be used to study dynamic responseof pile foundations when foundationsoils are non-linear and nonhomogeneous. The quasi-3Dtheory is adopted to the dynamicanalyses ofpile foundations. A number ofsolutionsfor elastic homogeneous response havebeen developed in order to validate the proposedquasi-3D method by comparing the results with the published elastic solutions based onfull-3D formulation.OUTLINE OF THE THESISChapter 2 gives a review of existing methods fordetermining dynamic soil pressureson rigid walls and presents objectives of the presentresearch study.Chapter 3 gives a simplified method ofanalysis for rigidwall-soil systems subjected tohorizontal dynamic loads assuming linearelastic response ofthe soil. Analytical solutionsof dynamic soil pressures against rigidwalls with homogeneous soil backfills are derivedfirst. Dynamic thrusts against rigid walls are determinedusing both sinusoidal motionsand earthquake motions as input. The results fromthe present analysis are validated bycomparison with the published results developedfrom full 2-D elastic response analysisby Wood (1973). Studies are made to examinethe accuracy of the response spectrummethod for determining peak dynamic thrustagainst rigid walls.Chapter 1. Introduction4Chapter 4 extends the method of analysis to dealwith rigid walls with an arbitrarynonhomogeneous soil profile. Analysesare performed to study the pattern of dynamicthrusts against rigid walls for nonhomogeneous soilprofiles. Finally the effect of soilnon-linearity on dynamic response of rigid wallsis explored by using the equivalent linear approach (Seed and Idriss, 1967).Chapter 5 gives a review for current methods of dynamic responseanalysis of pilefoundations and outlines the objectives of this thesis forthe analysis of pile foundations.Chapter 6 presents a quasi-3D finite element methodfor dynamic response analysisof pile foundations assuming elastic response of soiland piles. The proposed model isverified first against elastic solutions of single piles by Kayniaand Kausel (1982). Thenthe proposed model is calibrated using datafrom afull-scale vibrationtest on a single pile.Chapter 7 applies the proposed quasi-3D theory to the dynamicanalysis of elasticresponse ofpile groups. The proposed model for pile groups is verifiedfirst against elasticsolutions by Kaynia and Kausel (1982) and then against results ofa field vibration teston a 6-pile group by Crouse and Cheang (1987).Chapter 8 applies the proposed quasi-3D theory to the dynamicanalysis of pile foundations under earthquake loading taking the non-linear responseofthe soil into account.The non-linear finite element analysis is conducted in the timedomain. Procedures formodeffing non-linear soil response are also described inthis chapter.Chapter 1. Introduction5Chapter 9 describes the validation of the proposedmodel for non-linear dynamic response of pile foundations using data from centrifugetests of a single pile and a 2x2pile group. The variations of dynamic impedancesof pile foundations with time duringshaking are also evaluated and demonstrated forthe model pile foundations.Chapter 10 summarizes the developments describedin earlier chapters and presentsthe conclusions arising from the variousstudies.Part IDynamic Thrusts on Rigid Walls6Chapter 2Dynamic analyses of rigidwalls2.1 Literature reviewFor seismic design of a rigid wall it is importantto know the magnitude and distributionof seismic pressure on the wall inducedby earthquake motion. Probably theearliestresearches dealing with seismic inducedearth pressure on retaining structureswere thoseof Mononobe (1929) and Okabe(1926). The Mononobe-Okabe methodis the modification of Coulomb’s classic earth pressure theorywhich takes into account the inertiaforces caused by earthquake accelerations.Seed and Whitman (1970) made a detailedevaluation of the Mononobe-Okabe method.One of the basic requirements of applyingthe Mononobe - Okabe method is thatthe wall has to move sufficiently to createa limitequilibrium state in the backfill.This condition is not satisfied in most rigid wallcases.Several researchers have used elasticwave theory to derive seismic backfillpressureagainst a rigid wall. Matuo and Ohara(1960) obtained an approximate elasticsolution for the dynamic soil pressure ona rigid wall using a two-dimensionalanalyticalmodel. They simplified the problemby assuming zero vertical displacement inthe soilmass. This simplification leads to infinitelylarge wall pressure when Poisson’sratio ofthe soil is equal to 0.5 asin a fully saturated undrained backfill. Scott(1973) used a onedimensional elastic shear beam connectedto the wall by Winkler springs tomodel theproblem. The advantage ofthis model is that close-formsolutions can easily be obtained.7Chapter 2. Dynamic analysesofrigid walls8The disadvantage is that his solutionrequires a representative valueofthe Winkler springconstant. Scott used Wood’s(1973) solutions to define the characteristicsof the Winkler spring constant. However,the Winkler spring constant was determinedonly for thefirst-mode motion. For a wall withrelatively long backfill(L*/H>4),the accuracy ofScott’s first-mode solution deterioratesrapidly with the increase ofL*/Hratio.Wood (1973) made a comprehensivestudy on the behaviour of the rigid soil-retainingstructures subjected to earthquakemotions. His work is considered tobe one ofthe moreimportant contributions to understandingof this problem.Wood’s solution of rigid-wallproblem. In Wood’s solution,the soil is assumedto be an isotropic homogeneous elasticmaterial. The wall-soil system wasassumed tosatisfy the condition of planestrain. The analytical solutions were obtainedassuminga smooth contact between the walland the backfill; that is, the verticalboundaries areassumed to be free from shear stresses.Figure 2.1 shows the rigid-wall soil systemusedin Wood’s study and the associated boundaryconditions.The modal frequenciesWnm for the wall-soil system shown in Figure 2.1 are not explicitly expressed in Wood’s solution.For the homogeneous backfill, the modalfrequencyWnmis governed by2=r2—(2.1)Chapter 2. Dynamic analyses ofrigid walls9RIGID WALL PROBLEMFigure 2.1: Wall-soil system used in Wood’s study(after Wood, 1973)and32_r2’flfl(2.2)where V,V3 are the compression wave velocityand shear wave velocity ofsoil, r =and parameters c,/3 have particular valuesfor each mode. They must satisfy the following equation(2+;+(r2}sinhaH. sinh/3H —{4r2+‘2}coshaH.cosh/3H+4(r+/3)=O (2.3)The roots of the frequency equations were numerically evaluatedusing Newton’smethod on a digital computer. An iterativeprocess was used to compute the rootsindicated by sign changes in the frequency equations.Homogenous elastic soil(Plane strain)Chapter 2. Dynamic analyses ofrigid walls10It can be seen that the frequency solution is too complicated toapply in practice.Although graphic solutions of modal frequency weregiven by Wood, these solutions arelimited to particular values of Poisson’s ratio. Furthermore modalfrequencies are notavailable when the soil backfill is non-homogeneous,which is normally the case encountered in reality.The complex-amplitudes of steady-state wall force for a one-g-amplitudeharmonicbase forcing were presented in Wood’s study asF’(w) =Fnm(2.4)1(1)+2inmWflmwhereFnmis the static-one-g modal force per unit length. MathematicalexpressionofFnm is notavailable in Wood’s solution, but graphic resultsofFnm werepresented.The one-g static force per unit length of wall was obtained by applyinga static horizontal acceleration of one-g throughout the soil layer.The static force acting on the walldue to 1-g static horizontal loading,F8,is expressed as7H22135{2BsinhrH+C(2rH+k’+1)e+D(—2rH+k’+ 1)e’” — (C +D)(k’ + 1) — k2rH} (2.5)Where constants C,,D,B are parameters associated with theflthmode and properties of soil. Their detailed formulations can be found in Wood (1973).For aL*/Hratio of 5.0, the forceF8rcan be approximated satisfactorily using about20 terms in Eq.2.5. A smaller number of terms are required forL*/Hless than 5.0 andChapter 2. Dynamic analyses ofrigid walls11a larger number forL*/Hgreater than 5.0.Wood’s solution is quite complicated even under very idealisticconditions of a homogeneous soil layer under harmonic loading. Therefore, in engineeringpractice Wood’ssolution is applied approximately. The dynamic thrust is approximately takenas -yH2A(Whitman, 1991) for a horizontal acceleration Ag, which is the static force fora wall-soilsystem withL*/H=10and 11 = 0.4. The use ofthis expression must be justifiedin manycases especially at resonance when the fundamental frequencyof the wall-soil system isvery close to the predominant frequency of the excitation motion. At resonantconditionthe dynamic thrust are likely to be much greater than the static force.For earthquake type of loading, Wood (1973) states “In view of the uncertaintiesinherent in the estimation of earthquake-induced pressures on walls, computationof theresponse time-history is probably not warranted and approximate evaluationmay oftenbe satisfactory”. Wood proposed that the response spectrum methodis applied to estimate the peak dynamic thrust on the wall for earthquake loading.Although Wood’s solution for dynamic thrust on rigid walls is mathematicallycorrectfor harmonic loading, uncertainties and difficulties arise whenhis method is applied inpractice to the wall-soil system under earthquake loading. Significant errorsof dynamicthrusts may be caused by using the response spectrum method proposedby Wood.The other restriction of Wood’s solution is that his method is not capableof dealingwith rigid walls having arbitrarily nonhomogeneous backfills.In addition the effect ofsoil non-linearity on the dynamic thrust under strong motions cannot beassessed usingWood’s method.Chapter 2. Dynamic analyses ofrigid walls122.2 Objectives of current researchDue to the limitations of Wood’s solution in engineeringpractice, the current research istargeted to find an effective method for determiningthe dynamic thrusts on rigid wallssubjected to horizontal dynamic loads. A proposedmethod is used to analyze wall-soilsystems with nonhomogeneous soil profiles and tostudy the effect of soil non-linearityon dynamic response.The method of analysis is formulatedbased on simplified elastic wave equations. Theequations of motion are established consideringdynamic force equilibrium in the horizontal direction only. An analytical solution for dynamicpressures against rigid wallsis derived first for homogeneous backfihls.Under harmonic horizontal loading, peak dynamic thrusts against rigid walls are determined.For earthquake loading, time-history response of dynamicthrust is determined usingthe mode-superposition method. .The time-historyresponse of each mode is added up inphase. The commonly used Response Spectrummethod is also used to predict the peakdynamic thrust. The accuracy of the response spectrummethod is evaluated against therigorous mode-superposition method.The proposed method is then applied to the wall-soil systemswith nonhomogeneousbackfills. A finite element program SPAW(Seismic Pressures Against Rigid Walls) wasdeveloped to implement the analysis. The computationaltime and cost for time-historyanalysis is negligible using SPAW. The dynamicanalyses of nonhomogenous soil profilesChapter 2. Dynamic analyses ofrigid walls13are illustrated through two types of soilprofiles, soil profiles with linear and parabolicvariations ofshear moduli with depth. Analysesare performed for both harmonic loadingand earthquake loading assuming elasticresponse.The effect ofsoil non-linearity on dynamicthrusts against rigid walls was also studied.The equivalent linear approach (Seed andIdriss, 1967) is used to model non-linear soilresponse. The effect of the intensity of shakingon dynamic thrusts against rigid walls isinvestigated for soil profiles with parabolicvariations of shear moduli with depth.Chapter 3Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfihls3.1 IntroductionA method is proposed for determining the dynamic thrust against arigid soil-retainingstructure subjected to horizontal dynamic loads. The method isbased on simplified elastic wave equations. The equation of motion is derived from dynamicforce equilibrium inthe horizontal direction only. Two types of waves, shear waves and compression waves,are considered to define the horizontal motions of the backfills.The dynamic equation ofmotion is solved analytically taking into account the boundary conditions of the problem. The solution is applied first to obtainthe total thrustacting on the wall due to horizontal one-g static loading to estimate the accuracy of theapproximate solutions against Wood’s exact solutions from the 2-Dplane strain analysis.Simple explicit expressions for computing the dynamic thrusts are presented for bothsinusoidal motion and earthquake motion. Time history solutionsof the dynamic thrustsfor earthquake motions can be readily obtained using the mode superposition method.Studies were made to examine the different patterns of dynamicthrusts for harmonicloading and earthquake loading.Finally the response spectrum method is applied to obtainthe peak dynamic thrusts14Chapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills 15for earthquake motions. The accuracy of the responsespectrum method is also investigated.3.2 Dynamic analysis of rigid wall-soil systemFigure 3.1(a) shows the geometry ofthe problem andits boundary conditions. A uniformelastic soil layer is confined by two vertical rigidwalls at its two side boundaries and arigid base. The soil layer has a total length of 2L andheight of H. Subjected to horizontalseismic body force, the soil layer in the system generatesan antisymmetric field of horizontal normal stresses o with zero stresses atx = L. The original wall-soil problemcan be equivalently represented by half of its geometryusing this antisymmetric condition. The equivalent problem is shown in Figure3.1(b), and this is the physical modelthat will be analyzed. The ground accelerationis input at the base ofthe wall-soil system.The soil is assumed to be a homogeneous, isotropic, visco-elasticsolid with a shearmodulus G and Poisson’s ratio v. Theequations of dynamic force equilibrium for thebackfill in the horizontal and vertical directionsare written as50 9r 82u(3.1)---+--=(3.2)where o and o, are the normal stressesin the X and Y directions, respectively, andris the shear stress in the x-y plane. u and v arethe displacements in the X and Ydirections, respectively.Chapter 3. Dynamic Thrusts on Rigid Walls with UniformElastic Backfills16(a)Yôu/ôy=Ohomogeneous elastic soil(plane strain)H u=Ou=O(b)yôuIôy=Ohomogenous elastic soil(plane strain)H u=OôuIôxOU=OLFigure 3.1: Definition of rigid-wall problem(a) original problem (b) equivalent problemby using antisymmetric conditionChapter 3. Dynamic Thrustson Rigid Walls with Uniform Elastic Backfills17For the two-dimensional plane strain conditions,the stress components are related tothe displacements by2G ôu Ov= 1—2l1:1— v)— +v](3.3)2G 8v 8ua.y=(3.4)ôu Ov(3.5)Although the problem involvestwo displacement components u and v, onlythe horizontaldisplacement u is taken into accountin the analysis to simply the solution oftheproblem.The equation of dynamic force equilibriumin the horizontal direction Eq.3.1 isused inthe analysis. Considering various formsof approximation to the problem, the governingequation of the undamped free vibrationof the backfill in the horizontal direction can bewritten as82u c92u+G— = (3.6)and the normal stress o, is given by(3.7)wherepis the mass density ofthesoil backfill, t is time, 6 and /3 are functions ofPoisson’sratio v.Eq. 3.6 suggests that the dynamic responseofthe backfill is governed by two types ofwaves: the shear wave and the compressionwave. The compression wave in thebackfillproduces the dynamic earth pressureagainst the rigid wall given by Eq.3.7.Chapter 3. Dynamic Thrusts on Rigid Wallswith Uniform Elastic Baclthils18The precise expressions for 6 and /3 dependon the approximations used to model thewall-soil system. Three cases are examined andtheir corresponding expressions for 6 and/3 are given below.(i). v=0 assumption. In this case, thedisplacements in the vertical directionY areassumed to be zero.Using this assumption, o, andr,are obtainedfrom Eq. 3.3 and Eq. 3.5 as2(1—v) Elu— G— (3.8)1—2v ox9u(3.9)Substituting Eq. 3.8 and Eq. 3.9into Eq. 3.1 and comparing with Eq.3.6, one finds6/3= 2(1(3.10)1 — 2v(ii). o=O assumption. In this case,the normal stresses in the vertical directionYare assumed to be zero.Applying the assumption to Eq. 3.4, onefinds thatEly 1) Ott311Ely 1—vOxTherefore,o andOT1,/Oy are obtained fromEq. 3.3 and Eq. 3.5 as2 Elu= G— (3.12)1—i’ OxChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills191)313—1_vôx2L )Substituting Eq. 3.12 and Eq. 3.13 into Eq. 3.1 and comparing with Eq.3.6, one finds(3.14)(3.15)(iii). the proposed model. In this case, the shear stresses are modelledusing the shearbeam analogy(seeAppendix 1 for detail).In this model, the shear stressris given by— (3.16)The normal stress o is found by assuming o,=O in the backfill2 ott= (3.17)1—LI OXSubstituting Eq. 3.16 and Eq. 3.17 into Eq. 3.1 and comparing withEq.3.6, one finds21—tiThree different models yield three different expressionsfor the coefficient 8 in Eq.3.6. However the dynamic response of the wall-soil system for these threecases can berepresented by the same equation, Eq. 3.6. Therefore the general derivationof dynamicChapter 3. Dynamic Thrusts on RigidWalls with Uniform Elastic Baclcfills 20solutions proceeds from Eq. 3.6.Assume the displacement solution has the formu(x,y,t) = (A. sinamx + B cosamx)(C sinby+D cosby)Ymn(t)Applying the boundary conditionsu—0 at..y=0u=0 at..x=0The constants B and D are determinedto be zero.u(x,y,t) = sinamx sinby Ymn(t)= . COSam . sinby . Ymn(t)= E2C1bsinamxcosbyYmn(t)Applying the other two boundary conditionsOu—=0 at..z=Lauat..y=Hone obtainsam COS amL = 0cosbH = 0Chapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills 21thereforeb(2n—1)7r2H(2m — 1)iram—2LThe mode shape functions are written as4’mn(,y) =C1sinby . sina,x (3.20)and the displacement solution becomesu(x,y,t)Emn(,y) Ymn(t) (3.21)Substituting Eq. (3.21) into 3.6, one obtains—G(6a+b) Ymn(t)pmn(t)—a +b)=____=W?flflthe natural frequencies of the system are found tobe= +8a) (3.22)the frequency of the first mode isw1=4pH2(’8L2)(3.23)In the case of an undamped forced vibration subjectedto a ground accelerationi0(t),the governing equation becomes— (8G— +G—) = —püo(t) (3.24)Chapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfihls 22Substituting Eq. (3.21) into Eq. (3.24), multiplyingthe equation by the mode shapefunctions, and integrating over the domain, one obtainsthatJJpij%j(t)4)mndXdy+JJG(8a+bj)4ijYtj4’mndxdy = _iio(t)JJpmn(x,y).ckdyApplying the orthogonality conditions and recallingEq. (3.22), one obtainsJJpdxdy. mn(t) +JJpdxdyWYmn(t) =_ü0(t)JJpmn(X,y)Ymn(t) + Ymn(t) =_i10(t)mnwhere— ffpsin(a,mx) .sin(b1y)dx.dyffp sin (1mm) sin (by)dx4y16amn= 2(3.25)(2m — 1)(2n — 1)irLetYmn(t) = amn fmn(t)frnn(t)+W,nn fmn(t)=....ii0(t) (3.26)For a damped forced vibration of the wall-soil system,a constant modal dampingratio ). is introducedjL(t)+ 2iXWmn fmn(t)+W,nn fmn(t) = —i10(t) (3.27)For a given ground excitationiic(t) , a close-form solution of the displacement u isfound to beu(x,y,t) = EE sinamx Sjflbymn frnn(t)Chapter 3. Dynamic Thrusts onRigid Walls with Uniform Elastic Baclcfills23where fmn(t) is the time history solutionof Eq. (3.27) corresponding to a particularmodal frequencyWmn. It is noted that Eq. (3.27) is the standard damped vibrationequation of a single degreeof freedom system.The dynamic earth pressure acting on the wallis determined to be the normal stresso at x=O. The dynamic pressure distribution alongthe wall iso,(x,y,t)o ==13Gamamn sin(by) . fmn(t)The total dynamic thrust acting on thewall isP(t) =j(xyt) .dyP(t) = 13G. ammnf(t)P(t) = 3G.2(2n1)2L/H(3.28)The total dynamic moment acting at the baseof the wall isM(t)=1H,y,t)o dyM(t) =._amn.sin(bnH)f(t)(3.29)For a harmonic input üo(t)=. eict,the amplitude of the steady-state responsefmn(t) is found from Eq. 3.27 to bemax330Jmn —(2 —w2)+2i WChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backullls24For any excitation iio(t) the time history ofthe modal dynamic thrust associatedwith a particular mode is obtained using Eq.(3.28). The time-history of the dynamicthrust for the desired number ofmodes are then determinedusing the mode-superpositionmethod. Therefore the peak dynamic thrust acting on the wallcan be determined exactlyfor any type of input motion.For earthquake motion the peak modal thrust actingon the wall associated with themodal frequency Wmn could be determined using the pseudo-spectralvelocity Thepseudo-spectral velocity is derived from response spectral displacementby— omn-1vwhere is also the peak offmn(t)corresponding to an excitation frequencyWmn.From Eq. 3.28 the peak modal thrustFmnis determined as—G165[4nmn/32 2(.7r (2n— 1)L/HWmnEstimation of the peak dynamic thrust is made by combining the individualpeakmodal thrusts by some approximate method. Either the absolute summationor the rootsquare summation of the peak modal thrust is commonlyused.3.3 Static 1-g solution: Validation of modelIn the previous section, three models namely, v= 0, a-,=O, and the proposed model,were presented which yielded different expressions for the coefficient8 in Eq. 3.6. It isnecessary to examine the accuracy of solution provided by each model. Wood’srigoroussolution (Wood, 1973) provides a measure for evaluating the accuracy ofthe approximateChapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Baclcfills 25solutions. Wood’s solution for 1-g static horizontal loading is givenin Eq. 2.5.To allow a measure ofthe accuracy ofthe approximate solutions, a1-g static solutionis derived using the proposed approximate method. The 1-g static solution is thelimitdynamic solution when the period of the dynamic motion becomes infinitelylong. FromEq. 3.27 the static deflection produced by a 1-g static force is given byfrn.n(t) = (3.33)mnThe corresponding 1-g static thrust is obtained by substituting Eq. 3.33 into Eq. 3.28= G/3gE2162(3.34)ir (2n — 1)WmnL/HThe 1-g static moment acting at the base of the wall is obtained from Eq. 3.29M3= G/3gE2ammnsin(bH)(3.35)bnWmnThe total thrust against the wall due to 1-g static horizontal force is determinedbydoing a double summation for modes m and n from Eq. (3.34). A normalizedthrustratio is introduced and defined asTHRUST RATIOTOTAL . THRUST(3.36)pwhere Amax is the peak ground acceleration in m/sec2,ft/sec2or other consistentunit.Comparison of the accuracy of approximate solutions Theapproximate solutions are used to obtain the total 1-g static force for two wall-soil systems: onewith aChapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills 26semi-infinite backfill and the other one with a finitebackfill. The semi-infinite backfill isapproximated by using L/H=5.O and the finitebackfill is represented by using L/H=l.5.Results from the different analyses are compared in Figure3.2 (a) and 3.2 (b) for L/H=5.Oand L/H=1.5, respectively. Thefollowing observationsmay be made based on the results.The proposed model gives results that are in very good agreementwith Wood’sexact results for both L/H=5.O and L/Hz=l.5. Theapproximation of the proposed model works even better for walls retainingfinite backfill (L/H=1.5).Usually this model gives total force slightly lessthan the exact total force.The o,=O model yields results that are in very good agreement withthe exactresults for L/H=5.O. For wall-soil systems withL/H=5.O, the accuracy of theo,=O model is comparable to that of the proposedmodel. The difference between the two models is that theo-!,=Omodel overestimates the response butthe proposed model underestimates the response slightly.However for L/H=1.5, the o,=O model doesnot give as good results as theproposed model does. The solutions from the proposedmodel are much closerto the exact solutions than those from the o,=Omodel. The o,=O model mayoverestimate the total force by about18%. The proposed model only underestimates the total force by 4%.When the v=O model is applied, the accuracyof the solution is very goodprovided v < 0.3. As v exceeds 0.3, the solutionsstart to deviate from the exact solutions. For L/H=5.0, the accuracyof the solution from the v=0 modelbecomes unacceptable as v> 0.4.Recently Veletsos and Younan (1994) made studieson rigid walls with horizontallyChapter 3. Dynamic Thrusts on Thgid Walls withUniform Elastic Backfill5272.0(a) L/H=5.O—WOd exacto1 .5proposedmodel1.0f_Q50.1 0.20.3 0.40.5Poisson’sratio20_________r(b)o1 .5Woods exactproposed modeljl.0FII0.Q—0.1 0.5Poisson’sratioFigure 3.2: Comparis0 of theaccuracy ofapprom solutjo ford-w1 systems(a) L/1150(b)L/H15Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform ElasticBackfills 28semi-infinite backfills. The accuracy of the o,=O model andv=0 model were examinedin that paper against Wood’s rigorous solution. They reachedsimilar conclusions to theabove regarding the accuracy of these two models. Their conclusions weremade for thesemi-infinite backfills only, which is similarto the case with L/H=5.0.The studies presented conclude that the proposed model givesthe best approximationto solutions for the rigid-wall systems with infinite backfills andfinite backfills. Therefore, the proposed model will be used for all furtherstudies with 8 = /3 = 2/(1 — v).Static 1-g solution using 8 = 2/(1 — v) Additional analyseswere carried to studythe accuracy of the solution for the entire range ofL/H ratios. Figure 3.3(a) and Figure3.3(b) show the relationship between the normalized thrustratios and the L/H ratios forv = 0.3, 0.4 and 0.5. The results from Wood’s study (Wood,1973) are shown in Figure3.3(a), and the results from this study are shown in Figure 3.3(b).The two solutionsagree fairly well for the entire range ofL/H ratio. In general,the thrust ratios determinedfrom this study are about 5% less than that predicted by Wood. Although Wood’s solution that was obtained from a 2-dimensional planestrain analysis is more accurate thanthat from this study, the small amount of error in the thrustratio is compensated for bythe convenience of using the much simpler expression of thetotal thrust shown in Eq.