{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Civil Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Wu, Guoxi","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2009-04-15T21:32:41Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1994","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"This thesis deals with two important topics in soil-structure interaction: seismic earth\r\npressures against rigid walls and the seismic response of pile foundations. These two disparate problems are linked by a common method of solution which is an approximation\r\nto the response of the half-space, either linear or non-linear.\r\nThe approximate formulation permits analytical solutions against rigid walls when\r\nthe backfill is uniform and elastic. The solution agrees very closely with an existing exact solution. For elastic non-homogeneous backfills and for non-linear soil response the\r\napproximate formulation is expressed using the finite element method.\r\nAn efficient computer program SPAW has been developed to determine dynamic\r\nthrusts and moments against rigid walls for arbitrary non-homogeneous soil layers. Results of analyses show that the peak dynamic thrusts are larger for a uniform soil profile\r\nthan when the shear modulus of the soil varies linearly or parabolically with depth. The\r\nprogram SPAW also possesses the ability of modelling the effect of soil non-linearity on\r\ndynamic thrusts. Studies showed that an increase of peak dynamic thrust may be expected due to soil non-linearity, compared with results from a linear elastic analysis.\r\nA quasi-3D finite element method of analysis has been proposed to determine dynamic response of pile foundations subjected to horizontal loading. A computer program\r\nPlLIMP has been developed for the analyses of elastic response of pile foundations including the determination of pile impedances as a function of frequency. The analysis \r\nis conducted in the frequency domain. The program can analyze single piles and pile\r\ngroups in arbitrary non-homogeneous soil layers.\r\nAnother quasi-3D finite element computer program PILE3D has been developed for\r\nthe analysis of non-linear response of pile foundations in the time domain. The program\r\nis suitable of dynamic analyses of single piles and pile groups. The soil non-linearity\r\nduring shaking is modelled using a modified equivalent linear method. Yielding of the\r\nsoil is taken into account and there is a no-tension option controlling the analysis.\r\nThe proposed quasi-3D model has been validated using the elastic solutions from\r\nKaynia and Kausel (1982), Novak and Nogami (1977) and Novak (1974), Fan et al.\r\n(1991), data from full scale vibration tests of a single pile and a 6-pile group, and data\r\nfrom centrifuge tests of a single pile and a 2x2 pile group under strong shaking from\r\nsimulated earthquake. Excellent results have been obtained using the proposed method.\r\nTime-dependent variations of dynamic impedances of pile foundations during shaking\r\nhave been evaluated for the model pile foundations used in the centrifuge tests. The\r\nanalyses quantify the reduction in the stiffnesses of the pile foundations with the increased\r\nlevel of shaking. The translational stiffness \u0198\u0475\u0475 decreases the most during strong shaking;\r\nthe rotational stiffness \u0198\u03b8\u03b8 decreases the least. However, the damping of pile foundations\r\nincreases with the level of shaking.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/7158?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"Extent":[{"label":"Extent","value":"4557780 bytes","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/extent","classmap":"dpla:SourceResource","property":"dcterms:extent"},"iri":"http:\/\/purl.org\/dc\/terms\/extent","explain":"A Dublin Core Terms Property; The size or duration of the resource."}],"FileFormat":[{"label":"FileFormat","value":"application\/pdf","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/elements\/1.1\/format","classmap":"edm:WebResource","property":"dc:format"},"iri":"http:\/\/purl.org\/dc\/elements\/1.1\/format","explain":"A Dublin Core Elements Property; The file format, physical medium, or dimensions of the resource.; Examples of dimensions include size and duration. Recommended best practice is to use a controlled vocabulary such as the list of Internet Media Types [MIME]."}],"FullText":[{"label":"FullText","value":"DYNAMIC SOIL-STRUCTURE INTERACTION:PILE FOUNDATIONS AND RETAINING STRUCTURESByGuoxi 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 FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 10, 1994\u00a9 Guoxi WU, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. it is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of C-t IThe University of British ColumbiaVancouver, CanadaDate 0 c f. 7 (99DE-6 (2188)ABSTRACTThis thesis deals with two important topics in soil-structure interaction: seismic earthpressures against rigid walls and the seismic response of pile foundations. These two disparate problems are linked by a common method of solution which is an approximationto the response of the half-space, either linear or non-linear.The approximate formulation permits analytical solutions against rigid walls whenthe backfill is uniform and elastic. The solution agrees very closely with an existing exact solution. For elastic non-homogeneous backfills and for non-linear soil response theapproximate formulation is expressed using the finite element