(3.34). The simplicity in determination of the total thrust leadsto ready application inengineering.The thrust ratios plotted in Figure 3.3 are found to be independentofthe shear modulus G ofthe backfill. The total thrusts increase with the increaseof b/H ratio, but theyapproach steady values for L/H> 4. For L/H=5.0 and v = 0.4, the total thrust underChapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfihls291.2 -__________:(a) Woods1.00 -F—-a •a.Poisson’s ratio 0.30.6 oooo Poisson’s ratio 0.4D -Poisson’s ratio 0.5F-0.2 -1111111111111111111111iii iiii liiIiIIIiiIliiiii1 2 34 5L/H1.2 -:(b) author’s1.0:-0 -0.8.-...-.Pcsson’sratio0.3-oaoooPoisson’s ratio0.40.6 &Pöissöñ’dtio U.5D -F-0.2 -.0.0c_I_i I I i i II I I I I I I II I I I I II I I I I I I I I I II 15L/HFigure 3.3: Normalized thrust ratios for1-g static solution(a) Wood’s solution (b) author’s solutionChapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backilils3010.0Poisson’s ratio 0.3EnnnrPoisson’s ratio 0.4o.uPoisson’s ratio 0.56.0 —____ci)-c(• i_I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II I I I I I I1 2 3 45L/HFigure 3.4: Heights of thrusts due to1-g static horizontal force (wall heightH=lOm)1-g static horizontal force is estimated to be 1.0-yH2.For a lOm high wall and a backfillwith unit weight 7=19.6 kN/m3,the totalthrust is calculated to be 1960 kN forper unitwide wall. For values of ii other than0.3, 0.4 and 0.5, the 1-g static thrusts are easilyobtained using Eq. (3.34).The height of the resultant thrust above the baseof the wall due to the 1-g statichorizontal force is plotted in Figure3.4 against the L/H ratio for v = 0.3, 0.4 and0.5.The heights of resultant thrusts are identicalfor v = 0.3, 0.4 and 0.5. They remainconstant when Lull is greater than1.0. The heights of the resultant thrusts are about0.611 above the base of the wall.Chapter 3. Dynamic Thrusts on RigidWalls with Uniform Elastic Backlills 313.4 Dynamic thrusts under sinusoidal motionsDynamic amplification ofstructural response dependson the ratio ofthe frequency oftheinput excitation to thefundamental frequency ofthestructure. Resonant response occurswhen the excitation frequency matches the fundamental frequency.For wall-soil systems,a simple approach is to take the fundamental frequencyof the backfill w8 to approximatethe frequency of the system. This representation isespecially useful for wall-soil systemswith long backfills, where the fundamental frequency ofthe wall-soil system is very closeto the fundamental frequency of the backfill. However,for wall-soil systems with finitebackfills, the fundamental frequency of the combined wall-soilsystemWis more critical.Therefore, two frequency ratiosfRiandfR2are used to investigate the dynamicamplification of the wall-soil systems.The frequency ratiofRiis defined by the ratiobetween the angular frequency w of the input motionand the natural angular frequencyw3 of the fundamental mode of the infinite horizontal backfill.The frequency ratiofRiis quantitatively expressed asfRi= w/w8(3.37)For a uniform soil profile, w8 is determined by w8=V6ir/2H, in which V3 is the shearwave velocity of the backfill.The frequency ratiofR2is defined by the ratio between the angular frequency oftheinput motion (w) and the fundamental angular frequency(w11)of the wall-soil system.The frequency ratiofR2is expressed asfR2= (3.38)Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills321.2 — —-- - ——-“r01.0 -“ :-— --D05.0a,0.6 -L, 1.5a) .0.4 -0002--••___——--.1 10100 1000Number of modesFigure 3.5: Accuracy of solutions versus number of modes usedFor a uniform soil profile, the fundamental angular frequency w11 of the wall-soil system can be determined by using Eq. 3.23.Before proceeding with the analysis of dynamic amplification as a function of frequency ratio, studies were made first to examine the relationship between the accuracyof solutions and the number of modes used. Figure 3.5 shows the accuracy of dynamicthrust obtained using increasing number of modes in the dynamic solution. Analyseswere carried out at one randomly selected frequency for each L/H ratio. Frequenciescorresponding tofRi=1.0 andfRi= 1.14 were used for Lull = 5.0 and L/H = 1.5,respectively. For all analyses v = 0.4 and ) = 10% were used.The solutions converged very fast for the problems investigated. By just using thefirst mode, about 75% accuracy in the solutions was achieved for L/H=5.0 and 82% forL/H=l.5. Excellent accuracy (95%) was obtained by using the first 10 modes. AlthoughChapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills 33a much smaller number of modes could be used to compute the dynamicresponse adequately for engineering purpose, 600 modes were usedhere to get an ‘exact’ solution forassessing the accuracy of the results. From Eq.3.28, it can be seen that the solutionconverges very fast as the number of moden increases. The number of mode n requiredis usually much less than the number ofmode m requiredin the dynamic solution. Studies also showed that a larger number of moden is necessary for wall-soil systems withsmaller L/H ratios. Therefore a combination ofn=2 and m=300 was used for L/H=5.0,and another combination of n=6 and m=100 was usedfor L/H=1.5.The amplitudes of steady-state dynamic thrustswere determined using Eq. 3.28 andEq. 3.30 for harmonic excitations at differentfrequencies. The amplitudes of dynamicthrusts are normalized according to Eq. (3.36). The normalizedthrust ratios are plottedagainst the frequency ratiofRiin Figure 3.6(a) for L/H=5.0 and in Figure 3.6(b)forL/H=1.5. The dynamic thrusts increase very fast asthe excitation frequency approachesthe fundamental frequency of the wall-soil system.At resonant conditions, the peak dynamic thrusts are2.4pH2Amaz for L/H=5.0 and 3.OpH2Amaa, for L/H=1.5. Because thestatic thrusts are1.OpH2Amax for L/H=5.0 and 0.86pH2Ama,,, for L/H=1.5, their corresponding dynamic amplification factors are 2.4 forL/H=5.0 and 3.5 for L/H=1.5. Theresults suggest that the dynamic amplification forwall-soil systems with finite backfillsis larger than that for wall-soil systemswith semi-infinite backfills.The results also show that resonance occurs at a frequency ratiofRi= 1.05 forL/H=5.O and atfRi= 1.55 for L/H=1.5. Hence for two identical walls with identicalsoil properties, the fundamental frequency ofthe wall-soil systemwith L/H=1.5 is about1.45 times that with L/H=5.0.Chapter 3. Dynamic Thrusts on RigidWalls with Uniform Elastic Backfills340(I,DZI-4U)DF—i(a)L/H = 5.0 STEADY STATE RESPONSE3-21Poisson’s ratio = 0.4damping ratio 10%modes: N=2, M=30004111111111II I 11111 I II111111111111111111111111111110 1 2 34FREQUENCY RATIO, fRiE(b) L/H = 1.5 STEADYSTATE RESPONSE550 1(1111 (IjI III1111111( 111111(1111111! juhhhhhiII0 1 2 34FREQUENCY RATIO, fRiFigure 3.6: Normalized thrust ratios forsinusoidal motions (a) L/H=5.O (b) L/H 1.5Chapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills353.5 Dynamic thrusts under earthquake motionsDynamic thrusts are computed for wall-soil systemswith L/11=5.0 and L/H=1.5. Thewalls have a fixed height 11=10 m. The soil backfillhas Poisson’s ratio ii = 0.4, unitweight 7 = 19.6 kN/m3and a constant damping ratioA=10%.Two earthquake acceleration records were used in the analysis, theSOOE accelerationcomponent of the 1940 El Centro earthquake, and theS9OE acceleration component ofthe 1989 Loma Prieta earthquake recorded at Yerba Buena island.The peak accelerationof the El Centro input is 0.348g, but it was scaleddown to 0.07g to simulate the linearelastic response of the wall-soil system under asmall input motion. The peak acceleration of the Loma Prieta input is0.067g.The time histories of dynamic thrusts were computed usingEq. (3.28) by mode superposition method. As examples, time histories ofdynamic thrusts against rigid wallsare shown in Figure 3.7 for the El Centro input and in Figure3.8 for the Loma Prietainput. From Figure 3.7 (a) it is interesting to see that the highfrequency content of theinput motion has been filtered out when theinput motion passes througha relatively softbackfill with G=9810 kPa. For this case, the computed peak dynamicthrust is about182 kN/m.A typical time history of the height of dynamic thrust is illustratedin Figure 3.9.The most frequently occurring height of dynamic thrust is 0.6211above the base of thewall. In the region ofpeak thrust a heightof 0.611 would be seem a good value for design.The impact offrequency ratiosfRiandfR2on the dynamic thrusts against rigid wallsI I Jill 11111111111111111111111111111111111111124 6 8TiME(SEC)TIME(SEC)Figure 3.7: Time histories of dynamic thrusts using the El Centro input (a) L/H=5.O (b)L/H1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills36200100-0-—100.—200(a) L/H= 5.0-H=10 m, Poissons ratio 0.4shear modulus 9810kPa, damping 10%0zF(1)FC-)z>-z-F(I)zIC-)z>-0102001000—1—200- (b) L/H=1.5-:00--Lvj.vvvvv-vv- vvrH=10 m, Poisons ratio0.4shear modulus 39240kPa, damping 10%El Centro input(O.07g)1111111111111I liii Iliii 111111111111111111111111110 24 6 810Chapter 3. Dynamic Thrusts on Rigid Walls with UniformElastic BackililszF—C’)DC-)z>-0TIME (SEC)37Figure 3.8: Time histories of dynamic thrusts using the Loma Prieta input (a) L/H=5.O(b) L/H=1.5200-100—100--—20O(a) L/H=5.0H—lOm, Posson’s ratio 0.4,shear modulus 9810kPa; damping 10%Loma Prieta input011111111111111111 III 111111 III510 15TIME(SEC)200100‘‘III’20Ez-Iv)DFC-)z>-0(b) L/H=1.5I______0.—100-—200V’\JVV\J1) RJVVH=lOm; Poissons ratio 0.4;shear modulus 5805kPa; damping 10%Loma Prieta input111111! 111111 I1IIII1II1IIIIIII 111111110 51015 20Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform ElasticBackfills 38Table 3.1: Peak dynamic thrusts for walls with L/H=5.0 and 11=10m,G w8 Dynamic thrust (kN/m) Dynamic thrust (kN/m)(kPa) rad/sec (El Centro input)(Loma Prieta input)613.0 2.750 48.477.51360.0 4.096 54.2 -2453.0 5.501 74.2 79.34286.0 7.272 120.1 -6131.0 8.697 152.6 -9810.0 11.001 182.2 174.720020.0 15.716 132.8 159.561313.0 27.503 144.5 161.4is examined by varying the shear modulus G of thebackfill. The shear moduli of thebackfills vary from 613 kPa to 61313 kPa, which gives a distributionof the fundamentalfrequency of the backfill w8 from 2.75 to 27.503 rad/sec. The computedpeak dynamicthrusts are listed in Table 3.1 for wall-soil systems withL/H=5.0 and in Table 3.2 forL/H=1.5. The variation of shear moduli and the angular frequenciesw8 are also listedin these Tables. For the El Centro input, the maximum values amongthe peak dynamicthrusts are 182 kN/m for L/H=5.0 and 194.6 kN/m for L/H=1.5.For the Loma Prietainput, the maximum values among the peak dynamic thrusts are 174.7N/mfor L/H=5.0and 170.0 kN/m for L/H=1.5. These results suggestthat the peak dynamic thrusts arelittle dependent on the L/H ratio under earthquakemotions.The frequency ratiosfRiandfR2are determined using Eq. 3.37 and Eq. 3.38, respectively. In these equations the excitation frequency wof the input motion is required.For earthquake motions, the excitation frequency wis taken to be the predominant frequency of the earthquake. The predominant frequency of an earthquake motionis thefrequency at which the response spectral acceleration has the maximum valuein theChapter 3. Dynamic Thrusts on Rigid Wailswith Uniform Elastic Backfihls39Table 3.2: Peak dynamic thrustsfor walls with L/H=1.5 and H=10m,A=10%G Dynamic thrust (kN/m) Dynamic thrust(kN/m)(kPa) rad/sec (El Centro input)(Loma Prieta input)801.5 3.144 54.81090.0 3.66775.0 86.81905.0 4.848 116.0-2453.0 5.501 133.7126.44360.0 7.334 194.6170.05805.0 8.463 179.1145.49810.0 11.001 130.9151.039240.0 22.002 134.5143.912-___-L/H=5.0; H=10 m; Poissons rato 0.4;- shear modu!us 9810 kPa; damping 10 z;E10—fttQJOpt14tyJt+11rr1t1HEIGHT OF THRUST : 6.20 meters0— 1111111 I I 11111IIIIllillIjIl III111111111liiio 24 6 8 10TIME(SEC)Figure 3.9: A time history of the height of dynamicthrust, L/H=5.0Chapter 3. Dynamic Thrusts on RigidWalls with Uniform Elastic Backfills40acceleration spectrum. Therefore theexcitation frequencies of the El Centroinput andthe Loma Prieta input are determinedto be 11.64 rad/sec and 10.13 rad/sec,respectively.The peak dynamic thrusts are normalizedusing Eq. 3.36. The normalized thrustratios are plotted against the frequencyratiofRiin Figure 3.10 and against the frequency ratiofR2in Figure 3.11. Based on resultsfrom the limited number of analysesfor earthquake motions, it is suggestedthat the peak dynamic thrusts are1.30pH2Amafor L/H=5.0 and1.38pH2Amafor L/H=1.5. Because the static thrusts are1.0pH2Amafor L/H=5.0 and 0.86pH2Amfor L/H=1.5, their corresponding the dynamic amplification factors are about 1.3 forL/H=5.0 and 1.6 for L/H=1.5.A significant observation is made thatthe dynamic amplification under earthquakemotions due to resonance ismuch less than that under sinusoidal motions.Under sinusoidal motions, the dynamic amplificationfactors at resonance are 2.4 for L/H=5.0and3.5 for L/H=1.5. The dynamic amplification factorsunder earthquake motions are about50% of that under sinusoidal motions.3.6 Accuracy of the response spectrummethodTheresponse spectrum method is commonlyused to determine the responses ofstructuressubjected to earthquake motions.The response spectrum method whichadds modal values without taking the phasesof the modes into account is much simplerthan the directmode superposition method. Thereforeit is of interest to check on the accuracy withwhich the peak dynamic thrust canbe determined using the response spectrumapproach.Chapter 3. Dynamic Thrusts on Rigid Wallswith Uniform ElasticBaclcfills 414E(a) L/H=5.0Poisson’s ratio = 0.4damping ratio 10%0 3 modes: N=2, M=300sinusoidal motions00000I—Cr)DI0—1111111111111111lijI 111111111 11111111 I 1111111O1 2 34 5FREQUENCY RATIO, fRi4Poisson’s ratio = 0.4(b) L/H1.5damping ratio 10%• modes: N=6,M=1000sinusoidal motionsaearthquake motio(I)DFo 1 2 34 5FREQUENCY RATIO, fRiFigure 3.10: Normalized thrust ratios versusfRifor earthquake motions (a) L/H=5.0(b) L/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform ElasticBackfills03(1’D‘1-03I‘-2C,,D‘-1FREQUENCYRATIO,fR2Figure 3.11: Normalized thrust ratios versusfR2for earthquake motions (a) L/H=5.O(b) L/H1.5424(a) L/H=5.0Poisson’s ratio =0.4damping ratio10%modes: N=2,M=300sinusoidalmotions04— III(IIIIIlIIIIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII01 234FREQUENCYRATiO, fR2:Poisson’s ratio= 0.4:(b) L/H1.5 damping ratio10%:modes:N=6, M=100Esinusoidalmotions-i3ehquokemotions5501111111111I III I liiiI I III I 1111111I III I Ilill01 234Chapter 3. Dynamic Thrusts on Rigid Wails with Uniform ElasticBaclcfills 43The accuracy of the response spectrum methodis measured by a thrust factor Cpdefined asc=(3.39)where the total spectral thrust P31. is evaluatedusing the response spectrum methodby summation of peak modal thrustsFmn. Pmais the exact solution which is evaluatedusing the mode superposition method. The valuesofthe thrust factor Cp gives a measureof the accuracy of the response spectrum method.The peak modal thrustFmnis determined from Eq. 3.32 using the pseudo-spectralvelocitySof the input motion correspondingthe modal frequencyWmn. The pseudo-spectral velocities for the two selected accelerationrecords, the El Centro input and theLoma Prieta input, are shown in Figure 3.12(a) and3.12(b), respectively.Since the peak modal thrust only represents thepeak value for a particular mode,the determination of the total spectral thrust F8,,must be based on some form of modesummation. Summation of the absolute peak modalvalues (ABS) or the Square Root ofthe sum of the Squares of the peak modal values(RSS) are used.When the absolute summation (ABS) is used in the responsespectrum method, thetotal spectral thrust F8,,is determined byP51.= EY2Fmn (3.40)When the root square summation (RSS) is usedin the response spectrum method,the total spectral thrust P8,is determinedbyChapter 3. Dynamic Thrusts on Rigid Walls with Uniform ElasticBackfills 442.0-_(a)Ua)U)E 1 .5DAMPING RATIO 10%•- - - - DAMPING RATIO 5 %15>o10IUa, —0./ S SU,,05_______-o____ _______a,U,O.o — I I I II I I I III I I I I I I I I I I I I II I I III I I I I I I I I0.0 0.51.0 1.52.0PERIOD(SEC)0.5-(b)Ua,0.4EDAMPING RATIO10%DAMPING RATIO 5 %0.30-I-,U -.O2________I—— ——V.,0.1 Ia,U,0.0 — II I I I IjI I I I I I I I II0.00.51.0 1.52.0PERIOD(SEC)Figure 3.12: Pseudo-spectral velocities of (a) the El Centro input and (b) the LomaPrieta inputChapter 3. Dynamic Thrusts on Rigid Wallswith Uniform Elastic Backfills45=(3A1)Values ofthe thrust factor C1’were determined forboth sinusoidal motions arid earthquake motions. The earthquake motions are representedby the El Centro input and theLoma Prieta input. The studies were made forL/H=5.0 and L/H=1.5. 600 modes wereused for obtaining both P, andFmDiscussion of results When the RSS method was used, the relationshipbetween thethrust factor Cp and the frequency ratiofRiwas obtained and is shown in Figure 3.13.For harmonic loading the thrust factorCp changes very much with frequency. The response spectrum method could overestimate or underestimatethe peak dynamic thrustby 80% to 100%. For earthquake loading the thrust factorCF is usually greater thanone, mostly around 1.5. The response spectrum methodusually underestimates the peakdynamic thrust by as much as 50%. However the thrustfactor changes for a different soilprofile and a different frequency ratio. The uncertain variation ofCp makes it difficultto apply the response spectrum method for determiningthe actual peak dynamic thrust-ma2,in practice.When the absolute summation (ABS method) was used, therelationship between thethrust factor Cp and the frequency ratiofRiwas determined and is shown in Figure 3.14.For low frequency ratios, such asfRi< 0.8, the total spectral thrust P, obtaining fromthe ABS method agrees very well with the exactsolutionFma,under sinusoidal motions.However under earthquake motions, about20% overestimate of peak dynamic thrust isexpected for the same frequency ratio. For high frequency ratios,such asfRi> 1.5,the peak dynamic thrust may be overestimated by asmuch as 50% when the responseChapter 3. Dynamic Thrusts onRigid Walls with Uniform ElasticBaclcfihls 46o / ,,— Poissons ratio = 0.4C1 L/fT —damping ratio 10%- : modes: N=2, M=300 (RSS)o :° ° ° °data for earthquake motionsdata for sinusoidal motions.:::zo 1 2 34 5FREQUENCY RATIO, fRi0 / ./ Poisson’s ratio 0.4Sjb)L1H 1.5damping ratio 10%- modes:N=6, M=100 (RSS)o32= ° ° a adata for earthquake motionsdata for sinusoidal motions;D1raOOoO,ac— I I I liii111111111Ijii I Ililtil Ill liii I liii tillo 1 2 34 5FREQUENCY RATIO, fRiFigure 3.13: Variations ofthrust factor Cp versus frequency ratiofRi(RSS method) (A)L/H=5.O, (B) L/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls with UniformElastic Backfills474Poisson’s ratio = 0.4:(a) L/H = 5.0damping ratio 10%modes: N=2, M=300 (ABS)0odata for earthquake motionsdata for sinusoidal motions(I)D1-aO1 2 3 4 5FREQUENCY RATIO, fRicLo- Poisson’s ratio0.4(b) L/H = 1.5damping ratio 10%modes: N=6, M=100 (ABS)o32: a o odata for earthquakemotionsdata for sinusoidalmotionsC/)111111111111111111111111I IIIIIIIIIIIIIiIIIIItliii0 1 2 34 5FREQUENCY RATIO, fRiFigure 3.14: Variations ofthrust factorCp versus frequency ratiofRi(ABS method) (A)L/H=5.O, (B) L/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls withUniform Elastic Backfills 48spectrum method is used.Therefore it is recommended that the mode superpositionmethod be used in orderto accurately determine the peak dynamic thrusts against rigidwalls. The responsespectrum method may be used for approximately estimatingthe peak dynamic thrustsagainst rigid walls. The use of ABS method is suggestedwhen the response spectrummethod is selected, especially at low frequency ratios.Chapter 4Dynamic Thrusts on Rigid Walls with Non-homogeneousSoil Profiles4.1 IntroductionThe first objective ofthis chapter is to explore the effects oftypicalbackfill non-homogeneityon the magnitude of dynamic thrusts for elastic response. Two typesof soil profiles havebeen analyzed. They are profiles with linear and parabolic variationsof shear moduluswith depth.The second objective of this chapter is to evaluate the influenceof soil non-linearityon the magnitude and point of application of dynamic thrusts. Themoduli and dampingof soils are known to be strain dependent (Seed andIdriss, 1967). The equivalent linearelastic analysis developed by Seed and Idriss (1967)is used to simulate the soil non-linearresponse. Dynamic response characteristics such as magnitude ofdynamic thrust, fundamental frequency of the system, and amplification factors ofground acceleration aredetermined for different intensities of acceleration input.Elastic analysis. The undamped forced vibrationequation ofmotion of the backfill iswritten as82u 82u 82u— (8G— +G-)= —p’äo(t) (4.1)49Chapter 4. Dynamic Thrusts onRigid Walls with Non-homogeneousSoil Profiles 50lyôu/ôy=OH u = 0 (plane strain) G, H 8u13x = 0u0____XLFigure 4.1: A composition ofnon-homogenoussoil profilewhere 6 is equal to 2/(1-v),ii is Poisson’s ratio of soil, and üo(t)is the base acceleration caused by the earthquake.Figure 4.1 showsthe type of non-homogeneous soil profiles thatwill be analyzed inthis chapter. The backfill behind thewall is consisted oflayered soilswith different properties in each layer. The wallis considered to be rigid, and it doesnot move relative tothe base. The boundary conditionsfor this system are also shown in Figure4.1.Analytical solutions are in generalnot possible for nonhomogeneous backfills. Therefore, the finite element methodis employed to analyze dynamic responseof the wall-soilsystem.4.2 Finite elementformulation and its validationA finite element developed especiallyfor this study is shown in Figure4.2. The elementconsists of 6 nodeswith 6 horizontal displacement variables. Thedisplacement field has aChapter 4. DynamicThrusts on Rigid Walls withNon-homogeneousSoil Profiles51yIIU6 U5U4Ui U2xii I____________aFigure 4.2: A compositionof the finite elementused inSPAWlinear variation alongthe vertical directionand a quadratic variationalong the horizontaldirection.Let the displacementu be represented byu=EN2.u i=1,6The shape functionsN are given byN1 =N2 x(b—y)N3= xyN4=N5=N6= xy(a — x)Galerkin’s general procedureof weighted residualsis used to develop the stiffnessandmass matrices of thefinite element shownin Figure 4.2. The stiffness matrix[K] isgiven byChapter 4. Dynamic Thrusts on RigidWalls with Non-homogeneous Soil Profiles52[Kj=+ (4.2)The diagonal mass matrix of the elementis found to be[M1 =f{iiii44}(4.3)The stiffness and mass formulations shown in Eq.(4.2) and Eq. (4.3) are then appliedto every element in the system. Theglobal stiffness matrix [K] and the massmatrix [M]are assembled accordingly. The equationsof motion in matrix form are written[M]{’ii} + [C]{ri} + [K]{u} = —[M]{I}ii0(t)(4.4)where [C] is the damping matrix, and {I}is a column vector of 1.The natural frequencies of the systemare determined by analyzing the eigen values of the system. The damping matrixof each finite element is obtained accordingtothe desired degree of damping of the element.In this manner the damping matrix[C] isevaluated. A finite element programSPAW was developed based on these considerations.Validation ofF.E. method To validatethe reliability ofthe finite elementanalysis, itwas applied first to two uniform soil profilesfor which close-form solutions were obtainedin the previous chapter.The geometric mesh used in the finite elementanalyses is shown in Figure 4.3.Thismesh consists of3 elements in each layer with 20 layers. The horizontaldimensions ofthese elements are subjected to change proportionallyto the L/H ratio. This mesh hasChapter 4. Dynamic Thrusts on Rigid Wails with Non-homogeneousSoil Profiles 53.‘I__ -.LFigure 4.3: A finite element mesh used for dynamic analysesbeen used for all finite element analyses presentedin this chapter.Thefinite element method is applied first to auniform soil profile(L/H=5.0,H=10.Om)with G=9810 kPa and )L=10%. The fundamental frequency of this wall-soil system is1.75 Hz. This system is shaken by the SOOE acceleration componentof the 1940 ElCentro earthquake scaled to 0.07g. The timehistory of dynamic thrust against the wallcomputed using the finite element method is shownby the solid line in Figure 4.4(a). Thedashed line represents the closed form solutionin Figure 3.7. The agreement betweenthe two solutions is excellent over the entire time historiesof dynamic thrusts.The finite element method is next applied to a second uniform soilprofile (L/H=1.5,H=10.Om) with a higher shear modulus of 0=39240kPa and .\ = 10%. This wail-soilsystem is stiffer than the previous one. The fundamental frequency ofthis system is 2.54Hz. The time histories of dynamic thrustscomputed using the finite element methodand the close-form solution are shown inFigure 4.4(b). Here again excellent agreementis observed between the dynamic thrusts computedby the two methods of analysis.HChapter 4. Dynamic Thrusts on Rigid Walls withNon-homogeneous Soil Profiles 542001000—100—2002001000zF—C’)DF—C-)z>-0Iz-I—U)DI—C)z>-0:(a) L/H= 5.0from F.E. analysis•-- -- from close—formsolution-H=lOm, Poissonsratio 0.4;:Shear Modus9810 kPo; damping10%.— 11111111111111111111111111111111(11(liiij(j I I I I0 24 6810TIME(SEC):(b) L/H=1 .5from F.E. analysis.