method.An efficient computer program SPAW has been developed to determine dynamicthrusts and moments against rigid walls for arbitrary non-homogeneous soil layers. Results of analyses show that the peak dynamic thrusts are larger for a uniform soil profilethan when the shear modulus of the soil varies linearly or parabolically with depth. Theprogram SPAW also possesses the ability of modelling the effect of soil non-linearity ondynamic thrusts. Studies showed that an increase of peak dynamic thrust may be expected due to soil non-linearity, compared with results from a linear elastic analysis.A quasi-3D finite element method of analysis has been proposed to determine dynamic response of pile foundations subjected to horizontal loading. A computer programPlUMP has been developed for the analyses of elastic response of pile foundations including the determination of pile impedances as a function of frequency. The analysis11is conducted in the frequency domain. The program can analyze single piles and pilegroups in arbitrary non-homogeneous soil layers.Another quasi-3D finite element computer program PILE3D has been developed forthe analysis of non-linear response of pile foundations in the time domain. The programis suitable of dynamic analyses of single piles and pile groups. The soil non-linearityduring shaking is modelled using a modified equivalent linear method. Yielding of thesoil is taken into account and there is a no-tension option controffing the analysis.The proposed quasi-3D model has been validated using the elastic solutions fromKaynia and Kausel (1982), Novak and Noganii (1977) and Novak (1974), Fan et al.(1991), data from full scale vibration tests of a single pile and a 6-pile group, and datafrom centrifuge tests of a single pile and a 2x2 pile group under strong shaking fromsimulated earthquake. Excellent results have been obtained using the proposed method.Time-dependent variations of dynamic impedances of pile foundations during shakinghave been evaluated for the model pile foundations used in the centrifuge tests. Theanalyses quantify the reduction in the stiffnesses of the pile foundations with the increasedlevel of shaking. The translational stiffness decreases the most during strong shaking;the rotational stiffness k60 decreases the least. However, the damping of pile foundationsincreases with the level of shaking.UiTable of ContentsABSTRACT iiList of Tables viiiList of Figures ixAcknowledgement xv1 Introduction 1I Dynamic Thrusts on Rigid Walls 62 Dynamic analyses of rigid walls 72.1 Literature review 72.2 Objectives of current research 123 Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfllls 143.1 Introduction 143.2 Dynamic analysis of rigid wall-soil system 153.3 Static 1-g solution: Validation of model 243.4 Dynamic thrusts under sinusoidal motions 313.5 Dynamic thrusts under earthquake motions 353.6 Accuracy of the response spectrum method 40iv4 Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles4.1 Introduction4.2 Finite element formulation and its validation4.3 Linear elastic analyses with non-homogeneous soil profiles4.4 Equivalent linear simulation of non-linear response under earthquake loadsII Dynamic Analyses of Pile Foundations 665 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: Theory and Verification6.1 Introduction6.2 Dynamic analyses of pile foundations: formulation6.3 Pile head impedances6.4 Verification of the proposed model: pile head impedances . .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: Theory and Verification 1107.1 Introduction 1104949505561676776788080818689989999100100104V7.2 Rocking impedance of pile group 1107.3 Dynamic equation of motions in the vertical direction 1127.4 Determination of rocking impedance 1147.5 Elastic response of pile group: results and comparisons 1167.6 Full-scale vibration test on a 6-pile group 1227.6.1 Description of vibration and its testing results . 1227.6.2 Computed results using the proposed model 1258 Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 1328.1 Introduction 1328.2 Quasi-3D finite element analysis in the time domain 1338.3 Solution scheme for dynamic equation 1358.4 Non-linear analysis 1368.5 Features in dealing with yielding, tension 1418.6 Soil parameters required in PILE3D analysis 1428.7 Aspects relative to analysis of pile group 1459 Analyses of Centrifuge Tests of Pile Foundations 1489.1 Introduction 1489.2 Dynamic analysis of centrifuge test of a single pile 1489.2.1 Description of centrifuge test on a single pile 1489.2.2 Dynamic analysis of the single pile 1519.2.3 Non-linear pile impedances 1589.2.4 Computational times 1699.3 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 group 170vi9.3.3 Non-linear impedances of the 4-pile group 177III Summary and Suggestions for Future Work 17810 Summary and Suggestions for Future Work 17910.1 Dynamic thrusts on rigid walls 17910.2 Dynamic analyses of pile foundations 183Bibliography 188viiList of Tables3.1 Peak dynamic thrusts for walls with L\/H=5.0 and 11=10 m, )=10% . . . 383.2 Peak dynamic thrusts for walls with L\/H=1.5 and 11=10 m,A=10% . . . 394.3 Patterns of first natural frequencies for three 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 without the effect ofpile cap embedment 1287.6 Computed stiffness and damping of the transformer pile foundation . 