- - - - from close—form solutionui’:vvvvv‘ir’jVH=lOm, Poisson’sratio 0.4;-Shear Modulus39240 kPo;damping 10%-El Centro input(0.07g)—200—0 24 6810TIME(SEC)Figure 4.4: Comparisons ofdynamic thrustsbetween the F.E. method and the close-formsolution for uniform soils(a) L/H=5(b) L/H=1.5Chapter 4. Dynamic Thrusts on RigidWalls with Non-homogeneousSoil Profiles 55These comparative studies verify thatthe finite element formulation constructedforthis problem is a reliable technique forevaluating the dynamic response of thewall-soilsystem and that the mesh employed isan appropriate one.The finite element mesh used for therigid wall analysis has been embeddedin thecomputer program SPAW. There is noneed for other users to construct themesh. Thedynamic analyses can be performed byjust inputting the basic soil propertiesof eachlayer. The time history analysesonly take few minutes in a PC486 33MHzcomputer.4.3 Linear elastic analyses withnon-homogeneous soil profilesIn this analysis the shear modulusG and damping ratio ) for a given soil profile retainconstant values throughout theanalysis. They are not consideredto be functions ofstrains. ? =10% and v =0.4are used. The horizontal length of the soil layeris constrained at L/H=5.0. Two types ofsoil profiles are examined, a profile witha linearvariation of shear moduli with depthand a profile with a parabolic variationof shearmoduli with depth.Two types of motions, the sinusoidalmotion and the earthquake motion, are appliedto the two types of soil profiles.For the sinusoidal motion, the amplitudes ofthe steadystate dynamic thrusts are determined.For earthquake motion, the peak dynamicthrustsdeveloped during shaking are determined.Again the El Centro input is usedas the inputof earthquake motion in this study.Analyses were performed to examinethe relationship between the thrust ratio andChapter 4. Dynamic Thrusts on Rigid Walls withNon-homogeneous Soil Profiles56the frequency ratio. The frequency ratiofR2defined in Eq. 3.38 is used for this study.The thrust ratio has been defined in Eq.3.36 to be thTUSt/(pH2Ama),where isthe peak acceleration of the input motion in m/sec2,ft/sec2or other consistent unit.Figure 4.5 shows the relationship between the thrust ratioand the frequency ratiofR2for linear soil profiles. At resonance, the peak dynamic thrustsare1.56pH2Amaunder sinusoidal motions and1.OOpH2Amaa, under earthquake motions. Because the staticthrust is about 0.71pH2Ama,,their corresponding dynamic amplification factors are deterniined to be 2.2 for sinusoidal motions and 1.4 for earthquakemotions.Figure 4.6 shows the relationship between thethrust ratio and the frequency ratio forparabolic soil profiles. In general, the thrustratios are greater than those for linear soilprofiles by 20%. At resonance, the peak dynamic thrusts are1.87pH2Amax under sinusoidal motions and 1.18pH2JLunder earthquakemotions. Because the static thrust isabout 0.82pH2Amax, their corresponding dynamic amplificationfactors are determinedto be 2.3 for sinusoidal motions and 1.4 for earthquake motions.The dynamic response of parabolic soil profiles is compared withdynamic response ofuniform soil profile in Figure 4.7. For sinusoidal motions, the dynamicthrust ratios forparabolic soil profiles are less than those for uniform soil profilesin the frequency rangeoffR2< 2.0.The limited analyses conducted suggest that the dynamicthrust at resonance forsinusoidal motions are about60% greater than that for earthquake motions with samepeak acceleration. It should be noted that the earthquake motionsare represented bythe El Centro input only.Chapter 4. Dynamic Thrustson Rigid Walls with Non-homogeneousSoil ProfilesLinear soilprofile(a) SinusoidalSTEADY STATERESPONSEmotionsPoisson’s ratio = 0.4L/H=5.0, damping10%57Figure 4.5: Relationships betweenthrust ratio and frequencyratiofR2for linear soilproffles (a) sinusoidal motions(b) the El Centro input43-2-1—0U)DI—0IU)DI—04—1111111111111111111111111111111111111111111110 12 34FREQUENCY RATIO,fR23210(b) El Centro input PEAKDYNAMIC RESPONSEPoisson’s ratio = 0.4L/H = 5.0; damping 10%550IIIIIIIIIIIIIItIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1 23 4FREQUENCY RATIO,fR2Chapter 4. Dynamic Thrusts on Rigid Walls withNon-homogeneous SoilProfiles58Parobolc soilprofile4(a) SinusoidalSTEADY STATE RESPONSEmotions23Poisson’s ratio = 0.4L/H = 5.0; damping10%o 12 3 45FREQUENCY RATIO,fR24(b) El Centro inputPEAK DYNAMIC RESPONSE2‘Poisson’s ratio = 0.4L/H = 5.0; damping 10%(I)DO 12 3 45FREQUENCY RATIO,fR2Figure 4.6: Relationships betweenthrust ratio and frequency ratiofR2for parabolic soilproffles (a) sinusoidal motions(b) the El Centro inputChapter 4. Dynamic Thrusts on Rigid Walls withNon-homogeneous Soil Profiles594-Poissons ratio = 0.4o 3-L/H = 5.0; damping 10%Sinusoidal motions°°.Parabolic Soil Profilec::_:111111111111 IIIl,IIIIIIIIIIIIIItIIIII1I1II1IIII01 2 34 5FREQUENCY RATIO, fR2Figure 4.7: Comparison of dynamic thrust ratios for parabolicsoil profiles and uniformsoil profiles under sinusoidal motions (L/H=5)The other important aspect of dynamic thrust is the locationof the resultant thruston the wall. Typical time histories of heightsof dynamic thrusts for the three types ofsoil proffles are illustrated in Figure4.8. The results are obtained using the El Centroinput as input motion. For linear soil profiles dominant heightof dynamic thrust is atO.48H above the wall base. For parabolic soil profiles this height becomes0.5111 above thewall base. The uniform soil profile gives an average heightof 0.62H above the wall base.Generally the height ofdynamic thrust increases as the soil profilebecomes more uniform.Modal frequencies of wall-soil systems with different soil profiles areshown in Table 4.3. Essentially the natural frequencies of thewall-soil systems become more widelyspaced as the soil becomes more uniform.Chapter 4. Dynamic Thrusts on Rigid Walls withNon-homogeneous Soil Profiles8—6—4-2-0Figure 4.8: Typical time histories of heights of dynamic thrusts for three types of soilprofiles (H=lOm)Linear soil profiles60j 11 I108-6-4-2-10ItJAL ALIIAIAvvr,jv,1IvLiii AiiILiAAI II i Ii I i‘VS’i‘nA(rII[J1Tt24Parabolic soil profiles6 81 0EU)04,EU)z00’VIEU,00’VIITT1I!LAIALdLALLL‘r’iiy’I II24Uniform soilprofilesivv Vf1lyv F6 81086104—2—01’’’I‘i’ ‘I’’’‘.JIEF0 24 6810Time(sec)Chapter 4. Dynamic Thrustson Rigid Walls withNon-homogeneous SoilProfiles 61Table 4.3: Patterns offirst10thnatural frequencies forthree types of soil profiles(wa,rad/sec)frequencies frequenciesfrequenciesflthfreq. (linear profile)(parabolic profile) (constantprofile)1 11.311.35 11.702 13.114.2016.323 15.218.80 22.904 16.420.60 30.195 19.926.00 33.246 25.629.10 35.137 26.429.70 37.798 27.230.80 45.539 29.433.40 53.3610 33.238.50 55.154.4 Equivalent linearsimulation of non-linearresponse underearthquakeloadsA method of analysis formodelling the non-linearresponse of soil was proposedby Seedand Idriss (Seed& Idriss, 1967), which is designatedthe equivalent linear methodofanalysis. In this methoda set of elastic shear moduliand viscous damping ratioswhichare compatible witha measure of the effectiveshear strains inducedby an earthquakeare used to approximatethe hysteretic behavioursof soils. The equivalent linearelasticmethod is widely usedfor dynamic analysesin practice.The equivalent linearmethod for modelling nonlinearbehaviour is used in thisstudy.Since the level of shearstrain in each elementis not known at thebeginning of the analysis, strain compatiblemoduli and dampingratios are achievedby an iterative process.The effective strainused to determine moduliand damping is chosento be 0.65 of theChapter 4. Dynamic Thrusts on RigidWalls with Non-homogeneous SoilProfiles 62peak dynamic strain for the earthquaketype of motions. The damping ratioof the wall-soil system in each iteration isestimated by using an average valueof damping ratios forall soil elements in the system.Two wall-soil systems with parabolicvariation of shear modulus are analyzedusingthe equivalent linear technique. Onewall has a stiff backfill with a shearmodulus of132,000 kPa at the base of the backfill.The other wall has a backfill with ashear modulus of66,000 kPa at the base. The wallheight in each case is 11=10 m withL/H=5.0. Thewall-soil systems are shaken using theEl Centro acceleration record as the basemotion.The effects of non-linearity on the dynamicresponses of the systems are exploredusingincreasing levels ofinput acceleration.The peak accelerations ofinput motions vary from0.05g to 0.35g in increments of 0.05g.The data on shear strain dependent moduli and dampingpresented by Seed and Idriss(Seed & Idriss, 1970) were employed.At shear strain levels of0.0001,0.0005, 0.001, 0.005,0.01, 0.05, 0.1, 0.5, and 1.0 percent, the valuesofG/GmO.a, were selected as 100, 98.3,95.8, 84.3, 74.3, 43.0, 29.6, 10.9, 6.1percent, respectively. The values ofD/Dmaa, corresponding to the above strain levelsare the follows: 0.018, 0.055, 0.073,0.158, 0.275,0.457, 0.579, 0.891, 1.0. The maximumdamping ratioDmax is chosen to be 30%.Figure 4.9 shows the dynamicresponse of the stiff site. The intensityof shaking ismeasured by the peak accelerationof the input motion. The amplificationof groundacceleration shows its peak at an inputacceleration of 0.15g. The dynamicthrust ratio,the frequency ratio and the dampingratio increase steadily with the increaseofthe levelof shaking. The increasing level of shakingresults in reduction of shear moduli andconsequently reduction in the fundamentalfrequency of the wall-soil system. ThisresultsChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous SoilProfiles 63Parabolic soil profile G0= 132,000 kPo= •e.e. Damping ratios• 00000 Frequency ratios- O0DO Amplifaction of groundacceleration• AAAThrust ratiosn4zcocoooo.—--—--‘t .0.00.1 0.20.3 0.40.5Peak BoseAccelerationCg)Figure 4.9: Dynamic responses of a stiff,site due to non-linear effect,G0=132,000 kPain an increase in the frequency ratiofR2.The dynamic thrust ratio increases from 0.83for linear elastic response associated with low input accelerations to 1.04 for highly nonlinear response at the higher levels of input acceleration. The effect of strong shaking onthe dynamic thrust is clearly shown in Figure 4.10. At a base acceleration of0.35g thedynamic thrust increases 25% due to the non-linear effect.Figure 4.11 shows the dynamic response of the soft site. There is no clear indicationof resonant response such as seen in Figure 4.9. The dynamic thrust ratio is 1.233 at apeak base acceleration of 0.2g and remains almost constant up to an input accelerationof 0.35g. The dynamic thrust increases about 23% due to the non-linear effect. Theamplification factor ofground accelerations decreases from 3.0 to 2.2 with the increasinglevel of input acceleration.Chapter 4. Dynamic Thrusts on Rigid Wallswith Non-homogeneousSoil Profiles64Parabolic soil profile G0 132,000kPo1.5 -00000 NON—LINEAR ANALYSES--- LINEAR ELASTIC ANALYSESL/H = 5.0, H=lOm(I) Poisson’sratio 0.40El Centro inputI0.5—0.0 0.10.2 0.30.4 0.5Peak Base Acceleration( 9 )Figure 4.10: Effect of level of shaking onthe dynamic thrust, G0=132,000 kPaParabolic soil profile G0= 66,000kPo— *-*-*-.-* Damping ratios00000 Frequency ratiosD0DC Amplifoction of ground accelerationAA.&Thrust ratios:L/H=5.0, H=lOmZ:Poisson’s ratio0.4El Centro input00. 0.5Peak Bose AccelerationCg)Figure 4.11: Dynamic responses of a softsite due to non-linear effects,G0=66,000 kPaChapter 4. Dynamic Thrusts onRigid Walls with Non-homogeneous SoilProfiles 65The studies from equivalent linear analysesreveal that the dynamic thrustratiosusually increase with the levels of inputaccelerations. For the cases investigated,theincrease of dynamic thrust due to non-lineareffect is about 25% of the dynamicthrustobtained from a linear elastic analysis.Part IIDynamic Analyses of Pile Foundations66Chapter 5Dynamic analyses of pile foundationsDynamic soil-pile-structure interaction is a challenging areato geotechnical researchersand engineers. A very common example is the 3-Ddynamic analysis of a pile foundationfor a bridge abutment. The analysis involves modellingof soil-pile-soil interaction, theeffects of the pile cap, non-linear soil response, andin many cases incorporates seismically induced pore water pressures. Thereare many approaches to solving the dynamicresponse of pile foundation.Novak (1991) gave an extensive review of the more widely acceptedmethods of analysis for piles under dynamic loads. His study showed that pilegroup response can not bededuced from single pile response without taking pile-soil-pile interactioninto accountand that the dynamic characteristics ofpile groups are stronglyfrequency dependent andmay differ significantly from the characteristics of a single pile.5.1 Dynamic analyses of single pile responseAnalytical modelling of single pile response may be dividedinto two major categories,elastic continuum models coupling the soil and pile in a unifiedsystem and the lumpedmass-spring-dashpot models. The elastic continuummodels are mostly used for theanalysis of pile foundation subjected to low level excitationsuch as problems related67Chapter 5. Dynamic analyses ofpile foundations 68to machine foundations. The lumped mass models are formulated by separating the response ofpiles from the soil medium. The contribution ofthe soil medium to the dynamicresponse of pile foundation is taken into account by using empirically or analytically derived Winkler type springs and viscous dashpots along the pile shaft.Elastic continuum models The analytical approach that can model the interactionbetween the pile and soil using the theory of continuum mechanics is very difficult. Accurate mathematical solutions of the problem are not available even for the idealisticassumptions of linear elasticity or viscoelasticity, homogeneous soils and the pile beingwelded to the soil. Thus various approximate formulations have been developed. Anapproximate solution for the horizontal response of an endbearing pile in a homogeneoussoil layer was presented by Tajimi (1966). His formulation neglected the vertical component of the motion.The work of Novak, Nogami and their co-workers (Novak, 1974; Nogami and Novak, 1977; Novak and Aboul-Ella, 1978a,1978b; Novak et al., 1978; Novak and Sheta,1980,1982) is particularly significant in advancing solutions to the problem of an elasticbeam vibrating in a homogeneous or multi-layered elastic isotropic continuum subjectedto dynamic pile head loading. In 1974, Novak formulated a simple approach based onplane strain soil reactions. His formulation may be interpreted as a plane strain complextransmitting boundary attached directly to the pile. The solution was first presented fora homogeneous soil layer without any material damping. Material damping was laterincluded in closed form expressions for soil reactions in Novak et al. (1978). The formulation of the plane strain approach was further extended by Novak and Aboul-Ella(1978a,1978b) to include layered media. The computer program PILAY was formulatedChapter 5. Dynamic analyses ofpile foundations69for these types of solutions.In using Novak’s formulation much ofattention is focused on thepile head impedancefunctions. The impedances have a great influence on the response ofpile supported buildings and structures. The pile head impedances can be definedas the transfer functionsdescribing the ratios between the complex valued displacement responseat the pile headand the harmonic forces (or moments) applied at thepile head.Pile head impedances derived using elastic theory are mostappropriate for low levelshaking where the dynamic pile head forces induce essentially elasticstrains in the soilaround the pile. The pile head impedance is usually expressed in termsof complex shearmoduli, the real part of which represents the secantelastic stiffness of the soil and theimaginary part accounts for material (hysteretic) damping.Consequently the pile headimpedance has both a real component and an imaginary component.The real component represents the elastic stiffness ofsoil while theimaginary componentindicates energylosses due to wave propagation away from the pile (radiationdamping) and hystereticdamping in the soil. For low level of excitation hysteretic dampingis small and systemdamping is mostly dominated by radiation energy losses.The real component of pile head impedance derived usingNovak’s plane strain approach has been found to diminish as the frequency approacheszero (Novak and AboulElla, 1978a). This result is not realistic. Novak and Nogami havesuggested that planestrain soil reactions can be used provided low frequency correctionsare applied. Inthe computer program PILAY, a frequency cut-off is applied fordetermining pile headimpedances of single piles. Nogami and Novak use staticstiffness when the dimensionless frequency a = wro/V5 is less than about 0.3, where w is the excitationfrequency, r0Chapter 5. Dynamic analyses ofpilefoundations70is the pile radius and V8 is the effectiveshear wave velocity of soil in the depthrangewhere maximum pile bending occurs.Alternatively at lower frequencies, thepile headimpedance can be taken as constantand equal to that calculated at a suitabledimensionless frequency, such as a0.3.In PILAY analysis, constant dampingcoefficients are assumed when the dimensionless frequency a is less than0.3. That is, the damping coefficients are assumedto beindependent of frequency when a< 0.3. This assumption does not necessary hold especially when the frequency independent hystereticdamping is significant. Other methodsfor determining the dampingof pile foundations are given below. It should benoted,however, the Novak’s solutions wereprimarily intended for for machine foundationsforwhich a is usually greater than0.3.The imaginary component (damping)of pile head impedance, represents theenergy losses along the pile. For anequivalent viscously damped system, the viscousdashpot coefficient c (dampingcoefficient) is defined as the ratio of the dampingandthe frequency cjj = Cj/w. The equivalentviscous damping coefficients varywithdepth z along the pile becausethe hysteretic damping varies with pile defiectionsandstrain in the near field. Gazetas andDobry (1984) proposed a simple formulation forcomputing the damping coefficients.According to them, the damping coefficients consistof the radiation and hystereticdamping components, or ci., = C,.+The radiationdamping coefficients c,. given byGazetas and Dobry have been found in good agreementwith those derived by Novak etal. (1978) and Roesset and Angelides (1980).The expressions proposed by Gazetas and Dobryare frequency and depth dependent. For depthsgreater than 2.5 times the pile diameterthe radiation dashpot coefficient is given asChapter 5. Dynamic analyses ofpile foundations71cr =4rop8V{1+[(hl]125}()°75a°25(5.1)in which a is the dimensionless frequency a = wro/17, w isthe frequency ofexcitation,r0 is the pile radius and V8 is the freefield shear wave velocity. For shallower depths,theradiation dashpot coefficient is given asc,.=8ropsV8()075a_025(5.2)Recently Gazetas et al. (1993) proposeda simpler expression for the viscous dampingcoefficient along the pile. The frequencydependent radiation damping coefficientforvertical motions is expressed asTI ..,—0.25Cz—PsVs(Laoand for horizontal motionsc =6p8Vda°25(5.4)in which d is the pile diameter anda0 = wd/V3.These formulations were used in aBeam-on-Dynamic-Winkler foundation simplifiedmodel by Gazetas et al. (1993).The pile head impedances are often usedas foundation spring and dashpot parameters in the analysis of superstructures subjectedto earthquake loading. This type ofanalysis in which the pile foundationsare replaced by springs and dashpots is usuallycalled uncoupled analysis.The assumption generally made in an uncoupled analysisis that one may use the freefield surface accelerations as input intothe base of the superstructure. The assumptionOhapter 5. Dynamic analyses ofpile foundations72actually neglects the influence of foundation-groundmay have on the motions of the pilecap. The motions of the pile cap may differ significantly from themotions of the freefield surface due to kinematic interaction between thepile and soil.The influence ofkinematic interaction on pile head accelerationshas been studied byGazetas (1984) and Fan et al. (1991). In the latter publication,comprehensive studieswere made on the kinematic seismic response ofsingle piles and pile groups. The influence of kinematic interaction may become significantif the stiffness ratio between thepile and the soil is high, such as E/E8> 10, 000.An uncoupled superstructure analysis that neglects kinematicinteraction appears tobe valid provided the free field surface motions are dominatedby relatively low frequencywaves. The neglect of kinematic interaction generallyresults in an overestimate of dynaniic pile cap motions transmitted to the superstructure.The other difficulty in an uncoupled analysis lies in selectingappropriate equivalentelastic moduli of soil compatible with strains occurringduring a strong earthquake. Thereduction of soil stiffness and the increase of damping associatedwith a strong shakingare sometimes modelled crudely in these analysesby making arbitrary reductions in theshear moduli and arbitrarily increasing the viscousdamping. For this reason the resultsofthese studies have not proved very useful for the response ofpilefoundations to earthquake loading.The effect of soil non-linearity on pile head impedances ofsingle piles has been investigated for dynamic pile-head loads by Angelidesand Roesset (1981). A cylindricalChapter 5. Dynamicanalyses ofpile foundations731.21.0------‘(lbf/ft)0.6-21L__040.2d=4ft1t=2.5in.)N=lOcycles• F=horizontol force attop of pile (kips)0.00.0 0.5 1.0 1.5ao Z5 3.0FREQUENCY f (Hz)Figure 5.12: Variationof pile horizontal stiffness,k with force and frequencydue tosoil non-linearity(after Angelides andRoesset, 1981)region of soilsurrounding the pileis modelled by using toroidalfinite elements. Aconsistent boundarymatrix was placed at theedge ofthis core region.The equivalent linearmethod (Seed andIdriss, 1967) was usedto model the non-linearsoil response. Evenneglecting slippageand gapping, theydemonstrated a dramaticreduction of horizontalpile head stiffness byapplying harmonichorizontal force at thepile head (Figure 5.12).Similar studiesusing the program PILE3Ddescribed in chapter8 confirm the finding ofAngelides andRoesset (1981).The effect of soil non-linearityon pile head stiffnessissignificant andmust be taken intoaccount with appropriate accuracy.Lumped mass-spring-dashpotmodels. A morecomplex analysis of the seismicresponse of singlepiles which incoporatesthe non-linear response of soilis based on anapproach in whichpile foundation andthe superstructure are analyzedas a combinedsystem. The interactionbetween the pile and thenear field soil is modelled usinga seriesChapter 5. Dynamic analyses ofpile foundations74of non-linear Winkler springs derivedfrom full scale test measurements ornon-linearfinite element solutions (Yegian andWright, 1973; Arnold et al., 1977; Matlocket al.,1978a,1978b; Bea et al.,1984; Nogami and Chen,1987). The stiffness of one of thesesprings represents the combined stiffness of the strainsoftened, near field soil and theexterior free field soil whose properties are governedby the intensity of the earthquakeground motions. The methodofanalysis relating to the use of Winkler springsis usuallyreferred as lumped mass models.At large displacements, the response ofpile foundation is controlled by the non-linearcharacteristics of soil at high strain, pile separation(gapping), slippage and friction.Itis difficult to incorporate these factorsin a continuum model. Therefore lumpedmassmodels, such as these employed by Penzien (1970),Matlock et al. (1978a,1980) and others, have been used to model the pile response atlarge displacements. For example thedynamic pile analysis program SPASM(single pile analysis with support motion) wasformulated by Matlock et al. (1978a,1980)for realistic pile response analysis.Models of this type are versatile for analysis ofsingle piles. However difficulties existin relating the characteristics of the discrete elementsto standard geotechnical parameters of soil. Various non-linear resistance-deflectionrelationships known asp-y curvesand t-z curves have been proposed. Thesoil stiffness at a particular depth is establishedusing as input a non-linear soil resistance versuslateral pile deflection(p-y) curve where yrepresents the relative deflectionbetween the pile and the moving ground during shakingand p is the net soil resistance to the pile motion.Specification ofthe mathematical formofthe backbone p-y curves for both static andChapter 5. Dynamic analyses ofpile foundations75cyclic pile head loading of piles in sandand clay are available from several sources(American Petroleum Institute,1979; Stevensand Audibert,1986; Gaziogluand O’Neill,1984;Murchison and O’Neill, 1984). Theserecommendations have comefrom the results offull scale pile head loading tests. Extensivedata on thep-y curves and non-linear pile response were obtained by Yan (1990) usingthe hydraulic gradient similitude methodandby Gohl (1991) using the centrifuge testing.The most commonly used set of specificationfor constructingp-y curves is based on the recommendations of the AmericanPetroleumInstitute (1986). Mostly usedin offshore structures, thesep-y curves are available forclay and sand, and they makea difference between static loading andcyclic loading.However the validity of their use forearthquake analysis of piles has notbeen verified.In SPASM analysis, the response of the structurerelies on both the accuracy ofp-ycurves for representing the soilnon-linearity and the accuracy of time-historyinput offree field displacements. Verificationstudies of this method byGohi (1992) using datafrom centrifuge tests showed thatthe dynamic response of a