1297.7 Measured and computed resonant frequencies and damping ratios including the effect of pile cap embedment 1318.8 Relationship between Hardin and Drnevich constant k and plasticity indexPT (after Hardin and Drnevich, 1972) 1449.9 Parameters of dynamic impedances of single pile 167vi\u201dList of Figures2.1 Wall-soil system used in Wood\u2019s study (after Wood, 1973) 93.1 Definition of rigid-wall problem (a) original problem (b) equivalent problem by using antisymmetric condition 163.2 Comparison of the accuracy of approximate solutions for rigid-wall systems(a) L\/H=5.0 (b) L\/H=1.5 273.3 Normalized thrust ratios for 1-g static solution(a) Wood\u2019s solution (b)author\u2019s solution 293.4 Heights of thrusts due to 1-g static horizontal force (wall height H=lOm) 303.5 Accuracy of solutions versus number of modes used 323.6 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 Centro input (a) L\/H=5.0(b) L\/H=1.5 363.8 Time histories of dynamic thrusts using the Loma Prieta input (a) L\/H=5.0(b) L\/H=1.5 373.9 A time history of the height of dynamic thrust, L\/H=5.0 393.10 Normalized thrust ratios versus fRi for earthquake motions (a) L\/H=5.0(b) L\/H=1.5 413.11 Normalized thrust ratios versus fR2 for earthquake motions (a) L\/H=5.0(b) L\/H=1.5 423.12 Pseudo-spectral velocities of (a) the El Centro input and (b) the LomaPrieta input 44ix3.13 Variations of thrust factor Cp versus frequency ratio fRi (RSS method)(A) L\/Hz=5.0, (B) L\/H=1.5 463.14 Variations of thrust factor Cp versus frequency ratio fRi (ABS method)(A) L\/H=5.0, (B) L\/H=1.5 474.1 A composition of non-homogenous soil profile 504.2 A composition of the finite element used in SPAW 514.3 A finite element mesh used for dynamic analyses 534.4 Comparisons of dynamic thrusts between the F.E. method and the close-form solution for uniform soils(a) L\/H=5 (b) L\/11=1.5 544.5 Relationships between thrust ratio and frequency ratio fR2 for linear soilprofiles (a) sinusoidal motions (b) the El Centro input 574.6 Relationships between thrust ratio and frequency ratio fR2 for parabolicsoil profiles (a) sinusoidal motions (b) the El Centro input 584.7 Comparison of dynamic thrust ratios for parabolic soil profiles and uniformsoil profiles under sinusoidal motions (L\/H=5) 594.8 Typical time histories of heights of dynamic thrusts for three types of soilprofiles (H=lOm) 604.9 Dynamic responses of a stiff site due to non-linear effect,G0=132,000 kPa 634.10 Effect of level of shaking on the dynamic thrust,G0=132,000 kPa . . . 644.11 Dynamic responses of a soft site due to non-linear effects,Go=66,000 kPa 645.12 Variation of pile horizontal stiffness, k, with force and frequency due tosoil non-linearity (after Angelides and Roesset, 1981) 736.1 The principle of quasi-3D dynamic pile-soil interaction in the horizontaldirection 82x6.2 Finite element compositions for modelling horizontal motions 836.3 Pile head impedances 866.4 A pile-soil system used for computing impedances of single piles 906.5 Finite element modelling of single pile for computing impedances 916.6 Normalized stiffness k,,, and damping C,,,, versus a0 for single piles (E\/E31000, v =0.4, A=5%) 926.7 Normalized stiffness k,,8 and damping C,, versus a0 for single piles (E\/E3 =1000, ii =0.4, )=5%) 936.8 Normalized stiffness k88 and damping C89 versus a0 for single piles (E\/E8 =1000, v =0.4, A=5%) 946.9 Comparison of stiffness k,,,, and damping C with solutions by Novak andNogami (1977), Novak (1974) 966.10 Comparison of stiffness k,,,, for different mesh size 976.11 Pile foundation for analysis of kinematic response 986.12 Kinematic interaction factors versus a0 for E\/E8 = 1, 000 1016.13 Kinematic interaction factors versus a0 for E\/E3 = 10, 000 1026.14 The in-situ measured geotechnical data (after Sy and Siu, 1992) 1036.15 The layout of the full-scale vibration test on a single pile (after Sy andSiu, 1992) 1046.16 The soil parameters used in the analysis ( after Sy and Siu, 1992) . . . 1066.17 Finite element modelling of the expanded base pile 1066.18 An uncoupled system modeffing the horizontal motions of structure-pilecap system 1076.19 Amplitudes of horizontal displacement at the centre of gravity of the pilecap versus the excitation frequency 10837.1 The mechanism of rocking in a pile group 1117.2 The quasi-3D model in the vertical direction, Z 1137.3 A pile-soil system used for computing impedances of pile groups 1167.4 Comparison of dynamic interaction factor a with 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 rotational impedance K7\/A with solution by Kaynia and Kausel for 2x2 pile groups (E\/E8 = 1000, s\/d=5, A=N*Y2rk\u00b02 ) 1217.7 Idealized soil profile at Duwamish Substation ( after Crouse and Cheang,1987) 1237.8 Setup of a full-scale free vibration test 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 direction 1277.10 Response curves of the of transformer-pile cap system (a) NS direction (b)EW direction 1308.1 Hysteretic stress - strain relationships at 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 linear method 1408.4 Simulations of shear yielding and tension cut-off 1418.5 Comparison of damping ratios for sands and gravelly soils (after Seed etal., 1986) 145xii8.6 A diagram showing the representation of pile group supporting structure 1469.1 The layout of the centrifuge test for a single pile 1509.2 The prototype model of the single pile test 1509.3 Computed Fourier amplitude ratios (a) pile amplitude ratio (APH\/AFF)(b) free field amplitude ratio (AFF\/AB) (after Gohl, 1991) 1529.4 The finite element modeffing of centrifuge test 1539.5 The relationships between shear modulus, damping and the shear strainfor the loose sand 1549.6 The computed versus measured acceleration response at the free field . 