structureis sensitive to thetime-history input of free field displacements.The SPASM program underpredicts pilefiexural response. A key difficultyin using SPASM is the accurate determinationof freefield input motions to be usedalong the embedded length of the pile.The dampingproperties are determined separatelyby methods such as the one proposed byGazetasand Dobry (1984). The so-calledcoupled method in SPASM is actuallya semi-coupledmethod. The method only couplesthe super-structure with piles, but itdoes not couplepiles with their surrounding soils directly.Therefore the SPASM analysisis not applicable to analysis of pile groups.Chapter 5. Dynamic analyses ofpile foundations765.2 Dynamic analysis of pile groupsCurrently pile group stiffness and damping coefficients are widelyused in dynamic sub-structuring analysis of superstructure-pile foundation.The analysis of dynamic responseof pile group is limited to elastic response usinguncoupled multi-step analysis. Themethod of analysis (Gazetas et al., 1992) involves estimationof the dynamic foundation impedance and effective input motions applied tothe base of the superstructure.Dynamic sub-structuring analysis is generally carriedout using modal analysis incorporating equivalent elastic pile group stiffness and damping coefficients.The pile groupstiffness and damping coefficients necessary for theanalysis are evaluated using one ofthe following methods or a combination.A useful solution to the three-dimensional dynamic boundary-valueproblem has beendeveloped by Kaynia and Kausel (1982).Results from Kaynia and Kausel (1982) showthat dynamic stiffness and damping of pile groupare highly frequency dependent andmay significantly differ from that of a single pile. Both stiffnessand damping of a pilegroup can be either reduced or increased due to pile-soil-pileinteraction. They mayexhibit very sharp peaks or be affected even for verylarge pile spacings. The dynamiccharacteristics of a pile group may be explained bypile-soil interaction which dependson the ratio of the wave length to pile spacing. Athigher frequencies the waves propagating from a loaded pile in the group may be moving out of phaseat the location of anadjacent pile. The occurrence of this phase shift may resultin negative interaction coefficients which suggests that the stiffness of a dynamically loadedpile group may in factbe higher than the combined stiffness ofa single pile multiplied by the number of pilesin the group. However these analytical results are limitedto linear elastic response. Thesharp peaks in dynamic stiffness and damping of theelastic solution may be suppressedChapter 5. Dynamic analyses ofpile foundations77due to soil non-linearity.The concept of the dynamic interaction factor has been proposed by KayniaandKausel (1982) as an extension of the widely used static interaction factorapproach (Poulos, 1971, 1975, 1979). The dynamic interaction factor approach is an approximationofthe more rigorous pile group analysis. The use of dynamic interaction factorsavoids theheavy computing effort involved in a rigorous pile group analysis. Aset of interactionfactors is available for floating piles, homogeneous soil and a limited selectionof parameters in Kaynia and Kausel (1982) and for vertical vibration in linearly nonhomogeneoussoil in Banerjee (1987). El-Marsafawi et al. (1992a, 1992b) presentedapproximate procedures for estimating dynamic interaction factors based on boundary element analysis,the work of Kaynia (1982), Kaynia and Kausel (1982), Davieset al. (1985) and Gazetas(1991a, 1991b). These dynamic interaction factors are limitedto elastic response, andmostly for homogeneous soil.A procedure for estimating dynamic stiffness and damping of a pile group innonhomogeneous soil was developed and incorporated in a computer programDYNA3 (Novaket al.,1990). In DYNA3 analysis the Novak plane strain pile soil interactionapproachis used to determine stiffness and damping of each single pile, which is similarto thatemployed in PILAY analysis. The dynamic impedance of pile group is thendeterminedby considering the soil pile interaction (or group effect) based on the conceptof dynamicinteraction factors. The dynamic interaction factors used in DYNA3are the combinationof the static interaction factor by Poulos and Davies (1980) for vertical loadingand ElSharnouby and Novak (1986) for horizontal loading and the dynamic interactionfactorsby Kaynia and Kausel (1982). Although DYNA3 analysis can deal with nonhomogeneoussoil, the analysis is limited to linear elastic response and to the use of elasticdynamicChapter 5. Dynamic analyses ofpile foundations78interaction factors.The methods for direct group analysis ofpilefoundationsbased on a continuum modelare limited to linear elastic behaviourusing either boundary element or finite elementtechniques. The linear elastic assumption severelylimits the applicability of these models in describing response of pile groups to moderateto strong shaking where significantsoil non-linearity develops and changesthe extent of interaction between piles.Whilenon-linear 3-D finite element analyseshave been carried out for research purposes to examine pile to pile interaction under static lateralloading (Brown and Shie, 1991), thesemethods are rarely used in practice. Dynamic3-D finite element analyses of pile groupresponse incorporating non-linear soil response havenot been carried out to date.5.3 Objectives of this researchIn following chapters a continuum theoryfor analyzing dynamic response of single pilesand pile groups is presented. The proposed methodof analysis models the dynamic pile-soil-pile interaction as a fully coupled systemand also possesses ability of modellingsoilnon-linear response under strong earthquake loading.A simplified quasi-3D wave equation is proposedto describe the dynamic motionofsoil under horizontal shaking. The coupledequations of motions between the pileandsoil are solved using the finite element method.A finite element program PlUMP isdeveloped to compute pile head impedancesof single pile and pile group by applyingharmonic forces or moments at pile head. Analysesare carried out in the frequencydomain.Chapter 5. Dynamic analyses ofpile foundations79Studies are carried out to validate the applicability of the proposedquasi-3D modelfor simulating the elastic response ofpile foundations.Calibration of the proposed modelis made first against the elastic solutions by Kaynia and Kausel (1982).Verification ofthe proposed model is next conducted using data from full-scale vibrationtests on anexpanded base concrete pile and on a 6-pile group supportinga large transformer.Attention is then focused on extending the proposed model to incorporate non-linearsoil response under strong shaking. Dynamic analysis of pile foundation iscarried outin the time domain and the procedure of this analysis is incorporatedin a computerprogram PILE3D. The non-linear characteristics of soilis modelled by using a modifiedequivalent linear method of analysis. Also effectiveroutines are incorporated in PILE3Dto model the yielding of the soil and the gapping that may occur in thearea near thepile head.The capability ofthe quasi-3D model for simulating the non-lineardynamic responseof pile foundation subjected to earthquake loading is validated using datafrom the centrifuge tests on a single pile and a 2x2 pile group. Under strong shakingsoil non-linearityis significant and changes with time. The level of soil non-linearityalso varies in spaceat a certain time during shaking. Therefore the dynamicstiffness and damping of pilefoundation change with time. The variations of dynamic stiffness anddamping of pilefoundations during shaking are demonstrated for themodel pile foundations used in thecentrifuge tests.Chapter 6Elastic Response of Single Piles:Theory and Verification6.1 IntroductionIn this chapter, a quasi-3D finite element methodof analysis is proposed to determinethe dynamic response of pile foundations subjectedto horizontal loading. The proposedmodel is based on a simplified 3-D wave equation.The 3-dimensional dynamic responseof soil is simulated by displacements in the horizontalshaking direction. Displacements inthe vertical direction and in the horizontalcross-shaking direction are neglected. Therefore a quasi-3D wave equation is established.The finite element method is employed to solve the quasi-3Dwave equation in the3-D half-space domain. Elastic analysesare conducted in the frequency domain.Since the elastic solutions developed by Kaynia andKausel (1982) are the benchmarksolutions for the dynamic response of pile foundations,they solutions are used to calibrate the proposed model for elastic response.Dynamic impedances of single piles arecomputed and compared with those obtainedby Kaynia and Kausel (1982). Kinematicresponse of single piles is analyzed; and resultsare compared with those obtained byFan et al. (1991) who used solutions byKaynia and Kausel. Data from full-scale forcedvibration testing on a single pile are alsoused to validate the proposed model.80Chapter 6. Elastic Response ofSingle Piles: Theoryand Verification 816.2 Dynamic analyses of pile foundations: formulationUnder vertically propagating shear waves (Figure6.1) the soils mainly undergo sheardeformations in XOY plane except in the area nearthe pile where extensive compressiondeformations in the direction of shaking develop.The compression deformations alsogenerate shear deformations in YOZ plane, seeing Figure6.1. Under the light of theseobservations assumptions are made that dynamicmotions of soils are governed by theshear waves in XOY plane and YOZ plane, and the compressionwaves in the shakingdirection, Y. Deformations in the vertical direction and normal to thedirection of shakingare neglected. Comparisons with full 3-D elastic solutions confirmthat these deformationsare relatively unimportant for horizontal shaking.Let v represent the displacement of soil in the shaking direction, Y.The compressionforce is8G1.The shear force in XOY plane isG1,and the shear force in YOZplane is The two shear waves propagate in Z directionand X direction, respectively. The inertial force is p$; Applying dynamicforce equilibrium in Y-direction,the dynamic governing equation under free vibrationof the soil continuum is written as82v 82v 82vG’+ +Ps(6.1)whereG*is the complex shear modulus,Psis the mass density, and = 2/(1—v) fora Poisson’s ratio v. Since soil is a hysteretic material, the complexshear modulusG*isexpressed as = G(1+i2A), in which G is the shear modulus of soil, and Ais thehysteretic damping ratio of soil. The radiation damping will be included later.The displacement field at any point in each element is modelled bythe nodal displacements and appropriate shape functions. A linear displacementfield is assumed inChapter 6. Elastic Response ofSingle Piles: Theory andVerification82///// ///Y/////7////////////,/////,7//,.///////_/_40.Direction ofshakingFigure 6.1: The principle of quasi-3D dynamic pile-soil interaction in the horizontaldirectionstructural masszSoil7,,/‘shear‘shearI_____4CompressionYPile3-D finite elementsChapter 6. Elastic Response ofSingle Piles:Theory and Verification85156 221 54 —131221412131.3l254 131 156 —221—131_3l2—22l412The radiation damping is modelled usingvelocity proportional damping.Theing forceFd per unit length along the pile is given byFd=c,,-- (6.7)whereca, is theradiation dashpot coefficientfor horizontal motion.A simple expression for the radiation dashpotcoefficients c, which was proposedbyGazetas et al. (1993) and is givenin Eq. 5.4, is used in the analysis. Applyingthe sameprocedure as that used to obtain massmatrix, the radiation damping matrix[C’] for apile element is156 221 54 —131221412131_3l254 131 156 —221—131_3j2—2214j2The global dynamic equilibrium equationin matrix form is written as[M*j{i} + [C*j{)} + [K*]{v}= {P(t)}(6.9)in which {i},{i} and {v} are the nodal acceleration,velocity and displacement,respectively, and P(t) is the external dynamic loadsapplied.pEAl[M]ji— 420(6.6)damp-* Cr1pi1e (6.8)Chapter 6. Elastic ResponseofSingle Piles: Theory and Verification86__Kv4..0e=o =Figure 6.3: Pile head impedances6.3 Pile head impedancesThe impedances are definedas the complex amplitudes of harmonic forces(or moments) that have to beapplied at the pile head in order to generatea harmonic motionwith a unit amplitudein the specified direction (Novak,1991).The concept of translationaland rotational impedances is illustratedin Figure 6.3.The translational, the cross-coupling,and the rotational impedances of pile foundationsused in this analysis aredefined as•K: the complex-valued pile head shear force required to generateunit lateraldisplacement (v=1.O) at the pilehead while the pile head rotation is fixed (S = 0).•K8:the complex-valued pile head moment generated by the unit lateral displacement (v=1.0) at the pile head whilethe pile head rotation is fixed (S= 0)Chapter 6. Elastic Response ofSingle Piles: Theoryand Verification 87• K88:the complex-valuedpile head moment required to generate the unit pileheadrotation(1.0) while pile head lateral displacement is fixed(v=0)Since the pile head impedances K,,,,,K,,8,K88 are complex valued,they are usuallyexpressed by their real and imaginary parts as=k1+iC (6.10)=+i (6.11)in which and are the real and imaginary parts of the compleximpedances,respectively, and i = cj = Cj/w = coefficient of equivalent viscousdamping; andw is the circular frequency of the applied load. andC1 are usually referred as thestiffness and damping at the pile head. All the parametersin Eq. 6.10 are dependent onfrequency w.Determination of impedances K,,,,, K,,8 and K88 Pile head impedanceswill beevaluated as functions of frequency by subjecting the system to a seriesof harmonicloads. Under harmonic loading P(t)= Poeit,the displacement vector is of the formv=voet,and Eq.6.9 is rewritten as{[K] + i. w[C*]—w2[M]}{vo} ={P0} (6.12)or[K]global{VO} = {P0} (6.13)Chapter 6. Elastic Response ofSingle Piles: Theory and Verification88where[K1gzabo1[K*j+i w[C] — w2[.M] (6.14)According to the definition, impedances K,,,, andKue can be found by applying a unithorizontal displacement at the pile head under thecondition of a fixed pile head rotation.Eq.6.13 becomes0[K]gia&ai 1.0= K,,,,(6.15)0.0 K,,8wherev are the displacements of the nodes other than pile head.Dividing Eq. 6.15 byKu,,and eliminating the row of zero rotation, one obtainsIv/K,,l 1 0 1[K]9z0b = (6.16)IvJ(1.0)where v= 1/K,,,,. The moment at pile head M corresponding to v is also computed.This suggests that an easy alternative for determining pilehead impedances is toapply a unit horizontal force at the pile head andcalculate the complex displacement atthe pile head v. Therefore the pile head impedancesK,,,, and K,,9are determinedK,,,, = (6.17)K,,9= (6.18)Chapter 6. Elastic Response ofSingle Piles: Theoryand Verification 89Using the same principle, the rotational impedanceK88 can be determined by applyinga unit moment at the pile head under the conditionof a fixed pile head horizontal displacement. The rotational impedance K88is determined asK88 = (6.19)where 8 is the rotation at the pile head causedby the unit moment at the pile head.Because ofreciprocity principle, the cross-couplingimpedanceK8andK8are identical.6.4 Verification of the proposed model:pile head impedancesIn order to assess the accuracyof the proposed quasi-3D finite element approach, theimpedance functions for single piles aredetermined and compared with the analyticalresults by Kaynia and Kausel (1982). Analyseswere performed in the frequency domainfor elastic conditions. ImpedancesK8,K88 will be presented as functions of thedimensionless frequencya0,where a0 is defineda0 = (6.20)in which w is the angular frequency of the excitingloads (force and moment) at thepile head, d is the diameter of the pile, and V1is the shear wave velocity of the soilmedium. For a uniform soil profile with a shearmodulus G and a mass densityp, V8 iscomputed by V8 =It is found that for given values ofE/E8anda0,the ratiosK/(E8d),K8/(E3d2),andK98/(E8d3)hold unchanged for any soil modulusE5 of a uniform soil profile. Therefore the normalized impedancesK/(E5d),K8/(Ed2),andK88/(E8d3),are presentedChapter 6. Elastic ResponseofSingle Piles: Theory andVerification90single piles:LdEp/Es = 1,000L/d > 15 (floating pile)soil damping 5 %Poisson’s ratio 0.4Figure 6.4: A pile-soil system usedfor computing impedancesof single pilesas a function of the dimensionlessfrequency a0 when other parametersare given.Figure 6.4 shows a pile-soil systemand its relative parameters. A ratioof Er/ES =1, 000, which was used byKaynia and Kausel, is adoptedfor the analysis, where E andare the Young’s moduli of thepile and the soil, respectively. Thesoil medium has aPoisson’s ratio v =0.4 and a hysteretic damping ratio )L= 5%. A mass density ratiopp/pa=1.4 is used here.Due to symmetry, only halfof the full mesh is required to model the responseofpile-soil interaction.The halfmesh, shown in Figure 6.5, consists of1463 nodes and 1089elements. The useof half mesh reduces the size of the global matrixby a factor of 4.0with a correspondinglarge reduction in computational time. It tookabout 300 secondsto determine the dynamicimpedances for each frequencyusing a 486 PC computer.Chapter 6. Elastic Response ofSingle Piles: Theory and Verification91VI,/Figure 6.5: Finite element modeffingof single pile for computing impedancesWhen the symmetric conditionis applied, the Young’s modulus E and the massdensityPpof the pile should be reduced by a factor of twoin the case of a single pilewhich is bisected the axis of symmetry.This reduction is due to the fact that the centralpile has been shared evenly by theother half of the full mesh. For the same reason, theapplied loads (force and moment) at thepile head should be reduced by a factor of2.0.These adjustments are automatically includedin the program PlUMP.Discussion ofresults The normalizedquantitiesK/(E3d),K6/(E9d2),K98/(E3d)are presented as functions of dimensionless frequencya0.Since the impedances are complex quantities, their values are expressedin term of their real partsk1(stifFness) andimaginary partsC1(damping) accordingto Eq.6.l0. Hence the normalized stiffness anddamping are compared with those obtainedby Kaynia and Kausel (1982), and they areshown in Figure 6.6, 6.7, and 6.8, respectively.Chapter 6. Elastic Response ofSingle Piles: Theory andVerification 9212-10..- - -- Kaynia and Kausel (1982)8=••Proposed model6-4-.”Ef2-I I I I I I I I I II I I III I I I I I I I0.00 0.100.20 0.300.40dimensionlessfrequency,a012: Kayniaand Kausel (1982)10.: •Proposed model-0—IIIIliIIIIII1IIIIIIIllIiillIlIIIIIIIIJ0.00 0.100.20 0.300.40dimensionless frequency,a0Figure 6.6: Normalized stiffness k, and damping C versus a0 for single piles(E/E, 1000, ii =0.4, )=5%)Chapter 6. Elastic Response ofSingle Piles:Theory and Verification9312-10----- Kaynia and Kausel (1982)a aProposed modelN8-Cl) -a)C -4-2-0— I I I III I I I I I I I III I I I I I I I III I I I0.00 0.100.20 0.300.1dimensionless frequency,a012-Koynia and Kausel(1982)10 - •-°-- Proposed model0— i i I I I III I I I I I I I III I I I I I I II I I I I I0.00 0.100.20 0.300.40dimensionless frequency,a0Figure 6.7: Normalized stiffnessk9and dampingCueversus a0 for single piles(E/E5= 1000, ‘ =0.4, A=5%)Chapter 6. Elastic Response ofSingle Piles: Theory andVerification9440------ Koynia and Kausel (1982)Proposed model20-(I)1 :—I I I III I I I I I I I I I I I I I I I IIII I I I I I I I0.00 0.100.20 0.300.40dimensionless frequency,a020-- Kaynia and Kausel(1982)-Proposed modelcjio0—IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0.00 0.10 0.200.30 0.40dimensionless frequency,a.0Figure 6.8: Normalized stiffness k99and damping C versus a0 for single piles(E/E6= 1000, z’ =0.4, )=5%)Chapter 6. Elastic Response ofSinglePiles: Theory and Verification95The computed impedancesK8,andK88 agree well with those obtained by Kaynia and Kausel. Both solutionsshow that dynamic stiffnesses increaseslightly as thedimensionless frequency a0 increases.The computed quantities are slightlysmaller thanthose by Kaynia and Kausel, whichimplies their modelling ofthe pile-soilsystems resultsin stiffer response. The translationalstiffness computed by Kayniaand Kausel areabout 10% larger than those fromcurrent study. The other two stiffnessesk8 and k98show less sensitivity to the methodof computation. The differences ofk8 andk98 fromthe more exact solutions are about5%.However, the difference ofimpedance between the two solutionsis insignificant inpractice when soil non-linearityis an important factor. In many casesreduction of soilshear moduli with the increase of shearstrain is significant. Quantitatively modellingofreduction of shear moduli, especiallyin the near field of the pile,is important for thedetermination of pile head impedances.It will be shown later that the proposedmethodhas the ability to determinethe non-linear dynamic impedances asthe soil moduli decrease with the increase of shear strainunder strong shaking.Comparison with solutions by Novaket al. Using the mesh shown in Figure6.5,stiffness k,and dampingC were computed for a Er/ES ratio of 295.The results arethen compared to solutionspresented by Novak and Nogami (1977),and Novak’s approximate (Novak,1974). Figure6.9(a) and 6.9(b) show comparisons on stiffness anddamping, respectively. The valuesof dynamic stiffness computed by Novak and Nogamiare about 25% larger than thosecomputed by author. For a0> 0.3, values of dynamicstiffness from Novak’s approximatesolution are about 10% larger than those computedby author. For dimensionless frequencya0 less than 0.3, dynamic stiffness computedbyChapter 6. Elastic Response ofSingle Piles: Theory and Verification968-Novak and Nogami(1977)6 — —— Novak’s Approximate(1974)42:0.00 0.100.20 0.300.40dimensionless frequencya08-Novak and Nogami(1977)6- — —— Novak’s Approximate(1974)•Proposed modelIo0— i i i III I I I I I I I I I I I I I II I I III I I I I0.00 0.10 0.200.30 0.40dimensionless frequencya0Figure 6.9:Comparison of stiffness and damping C with solutionsby Novak andNogami (1977), Novak(1974)Chapter 6. Elastic Response ofSingle Piles: Theory and Verification978-6-___Fine mesh with 1 089 elementsCoarse mesh with 709 elements4----- -------- -.2-I I I I I I I I I I I I I I I I II I I I I I I I III I I I I I I I I0.00 0.100.20 0.30 0.40dimensionless frequency,a0Figure 6.10: Comparison of stiffness for different mesh sizeNovak’s approximate method diminishes with the decrease of frequency.The dampingfrom Novak and Nogaini or Novak’s approximate method is normallylarger than damping computed in the present study (Figure 6.9(b)).Effect ofthe number offinite elements The number offinite elements used in theanalysis has some influence on the impedances computed using the proposed method.Theoretically accuracy ofthe results increases with the number of finite elements. Especially at a high frequency (such as a0> 0.3), the number of finite elements needs to bevery large to capture the possible number of modes that are significant to the responseof pile foundations at that frequency.Figure 6.10 shows a comparison of the dynamic stiffness computedby two differentmeshes. It is clear that some differences exist between results from the two meshes aroundChapter 6. Elastic ResponseofSingle Piles: Theory and Verification98______pile head motion free field motionsingle free-head piles:LEp/Es = 1,000 or 10,000dLid = 201 .5Lsoil damping 5%Poisson’s ratio 0.4-4base motionFigure 6.11: Pile foundation foranalysis of kinematic responsea0=O.3. A finer mesh with more finite elements isbetter to represent the dynamic response accurately.6.5 Verification of the proposed model:kinematic interactionA pile-soil system shown in Figure6.11 is subjected to a harmonic displacement vbei1tat its rigid base. The dynamic responseat the pile head may be same as or very closeto the dynamic response at the free field surfaceif the pile is very flexible. However inmany cases the dynamic responseat the pile head differs significantly from the responseat the free field surface becausepiles are generally much stiffer than soil and thus modifysoil deformations. This typeof interaction between piles and soils is called kinematicinteractionChapter 6. Elastic Response ofSingle Piles: Theory and Verification996.5.1 Kinematic interactionfactorsThe dynamic motions at the free fieldsurface and at the pile head are differentdue to thekinematic interaction between thepile and the soil. Let the harmonic displacementsatthe free field surface be representedbyv11eif1t, and at the pile head byvetand&peiat,in which v, and 8, are the complex amplitudesof the translational displacement and therotational displacement, respectively.Absolute values of complex amplitudesof harmonic displacements are used fordetermining the kinematic interaction factors.The kinematic interaction factors I,. andI, aredefined after Gazetas (1984) as=(6.21)= d(6.22)in which Ui,, U1Iand are the absolute values of thecomplex amplitudes v,, V11and,respectively; and d is the diameterof the pile.6.5.2 Computed kinematic interactionfactorsThe kinematic interaction factors areobtained for pile-soil systems with a flexible pile(E/E8= 1, 000), and with a stiffpile (E/E, = 10, 000). The other parameters of thesystems are shown in Figure 6.11.The accuracy ofthe quasi-3D finite element methodis checked against the boundaryintegral method developed by Kaynia and Kausel(1982) and used by Fan et al. (1991).Chapter 6. Elastic Response ofSingle Piles:Theory and Verification 100The computed kinematic interaction factorsI, and‘ç1as functions of the dimensionless frequency a0 are plotted in Figure6.12 forE/E8=1,000 and in Figure 6.13 forE/E=10,000 together with the interaction factorsobtained by Fan et al. (1991). Acomparison of the two sets of factorsshows that there is very good agreement betweenthe quasi-3D solutions and the boundary elementsolutions.The kinematic interaction becomes more significantwhen the stiff pile is placed inthe soil. ForE/E8=10,000, the response of the pilehead is significantly reduced whenthe dimensionless frequency a0 is greater than0.25. At a0 = 0.35, the amplitude ofthe translational displacement at the pile headis only 45% of the amplitude of thedisplacement at the free field surface.6.6 Verification of the proposed model:forced vibration testingDynamic vibration testing of an expandedbase concrete pile was conducted and reportedby Sy and Siu (1992). The vibration testwas carried out by applying very low harmonicloads at the structural mass, whichgenerated elastic response in the system. This provides an opportunity to validate the quasi-3D modelfor elastic response.6.6.1 Description of site conditionand test resultsThe testing site is located in the Fraserriver delta south of Vancouver. The soil profileat the testing site consists of 4m of sand and gravel fill overlying a 1 m thick silt layerover fine grained sand to 40m depth. A seismic cone penetration test (SCPT88-6)was conducted 0.9 m from the test pile location.In addition a mud-rotary drill hole(Chapter 6. Elastic Response ofSingle Piles: Theory and Verification1012.0• --Fan et al. (1991)(a) - -- Proposed model‘palo• Ep/Es = 1000_D1.5 -0o10-CDC -0 -0.5 -I.1).90.0 — I I III I I I I I I III I I I I I III I I I I I I I I I I I I I I0.0 0.1 0.2 0.30.4 0.5dimensionless frequency0.5 -(b)Fan etal.(1991)0.4- ..01..Proposed modelEp/Es = 10000.3 -____4-’C.)CD0.2 -C.)CD0.14-’.90.0 — 1Ti I I III I I I I III I I I I I I I I I I I I0.0 0.1 0.2 0.30.4 0.5dimensionless frequency a0Figure 6.12: Kinematicinteraction factors versus a0 for Er/ES = 1, 000Chapter 6. Elastic Response ofSinglePiles: Theory and Verification1022.0 -(a)-----Fanetal. (1991)....Proposed model1.5- Ep/Es = 10,000o1.0 -‘4-C00.5 -0.0 - I III I I I I I I I I I I I I I I III I (I I I I I I0.0 0.1 0.2 0.30.4 0.5dimensionless frequency a00.5 -(b)Fan etal. (1991).Proposed model- Ep/Es = 10,0000tO.3-‘IC.20 2 -C.)C’,00.1 -0.0— IiIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0.0 0.1 0.2 0.30.4 0.5dimensionless frequency a0Figure 6.13: Kinematicinteraction factors versus a0 for Er/ES 10,000Chapter 6. Elastic Response ofSingle Piles: Theory and Verification103CPT Oc (bar) SPT N (blows/O.3m)Vs (m/s)0 125 250 0 10 20 30 40 0 100 200 3000—iiIi — iiiiIiiiitiiIii, — iiIiitiIiiit:—,A’... 0•• 4..-.--- 0•IC p- - •-. 0•- 0 •r115-- -d0 •0I—00—- I — 0•w 20- - -0025--• . :30-:SCPT 88—B•. •....DH 88—2- — -Figure 6.14: The in-situ measured geotechnical data (after Sy and Siu, 1992)DH88-2) was carried out 2.4 m from the test pile location. The measured in-situ shearwave velocity data are presented in Figure 6.14, together with the cone penetration test(CPT) data and the Standard Penetration Test (SPT) data.The layout of the pile test is shown in Figure 6.15. The pile is an expanded baseconcrete pile (Franki-type), which had a nominal 510 mm diameter shaft down to 7.6m depth with an estimated 0.93 m diameter spherical base. In order to perform thevibration test, the top of the cast-in-situ concrete shaft was extended above the groundsurface, and a structural mass consisting of 1.6 m cube of reinforced concrete was thenformed on top of the pile. The final length of the additional pile shaft was 1.37 m with150 mm above the ground surface.Sinusoidal sweep testing was carried out for determining the fundamental frequencyChapter 6. Elastic Response ofSingle Piles: Theory and Verification104FFI—ELECTROMAGNETIC SNAJ<ER_________ACCELEROMETER LOCATION FOR43Omnj_ACCELEROMCFER LOCATION FORI CONCRETE STRUCTURAl. MASS-:ROCKING EXCITATIONIIBm 1.6m150mm I ACCELEROMETER LOCATION FORHORIZONTAL EXCITATIONI__________510mm SQUARE REINFORCED1 .37mCONCRETE SECTION510mm DIR. CONCRETESHAFT WITH 8—20mm DIA.REINFORCING BARS6.4mxU___930mm DLA. SPHERICAl.CONCRETE BASEFigure 6.15: The layout ofthe full-scale vibrationtest on a single pile (after Sy andSiu,1992)and damping ratio of the system.Harmonic loads with an amplitude of 165 N was applied horizontally at the center ofthe shaker(Figure 6.15). The shaker is located at 2.03m above the pile head. The measured fundamental frequencyof the structure-pile-soilsystem was 6.5 Hz. The dampingratio was determined to be 4%. The damping ratiowas calculated from the measuredresponse curve using the bandwidth method (Cloughand Penzien, 1975).6.6.2 Computed results usingthe quasi-3D modelThe structural properties of thepile cap and the test pile used in the analysis are presented in Table 6.4. The shear wave velocity(V6)1unit weight and damping ratio (D3)used in the analysis are shown inFigure 6.16. According to Sy and Siu (1992), exceptfor the top 1.2 m depth, an upperbound of the measured V, values was used to accountfor the effect of soil densificationcaused by pile installation. However V6 values at theChapter 6. Elastic Response ofSinglePiles: Theory and Verification105Table 6.4: Structural propertiesof pile cap and test pile (afterSy and Siu, 1992)ParameterIUnitIValue. PILE CAP AND SHAKERMassMg 10.118 -_Mass moment of inertiaMg in2 4.317Height to center of gravityin 0.8, TEST_PILETop 1.37m : axial rigidity (EA)MN 6350Top 1.37m : flexural rigidity (El)MN in2 1411.37-7.77m : axial rigidity (EA)MN 5150I.37-7.77m flexural rigidity (El)tIN in2 92Base axial rigidity (EA)tIN 14,720Base : flexural rigidity (El)MN in2 800Material damping ratIo0.01Poisson’s ratio0.25upper 1.2 m were reduced since theoriginal soil around the extended pileshaft sectionwas replaced by the loose backfill.Poisson’s ratio v 0.3 was assumedfor all soil layers.Figure 6.17 shows the 3-Dfinite element model used for obtaining thepile headimpedances. The finite element modelconsists of 1225 nodes and 889 elementswith onebeam element above the ground surfacerepresenting the pile segment above the ground.The expanded concrete base wasmodelled by a solid element rather thana beam elementin the finite element analysis.The dynamic impedances K,,9,andK88 were obtainedat the pile head.After the dynamic impedancesof the pile foundation have been determined,the dynamic response of the pile capcan be obtained by performing a structural analysis. Thetranslational and rotationalresponse of the pile cap are obtained byusing the dynamic0DEPTH(m)0)11,11111.1CD I-. CD CD CD CD 0 0 CD CD p CD p Ci) CD CD0IIUqCD I cj CD U) 0 2 p CD CD Ci) Ci) CD -S. CD pp Cl) -S. U)np CD (ID p C,) -S. ICD_l_IIIIIIIII1_IlIIII,i1%)-I_n 0—.C3p Cl) 0 Cl) CD 0 CD pC.) p 0-Cl) K’I_______IIIIIIIIIIIIIID 0Chapter 6. Elastic Response ofSingle Piles: Theory andVerification 107Figure 6.18: An uncoupled system modelling the horizontal motionsof structure-pile capsystemsolution of a two-degree of freedom system (Figure6.18). Under harmonic loads, thetranslational displacement amplitude v, and the rotational displacementamplitude 6,, atthe pile head are computed according to thefollowing equation2m mh9Iv,,+iC k9+iC9f 1 1‘o—w çmh9(8Jk9+iC9k99+iC99(6,,J (M0(6.23)where m is the mass of the pile capand shaker,h9is the height of the centre ofgravity to the pile head, and .1 is the massmoment of inertia at the centre of gravity;C are the stiffnesses and dampings at the pile head; P0 and M0are amplitudes ofthe harmonic external force and moment,respectively, applied at the pile head.The quantities v and 6,, are determined using the testing loads.During the test harmonic horizontal loads with amplitude of165 N was applied at the shaker, which caused2DF systemChapter 6. Elastic Response ofSingle Piles: Theory and Verification108(expanded base pile)measuredE‘ :frequency:6.5HzI I I I II I I2 61014frequency, HzFigure 6.19: Amplitudes of horizontal displacement at the centre of gravityof the pilecap versus the excitation frequencya moment of 335 N.m at the pile head. ThereforeP0=165 N and M0 335 N.m wereused in Eq. 6.23 for obtaining v, and 8. The horizontal displacement amplitude at thecentre of gravity of the mass can now be calculated byvcg = v, + 8, (6.24)The analyses were carried out at different frequencies w. The computed horizontaldisplacement amplitude at the center of gravity of the mass versus frequency w is shownin Figure 6.19.Discussion of results Very clear and pronounced peak response is observed for thehorizontal motion. Maximum horizontal displacement at the center ofgravity of the pilecap occurs at an excitation frequency around 6.67 Hz compared to a measured resonantfrequency of 6.5 Hz. The damping ratio is evaluated from the response curve in FigureChapter 6. Elastic Response ofSingle Piles: Theoryand Verification 1096.19 using the bandwidth method. The computeddamping ratio is 6% compared to ameasured damping ratio of 4%. This analysis demonstratesthat the proposed model hasthe capability of modelling the dynamic responseof single piles.Chapter 7Elastic Response of Pile Groups: Theory andVerification7.1 IntroductionThe quasi-3D model applied in the previouschapter to single piles is also applicable tothe analysis of elastic response of pile groupsunder horizontal excitation. However, thehorizontal displacement is coupled withrocking of the group. The rocking impedanceofpile group is the measure of the resistance torotation of the pile cap provided by theresistance of each pile in the group to verticaldisplacements.In this chapter, the determination of the rocking impedanceof a pile group is formulated first by applying the quasi-3D model in thevertical direction. For the verificationof the proposed model, dynamic impedances of a 2x2pile group are computed and compared with those by Kaynia and Kausel(1982). Finally, results of a full-scale vibrationtest on a 6-pile group are used to verify the proposedmodel.7.2 Rocking impedance of pile groupThe rocking impedance of a pile group reflectsthe resistance of the pile group to therotation of the pile cap when piles are attachedto a pile cap. If the piles can be considered pinned to the pile cap the rotationof the pile cap does not cause moments at pileheads, but it does induce vertical axialforces at pile heads as shown in Figure 7.1. The110Chapter 7. Elastic ResponseofPile Groups: Theory and Verification111.-.-.-..-.-.-.-.-.-.-.-.-.- -------.--- .-- - --.-.----- -.-.-- ---- - -.-------.-------.-.-rigid baseFigure 7.1: The mechanism of rockingin a pile grouprocking impedance K,..of a pile group is defined as the summationof moments aroundthe centre of rotationof the pile cap. These moments are causedby the axial forcesatall pile heads requiredto generate a harmonic rotation withunit amplitude at thepilecap. This definition isquantitatively expressed asKr,. . F whefl...cap 1.0 (7.1)where r are distances betweenthe centre of rotation and the pile headcentres, and Fare the amplitudes ofaxial forces at the pile heads.In the analysis, thepile cap is assumed to be rigid. For a unit rotationof the pilecap, the vertical displacementsw’ at all pile heads can be easily determinedaccording to their distances from thecenter of rotation r. Now thetask is to determine theaxial forces F at the pileheads which are required to generatethese vertical displacements w’. The quasi-3Dmodel is applied in the vertical directionto accomplish this task.centre of0pile cap pilerrjChapter 7. Elastic Response ofPile Groups: Theory and Verification1127.3 Dynamic equation of motions in the vertical directionUnder a vertically propagating compression wave, the soilmedium mainly undergoescompression deformations in the vertical direction. In the two horizontal directions,shearing deformations are generated due to the internal friction of thesoil. Althoughcompressions occur in the two horizontal directions, assumptions aremade that the normal stresses in the two horizontal directions are small and can beignored. Thereforethe dynamic motions of the soil are governed by the compressionwave in the verticaldirection and the shear waves propagating in the two horizontaldirection X and Y asshown in Figure 7.2.By analogy to the principle used in the previous chapter, the quasi-3D wave equationof soil in the vertical direction is given byG—+G*4+6ZG4=p$(7.2)whereG*is the complex shear modulus,P8is the mass density of soil, and 6 is afunction of Poisson’s ratio v. Based on assumptions that normal stresses in the twohorizontal directions X and Y are zero, it is determined6= 2(1+1?).The stiffness matrix [K]801 and mass matrix [M]801 are evaluated from Eq. 7.2 foreach soil element as described earlier.Under a vertical propagating compression wave, the undamped free vibration equation of motion for a pile element is given byChapter 7. Elastic Response ofPile Groups: Theory and Verification 113I /7/ / / /“__7/ // /3-D finite elementsy//Z///////// 7/7-,7.7_,7-77.7;71[K]piie— EA1 —11—[—1 1j[M]pjie— plA F2 116[ 2JoilPileFigure 7.2: The quasi-3D model in the vertical direction, Z82w 82w=pA-- (7.3)where EA is the compression rigidity of the pile, and p, is the mass density of the pile.The stiffness matrix[K]pjieandthe mass matrix [M]jie ofthe pile element are given byand(7.4)(7.5)The radiation damping under vertical motion is also modelled using velocityproportional damping. The radiation dashpot coefficient c2,which was proposed by Gazetasetal. (1993) and is given in Eq. 5.3, is used in the analysis. The radiation dampingmatrixChapter 7. Elastic Response ofPile Groups: Theory and Verification114for a pile element is given byci2 1[C]piie = (7.6)12The global dynamic equilibrium equation in matrixform is given by[M]{’th} + [C]{tb} + [K]{w} ={P(t)} (7.7)in which {zui}, {zi} and {w} are the nodal acceleration, velocityand displacement, respectively, and P(t) is the external dynamic loads applied.7.4 Determination of rocking impedanceIn order to evaluate the rocking impedance of pilegroup, harmonic forces P(t)= Poeitare applied, which generate harmonic displacementsw= woeit.Therefore, Eq.7.7 isrewritten as[K]global{WO} {P0} (7.8)where[K]globaj = [K] +i w[C] — w2[M](7.9)Since the vertical displacements at the pile heads {wç,wç,•,•,WPm}Tare known, theaxial forces {F1,F2,, ,Fm}T at the pile heads are determinedbyChapter 7. Elastic Response ofPile Groups: Theory and Verification1150W2 0Knm0 (7.10)Kmn KmmF1WFmWhereKnm, Kmn, and Kmm are sub-matrices of the global matrixKgio&ai,and{w1,w2,•,•,w}Tare vertical displacements at nodesother than the pile heads. The pilehead axial forces {F1,F2,,Fm}Tcan be determined ifthe displacements {wi,w2,.,.,wn}Tare known. Applying thematrix separation technique to Eq. 7.10yieldsWi[K]+ [Knml= {0}(7.11)WnandWiF1Wi[Kmnj +[Kmm] • (7.12)FmWnAfter the displacement vector{w1,w2,.,.,w}Tis computed from Eq.7.11, the pilehead axial force vector {F1,,Fm} is then determined using Eq. 7.12 . Now the rockingimpedances of the pile group are evaluatedusing Eq. 7.1. The procedure for computingrocking impedances of pile groupis incorporated in the computer program PlUMP.Chapter 7. Elastic Response ofPile Groups: Theoryand Verification116F____________________ _________________2x2 pile group:LEp/Es = 1,000ds/d=5.0L/d > 15 (floating pile)soil damping 5 %Poisson’s ratio 0.4S•IFigure 7.3: A pile-soil system used for computingimpedances of pile groups7.5 Elastic response of pile group:results and comparisonsThe dynamic impedances of a 4-pile (2x2) groupwith s/d=5.0 are presented, in which sis the centre to centre distanceof two adjacent piles and d is the pile diameter. A rigidpile cap is rigidly connected tothe four pile heads as shown in Figure 7.3. A stiffnessratio of the pile and soil E/E2= 1000 is used, and a mass density ratioPa/Pp0.7 isapplied.The dynamic impedancesof pile groups were obtained by Kaynia and Kausel (1982)for a half-space soil medium.In approximating the half-space soil medium using thefinite elements, a rigid baseis assumed at a depth of 5L (L=length of pile) beyond thetip of the pile. The dynamicimpedances are evaluated at the bottom of the pile cap.Chapter 7. Elastic Response ofPile Groups: Theoryand Verification 117Discussion of results In order to show thepile group effect, dynamic impedances ofthe pile group are normalized to the static stiffnessof the pile group expressed as thestiffness of a single pile times the number of pilesin the group. The normalized dynamicimpedances ofthe pile group, which are called the dynamic interactionfactors, are therefore defined asa,,,,N.kV°V(7.13)a,,9N.k,,°9(7.14)K88a99=(7.15)LV96in which k,,,,k,,°9,k906 are static stiffnesses of a single pile identical tothose in the pilegroup that is placed in the same soil medium, and N is the numberof piles in the pilegroup(N=4 for a 4-pile group). K99 in Eq. 7.15 is the individualrotational impedanceat the head of each pile due to the geometrical andmaterial properties of the pile.In order to present the results graphically, the complex-valued dynamicinteraction factors a23 are separated by their real parts a(stiffness)and imaginary partsa(damping). The computed dynamic interaction factora(stiffness) is compared inFigure 7.4(a) with that by Kaynia and Kausel (1982). Very goodagreement is observedbetween the two solutions for dimensionless frequencya0 < 0.28. For a0 > 0.28, thecomputed values are about 25% higher than those byKaynia and Kausel. The computedinteraction factors a(damping) are in good agreement with thoseby Kaynia and Kausel(Figure 7.4(b)).The dynamic interaction factors a,,,,, anda99 of the pile groupare shown in Figure 7.5(a) for stiffness component and in Figure 7.5(b) for dampingcomponent. TheChapter 7. Elastic Response ofPile Groups: Theoryand Verification 118results show that the translationalstiffness (or cx) shows the greatest effectofgrouppile-soil interaction; whereas the rotationalstiffness k88 (oraee) shows the least effect(Figure 7.5(a)). However their corresponding dampingcomponents show the reversetrend (Figure 7.5(b)). For a0< 0.3 the stiffness interaction factorsave, and cxee arein the range of 0.6, 0.7 and0.9, respectively.Because the piles are rigidly connectedto the pile cap at the pile heads, the totalrotational impedance of the pile groupconsists of both the rocking impedanceKrof the pile group and the rotationalimpedance K98 at the head of each pile= K.+K89 (7.16)Following the notation used by Kayniaand Kausel, the total rotational impedanceof thepile group K7 is normalized as K/(N. r?in whichk°is the static verticalstiffness of a single pile placed in the same soilmedium.The normalized quantity K’/(Nk) is compared in Figure 7.6 with thatby Kaynia and Kausel. Very goodagreement between the two solutions isseen for thestiffness component (Figure7.6(a)). Good agreement between the two solutionsfor thedamping components also existsfora0 < 0.3 (Figure 7.6(b)). Fora0>0.3, the computeddamping component is about25% less than that by Kaynia and Kausel.It has been shown that the proposedmodel can well simulate the dynamic characteristics of pile groups. The conclusionis drawn from comparisons with analyticalresultsby Kaynia and Kausel. To further verify theapplicability of the proposed modelforsimulating the elastic response of pile group, fieldvibration tests of a 6-pile group areChapter 7. Elastic Response ofPile Groups:Theory and Verification1193-(a) stiffness component2-Kaynia and Kausel (1982)- Proposedmodel.22x2 pile group; s/d=5C 1-I.90-0.00 0.100.20 0.300.40dimensionless frequencya03-:(b) damping component2-04-,C)0‘4—— 0— i I I I III I I I I I I I III I I I I I I I I I I II I0.00 0.100.20 0.300.40dimensionless frequency,a0Figure 7.4: Comparison of dynamicinteraction factor a.,, with solution by Kaynia andKausel for 2x2 pile groups(E/E3= 1000,s/d 5.0)Chapter 7. Elastic Response ofPile Groups:Theory and Verification1203-(a) stiffness component-TranslationCross—coupling2-Rotation2x2 pile group; s/d=50 -C)0oz:0a)-90—0.000.10 0.200.30 0.40dimensionless frequency,a03-(b) damping component2-U’0-I-,C)0c 1-0C-)00- .. .. ::.-90.00 0.100.20 0.30dimensionless frequency,a0Figure 7.5: Dynamic interactionfactors a, a66 versus a0 for 2x2 pile groups(E/E, = 1000,s/d= 5.0)Chapter 7. Elastic Response ofPile Groups:Theory and Verification1214-(a) stiffness componentKaynia and Kausel (1982)3Proposed model- 2x2 pilegroup; s/d=52--.-1—0—iiiiiiiiiitiiii,iiiiiiiiiiiiiiiiiiiiiii0.00 0.100.20 0.300.40dimensionless frequency,a04-- (b) dompng component3-2-1-__________________-0-iiiiiijtI1IlIIIIIIIIIIIIIIIIIIJIIIIIt0.00 0.10 0.200.300.40dimensionless frequency,a0Figure 7.6: Comparison of normalizedtotal rotational impedance K7/A with solutionby Kaynia and Kauselfor 2x2 pile groups (E/E3= 1000, s/d=5, AN * Th’?k°)Ohapter 7. Elastic Response ofPile Groups: Theory and Verification122analyzed in the next section.7.6 Full-scale vibration test on a 6-pile groupA quick release horizontal vibration test was performed on a full-scale pile groupfoundation of a large transformer bank (Bank 79) located at the Duwamish substation,Seattle,Washington. Test data and analytical results were reported by Crouseand Cheang(1987). The foundation of the transformer consists of a pile cap with6 vertical pilesembedded in 40 ft of loose saturated, sandy soils overlying stiff soil.The transformer-pile cap system is analyzed using the proposed quasi-3D finiteelement method of analysis. Results of this analysis are used to verify the applicabilityofthe proposed model for pile group.7.6.1 Description ofvibration and its testing resultsThe transformer and foundation system has been described in detailin Crouse andCheang (1987) and are briefly summarized below. The soil profileat the location oftransformer bank consists ofmostly loose to medium dense sand tosilty sand, with somedense sand or gravelly sand layers, overlying very dense gravellysand glacial till at 12.2m depth. The ground water table was at a depth of 3.7 m. The in-situ shearwave yelocities (V3)measured from a downhole seismic survey inthe sand to silty sand depositsare 125 m/sec in the upper 3.7 rn and 165 rn/sec below3.7 m depth. Figure 7.7 showsthe idealized soil proffle at Duwaniish Station according to Crouse andCheang (1987).Response of Pile Groups:LAYERDEPTH0THICKNESSII,) fpI)4’ 2 2204’ 666010 4 1011102’ 13 13682’ 15 14634’ 18 160620— 5 22.5 1820S3’ 26.5 201030 4’ 30 21774’ 34 23676’ 39 26054050Theory and Verification‘m. •7(pit) (10i.sI) (l) Ii79.2 6 110 30237.6 6399.6 6492.5 10526.7578.2655.2723.6 129 40783.1 119 30852.1 119 30937.8 11930Figure 7.8 shows the transformer bank and the pile foundation.The transformer,weighing 326 kip is anchored to concrete pedestals which is a continuouspart of the pilecap. The pile cap has a dimension of 13.4 ft by 8.00 ft. The pile cap is embeddedbeneaththe ground surface as shown in Figure 7.8. The pile foundation consists of6 vertical, 12inches O.D. by 0.172 inch wall thickness, concrete filled steel pipe piles.These piles arespaced at 4.67 ft and 5.00 ft centre to centre in the X and Y directions,respectively. Allthe piles are extended into the very dense glacial till layer at40 ft depth. The compositecompressional rigidity (EA) and the flexural rigidity (El) of the each concrete filled pipepile are 5.1 x 1081b and 4.24 x1071b.ft2,respectively, where E= Young’s modulus, A:=cross section area, and 1= bending moment ofinertia.123Chapter 7. ElasticSANDToSILTYSAND iGRAVELLY ::‘.SANDSANDToSILTYSANDGRAVELLYSANDGLACIALTILLIFigure 7.7: Idealized1987)YM80LS• VERTICAL CONFINING STRESSSHEAR STRENGTHLOWSTRAIN SHEAR MODULUS-TOTAL DENSITY• - INTERNAL FRICTION ANGLEsoil profile at Duwamish Substation(after Crouse and Cheang,The quick-release free vibration test was conducted by Crouse and Cheang (1987).Chapter 7. Elastic Response ofPile Groups: Theoryand VerificationPLAN-ELEVATIONPILE CP0SS.SECTION1—CONCRETEt 0.172”12 0.0.rutFOUNDATION PLAN124Figure 7.8: Setup of a full-scale free vibration teston a 6-pile group (after Crouse andCheang, 1987)‘CI?4T10 0 ola.si—Io0I ‘III——Izi.—I 2!.