1559.7 The computed versus measured acceleration response at the pile head . 1569.8 The computed versus measured displacement response at the top of thestructure 1569.9 The computed versus measured moment response at the soil surface . 1579.10 The computed versus measured moment response at depth D=3m . . . 1579.11 The computed versus measured moment distribution of the pile at peakpile deflection 1589.12 3-D plots of the distribution of shear moduli at t=12.58 sec 1599.13 3-D plots of the distribution of shear moduli at t=17.11 sec 1609.14 Variation of stiffnesses k,k9,k98 of the single pile at f=1.91 Hz . . . 1639.15 Variation of stiffnesses k and k98 with time under different excitationfrequency 1639.16 Variation of translational damping C versus time under different frequency 1649.17 Variation of hysteretic dampings C,, C6 and C of the single pile . . . 1659.18 Variation of radiation damping constant at different frequencies . . 166xlii9.19 Comparison of dynamic stiffnesses of pile foundations with full structuralmass and without structural mass 1689.20 The layout of centrifuge test for 4-pile group (after Gohi, 1991) 1719.21 Finite element modelling of the 2x2 pile group 1739.22 The relationships between shear modulus, damping and the shear strainfor the dense sand 1749.23 The computed versus measured acceleration responses at pile cap . . 1759.24 The computed versus measured displacement at top of structural mass 1759.25 The computed versus measured moment at depth D=2.63 m 1769.26 Distribution of moments at peak pile cap displacement 1769.27 Variation of stiffnesses k8 of the 4-pile group at f=1.91 Hz 177xivAcknowledgementThe author sincerely thanks his research supervisor, Professor W.D. Liam Finn, for hisguidance, suggestions and encouragement during the course 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 members of the supervisorycommittee and for reviewing the manuscript. Appreciation is also extended to otherfaculty members in the Dept. of Civil Engineering for offering many excellent courseswhich established the cornerstone of this research.The postgraduate fellowship awarded by the Canadian National Science and Engineering Research Council and the research assistantship provided by the University ofBritish Columbia are gratefully acknowledged.Finally, the author would like to thank his wife Liffian for her love and faithful supportover the course of preparation of this thesis. This thesis is dedicated to her and our sons,Galen and Allan.xvChapter 1IntroductionThis thesis deals with two important topics in soil-structure interaction: seismic earthpressures against rigid walls and the seismic response of pile foundations. These twodisparate problems are linked by a common method of solution based on an approximation to the response of the half-space, either linear or non-linear. In the case of the pilefoundations, the piles are modelled as linear beam element inclusions in the half space.The rigid wall solution has important applications: seismic earth pressures againstdeep basement walls, buried containment structures and the wingwalls of dams. Theseismic analysis of pile foundations remains a challenging problem both in engineeringpractice and in research. The action of pile foundations is a key element in evaluating properly the response of pile-supported buildings, bridges and offshore platforms toearthquake loading. Characterization of the stiffness and damping of pile foundations isa complex major task for large bridges with multiple points of seismic input. The methoddeveloped in this thesis for solving these two major soil-structure interaction problems iscapable of simulating important features of seismic interaction.The methodology used for solving the rigid-wall problem is essentially a 2-D planestrain application of the quasi-3D theory developed for 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 excited in a 3-D half-space by shear waves1Chapter 1. Introduction 2propagating vertically is governed primarily by compression waves propagating in theprincipal shaking direction and shear waves propagating in the two other directions.Other types of waves in the 3-D half-space are ignored in the analysis because they areassumed to be less significant. This assumption will be validated later using elastic solution based on a full 3-D formulation.In the seismic analysis and design of rigid-wall systems, the basic challenge is to evaluate the magnitude and distribution of dynamic soil pressures against the walls inducedby ground shaking. In addition to the evaluation of dynamic soil pressures against rigidwalls with elastic homogeneous soil backfills, there are two important issues relative torigid walls. The first issue is the accurate evaluation of dynamic soil pressures againstrigid walls with arbitrary non-homogeneous backfills. The second issue is the appropriatemodelling of soil non-linearity under relatively strong shaking. Therefore, a portion ofthis thesis is devoted to developing a cost-effective method for dealing with arbitrarynon-homogeneous soil proffle and soil non-linearity.In the first part of this thesis, the 2-D plane strain formulation of the quasi-3D theoryis used to obtain solutions of dynamic soil pressures against rigid walls with uniformbackfills. Then the theory is implemented into an effective finite element program. Thefinite element method of analysis is used to explore the effect of soil non-homogeneity ondynamic pressures on rigid walls. Effect of soil non-linearity under strong shaking is alsoinvestigated.The second part of this thesis is devoted to dynamic response of pile foundationssubjected to horizontal shaking. Pile foundations are widely used in civil engineeringChapter 1. Introduction 3works. Pile-group foundations are used to support important structures such as high-rise buildings, bridges, and large power transmission towers. When these pile-supportedstructures are located in a seismic active zone, concerns arise 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 which are restricted to elastic response.This thesis describes the development of a cost-effective and reliable numerical procedure which can be used to study dynamic response of pile foundations when foundationsoils are non-linear and nonhomogeneous. The quasi-3D theory is adopted to the dynamicanalyses of pile foundations. A number of solutions for elastic homogeneous response havebeen developed in order to validate the proposed quasi-3D method by comparing the results with the published elastic solutions based on full-3D formulation.OUTLINE OF THE THESISChapter 2 gives a review of existing methods for determining dynamic soil pressureson rigid walls and presents objectives of the present research study.Chapter 3 gives a simplified method of analysis for rigid wall-soil systems subjected tohorizontal dynamic loads assuming linear elastic response of the soil. Analytical solutionsof dynamic soil pressures against rigid walls with homogeneous soil backfills are derivedfirst. Dynamic thrusts against rigid walls are determined using both sinusoidal motionsand earthquake motions as input. The results from the present analysis are validated bycomparison with the published results developed from full 2-D elastic response analysisby Wood (1973). Studies are made to examine the accuracy of the response spectrummethod for determining peak dynamic thrust against rigid walls.Chapter 1. Introduction 4Chapter 4 extends the method of analysis to deal with rigid walls with an arbitrarynonhomogeneous soil profile. Analyses are performed to study the pattern of dynamicthrusts against rigid walls for nonhomogeneous soil profiles. Finally the effect of soilnon-linearity on dynamic response of rigid walls is explored by using the equivalent linear approach (Seed and Idriss, 1967).Chapter 5 gives a review for current methods of dynamic response analysis of pilefoundations and outlines the objectives of this thesis for the analysis of pile foundations.Chapter 6 presents a quasi-3D finite element method for dynamic response analysisof pile foundations assuming elastic response of soil and piles. The proposed model isverified first against elastic solutions of single piles by Kaynia and Kausel (1982). Thenthe proposed model is calibrated using data from a full-scale vibration test on a single pile.Chapter 7 applies the proposed quasi-3D theory to the dynamic analysis of elasticresponse of pile groups. The proposed model for pile groups is verified first against elasticsolutions by Kaynia and Kausel (1982) and then against results of a field vibration teston a 6-pile group by Crouse and Cheang (1987).Chapter 8 applies the proposed quasi-3D theory to the dynamic analysis of pile foundations under earthquake loading taking the non-linear response of the soil into account.The non-linear finite element analysis is conducted in the time domain. Procedures formodeffing non-linear soil response are also described in this chapter.Chapter 1. Introduction 5Chapter 9 describes the validation of the proposed model for non-linear dynamic response of pile foundations using data from centrifuge tests of a single pile and a 2x2pile group. The variations of dynamic impedances of pile foundations with time duringshaking are also evaluated and demonstrated for the model pile foundations.Chapter 10 summarizes the developments described in earlier chapters and presentsthe conclusions arising from the various studies.Part IDynamic Thrusts on Rigid Walls6Chapter 2Dynamic analyses of rigid walls2.1 Literature reviewFor seismic design of a rigid wall it is important to know the magnitude and distributionof seismic pressure on the wall induced by earthquake motion. Probably the earliestresearches dealing with seismic induced earth pressure on retaining structures were thoseof Mononobe (1929) and Okabe (1926). The Mononobe-Okabe method is the modification of Coulomb\u2019s classic earth pressure theory which 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 that the wall has to move sufficiently to create a limitequilibrium state in the backfill. This condition is not satisfied in most rigid wall cases.Several researchers have used elastic wave theory to derive seismic backfill pressureagainst a rigid wall. Matuo and Ohara (1960) obtained an approximate elastic solution for the dynamic soil pressure on a rigid wall using a two-dimensional analyticalmodel. They simplified the problem by assuming zero vertical displacement in the soilmass. This simplification leads to infinitely large wall pressure when Poisson\u2019s ratio ofthe soil is equal to 0.5 as in a fully saturated undrained backfill. Scott (1973) used a onedimensional elastic shear beam connected to the wall by Winkler springs to model theproblem. The advantage of this model is that close-form solutions can easily be obtained.7Chapter 2. Dynamic analyses of rigid walls 8The disadvantage is that his solution requires a representative value of the