—Chapter 7. Elastic Response ofPile Groups:Theory and Verification125A sling was attached to the transformer,and then pulled and quickly releasedto let thestructure vibrate freely. The motionswere recorded at various locationson the foundation. Tests were performed in both principalhorizontal directions, NS or Y-axisand EWor X-axis, of the transformer foundation.The resonant frequencies and dampingratios of the transformer foundationsystemwere determined from the recorded timehistories of transient vibrationsby Crouse andCheang (1987). The measured fundamental frequenciesin NS and EW directions are3.8Hz and 4.6 Hz, respectively. The measured dampingratios in the NS and EW directionsare 6% and 5%, respectively.7.6.2 Computed resultsusing the proposed modelIn present analysis the soil profile shownin Figure 7.7 is used except the shear modulusdistribution modified bySy (1992) is adopted. According toSy (1992), a correction onmeasured shear wave velocity was madeto account for the soil densificationdue to pileinstallation. An increase of4% in low strain shear modulusGma values was applied tothe measured free field values for thiscorrection. Poisson’s ratio ii= 0.3 and materialdamping ratio )=5% are used for all soil layers in present analysis.Figure 7.9(a) and 7.9(b) showthe finite element models used in the analysisfor obtaming dynamic impedances inNS direction and EW direction, respectively.Dynamicimpedances of the pile foundationare computed covering an excitation frequencyrangefrom 0 Hz to 6 Hz. At thisstage of the analysis, the effect of pile cap embedmentisnot taken into account. The computedimpedances of the pile group correspondingtothe excitation frequencies at 3.74 Hz inNS direction and at 4.63 Hz in EW direction areChapter 7. Elastic Response ofPileGroups: Theory and Verification126listed in Table 7.6. The stiffnessand damping values shown in this Tableare referencedto the bottom of the pile cap.In order to determine the resonant frequenciesof the transformer-pile cap system,dynamic response of the systemis computed by subjecting the system tohorizontal harmonic forced excitation at different frequencies.Figure 6.18 shows the structural massand its supporting dynamic impedances(springs and dashpots). Eq. 6.23 is usedagainfor obtaining dynamic response ofthe structural mass at a particular excitationfrequency.Harmonic force and moment wereapplied at the centre of gravityof the transformer-pilecap system. According to Crouseand Cheang, the transformer andpile cap have a totalmass of 1.13 x i0 slugs. The heightof centre of gravity ofthe system is 10.9 ft abovethepile head. The momentsof inertia are 4.38 xlO5slug.ft2and 5.46 xlO5slug.ft2aboutaxes in the NS directionand in the EW direction, respectively.The response curves of the transformer-pilecap system are shown in Figure7.10 indashed lines. The resonant frequencyof the system is the excitation frequencyat whichthe peak dynamic response ofthe system occurs. The damping ratiosare determinedusing the the bandwidth method fromthe response curves. The computed resonantfrequencies and damping ratios are shownin Table 7.5. Results are presented togetherwiththe measured values.Effect of pile cap embedmentIn the previous analysis the effect ofpile cap-soil interaction was not included. Accordingto Crouse and Cheang, gaps between thepile capand the soil underneath it mayexist due to settlement of soil. Also conventionallytheeffect ofpile cap-soil interactionhas been included by considering the soil reactionactingChapter 7. Elastic Response ofPile Groups:Theory and Verification127// / //////N///[/////////// / /,///jY//////A/ I A‘4,/ 1//A/Figure 7.9: 3-D finite elementmodels of the 6-pile foundation (a)NS direction, (b) EWdirection(a)/ / //////E/////h//00I’ll’__I(b)/// ////A//4/ /1///___t1IIChapter 7. Elastic Response ofPileGroups: Theory and Verification128Table 7.5: Computed resonantfrequencies and damping ratios withoutthe effect of pilecap embedmentResonant frequencies (Hz) DampingratiosComputed Measured ComputedMeasuredNS EW NS EW NS EWNS EW3.50 4.37 3.8 4.600.08 0.06 0.06 0.05on the vertical sides of the pilecap (Prakash and Sharma, 1990& Novak et al.,1990).Therefore in this analysis only the soilreactions acting on the vertical sides ofthepilecap are included and the soil reactionsacting on the base of the pile capare not considered. The side reaction due to pilecap embedment usually result in increased foundationstiffness and damping. Accordingto Beredugo and Novak (1972), the foundationstiffnessand damping due to pile cap embedmentcan be determined using theplain strain soilmodel.The rectangular pile cap had an areaof 106.67ft2,an equivalent radius of5.83 ft.The embedment depth of the pilecap was 1.75 ft, and the shear modulus of soilat thatdepth was 6.0 x iO psf. Based on thesedata, using Beredugo and Novak’s solution,thestiffness and damping dueto pile cap embedment are determined at the bottomof thepile cap asTranslationk= 4.20 x106lb/ftC = 1.33 x iO.lb/ftCross-couplingChapter 7. Elastic Response ofPile Groups:Theory and Verification 129Table 7.6: Computed stiffness and damping ofthe transformer pile foundationN-S (f=S.74 Hz) E-W(f=4.6S Hz)with without withwithoutembed. effect embed. effect embed.effect embed. effectk(lb/ft) 2.46e-b7 2.04e+7 2.55e+72.13e+7k8(lb/rad) -2.21e+7 -2.57e+7 -2.43e+7-2.79e+7k’(lb.ft/rad) 1.34e+91.25e-b9 2.59e-l-9 2.50e+9C(lb/ft) 10.Oe+6 7.05e+6 8.81e-l-65.O5eH-6Ce(lb/rad) -5.lOe+6 -7.62eH-6 -1.97e-f-6 -5.25e--6C”(1b.ft/rad) 9.20e-l-76.91e+7 1.74e+81.44e+8k8= 3.68 x106lb/rad= 1.16 x iO.ib/radRotation= 9.34 x 1W lb.ft/rad= 1.03 x106w lb.ft/radAfter adding the stiffness and damping due to pilecap embedment to these ofthe pilegroup, the combined stiffness and damping of the transformerfoundation are determinedand they are shown in Table 7.6 at the resonantfrequencies.The dynamic response of the transformer-pilecap system is obtained including theeffect of pile cap embedment. Figure7.10 shows the response curves of the transformerpile cap system when the effect of pile cap embedmentis included. They are comparedwith those when the embedment effect is notincluded. It can be seen that the pilecapembedment results in increasedstiffness and damping of the transformerfoundation.Chapter 7. Elastic Response ofPile Groups: Theory andVerificationC,)C0aCr,a)+w&+w&&N—S direction(6—pile group)130without embedment effect_____with embedment effectc\JS.pI’‘II II IJ,• I• II I• I,,Iimeasuredfrequency— 11111111111111111110 24 6 8frequency, HzE—W direction(6—pile group):without embedmenteffectwith embedment effect- measured•‘ frequencyEJI I I III I I I I I II I I I0 24 6 8frequency, HzFigure 7.10: Response curves of the of transformer-pile cap system (a) NS direction (b)EW directionChapter 7. Elastic Response ofPile Groups: Theoryand Verification 131Table 7.7: Measured and computed resonant frequenciesand damping ratios includingthe effect of pile cap embedmentNatural frequencies (Hz) Damping ratiosComputed Measured Computed MeasuredNS EW NS EW NS EW NS EW3.74 4.63 3.8 4.60 0.090.09 0.06 0.05Thecomputed and measured resonant frequenciesand damping ratios ofthe transformer-pile cap system are given in Table 7.7. The computed resonantfrequencies including theeffect of pile cap embedment are 3.74 Hz and 4.63Hz in the NS direction and in the EWdirection, respectively. The corresponding measuredresonant frequencies are 3.8 Hz and4.6 Hz. The computed resonant frequenciesmatch very well with the measured frequencies in both principal directions. However, the computeddamping ratios are about50%to 80% higher than the measured damping ratios.Chapter 8Non-Linear Analysis of Seismic Soil-Pile-StructureInteraction8.1 IntroductionIn this chapter the quasi-3D finite element method of analysis describedin chapter 6and 7 is used to model dynamic response of pile foundationssubjected to earthquakeloading. Since earthquake excitation is a random process,the non-linear finite elementanalysis is conducted in the time domain. The useof time-domain analysis makes itpossible to model the variations of soil properties with time under earthquakeloading.Therefore, adjustments in the proposed model are made first to accommodatethe time-domain analysis. Then studies are focused on modellingnon-linear response of the soilunder earthquake loading.A finite element program PILE3D has been developed for dynamic analysisof pilefoundation under earthquake loading. In PILE3D, the shear stress-strainrelationship ofsoil is simulated to be either linear elastic or non-linear incrementallyelastic. When thenon-linear option is used, the shear modulus and the hysteretic dampingare determinedusing a modified equivalent linear approach based on the levelsof dynamic shear strains.Features such as shear yielding and tension cut-off are incorporated in PILE3Dalso. Thedynamic response of pile groups can be effectively modelledusing PILE3D.132Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction1338.2 Quasi-3D finite element analysis in thetime domainThe basis ofthe quasi-3D finite element method ofanalysis for pile foundations has beengiven in Chapter 6 and Chapter 7. In these chapters, dynamic analyses wereperformed inthe frequency domain. To accommodate the time-domainanalysis presented herein, someadjustments are required in the formulation of the global dynamic equilibriumequationgiven in Eq. 6.9.The adjustment is made first to the formulation of the mass matrix.For dynamicanalyses in the time domain, the use of a diagonalmass matrix can save both computational time and space. Thus the diagonal mass matrix formulations areused in PILE3Dto construct[M]eiem for both the soil element and the beam element. The diagonal massmatrices for the soil element and the beam elementare given by[M]80,= PSvol{1.0, 1.0,1.0,1.0,1.0,1.0, 1.0, 1.0} (8.1)[Mlbeam = pAl{1/2, 1/78, 1/2, 1/78} (8.2)The adjustment is made next to the formulation of stiffness matrix anddamping matrix. In Eq. 6.9, the stiffness matrix[K*]is formulated using the complex shear modulus= G(1+i . 2)i). In the time-domain analysis, the stiffness matrix[K] is formulatedusing the real shear modulus C. The hysteretic dampingratio .\ of soil is included usingequivalent viscous damping and its formulation is givenbelow.A procedure for estimating damping coefficients for each individual elementproposedby Idriss et al. (1974) is employed in PILE3D. The main advantage ofthis procedureis that a different degree of damping can be applied in each finite elementaccording itsshear strain level. The damping is essentially a Rayleigh-type damping,assuming theChapter 8. Non-Linear Analysis of Seismic Soil-Pile-StructureInteraction 134damping is contributed one half by mass and theother half by stiffness. In each timeperiod LT the global stiffness matrix [K] is computedbased on the individual shearmodulus in each element, and the global mass matrix[M} is always constant through thetime domain. The fundamental natural frequencyw1 of the pile-soil system is obtainedby solving the corresponding eigenvalue problem.The fundamental frequency of the pile-soil systemw1 is then applied to every soilelement in the system. The damping matrix for a soil elementis given by[C]eiem [M]eiem -+- /3. [K]eiem (8.3)in whicha= ‘e1em= elem/LL1and )telem is the hysteretic damping ratio of soil in the elementand is determinedbased on the level of shear strain in the element.The global mass matrix [Mj, the global stiffness matrix[K] and the global dampingmatrix [C] are assembled fromeach individual finite element. Therefore under earthquakeloading, the global dynamic force equilibrium equation in matrixform is given by[M]{’ü} + [C]{’ô} + [K]{v} = —[M]{I} i,(t)(8.4)in whichib(t) is the base acceleration, and {I} is a column vector of 1. {i}, {z)} and{v} are the relative nodal acceleration, velocity and displacement,respectively.Chapter 8. Non-Linear Analysis of SeismicSoil-Pile-Structure Interaction1358.3 Solution scheme for dynamicequationThere are two methods for solvinga dynamic equation, the mode-superpositionmethodand the direct step-by-step integrationmethod (dough and Penzien,1975;Newmark,1959;Wilson et al.,1973).The mode-superposition method is veryuseful when a linear system isto be analyzed. This method requires the evaluationof the vibration modal frequencies andtheircorresponding modal vectors. It basicallyuncouples the response of the system,andevaluates the response of eachmode independently of others. Themain advantage ofthis approach is that the dynamicresponse of a system canbe evaluated by consideringonly some vibration modes even insystems that may have many degreesoffreedom; thusthe computational efforts maybe significantly reduced. However the mode-superpositionapproach is not applicable tonon-linear systems.The direct step-by-step integration methodis applicable to both linear and non-linearsystems. The non-linear analysisis approximated by analyses of a successionof differentlinear systems. In other words,the responses ofthe system are computedfor a short timeinterval assuminga linear system having the same properties determinedat the start ofthe interval. Before proceedingto the next time interval, the propertiesare determinedso that they are consistent withthe state of displacement and stress atthat time.The direct step-by-step integration proceduredeveloped by Wilson et al. (1973)wasemployed in PILE3D to solvethe dynamic equation Eq.8.4.Chapter 8. Non-Linear Analysis of SeismicSoil-Pile-Structure Interaction136The equation Eq.8.4 is solvedby an incremental form[Mj{} + [Cj{ii} + [K]{v} = —[M]{I}z..i,(t) (8.5)Since Eq. 8.5 is used for solving theincremental values of dynamic response,thedynamic equilibrium should be ensuredby checking equation Eq. 8.4 after eachstep ofintegration. During the dynamic analysisunbalanced force{L.P}unbal is computed aftereach integration step{J.P}unbal = — [C]{)} — [K]{v} — [M]{I}.(t) (8.6)This unbalanced force is then added to theright hand side of equation Eq.8.5 in thenext step of integration to satisfy dynamicequilibrium.8.4 Non-linear analysisTypical relationships between shear stressand strain at different strain amplitudesunderdynamic cyclic loading is shown in Figure8.1. Firstly the shear stress increaseswith theshear strain non-linearly. Secondly the loading-unloadingcurve forms a hysteresis ioop.The shape of the T— 7curve determines the degree of reductionof shear modulus withshear strain, and the area of the hysteresisioop represents the amount of strain energydissipated during the cycle. The dissipatedenergy implies the degree of materialdamping at this strain level.In a dynamic analysis involving hysteretic non-linearity,a rigorous method ofanalysisin modelling the shear stress-strain behaviouris to follow the actual loading-unloadingreloading curve. This method has been successfullyapplied in 1-D ground motionanalyses (Finn et al., 1977, Lee and Finn,1978) and 2-D plane strain analyses (Finnet aL,1986). However this method requiresupdating the tangential shear modulus forall soilChapter 8. Non-Linear Analysis of SeismicSoil-Pile-Structtire Interaction137stressstrainFigure 8.1: Hysteretic stress - strain relationshipsat different strain amplitudeselements and building up stiffnessmatrix in every time step of integration.This procedure is too time-consuming for 3-D analysis.The equivalent linear method is employedin PILE3D to model the soil non-linearhysteretic behaviour. The equivalentlinear method was initially proposed bySeed andIdriss (1967), and it has been widelyaccepted and used in soil dynamic analyses. Thespirit of this method is that the hysteretic behaviourof soil can be approximated by aset of effective shear moduli and viscousdamping which are compatible withthe levelsof shear strains. Figure 8.2(a) showstypical relationships between the ratioG/Gmaxeffective shear modulusG over the shear modulus at very low strainGmax,and theeffective shear strain. Figure 8.2(b)shows typical relationships between damping ratioand the shear strain.G2(strain)2Chapter 8. Non-Linear Analysis of SeismicSoil-Pile-Structure InteractionIISl..er StroIn. r138Figure 8.2: Relationships between shear moduli,damping ratios and shear strains (afterSeed and Idriss, 1970& Seed et al.,1986):E.,hrta/In F14*3end 4a?____0i0- :0-0 10.1IjCEoWeissman and Hart 0961) 0 —— 0• Hardin (1965)o24 — o Drnevich. Hall and Richort 0966 -_______________0 Matnishito. Kishido and Ky00967)• Silver and Seed (1969)—Donovan (1969)20— V Macdin and Drnevch (1970)— 7V Kishido and Takono 0970)7/1€,_—.‘.—i.I? ‘k/6 .‘,-v‘cito-- K,-.to-aShea: Swain-percentChapter 8. Non-Linear Analysis ofSeismic Soil-Pile-StructureInteraction 139The equivalent linear method has been appliedin the computer code SHAKE (Schnable et al., 1972) for 1-D ground motion analysesand QUAD-4 (Idriss, 1974) for 2-Dplane strain analyses. Here the proceduresof this method are incorporated in the quasi-3D dynamic pile-soil interaction analysis. Howeverthey are used with some modifications.In SHAKE, analysis is performed using initial shear moduliand dampings for theentire input acceleration record; then a second analysisis performed for the same acceleration record using constant shear moduli and dampingswhich are compatible tothe effective shear strains obtained from the firstanalysis. In other words, the shearmodulus and damping are determined accordingto the shear strain using curves such asthose shown in Figure 8.2. Iteration processis used to achieve the compatibility betweenmodulus, damping and shear strain. Analyses ceasewhen the differences between themoduli and dampings in the two subsequent analysesare with the desired given values.There is a disadvantage ofthe equivalent linear method usedin SHAKE. A set ofconstant values of shear modulus and damping is usuallynot appropriate to represent thenon-linear behaviour of soil within the whole time domain. Especiallyunder earthquaketype of loading the level of shear strain usually changesvery much from the beginning,through the middle to the end of the time domain.The common-used criterion of takingthe effective shear strain equalto 65% of the maximum shear strain may not applicablein many cases.A procedure of applying the equivalent method based on the periodicallevel of shearstrain is proposed to overcome this problem. This procedurerequires no iteration process. Figure 8.3 shows the idea of this method. Inthis procedure an assumption is madethat the effective shear moduli and damping in aperiod of time /.T, remain constantChapter 8. Non-Linear Analysis ofSeismicSoil-Pile-Structure Interaction140responses, Atimeti ti+1 tjFigure 8.3: The principle of modified equivalentlinear methodand are determined by the peak shear strain in theprevious period T_1based onrelationships such as those shownin Figure 8.2. The shear moduli and dampingratios inall soil elements are then determinedfor the period of time. PILE3D has the capabilityallowing the use of different curvesinput by users.The selection of the length of timeperiod is based on the fundamental frequency ofthe input earthquake motion, and this lengthcan be selected by users. The lengthoftime period is selected neither tooshort to take too much computational time nor toolong to lose the accuracyof the analysis. A length between 0.4 second and1.0 second isgood for the earthquake recordsused in the analysis presented in the thesis.AA,AAiAtj+1Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-StructureInteraction 141shear stress rnormal stress uTf___________arstrainrna1strainTf7shear yieldingtension cut-offFigure 8.4: Simulations of shear yielding and tension cut-off8.5 Features in dealing with yielding, tensionApart from the non-linear shear stress-strain relationship, shearyielding of soil materialoccurs when the shear stress exceeds the shear strength of thematerial. When yielding occurs the soil element may develop significantly large shear deformationunder verysmall increment of shear stress. In other words the shear modulusof the soil is significantly reduced due to shear yielding. The shearyielding is prominent near the pile head.A numerical procedure is included in PILE3D so that the shear modulus is reducedto a very smafl value when shear yielding occurs. Figure 8.4(a) shows theprinciple ofthis procedure. The shear stress used for this purpose is the maximum shearstress inthe vertical plane; while the shear strengthr1is input by users according to the type ofsoil. The shear strength r1 may be determinedby c+o0. tanq5(c is the cohesion,qisthe friction angle, and is the effective overburden stress).Chapter 8. Non-Linear Analysis ofSeismic Soil-Pile-StructureInteraction 142On the other hand, cracking or tensile failure may occur when the totallateral normal stress of a soil element exceeds the tensile strengthof soil. For sands any tensilestress developing may lead to a tensilefailure. When a tensile failure occurs soil hasno resistance. A no-tension model shown in Figure8.4(b) is used to accommodate thisphenomenon. One of the criteria built in PILE3Dfor checking tension is0dynamc> O + 0j (8.7)whereo is the static lateral stress of the soil element,and 0j is the tensile strengthof soil, and0dynamicis the dynamic lateral stress computedduring the analysis.The options of checking shear yielding and allowing no-tension to developin soils areavailable only when a dynamic step-by-step integration procedure is used.These featuresovercome the difficulties that may be encountered in a pure elasticanalysis. In a pureelastic analysis no controls on the shear yielding or thetensile failure can be enforced.The application of these features has been found veryeffective in eliminating the over-stiffness of soil in the zone near the pile.8.6 Soil parameters required in PILE3D analysisThe method was developed so that fundamental soil parametersare used.Shear modulus of soil The key parameter requiredfor this analysis is the low-strainshear modulus of soil. The low strain shear modulus is also called themaximumshear modulus. TheGmax can be determined accurately by measuring the shear waveChapter 8. Non-Linear Analysis ofSeismic Soil-Pile-StructureInteraction 143velocity V3 of soil, thenGmax= pV2= 7/g V2 (8.8)wherep isthe mass density of soil, and-y is the unit weight of soil and g is the gravityacceleration, 9.81 rn/sec2.When data on shear wave velocity are not available, empirical equationsmay beapplied to estimate the low strain shear modulusGmax. A useful empirical equationproposed by Hardin and Drnevich (1972) is of the formGm= 32O.8(2.9÷(°(OCR’ (8.9)in whiche void ratio;OCR = overconsolidated ratio;k = a constant dependent on the plasticity of the soil;Pa = atmospheric pressure, 101.3 kPa;= current mean normal effective stress.The Hardin and Drnevich equation is applicable for both sands andclays. For claysit is recommended that the equation be used when the void ratio isin the range of 0.6to 1.5. The variation of constant k with plasticity index PTis given in Table 8.8.For most practical purposes, Seed and Idriss (1970) proposed anotheruseful expression for estimatingGmax of granular soils(sands and gravels)Chapter 8. Non-Linear Analysis ofSeismic Soil-Pile-StructureInteraction 144Table 8.8: Relationship between Hardin and Drnevichconstant k and plasticity index PT(after Hardin and Drnevich, 1972)PT 0.0 20 40 60 80 >100k 0.0 0.18 0.30 0.41 0.480.50Gma, =1000(k2)maw(ø)°5inpsf units (8.10)The coefficients(k2)ma was found to vary from about 30 for loose sands to about 75for dense sands. Values of(k2)maa for relatively dense gravels are generally in the rangeof about 80 to 180.A useful relationship between(k2)maa, and SPT (N1)6ovalues was proposed by Seedet al. (1986)(k2)ma = 20(N1) (8.11)For clays the maximum shear modulus may be calculated basedon the undrainedshear strength, S,, using the equationGmcut = Kctay Su (8.12)in whichKday is aconstant for a given clay. From Seed and Idriss (1970)the typicalvalues ofKciay varybetween 1000 to 3000.Damping ratio and other parameters The second inputrequired in the analysis isthe relationship between damping ratio and the shear strainof soil. Although variationChapter 8. Non-Linear Analysis ofSeismic Soil-Pile-StructureInteraction145j0-43 - 3ioCyclic Shear Strain trFigure 8.5: Comparison of damping ratios for sandsand gravelly soils (after Seed et al.,1986)may be expected for specific soilmaterial, typical values of damping ratios as functionof shear strain were given by Seed et al.(1986) for sands and gravelly soils, and they areshown in Figure8.5. Other parameters required in the analysis are Poisson’s ratio andshear strength of soil.8.7 Aspects relative to analysisof pile groupThe program PILE3D has the capability ofsimulating the dynamic response of a pilegroup supporting a rigid pile cap. Analysisis done in a fully coupled manner. The rigidpile cap is represented by a concentrated massat the centre of gravity of the pile cap,and the mass is rigidly connected to the piles by amassless rigid bar.Figure 8.6 shows the principle for modelling a pilecap in PILE3D. The pile cap ismodelled as a rigid bodyin the analysis. The motions of the pile cap are represented20‘.. I I I0 Oato lot Gravels and Gravelly SoálSINs invhqahon— Averoq. ‘Vlue for Sands.—;-A —j— --Uirand .ower eouncsIdriss/p ,‘— ,EEE°E--.———10_IChapter 8. Non-Linear Analysis ofSeismicSoil-Pile-Structure Interaction146rocking impedanceFigure 8.6: A diagram showingthe representation of pile group supportingstructureby the motions of the pile nodes whichare connected to the bottom of the pile cap.These nodes are called the pile headnodes. The pile head nodes have identical defiections (translations and rotationsin the vertical plane) as the pile cap does. Thereforeaprincipal restrainingpile head node can be used to represent the motionsof all pile headnodes or of the pile cap. In thefinite element analysis the translation androtation ofall pile head nodes in the group arerestrained to the principal pile head node.In thisway, identical defiectionsof all pile head nodes are achieved. The massof the pile capis connected to the principal pilehead node so that all pile head nodes share thesamemass, stiffness and defiections.On the other hand, under seismicloading the pile cap rotation is primarily resistedby the rocking impedance of the pile group.The rocking impedance is induced by thevertical resistance of the piles.In the proposed quasi-3D analysis the rocking impedancecannot be included directly becausethe vertical and horizontal motions are uncoupled.Chapter 8. Non-Linear Analysis ofSeismic Soil-Pile-Structure Interaction147The rocking impedance of the the pile group is includedin the analysis by using thefollowing procedures.The rocking stiffness and damping is updated at selected times during thehorizontalmode analysis by using PILE3D in the vertical mode. The procedurefor computingrocking impedanceKrr has been given in chapter 7. The rocking impedance is computedusing the current values of strain dependent shear moduli and damping ratioof soil. Thecurrent rocking impedance (stiffness and damping) is then transferredto the pile cap asrotational stiffness and damping.The rocking impedance of pile group is important to the dynamic response underhorizontal shaking, and it has been properly treated in the dynamic analysis ofpile groupusing PILE3D.Chapter 9Analyses of Centrifuge Tests of Pile Foundations9.1 IntroductionIn this chapter the proposed quasi-3D finite element methodof analysis is used to analyze the non-linear response of model pile foundations subjectingto horizontal loadingin centrifuge tests. Centrifuge tests on a single pileand a 2x2 pile group are analyzedusing the computer program PILE3D. The computed resultsare compared with thosemeasured in the centrifuge tests. The ability of theproposed model for simulating thenon-linear response of pile foundations is evaluated.During strong shaking, the shear modulusand damping ratio of soil medium changewith time, which causes corresponding changesin the dynamic impedances of pile foundations. These variations in dynamic impedanceswith time during strong shaking areevaluated for the model pile foundations. This is thefirst time that the time-histories ofdynamic impedances have been calculated.9.2 Dynamic analysis of centrifugetest of a single pile9.2.1 Description of centrifugetest on a single pileA centrifuge test on a single pile was carried out atthe California Institute of Technology (Caltech) by B. Gohi (1991). Detailed dataon the centrifuge test are given by Gohi148Chapter 9. Analyses of Centrifuge Tests ofPile Foundations149(1991). Details of the test may also be found ina paper by Finn and Gohi (1987).Acentrifuge acceleration of 60g was used for thetest.Figure 9.1 shows the soil-pile-structure systemused for the single pile test. The effectof the super-structure was simulated by clampinga rigid mass at the pile head. Thepile head mass was instrumented using a non-contactphotovoltaic displacement transducer and an Entran miniature accelerometer.The locations of the accelerometer andlight emitting diode (L.E.D.) used by the displacementsensor are shown in Figure 9.1.The pile head displacements were measured with respectto the moving base of the soilcontainer. The prototype parameters of the singlepile test is shown in Figure 9.2. Themodel pile has a unit weight of 74.7 kN/m3.Theprototype pile has an outer diameterdouter 0.5712 m and an inner diameter dinner = 0.5412 m. The flexural rigidity of theprototype pile is 172,614 kN.m2.The sand used for the centrifuge test was a loosesand with a void ratio co = 0.78and a mass densityp = 1.50Mg/rn3.The friction angle of the sand was determined tobe300.Gohi (1991) has showed that the low strain shearmoduli of the sand foundationvary as the square root of the depth, and thatthey can be quantitatively evaluated byusing the Hardin and Black (1968) equation()—e01I NO.5—V‘ +e0where e0is the in-situ void ratio ofthe sand andois the mean normal effective confiningstress in kPa. The mean normal effective confiningstress is computed from the effectivevertical stress oChapter 9. Analyses of Centrifuge Tests ofPile FoundationsLED.