Winkler springconstant. Scott used Wood\u2019s (1973) solutions to define the characteristics of the Winkler spring constant. However, the Winkler spring constant was determined only for thefirst-mode motion. For a wall with relatively long backfill (L*\/H >4 ), the accuracy ofScott\u2019s first-mode solution deteriorates rapidly with the increase of L*\/H ratio.Wood (1973) made a comprehensive study on the behaviour of the rigid soil-retainingstructures subjected to earthquake motions. His work is considered to be one of the moreimportant contributions to understanding of this problem.Wood\u2019s solution of rigid-wall problem. In Wood\u2019s solution, the soil is assumedto be an isotropic homogeneous elastic material. The wall-soil system was assumed tosatisfy the condition of plane strain. The analytical solutions were obtained assuminga smooth contact between the wall and the backfill; that is, the vertical boundaries areassumed to be free from shear stresses. Figure 2.1 shows the rigid-wall soil system usedin Wood\u2019s study and the associated boundary conditions.The modal frequencies Wnm for the wall-soil system shown in Figure 2.1 are not explicitly expressed in Wood\u2019s solution. For the homogeneous backfill, the modal frequencyWnm is governed by2 =r2\u2014(2.1)Chapter 2. Dynamic analyses of rigid walls 9RIGID WALL PROBLEMFigure 2.1: Wall-soil system used in Wood\u2019s study (after Wood, 1973)and32_r2\u2019flfl (2.2)where V, V3 are the compression wave velocity and shear wave velocity of soil, r =and parameters c, \/3 have particular values for each mode. They must satisfy the following equation( 2+;+(r2}sinhaH. sinh\/3H \u2014{4r2+\u20182 }coshaH.cosh\/3H+4(r+\/)=O (2.3)The roots of the frequency equations were numerically evaluated using Newton\u2019smethod on a digital computer. An iterative process was used to compute the rootsindicated by sign changes in the frequency equations.Homogenous elastic soil(Plane strain)Chapter 2. Dynamic analyses of rigid walls 10It can be seen that the frequency solution is too complicated to apply in practice.Although graphic solutions of modal frequency were given by Wood, these solutions arelimited to particular values of Poisson\u2019s ratio. Furthermore modal frequencies 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-amplitude harmonicbase forcing were presented in Wood\u2019s study asF\u2019(w) =Fnm(2.4)1(1)+2inmWflmwhere Fnm is the static-one-g modal force per unit length. Mathematical expressionof Fnm is not available in Wood\u2019s solution, but graphic results of Fnm were presented.The one-g static force per unit length of wall was obtained by applying a 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\u2019 + 1)e+D(\u20142rH + k\u2019 + 1)e\u2019\u201d \u2014 (C + D)(k\u2019 + 1) \u2014k2rH} (2.5)Where constants C,, D, B are parameters associated with the flth mode and properties of soil. Their detailed formulations can be found in Wood (1973).For a L*\/H ratio of 5.0, the force F8r can be approximated satisfactorily using about20 terms in Eq.2.5. A smaller number of terms are required for L*\/H less than 5.0 andChapter 2. Dynamic analyses of rigid walls 11a larger number for L*\/H greater than 5.0.Wood\u2019s solution is quite complicated even under very idealistic conditions of a homogeneous soil layer under harmonic loading. Therefore, in engineering practice Wood\u2019ssolution is applied approximately. The dynamic thrust is approximately taken as -yH2A(Whitman, 1991) for a horizontal acceleration Ag, which is the static force for a wall-soilsystem with L*\/H=10 and 11 = 0.4. The use of this expression must be justified in manycases especially at resonance when the fundamental frequency of the wall-soil system isvery close to the predominant frequency of the excitation motion. At resonant conditionthe dynamic thrust are likely to be much greater than the static force.For earthquake type of loading, Wood (1973) states \u201cIn view of the uncertaintiesinherent in the estimation of earthquake-induced pressures on walls, computation of theresponse time-history is probably not warranted and approximate evaluation may oftenbe satisfactory\u201d. Wood proposed that the response spectrum method is applied to estimate the peak dynamic thrust on the wall for earthquake loading.Although Wood\u2019s solution for dynamic thrust on rigid walls is mathematically correctfor harmonic loading, uncertainties and difficulties arise when his method is applied inpractice to the wall-soil system under earthquake loading. Significant errors of dynamicthrusts may be caused by using the response spectrum method proposed by Wood.The other restriction of Wood\u2019s solution is that his method is not capable of dealingwith rigid walls having arbitrarily nonhomogeneous backfills. In addition the effect ofsoil non-linearity on the dynamic thrust under strong motions cannot be assessed usingWood\u2019s method.Chapter 2. Dynamic analyses of rigid walls 122.2 Objectives of current researchDue to the limitations of Wood\u2019s solution in engineering practice, the current research istargeted to find an effective method for determining the dynamic thrusts on rigid wallssubjected to horizontal dynamic loads. A proposed method is used to analyze wall-soilsystems with nonhomogeneous soil profiles and to study the effect of soil non-linearityon dynamic response.The method of analysis is formulated based on simplified elastic wave equations. Theequations of motion are established considering dynamic force equilibrium in the horizontal direction only. An analytical solution for dynamic pressures 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 dynamic thrust is determined usingthe mode-superposition method. .The time-history response of each mode is added up inphase. The commonly used Response Spectrum method is also used to predict the peakdynamic thrust. The accuracy of the response spectrum method is evaluated against therigorous mode-superposition method.The proposed method is then applied to the wall-soil systems with nonhomogeneousbackfills. A finite element program SPAW (Seismic Pressures Against Rigid Walls) wasdeveloped to implement the analysis. The computational time and cost for time-historyanalysis is negligible using SPAW. The dynamic analyses of nonhomogenous soil profilesChapter 2. Dynamic analyses of rigid walls 13are illustrated through two types of soil profiles, soil profiles with linear and parabolicvariations of shear moduli with depth. Analyses are performed for both harmonic loadingand earthquake loading assuming elastic response.The effect of soil non-linearity on dynamic thrusts against rigid walls was also studied.The equivalent linear approach (Seed and Idriss, 1967) is used to model non-linear soilresponse. The effect of the intensity of shaking on dynamic thrusts against rigid walls isinvestigated for soil profiles with parabolic variations 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 a rigid soil-retainingstructure subjected to horizontal dynamic loads. The method is based on simplified elastic wave equations. The equation of motion is derived from dynamic force 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 of motion is solved analytically taking into account the boundary conditions of the problem. The solution is applied first to obtain the total thrustacting on the wall due to horizontal one-g static loading to estimate the accuracy of theapproximate solutions against Wood\u2019s exact solutions from the 2-D plane strain analysis.Simple explicit expressions for computing the dynamic thrusts are presented for bothsinusoidal motion and earthquake motion. Time history solutions of the dynamic thrustsfor earthquake motions can be readily obtained using the mode superposition method.Studies were made to examine the different patterns of dynamic thrusts for harmonicloading and earthquake loading.Finally the response spectrum method is applied to obtain the peak dynamic thrusts14Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 15for earthquake motions. The accuracy of the response spectrum method is also investigated.3.2 Dynamic analysis of rigid wall-soil systemFigure 3.1(a) shows the geometry of the problem and its boundary conditions. A uniformelastic soil layer is confined by two vertical rigid walls at its two side boundaries and arigid base. The soil layer has a total length of 2L and height of H. Subjected to horizontalseismic body force, the soil layer in the system generates an antisymmetric field of horizontal normal stresses o with zero stresses at x = L. The original wall-soil problemcan be equivalently represented by half of its geometry using this antisymmetric condition. The equivalent problem is shown in Figure 3.1(b), and this is the physical modelthat will be analyzed. The ground acceleration is input at the base of the wall-soil system.The soil is assumed to be a homogeneous, isotropic, visco-elastic solid with a shearmodulus G and Poisson\u2019s ratio v. The equations of dynamic force equilibrium for thebackfill in the horizontal and vertical directions are written as50 9r 82u(3.1)--- + -- =(3.2)where o and o, are the normal stresses in the X and Y directions, respectively, andr is the shear stress in the x-y plane. u and v are the displacements in the X and Ydirections, respectively.Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 16(a)Y \u00f4u\/\u00f4y=Ohomogeneous elastic soil(plane strain)H u=O u=O(b)y\u00f4uI\u00f4y=Ohomogenous elastic soil(plane strain)H u=O \u00f4uI\u00f4xOU=OLFigure 3.1: Definition of rigid-wall problem (a) original problem (b) equivalent problemby using antisymmetric conditionChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 17For the two-dimensional plane strain conditions, the stress components are related tothe displacements by2G \u00f4u Ov= 1\u2014 2l1:1\u2014 v)\u2014 +v] (3.3)2G 8v 8ua.y= (3.4)\u00f4u Ov(3.5)Although the problem involves two displacement components u and v, only the horizontaldisplacement u is taken into account in the analysis to simply the solution of the problem.The equation of dynamic force equilibrium in the horizontal direction Eq.3.1 is used inthe analysis. Considering various forms of approximation to the problem, the governingequation of the undamped free vibration of the backfill in the horizontal direction can bewritten as82u c92u+ G\u2014 = (3.6)and the normal stress o, is given by(3.7)where p is the mass density of the soil backfill, t is time, 6 and \/3 are functions of Poisson\u2019sratio v.Eq. 3.6 suggests that the dynamic response of the backfill is governed by two types ofwaves: the shear wave and the compression wave. The compression wave in the backfillproduces the dynamic earth pressure against the rigid wall given by Eq. 3.7.Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclthils 18The precise expressions for 6 and \/3 depend on the approximations used to model thewall-soil system. Three cases are examined and their corresponding expressions for 6 and\/3 are given below.