—_c1__[216.5L_......b...J-_—_ AccelerometerSoil surface —.. +No.2 iNo.4Location ofStrain gaugesNo. 6No. 3Base of centrifuge — — Pile tipbucket_________________________________4seismic motion150No. I+ No.5• Axial strain gauge+ No.7No.8 +Figure 9.1: The layout of the centrifuge test for a single pileI Istructural mass:m = 53.2 kN.sec/mIcg53.11 kN.se.rhcg=0.99 m0.99 m1’12 mloose sanddepositpile:El = 172,614 kN.n?L = 12.89 md = 0.5712 mFigure 9.2: The prototype model of the single pile testChapter 9. Analyses of Centrifuge Tests ofPileFoundations 1511-1-2K00mO•j (9.2)Using e0 0.78 and a lateral stress coefficient K00.4 for the loose sand, Gohi showedthat measured shear wave velocities and those computed by usingthe Hardin and Blackequation are in good agreement.A horizontal acceleration motion is input at the baseof the system. The peak acceleration of the input motion is 0.158g. During the centrifuge test,accelerations at the freefield surface and at the pile head and displacements atthe top ofthe super-structure wererecorded. Dynamic moments at the selected locationsalong the pile shown in Figure 9.1were also recorded during the test.The computed Fourier amplitude ratios of the pilehead response and the free fieldmotion with respect to the input motion are givenin Figure 9.3(a) and Figure 9.3(b),respectively. The natural frequency of the freefield acceleration is estimated to be 2.75Hz, and the fundamental frequency of the pile to be 1.1 Hz.The period of the pileresponse is much longer than the period of the free field motion.9.2.2 Dynamic analysis of the single pileThe centrifuge test of single pile is analyzed at the prototypescale. Figure 9.4 showsthe finite element model used in the analysis. The sand depositis divided into 11 layers. A decreasing thickness of layer is used towardthe soil surface. This arrangementwould allow more detailed modelling ofthe stress and strain field where lateral soil-pileChapter 9. Analyses ofCentrifuge Tests ofPile Foundations152A - -__A AV -..--.T!f-w’’?‘. .‘‘jA-0 2 4 6 8Frequency (Hz) -Frequency (Hz)Figure 9.3: Computed Fourieramplitude ratios (a) pileamplitude ratio (APH/AFF) (b)free field amplitude ratio (AFF/AB)(after Gohi, 1991)1.1 Hz+86-4.2-00II,00E0Ca)4-E64-2f = 2.75 Hz‘--00 24 6 810Chapter 9. Analyses of CentrifugeTests ofPile FoundationsGEO .SCRL0I I1530153/IIIRA.SYMMETRY I I’HSHAKING DIRECTIONFigure 9.4: The finite element modeffing of centrifuge testinteraction is strongest. Thepile is modelled using 15 beam elements including5 elements above the soil surface. Thesuper-structure mass is treated as a rigid body, andits motion is represented bya concentrated mass at the centre of gravity. A verystiffbeam element with fiexural rigidity1000 times that of the pile was used to connect themass and the pile head. The motionof any point in the rigid super-structure can bedetermined according to its geometric relationshipto the reference point, the pile head.The present finite element mesh consistsof 666 nodes and 456 elements.The finite element analysis wascarried out in the time domain. The non-linear analysis was performed to account forthe changes of shear moduli and damping ratios dueto dynamic shear strains. Accordingto Gohi (1991), the shear-strain dependency of theshear modulus and dampingratio used in the analysis is shown in Figure 9.5. The maid-mum shear modulusGm was calculated according to Eq. 9.1. The maximum dampingratio of the loose sand was taken as25% following Gohi (1991).//AXIS OF ////W/ /Chapter 9. Analyses of Centrifuge Tests ofPile Foundations154LllLEiZIfl:* DAPI11GI I IIIllI —i—i.•—••••••••—•—•i••••••——t—•——i•— I I 111110.0010.010.11SHEAR STRAIN(%)Figure 9.5: The relationships between shear modulus, damping andthe shear strain forthe loose sandResults of analysis The computed and measured accelerations in thefree field andat the pile head are shown in Figure 9.6 and Figure 9.7, respectively. Thereis goodagreement between the measured andthe computed accelerations.Thecomputed and measured time histories ofdisplacements atthe top ofthe structureare plotted in Figure 9.8. The computed displacements are smaller than the measureddisplacements in the first 10 sec ofmotion. However,the computed peak displacement is56 mm compared to the measured peak displacement of 67 mm, with an error of about16%. The frequency content of the displacement response has been captured satisfactorily by the analysis.H0.50.L...LL0.2500.00__.**_***10.0001wIIThe computed time histories ofmoments in the pile at the soil surface and at a depthChapter 9. Analyses of CentrifugeTests ofPile Foundations1550.3MEASURED• COMPUTEDii01z • •.4 4•I—$1 .I • a •••4ck . ‘StI aLi_i‘aw:().C)—0.3i ii i i i i i i i0.00 10.0020.00 30.00TIME(SEC)Figure 9.6: The computedversus measured accelerationresponse at the free fieldof 3 m (near point ofmaximum moment) are plotted againstthe measured time historiesin Figure 9.9 and Figure 9.10,respectively. Satisfactory agreementbetween the computed and the measured momentsis observed in the range oflargermoments.The distributions ofthe computed and measured bendingmoments along thepile attheinstant ofpeak pile deflectionare shownin Figure 9.11. Thecomputed moments agreevery well with the measuredmoments. The bending momentsincrease to a maximumata depth of 3.5 diameters,and then decrease to zero at a deptharound 12.5 diameters.The moments along the pilehave same signs at any instanttime, suggesting that theinertial interactioncaused by the pile head mass was dominantand the pile was vibratingin its first mode. Themeasured peak moment is 325 kN.m,whereas the computed peakmoment is 344 kN.mwith an error of6% overestimate.Chapter 9. Analyses of CentrifugeTests ofPile Foundations1560.10z—0.00—0.100.300.200.10z00.00—0.10< —0.20—0.300.00 10.0020.00TIME(SEC)Figure 9.7: The computed versusmeasured acceleration response atthe pile head30.00TIME(SEC)Figure 9.8: The computedversus measured displacement response at the topof thestructureChapter 9. Analyses ofCentrifuge TestsofPile FoundationsFigure 9.9: The computed versusmeasured moment response at thesoil surfacezFz0zzaD400157z‘ 200Fzw0z—20CzbJaD—40C0.00 10.0020.00TIME(SEC)MEASURED:------- COMPUTED.I I I I I I30.005002500—250—5000:MEASURED-COMPUTEDI I I I I I I I III I I I I I I I.00 10.00 20.00 30.00TIME(SEC)Figure 9.10:The computed versus measured moment response at depthD=3m: SOILSURFACA•qboeoeoCOMPUTEDo°°°MEASUREDI I I I I I IBENDING MOMENT (KN.M)Figure 9.11: The computed versusmeasured moment distribution ofthepile at peak piledeflectionUnder earthquake loads theshear moduli and damping ratios of soilvary with boththe time and the location.The proposed non-linear analysis is capableof tracing thevariation of soil properties atany instant during shaking. As examplesof this feature,distributions of soilshear moduli at depths of 0.25 m and 2.10in are plotted in Figure9.12 at an instant t=12.58 sec,and in Figure 9.13 at another instantt=17.11 sec. It isseen that at a certain depthsuch as 2.10 m the soil shear moduli inthe near field of thepile are much less thanthe shear moduli in the far field.At any instant during shaking, aset of soil properties are determined for eachsoilelement. Therefore thedynamic impedances of the pile foundationcan be evaluatedcorresponding to soil propertiesat this instant. The variation of dynamic impedancesofpile foundations with time duringshaking is evaluated. These non-linear pile impedancesChapter 9. Analysesof Centrifuge Tests ofPile Foundations158200—2-—4-—6--—8-—10-—12—400 1009.2.3 Non-linearpile impedancesChapter9. Analyses of Centrifuge Tests ofPile Foundations159single pile at 12.58 secinitial shear modulus 12945 kPa(a) atdepth 0.25 minitial shear modulus 36610 kPa(b) atdepth 2.10 mFigure 9.12: 3-D plotsof the distribution of shear moduli at t=12.58 secChaPt&9.AnalYsesoftfugeTestsof PileFoundations160sirg pi’e at17.llsecjnitia shear mod0l1294 kPa(a) atdepth 0.25minitial shear modU536610kpaC.2(J)(b)atdepth2.l°mjgute9.13:3-D plots of thedistnbutbonofsheatoduliatt1711SecChapter 9. Analyses of Centrifuge Tests ofPile Foundations161include effect of pile-soil kinematic interaction andinertial interaction from the superstructure.Dynamic impedances under an earthquakemotion The concept of dynamicimpedance is formulated to reflect the complexforce-deflection relationship ofa pile foundation under harmonic pile head loading ata specific excitation frequency. This conceptis well suited for the dynamic analysis of a machinefoundation which is normally excitedat specific frequencies. However under earthquakeexcitation, the pile foundations donot usually vibrate at a constant frequency. Strictlyspeaking, the concept of dynamicimpedance does not apply directly in dynamicanalyses of pile foundations involvingearthquake motions.Conventionally dynamic impedances are usedin sub-structuring analysis of pile supporting structures. Pile head impedances are usuallycomputed at the dominant frequency of excitation appropriate to an earthquake motion. Innormal design prior to anearthquake event, it would appear logicalto select the fundamental frequency of the pileas the frequency ofinterest, provided the earthquake motions areexpected to contain significant frequency contents around the fundamental frequencyofthe pile. An alternativetechnique is to explore variations ofdynamic stiffnessand damping ofthe pile foundationin the frequency range of an earthquakemotion, such as from 0 Hz to 10 Hz. The latterapproach is used in this study to lookinto the characteristics of dynamic stiffness anddamping as functions of frequency.The dynamic impedances (stiffness and dampingCi,) of the single pile are computed using program PlUMP. These impedancesare determined at the ground surface.Chapter 9. Analyses of Centrifuge Tests ofPile Foundations162Excitation frequencies of 1.91 Hz, 6.0 Hz and 10Hz are selected to explore the effect ofexcitation frequency on dynamic impedances of pilefoundations. Frequency is limited tothe 0-10 Hz range considered appropriate for seismicloading.Stiffnesses of the pile foundation during shaking At the excitationfrequencyf=1.91 Hz, the dynamic stiffnesses of the pile decrease dramatically as thelevel of shaking increases (Figure 9.14). The dynamic stiffnesses experienced theirlowest values inthe 10 to 14 seconds range when the maximum accelerations occurredat the pile head.It can be seen that the translational stiffnessku,, decreased more than the rotationalstiffnesskeeor the cross-coupling stiffness k. At their lowest levels,decreased to20,000 kN/m which is only 13.8% of its initial stiffness of 145,000 kN/m. k,,6 decreasedto 45,000 kN/rad which is 36% of its initial stiffness of125,000 kN/rad. ke showed theleast effect of shear strain. It decreased to 138,000 kN.m/rad whichis 63.6% ofits initialstiffness of 217,000 kN.m/rad. The stiffnesses rebounded when thelevel of shaking decreased with time. Representative values of the pile stiffnessesk,,,,,k,,9andk66 for the usein a structural analysis might be selected as 40,000 kN/m,65,000 kN/rad and 160,000kN.m/rad, respectively, on the basis of the time histories shown inFigure 9.14. Thesestiffnesses are 27.6%, 52% and 73.7% of their original stiffnesses.The variations of translational stiffness k,,,, and rotational stiffness k66 withtime atdifferent excitation frequencies are shown in Figure 9.15. It can be seen thatthe excitation frequency has little influence on the dynamic stiffness of the pilefoundation forfrequency less than 10 Hz. The dynamic stiffnesses of the pile foundationmay be considered independent of frequency under seismic loading.Chapter 9. Analyses of CentrifugeTests ofPile Foundations163250000 -200000kee(kN.m/rd)150000-:100000k8(kN/rczct50000-k (kN/m)o—1111111111 1111111I111111110 5 1015 20 2530TIME(SEC)Figure 9.14: Variation of stiffnessesk9,k99 of the single pile at f=1.91 Hz250000 -DDDfrequency 1.91 Hzafrequency 6.00 Hz200000frequency 10.0 Hz150000-s5I I I III I I I I I I II II I I I III I0 5 1015 20 25 30TIME(SEC)Figure 9.15: Variation of stiffnessesku,., and k with time under differentexcitationfrequencyChapter 9. Analyses of Centrifuge TestsofPile Foundations16450000frequency 1.91 Hzfrequency 6.00 HzI frequency10.0 Hz40000 -30000AA/20000-100001.1111111111111111I111111111I I I0 5 10 15 20 25 30TIME(SEC)Figure 9.16: Variation oftranslationaldamping C,,, versus time under differentfrequencyDampings of the pile foundationduring shaking The variatin of damping withthe excitation frequency is different fromthat of the stiffness. The values of dampingsusually increase with the excitationfrequency due to the frequency-dependent radiationdamping. Figure 9.16 shows typicalvariations of the translational dampingC,,,, versustime as the excitation frequencyincreases. It is seen that the amount of incrementaldamping due to the change of frequencyis roughly proportional to the amount ofincremental frequency. Same patternsare observed for the cross-coupling dampingC,,9 andthe rotational dampingC98.Under theseobservations and due to the fact that hystereticdamping is independent offrequencyw, the following expression is proposed to representdampings C3 of pile foundations underseismic loadingC, — C2+Rw°75 (9.3)/AqJf/where C represents thefrequency-independent hysteretic damping, andR.,, is theChapter 9. Analyses of Centrifuge TestsofPile Foundations16525000 -_________________________°-°° translationcross—couplingrotation20000C•g15000-Q.. IIIIIIIIIIIIIIIIIIIIIIIII;III0 5 10 15 20 25 30TIME(SEC)Figure 9.17: Variation of hystereticdampings C,C9andC of the single pileradiation damping constant,w is the excitation frequency w = 2irfThe hysteretic damping C were determinedby using very low excitation frequency(such as f=0.01 Hz) when computingimpedances. Figure 9.17 shows the variationsofhysteretic damping contributionsC, (translation), C (cross-coupling) andC (rotation) with time. The rotational hystereticdamping is less sensitive to the changeofshaking level than the othertwo hysteretic dampings. Representative valuesof 12,000kN/m, 11,000 kN/rad, and12,000 kN.m/rad may be selected to representthe hystereticdampings of Crn,,C,,9andC99,respectively.The radiation damping constantsare computed using Eq. 9.3. As expected theradiation damping constantsH1,fail into a relatively narrow zone in the time domainwhen the frequencychanges from 1.91 Hz to 10 Hz. Figure 9.18 showsthe translationalradiation damping constantR,,,, with time for the three frequencies analyzed. A value of>>-2000.g1000C0.900850 ofR, is typical for representing the translational radiationdamping. Applying thesame concept, a value of 600 is found typical forbothR1,6andR60.The results of studies on pile impedances are summarizedin Table 9.1. Since thedamping C3 are functions of excitationfrequency the resulting damping coefficients(c,j = Cq/w) are different at different excitationfrequencies. For instance at w = 12rad/sec (f=1.91 Hz), the damping coefficients c,c,8,C89 have values of 1457 kN.sec/m,1239 kN.sec/rad and 1322 kN.m.sec/rad, respectively. However thecorresponding damping coefficients change to 741 kN.sec/m, 604 kN.sec/rad and635 kN.m.sec/rad if theexcitation frequency changes to31.41 rad/sec (f = 5 Hz).Effect of structural masson pile impedances In the analysis described early theeffect of the structural mass onthe dynamic impedances has been included. The heavystructural mass would significantlyincrease the level of non-linearity of soil in the nearChapter 9. Analyses of Centrifuge Tests ofPileFoundations3000166:frequency 1.91 Hz: -- frequency 6.00 Hz:000Dfrequency 10.0 Hz.I—I I I I I I III I I III I I III III I I I I0 5 10 15 20 25 30TIME(SEC)Figure 9.18: Variation of radiation dampingconstant R. at different frequenciesChapter 9. Analyses of Centrifuge Tests ofPile Foundations167Table 9.9: Parameters of dynamic impedances of single pileAverage Minimum Hysteretic Radiation dampingstiffness stiffness damping C constantR3k(kN/m) 40,000 20,000C(kN/m) 12,000 850ke(kN/rad) 65,000 45,000C9(kN/rad) 11,000 600kee(kN.m/’rad) 160,000 138,000Cee(kN.m/rad) 12,000 600field, and thus significantly affect the dynamic impedances. It is important to includethe structural mass at the pile head in the analysis so that the non-linear behaviourofsoil can be adequately modelled.Studies were conducted to explore the effect ofstructural mass on dynamic stiffnessesof pile foundations. The centrifuge tests presented early were re-analyzed by not takingthe structural mass into account. A set of time-dependent shear moduli and dampingratios of soil were obtained. Dynamic impedances of the pile foundation were computedagain using the new sets of soil properties. The effect of the structural masson dynamicstiffnesses of the pile foundation is illustrated in Figure 9.19. The translational and rotational stiffnesses are used to display this effect.The dynamic stiffnesses ofthe pile foundation with the full structural mass are muchless than those without structural mass. This is because soils in the near field of thepileare much more strongly mobilized when the structural mass is present during shaking.Chapter 9. Analyses ofCentrifuge Tests ofPileFoundations168translational stiffness k,,%kN/m)200000o o o o ofull structural massno structural mass150000 -(I)U)w100000UU(I)50000,IIIlIIIIIIIIIIIIIIIIIIIIIIII0 5 1015 20 25 30TIME(SEC)rotational stiffnessk00(kN.m/rad)250000oOo 0full structural massno structural mass200000ci)UzUU“150000100000 I I I I I I I I I I I I I I I(III I I I0 5 10 1520 25 30TIME(SEC)Figure 9.19:Comparison of dynamicstiffnesses of pile foundations withfull structuralmass and withoutstructural massChapter 9. Analyses of Centrifuge Tests ofPileFoundations 1699.2.4 Computational timesThe non-linear analysis was carried out in thetime domain. The average CPU time usinga PC-486(33MHz) computer needed to completeone step integration is 7.0 sec for thefinite element grid shown in Figure 9.4, and3 hours of CPU time are required for an inputrecord of 1550 steps. The computation timeis much shorter for a linear elastic analysis when the shear moduli of soil foundation remainconstant through the tMime domain.The average computational time for computing thedynamic impedances usingPlUMPis 50 seconds for one set of soil properties. The totalcomputational time required to generate curves as shown in Figures 9.14 is about30 minutes.9.3 Dynamic analysis of centrifuge test ofa pile group9.3.1 Description of centrifuge teston a 4-pile group (2x2)A centrifuge test on a 4-pile (2x2)group was conducted by Gohi and reported by Gohl(1991) and Finn and Gohl (1987). The test setupis shown in Figure 9.20. The pileswere set in a 2x2 arrangement at a centreto centre spacing of 2 pile diameters or 1.14m. The properties of piles in the groupare same as those of the single pile. Two of thefour piles were instrumented. The pilesin the group were rigidly clamped to a stiff pilecap, and four cylindrical masses werebolted to the cap at locations shown in the Figure9.20. The top of the pile cap was the location wherethe four pile heads were clamped tothe structural mass. A pile cap accelerometer anda displacement L.E.D. were placed atlocations shown in the figure. The displacementL.E.D. was located 46 mm (2.76 m inprototype scale) above the four pileheads.Chapter 9. Analyses of Centrifuge TestsofPile Foundations170After being converted to the prototype scale,the pile cap has a mass of 220.64kN.sec2/mand a mass moment ofinertiaabout centre ofgravityI= 715.39 kN.sec2.m.The centre ofgravity was 0.96 m above the pile-head.The piles had a free standing lengthof 1.21 m above the soil surface.The sand used for the pile-group test was a drydense sand with a void ratioe0=0.57and a mass densityp = 1.70Mg/rn3.The friction angle of the dense sand was 45°. Gohl(1991) showed that the small strain shear modulusGmcza,can be evaluated using theHardin and Black equation (Eq.9.1) with a lateral stress coefficient K0=0.6.The four pile group was shaken by a simulated earthquakeacceleration motion. Peakaccelerations of up to 0.14g were appliedto the base of the foundation and were dominated by frequencies in the range of0 to 5 Hz. The free field accelerations were stronglyamplified through the sand deposit tovalues of up to 0.26g at the surface. Pilecapaccelerations of up to0.24g and displacementsup to 60 mm were recorded duringthetest. Residual displacements ofup to 10 mm remained at the end of earthquakemotion.9.3.2 Dynamic analysis of the pilegroupIn the analysis, the pile cap was treatedas a rigid body. Hence all pile heads, whichwererigidly connected to the bottom ofthe pile cap, bad identical translational and rotationaldefiections as the pile cap.In the finite element analysis, restrained nodesare used toimpose the identical motions on the nodes.This technique ofrestraining nodes hasbeendescribed in the earlier chapter and wasapplied in the analysis of pile groups.Chapter 9. Analyses of CentrifugeTests of Pile Foundations171PILE CAPCAP MASSESHEAD CLAMPACCELEROMETERSURFACEc PILE CAP MASS NO. 2 (DISPLACEMENTLED LOCATED ON TOP OF MASS)CYLINDRICALMASS BOLTEDTO PILE CAP(14.3mm0, 38.1mm HIGH)CAP MASS NO.3ACCELEROMETERSLOTS TO ALLOWVARIABLE PILESPACINGS0 20mmI -ISCALESIDE VIEWFigure 9.20: The layout ofcentrifuge test for 4-pile group (after Gohi, 1991)Chapter 9. Analyses of Centrifuge Tests ofPileFoundations 172Another important issue in the analysisof pile groups is to incorporate rocking stiffness and damping into the analysis of horizontalmotions. In the proposed quasi-3Danalysis, the horizontal motions are uncoupledwith the vertical motions. The rockingstiffness induced by the vertical resistanceof piles can not included directly in the quasi-3D analysis of horizontal motions. However therocking impedances (resistance) maysignificantly restrain the rotational motions of thepile heads, and the effect of rocking impedances on the horizontal motionsmust be taken into account. The rockingimpedances may be even larger thanthe rotational impedances from the pileheads, especially when the pile spacing is large.The rocking impedances of the pile cap are includedin the analysis of pile groups.They are computed by performing a separate quasi-3Danalysis in the vertical mode during the analysis in the horizontal mode. Thisimpedance calculation is made using thecurrent values of strain dependent shearmoduli and damping ratios of soil. The theoryand procedures for computing such rocking impedanceshave been given in chapter7.After the rocking impedances were computed foreach time period, they are transferredto the pile cap as rotational stiffnessand damping.The finite element mesh used for thepile group analysis is shown in Figure 9.21. Themesh used in the present analysis has947 nodes and 691 elements. The sand foundationwas modelled by 11 horizontal layerswith a smaller thickness toward the sand surface.Each pile was modelled using 15 beam elementsincluding 5 elements for the part abovesoil surface. A very stiff masslessbeam element was used to connect the structural massto the pile heads.The analysis was carried out using the non-linearoption to simulate the changes ofChapter 9. Analyses of Centrifuge Tests ofPile Foundations173GEO.SCRLEd—lLrA,LJH-4SHAKING DIRECTIONFigure 9.21: Finite element modelling of the 2x2 pile groupshear moduli and damping of soil with the shear strain. According toGohi (1991), theshear-strain dependency of current shear moduliG and damping ratio D of the densesand are shown in Figure 9.22. The small strain shear moduliGm are functions of theoverburden stresses and are estimated according toEq. 9.1. The maximum hystereticdamping ratios Dm of the sand foundation are taken as 25% following Gohi (1991).Results ofanalysis Fig.9.23 shows the computed acceleration response at the pile capversus the measured acceleration response.There is fairly good agreement between themeasured and the computed accelerations. The computed peak acceleration at the pilecap is O.23g which agrees very well with the measured peak acceleration of 0.24g.The computed displacement at the top ofthe structural mass matches fairly well withthe measured displacement in the first 11 secs of motion (Figure 9.24). The computeddisplacement response did not show any residual displacement; whereas the measuredChapter 9. Analyses ofCentrifuge Tests ofPile Foundations174075 -MODULUS DAMPING050- -x -025 -c.cc ;10.0001 0.0010.01 0.11SHEAR STRAIN (%)Figure 9.22: The relationships between shear modulus, damping and theshear strain forthe dense sanddisplacement response showed a permanent residual displacement ofabout10 mm at theend ofearthquake motion. This is because the analysis is carried out usingthe equivalentlinear elastic approach.The computed moment time history in the instrumented pile at a depthof 2.63 min the area of maximum moment is plotted against the measured momenttime historyin Figure 9.25. There is good agreement between the measured and the computed moments. The distribution of computed and measured bending moments along the pileat the instant of peak pile cap displacement are shown in Figure 9.26. The computedmoments agree reasonably well with the measured moments, especially in the range ofmaximum moments. The computed peak moment is 203 kN.m compared to a measuredpeak moment of 220 kN.m.Chapter 9. Analyses of Centrifuge Tests ofPile Foundations—0.0750 5 10 1520 25 30TIME(SEC)1750.40.2z.00.0w-JwC-)-0.2—0.4-MEASUREDCOMPUTED4MbøI I I III I I I I I I III I I III I I II0 510 15 2025TIME(SEC)Figure 9.23: The computed versus measuredacceleration responses at pile cap300.0500.025—0.000—0.025—0.050 -MEASUREDCOMPUTEDi——-t----- ---—iIIFigure 9.24: The computed versus measureddisplacement at top of structural massChapter 9. Analyses of Centrifuge Tests of Pile Foundations176MEASUREDCOMPUTED—. 200zUiu0 U0 -:—100‘. .,.‘ ‘.‘.—200—300 i i0TIME(SEC)Figure 9.25: The computed versus measured moment at depthD=2.63 m111110 200 400MOMENT (KN.M)30011111111 I LIII11111111111IIII510 15 20 25300w20—2-—4--—6——8——10_—400 —200BENDING:__-.MEASUREDQ o.€COMPUTEDFigure 9.26: Distribution of moments at peak pile cap displacementChapter 9. Analyses of Centrifuge Tests of PileFoundations1776.OE+5 —4.OE+5 —2.OE+50.OE—t—0—0 5 10 15 2025 30TIME(SEC)Figure 9.27: Variation of stiffnessesk8of the 4-pile group at ft=1.91Hz9.3.3 Non-linear impedances of the 4-pilegroupApplying the same technique as used in the analysis ofsinglepile, dynamic impedances ofthe 4-pile group are evaluated. To illustrate the resultsofthis analysis, the translationalstiffness k and the cross-coupling stiffness k06of the 4-pile group are presented.The variations of stiffnesses andk9of the 4-pile group with time are shown inFigure 9.27 at an excitation frequency f=1.91 Hz.The dynamic stiffnesses of the 4-pilegroup k1,andk9 were reduced dramatically at times when strong ground motions occurred. decreased to 80,000 kN/mfrom its initial stiffness of 460,000 kN/m; whereask6 decreased to 160,000 kN/rad from its initial stiffness of 420,000 kN/rad. The stiffnesses k and !c were reduced to about17% and 38% of their initial stiffnesses.Part IIISummary and Suggestions for Future Work178Chapter 10Summary and Suggestions forFuture Work10.1 Dynamic thrusts on rigid wallsApproximate methods for determining the dynamicthrusts against rigid walls subjectedto horizontal dynamic loads were presented. Analyticalsolutions were obtained for thedynamic thrusts against rigid walls.Calibration was made by comparing the approximate1-g static solutions with Wood’sexact l-g static solution. Results showed that theproposed model using the shear beamanalogy produces best approximation to solutionof the rigid-wall problem. The computed total static thrusts are about5% less than those from Wood’s solution for anyLull ratio, in which H is the height ofthe wall andL is the halflength ofthe soil backfillconfined by two vertical rigid walls.Dynamic analyses have been performedfor wall-soil systems with uniform backfillsand subjected to both sinusoidal motionand earthquake motion. The wall-soil systemswith semi-infinite backfihls are approximatedby using L/H=5.O; the wall-soil systemswith finite backfills are represented by using L/H=1.5.Under sinusoidal motion, at resonant conditions thepeak dynamic thrusts are2.4pH2Amaa,for L/H=5.O and3.OpH2Amfor L/H=1.5. The static thrusts are1.OpH2Ama, for179Chapter 10. Summary and Suggestions for FutureWork 180L/H=5.O and0.86pH2Amaa, for L/H=1.5. Therefore, their corresponding dynamic amplification factors are 2.4 for L/H=5.0 and3.5 for L/H=1.5. These results stronglysuggest that the use of a static force for dynamicloading may result in serious underestimate of the dynamic thrusts against rigid walls.The earthquake motions are represented by the scaled records ofthe El Centro andthe Loma Prieta ground motions. The resultsof analyses show that the peak dynamicthrusts are1.3OpH2Amaa,for L/H=5.0 and1.38pH2Amafor L/H=1.5. Their corresponding dynamic amplification factors are about 1.3 forL/H=5.0 and 1.6 for L/H=1.5. Thedynamic amplification factors under earthquakemotions are about 50% of that undersinusoidal motions.Finite element formulations for evaluating the dynamicthrusts against rigid wallswith non-homogeneous soil profiles have beenpresented. Comparisons with close-formsolutions showed that the dynamic thrusts can be predicted accuratelyusing the proposed finite element method of analysis.An efficient computer program SPAW has beendeveloped for determining the dynamic thrusts and moments against rigid walls. Theprogram was designed for dynamicanalysis of rigid walls with arbitrary non-homogeneoussoil layers under sinusoidal motions and earthquake motions. A finiteelement mesh necessary for the analysis has beenembedded in SPAW so that only the properties ofsoil layers are required for input. Thecomputational time for a dynamic time-history analysisis only few minutes in a PC48633MHz computer.Chapter 10. Summary and Suggestionsfor Future Work 181For soil profiles with linear variation of shearmodulus with depth, the analyses revealed that the peak dynamic thrustat resonance isL56pH2Amaa,for sinusoidal motions.The peak dynamic thrust reduces to1.OpH2Aa,,for the scaled El Centro input. For soilprofiles with parabolic variation of shearmodulus with depth, the peak dynamic thrustat resonance is1.87pH2Amafor sinusoidal motions. The peak dynamic thrust reducesto1.l8pH2Amax for the El Centro input.Because the static force is aboutO.7lpH2Amaxfor linear soil profiles and0.82pH2Amaxfor parabolic soil profiles. The corresponding dynamicamplification factors are 2.3 forsinusoidal motions and 1.4 for earthquake motions.The dynamic thrust at resonance forsinusoidal motions are about60% greater than that for earthquake motions with samepeak acceleration.Results from the equivalent linear elasticanalyses show that the non-linearityof soilshas significant effects on the dynamic thrusts.For parabolic soil proffles the peak dynamic thrust of a wall-soil system under strongshaking (0.35g) due to non-linear effectis about 25% higher than that obtainedusing the linear elastic analysis. This result wasobtained using the scaled El Centro inputas the base motion. For wall-soil systems thathave different soil profiles or that aresubjected to other earthquake motions, the finiteelement program SPAW can be readilyapplied to evaluate the dynamic response of thesystems.The conclusions made on the studyof dynamic thrusts on rigid walls are presentedas follows:Chapter 10. Summary and Suggestions for FutureWork 1821. The proposed simplified wave equation has beensuccessfully applied to determinethe dynamic thrusts and moments acting on rigid walls.It implies that the dynamicmotions ofsoil structure under horizontal motionsare mainly governed by the wavesin the horizontal directions.2. It is recommended that the mode superpositionmethod be used to determine thepeak dynamic thrusts against rigid walls. The useof the absolute summation ofmodal thrust is suggested when the response spectrummethod is selected, especiallyat low frequency ratios.3. When wall-soil systems are subjected to dynamic loading,the total thrusts computed using the 1-g static solution should not beused to represent the peak dynamicthrust. The use of static solutionmay cause significant error of dynamic thrust especially at resonance. A dynamic analysis is requiredunder dynamic loading.4. The height of the resultant dynamic thrust isgenerally suggested to be at 0.611above the base of the wall for awall height of H. However for non-homogeneoussoil profiles, such as linear soil profiles or parabolic soilprofiles, the height maydecrease to about 0.511.5. Thepeak dynamic thrusts become large when the soil profilebecomes more uniform.Under sinusoidal motions, the peak dynamic thrustsat resonance are1.56pH2Am,1.87pH2Amacj, and 2.4OpH2Amax for linear, parabolic and uniform soil profiles, respectively. For scaled El Centroinput, the peak dynamic thrusts are 1.00pH2Amam,1.l8pH2Amczx and 1.3OpH2Amaz for linear, parabolic and uniform soil profiles, respectively. These results are obtained forwall-soil systems with L/H=5.0 andA=10%.6. The effect of soil non-linearity on the dynamic thrust seemsto be significant forChapter 10. Summary and Suggestionsfor Future Workt83rigid walls. A 25% increase of peakdynamic thrust may be expectedfor wall-soilsystems subjected to earthquake motionswith peak accelerations in the orderofO.35g, compared to results from a linear elastic analysis.Suggestions For future research, studiesshould be focused on includingthe effectsof seismic pore water pressuresin a saturated backfill on thedynamic thrusts. Thedevelopment of pore water pressure maysignificantly reduce the shear moduliof soils.Also one should be aware ofthatthe mechanism ofliquefactionin soil layers confined byrigid walls is different from thatin a free field. The shear strainsof soils are much lessespecially in the area near the wall.10.2 Dynamic analyses ofpile foundationsA quasi-3D finite element methodof analysis has been proposed to determinedynamicresponse of pile foundations subjectedto horizontal loading. The proposedmodel isbased on a simplified 3-Dwave equation. The 3-dimensionaldynamic response of soilis simulated by displacementsin the horizontal shaking direction. Displacementsin thevertical direction and in the horizontalcross-shaking direction are neglected.The use ofthe proposed simplified wave equationgreatly saves the computing space andcomputing time for the finite elementanalysis. However it maintains adequateaccuracy in themodelling of the dynamic responseof pile foundations.The proposed quasi-3D theoryis first incorporated using the finiteelement methodin the frequency domain. Thisformulation is used for the analysisof elastic responseof pile foundations. Elastic solutionsof Kaynia and Kausel (1982) have beenused tovalidate the proposed modelfor elastic response. Dynamic impedancesof single pilesand 2x2 pile groups have beencomputed and compared with those obtainedby KayniaChapter 10. Summary and Suggestions for FutureWork 184and Kausel (1982). Also kinematicinteraction factors of single piles havebeen computedand compared with those obtained by Fanet al. (1991) who used solutions by Kayniaand Kausel. The computed results fromthe proposed quasi-3D modelagreed well withthe results by Kaynia and Kausel.Full scale vibration tests of a singlepile and a 6-pile group have been analyzedusing the proposed quasi-3D model. Forthe single expanded base pile, the computedfundamental frequency of the pile cap systemwas 6.67 Hz, which agreed well with themeasured fundamental frequency of6.5 Hz. The computed damping ratio is6% compared to a measured damping ratio of4%. For the 6-pile group supporting a transformerbank, the computed fundamental frequenciesofthe transformer-pile cap were 3.74Hz and4.63 Hz in the NS and EW directions, respectively,which agreed well with the measuredfundamental frequencies of3.8 Hz and 4.6 Hz in the corresponding two directions.Thecomputed damping ratios of the system inthe two directions were9% and 9%, comparedto measured damping ratios of6% and 5%. The damping ratios were overestimated usingthe proposed model.The proposed quasi-3D method is also formulatedin the time domain using the finiteelement method. This formulation is targetedfor the analyses of non-linear response ofpile foundations under earthquakeloading. The time-domain analysis allows modellingthe variations of soil properties with timeunder earthquake loading.The procedures of non-linear time-domain analysisare incorporated in the computerprogram PILE3D. In PILE3D,the shear stress-strain relationship of soil is simulatedtobe either linear elastic or non-linear elastic.When the non-linear option is used, theshear modulus and the hysteretic dampingare determined using a modified equivalentChapter 10. Summary and Suggestionsfor Future Work 185linear approach based on the computedlevels of dynamic shear strains. Features suchas shear yielding and tension cut-offare incorporated in PILE3D as well. The dynamicresponse of pile groups can also be effectively modelledusing PJLE3D.Centrifuge tests of a single pile and a 2x2pile group have been analyzed using theproposed quasi-3D finite element methodofanalysis. The ability ofthe program PILE3Dto model the dynamic response of thepile supported structures under moderately strongshaking has been proven adequate for engineering purpose.These studies suggest that theshear-strain dependent shear moduli and dampingratios of soil is modelled adequatelyin the analysis by using the modified equivalent linearapproach. Also the approximatemethod for modelling the shear yieldingand the tension cut-off seems to yield satisfactory results.The other important feature of the proposedmethod is that the time histories ofdynamic impedances of single piles and pile groupscan be computed. This is the firsttime that this has been done.The results of analyses showed that stiffnessesof the pile foundations decrease withthe level of shaking; the dampings of thepile foundations increase with the level of shaking. In a seismic event, the translational stiffnesskr,,decreases the most due to the shearstrain dependency of the stiffness;the rotational stiffness k99 shows the least effect ofshear strain. At the moderately strong shaking ofthe centrifuge tests, the translationalstiffness would decrease to as much as15% of its initial stiffness; however the rotationalstiffness decreases to about60% ofits initial stiffness. The variation of the cross-couplingstiffness k,,6 is around50% of its initial stiffness.Chapter 10. Summary and Suggestions for Future Work 186The dynamic stiffnesses of pile foundations are little affected by the excitation frequencies in the range of earthquake loadings,f10Hz. However the damping usuallyincreases with excitation frequency. The damping can be representedby the hystereticand radiation damping components. The coefficient of equivalent viscous damping canbe determined according to the damping at any specific frequency. Generally viscousdamping coefficients decrease with excitation frequency.The quasi-3D finite element method has been proven to be a very cost-effectivemethod. Using a PC-486 (33MHz) computer, 3 hours of CPU time are required toconduct the non-linear analysis of a single pile for an input record of about 30 seconds.For the same length of input record, 30 minutes of CPU time is required for computingthe pile impedances associated with the time-dependent soil properties.Studies presented in this thesis lead to general conclusions on dynamic response ofpile foundations as follows:1. The proposed quasi-3D finite element method ofanalysis is an effective approach fordetermining elastic response of pile foundations and analyzing non-linear responseof pile foundations under earthquake loading. The simplified 3-D wave equationsof motion greatly reduce the computing time of the finite element analysis. Theerror ofresults caused by the quasi-3D model is minor especially at low frequenciessuch as those in an earthquake motion.2. A quasi-3D finite element program PlUMP has been developed for analyses ofelastic response ofsingle piles and pile groups. Analyses are performed in the frequencydomain. The proposed model has been vaJidated using the elastic solutions fromChapter 10. Summary and Suggestions for FutureWork 187Kaynia and Kausel (1982) and data from full scale vibration testsof a single pileand a 6-pile group.3. A quasi-3D finite element program PILE3D has been developedfor the analysesof non-linear response of single piles and pile groups. Analysesare performed inthe time domain. The characteristics of soil non-linearity with strains is modelledusing the modified equivalent linear approach. Other features suchas the modellingof shear yielding and tension cut-off are incorporatedin PILE3D. The program isapplicable for analyses of single piles as well as pile groups.4. The proposed model for modelling non-linear response of pilefoundations underearthquake loading has been validated using datafrom centrifuge tests of a singlepile and 2x2 pile group. Satisfactory results have been obtained.5. Time-dependent variations of dynamic impedances of pile foundations duringshaking can be evaluated and have been demonstrated for the modelpile foundationsused in the centrifuge tests. The results of analyses showed that stiffnesses ofpilefoundations decrease with the level of shaking. The translationalstiffness k,,1,decreases the most at high strain level; the rotational stiffness k98 decreases theleast.However, the damping of pile foundations increases with the levelof shaking.Suggestions for future work For future research on dynamicresponse of pile foundations, studies should be focused on including the effectsof seismic pore water pressureson the response of pile foundations. This next step is very important becausepile foundations are often used at potentially liquefiable sites.Bibliography[1] American Petroleum Institute (1986). “ RecommendedPractice for Planning, Designing and Constructing Fixed Offshore Platforms,” 16th edition,Dallas, TX.[2] Angelides, D.C., and Roesset J.M., (1981).“Non-linear Lateral Dynamic Stiffnessof Piles,” ASCE, Jour. of the Geotech. EngineeringDivision, vol. 107, no. GT11,pp.1443-1460[3] Arnold, P., Idriss, I.M., Reimer, R.B., Beebe, K.E., and Marshall,P.W. (1977).A Study of Soil-Pile-Structure Systems in Severe Earthquakes,”OTC 2749, 9thOffshore Technology Conference, Houston, Texas,pp.189-198.[4] Bea, R.G., Litton, R., Nour-Omid, S. and Chang,J.Y. (1984). “A Specialized Design and Research Tool for the Modelling of Near-Field SoilInteractions,” OTC4806, 16th Offshore Technology Conference, Houston, Texas,pp.249-252.[5] Brown, D.A. and Shie, C.F. (1991). “Modification ofp-y Curves to Account forGroup Effects on Laterally Loaded Piles.” GeotechnicalEngineering Congress 1991,ASCE, Geotechnical Special Publication No. 27, vol. 1,pp.479-490.[6] Clough, R.W. and Penzien, J., (1975). Dynamics ofStructures. McGraw-Hill BookCompany.[7] Crouse, C.B. and Cheang, L., (1987). “ Dynamic Testingand Analysis of PileGroup Foundations.” Dynamic Response of PileFoundations - Experiment, Analysis and Observation, ASCE Geotech. Special PublicationNo. 11, 79-98.[8] Davies, T.G., Sen, R., and Banerjee, P.K. (1985). “ Dynamic Behaviourof PileGroups in Inhomogeneous Soil,” J. Geotech. Eng., ASCE,Vol. 111, No. 12,pp.1365-1379.188Bibliography189[9] El-Marsafawi, H., Kaynia, A.M., and Novak, M.(1992a). “Interaction Factors andthe Superposition Method for Pile Group Dynamic Analysis,”Research Report,GEOT-1-1992, Univ. of Western Ontario, London,Ontario.[10] El-Marsafawi, H., Kaynia, A.M., andNovak, M. (1992b). “ The SuperpositionApproach to Pile Group Dynamics,” Geotechnical Special PublicationNo. 34, ASCE,New York, N.Y.,pp.114-136.[11] El Sharnouby, B. and Novak, M., (1986). “FlexibilityCoefficients and InteractionFactors for Pile Group Analysis.” Can. Geotech.J., 23, 441-450.[12] Fan, K., Gazetas,G., Kaynia, A., Kausel, E., and Shahid, A., (1991). “KinematicSeismic Response of Single Piles and Pile Groups.”Jour. of Geotech. EngineeringDivision, ASCE, Vol 117(12), 1860-1879.[13] Finn, W.D. Liam, Lee, K.W. and Martin,G.R. (1977).” An Effective Stress Modelfor Liquefaction”, Journal of the Geotech. EngineeringDivision, ASCE, June, 517-533.[14] Finn, W.D. Liam, and Gohi, W.B.,(1987).” Centrifuge Model Studies of Pilesunder Simulated Earthquake Loading,” DynamicResponse of Pile Foundations -Experiment, Analysis and Observation,ASCE Geotech. Special Publication No.11, 21-38.[15] Finn, W.D. Liam, M. Yogendrakumar,N. Yoshida and H. Yoshida. (1986). “TARA3: A Program for Nonlinear Staticand Dynamic Effective Stress Analysis,” SoilDynamics Group, University of British Columbia,Vancouver, B.C., Canada.[16] Gazetas, G. (1984). “SeismicResponse of End-Bearing Piles,” Soil Dynamics andEarthquake Engineering, Vol.3, No.2,pp.82-94.[17] Gazetas, C., Fan, K. and Kaynia,A. (1993). “Dynamic Response of Pile Groupswith Different Configurations,” Soil Dynamicsand Earthquake Engineering, No.12,pp.239-257.Bibliography190[18] Gazetas, G. and Dobry, R., (1984).“Horizontal Response ofPiles in LayeredSoils,”ASCE, Jour. of the Geotech. Engineering Division,Vol. 110, No.1, pp.20-41[19] Gazetas, G. and Makris, N. (1991a).“Dynamic Pile-Soil-Pile Interaction.Part I:Analysis of Axial Vibration,” EarthquakeEngineering Structure Dynamics,Vol.20, No.2,pp.115-132.[20] Gazetas, G., Fan, K., Kaynia, A.M.and Kausel, E. (1991b). “ Dynamic Interaction Factors for Floating Pile Groups,”J. Geotech. 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Div., ASCE,98(7),667-692.[25] Hardin, B.O., and Black,W.L., (1968).” Vibration Modulus of Normally Consolidated Clay,” ASCE, J. Soil Mechanicsand Foundations Division, Vol. 94,353-369.[26] Idriss, I.M., et al., (1973).” A Computer Programfor Evaluating the Seismic Response of Soil Structures by Variable DampingFinite Element Procedures,” ReportBibliography191No. EERC 73-16, Earthquake Engineering ResearchCentre, University of California, Berkeley, Calif., July.[27] Idriss, I.M., Seed, H.B., and Serif, N., (1974).“Seismic Response by Variable Damping Finite Elements” Jour. of Geotech. EngineeringDivision, ASCE, 100(1), 1-13.[28] Kaynia, A.M., (1982). “ Dynamic Stiffness andSeismic Response of Pile Groups,”MIT Research Report R82-03, Cambridge, MA, USA.[29] Kaynia, A.M. and Kausel, E., (1982). “DynamicBehaviour of Pile Groups.” Proc.of Conf. on Numerical Methods in Offshore Piling,Univ. of Texas, Austin, Texas,USA, 509-532.[30] Lee, M.K.W., and Finn, W.D. Liam (1978). “DESRA-2: DynamicEffective StressResponse Analysis of Soil Deposits with Energy TransmittingBoundary IncludingAssessment of Liquefaction Potential,” Soil MechanicsSeries Report No. 38, Dept.of Civil Engineering, University of British Columbia, Vancouver,Canada.[31] Matlock, H., Foo, S.H.C., and Bryant L.M.,(1978a). “ Simulation of Lateral PileBehaviour Under Earthquake Motion,” Proc.Earthquake Engineering and Soil Dynamics, ASCE Specialty Conference, Pasadena,Calif.,pp.601-619.[32] Matlock, H., Foo, S.H.C., and Cheang L.C., (1978b). “ Exampleof Soil Pile Coupling Under Seismic Loading,” proc. 10th Annual OffshoreTechnology Conference,Houston, Texas, paper no. OTC 3310.[33] Matlock, H., and Foo, S.H.C. (1980) “ Axial Analysisof Piles Using a HystereticDegrading Soil Model,” Proc. Tnt. Symp. Numer.Methods Offshore Piling, Institute of Civil Engineers, London,pp.127-133.[34] Matuo, H., and Ohara, 5. (1960) “ Lateral Earthquake Pressureand Stabilityof Quay Walls During Earthquakes,” Proceedings,Second World Conference onEarthquake Engineering, Vol. 2.Bibliography192[35] Mononobe, N., and Matuo, H. (1929)“ On the Determination of EarthPressureduring Earthquakes,” Proceedings,World Engineering Conference,Vol 9.[36] Murchison, J.M. and O’Neil M.W.(1984). “ Evaluation of P-Y RelationshipsinCohesionless Soils,” from “Analysisand Design of Pile Foundations,”proc. of asymp. sponsored by the ASCE GeotechnicalEngineering Division, ASCE NationalConvention, San Francisco, Calif.Oct. 1-5, 1984,pp.174-192.[37] Newmark, N.M., (1959). “ A Methodof Computation for StructuralDynamics,”Jour. of the Engineering Mechanics Division,ASCE, Vol. 85, EM3, July.[38] Nogami, T. and Chen, H.L., (1987). “Predictionof Dynamic Lateral ResponseofNon-linear Single Pile by Using WinklerSoil Model,” Dynamic Response of PileFoundations - Experiment, Analysisand Observation, ASCE Geotech. 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(1973). “Lateral Soil Resistance-DisplacementRelationships for Pile Foundations in Soft Clays,”OTC 1893, 5th Offshore TechnologyBibliography196Conference, vol. 2, pp.663-671.AppendixiDerivation of Equations for the Proposed Model: Rigid-Wall ProblemThe proposed model is an extension of the classic shear beam model to include horizontal normal stresses in the direction of shaking.The shear stress is given by8u ôvTy_G(ã_+_)(.1)In the shear beam = 08u= (.2)The normal stress o in the direction of shaking is determined using the assumption=02G ôv uo.y=l_2’1”8yth10ThereforeEly vEluEly 1—vElxThe normal stress o in the direction of shaking is given by2G Elu Ely12VE(hL)Elx+MElyl(.5)Substituting for2 Elu= (.6)1—ti O3197198Substituting Eq. .2 and Eq. .6 into the equilibriumequation(7)oxoyPOt2one finds that2 02u 82u 02u1 —+G=(.8)Therefore by comparison with Eq.(3.6) and (3.7)(.9)Eq. .8 satisfies equilibrium but does not satisfysecond order compatibility.


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