(i). v=0 assumption. In this case, the displacements in the vertical direction Y areassumed to be zero.Using this assumption, o, and r, are obtained from Eq. 3.3 and Eq. 3.5 as2(1\u2014v) Elu\u2014 G\u2014 (3.8)1\u20142v ox9u(3.9)Substituting Eq. 3.8 and Eq. 3.9 into Eq. 3.1 and comparing with Eq.3.6, one finds6 \/3= 2(1(3.10)1 \u2014 2v(ii). o=O assumption. In this case, the normal stresses in the vertical direction Yare assumed to be zero.Applying the assumption to Eq. 3.4, one finds thatEly 1) Ott311Ely 1\u2014vOxTherefore, o and OT1,\/Oy are obtained from Eq. 3.3 and Eq. 3.5 as2 Elu= G\u2014 (3.12)1\u2014i\u2019 OxChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 191)313\u2014 1_v\u00f4x2 L )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 modelled using the shearbeam analogy(see Appendix 1 for detail).In this model, the shear stress r is given by\u2014 (3.16)The normal stress o is found by assuming o,=O in the backfill2 ott= (3.17)1\u2014LI OXSubstituting Eq. 3.16 and Eq. 3.17 into Eq. 3.1 and comparing with Eq.3.6, one finds21\u2014tiThree different models yield three different expressions for the coefficient 8 in Eq.3.6. However the dynamic response of the wall-soil system for these three cases can berepresented by the same equation, Eq. 3.6. Therefore the general derivation of dynamicChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclcfills 20solutions proceeds from Eq. 3.6.Assume the displacement solution has the formu(x,y,t) = (A. sin amx + B cosamx)(C sin by + D cosby) Ymn(t)Applying the boundary conditionsu\u20140 at..y=0u=0 at..x=0The constants B and D are determined to be zero.u(x, y,t) = sin amx sin by Ymn(t)= . COSam . sin by . Ymn(t)= E2C1b sin amx cosby Ymn(t)Applying the other two boundary conditionsOu\u2014=0 at..z=Lauat..y=Hone obtainsam COS amL = 0cos bH = 0Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 21thereforeb(2n\u20141)7r2H(2m \u2014 1)iram\u20142LThe mode shape functions are written as4\u2019mn(,y) =C1sinby . sin a,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\u2014G(6a + b) Ymn(t) pmn(t)\u2014a + b)=____= W?flflthe natural frequencies of the system are found to be= + 8a) (3.22)the frequency of the first mode isw1 = 4pH2(\u20198L2) (3.23)In the case of an undamped forced vibration subjected to a ground acceleration i0(t),the governing equation becomes\u2014 (8G\u2014 + G\u2014) = \u2014p\u00fco(t) (3.24)Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfihls 22Substituting Eq. (3.21) into Eq. (3.24), multiplying the equation by the mode shapefunctions, and integrating over the domain, one obtains thatJJ pij%j(t)4)mndXdy+JJ G(8a+bj)4ijYtj4\u2019mndxdy = _iio(t)JJ pmn(x, y).ckdyApplying the orthogonality conditions and recalling Eq. (3.22), one obtainsJJ pdxdy . mn(t) + J J pdxdy WYmn(t) = _\u00fc0(t)JJ pmn(X, y)Ymn(t) + Ymn(t) = _i10(t) mnwhere\u2014 ffpsin(a,mx) . sin(b1y)dx.dyf f p sin (1mm) sin (by)dx4y16amn = 2 (3.25)(2m \u2014 1)(2n \u2014 1)irLet Ymn(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) = \u2014i10(t) (3.27)For a given ground excitation iic(t) , a close-form solution of the displacement u isfound to beu(x, y, t) = EE sin amx Sjfl by mn frnn(t)Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclcfills 23where fmn(t) is the time history solution of Eq. (3.27) corresponding to a particularmodal frequency Wmn. It is noted that Eq. (3.27) is the standard damped vibrationequation of a single degree of freedom system.The dynamic earth pressure acting on the wall is determined to be the normal stresso at x=O. The dynamic pressure distribution along the wall iso,(x,y,t)o == 13Gamamn sin(by) . fmn(t)The total dynamic thrust acting on the wall isP(t) = j(xyt) .dyP(t) = 13G. am mnf(t)P(t) = 3G.2(2n1)2L\/H(3.28)The total dynamic moment acting at the base of the wall isM(t)= 1H,y,t)o dyM(t) = ._amn.sin(bnH)f(t) (3.29)For a harmonic input \u00fco(t) = . eict, the amplitude of the steady-state responsefmn(t) is found from Eq. 3.27 to bemax330Jmn \u2014 (2 \u2014 w2) + 2i WChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backullls 24For any excitation iio(t) the time history of the modal dynamic thrust associatedwith a particular mode is obtained using Eq. (3.28). The time-history of the dynamicthrust for the desired number of modes are then determined using the mode-superpositionmethod. Therefore the peak dynamic thrust acting on the wall can be determined exactlyfor any type of input motion.For earthquake motion the peak modal thrust acting on the wall associated with themodal frequency Wmn could be determined using the pseudo-spectral velocity Thepseudo-spectral velocity is derived from response spectral displacement by\u2014 omn-1vwhere is also the peak of fmn(t) corresponding to an excitation frequency Wmn.From Eq. 3.28 the peak modal thrust Fmn is determined as\u2014G16 5[4nmn\/3 2 2 (.7r (2n\u2014 1) L\/HWmnEstimation of the peak dynamic thrust is made by combining the individual peakmodal thrusts by some approximate method. Either the absolute summation or the rootsquare summation of the peak modal thrust is commonly used.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 coefficient 8 in Eq. 3.6. It isnecessary to examine the accuracy of solution provided by each model. Wood\u2019s rigoroussolution (Wood, 1973) provides a measure for evaluating the accuracy of the approximateChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclcfills 25solutions. Wood\u2019s solution for 1-g static horizontal loading is given in Eq. 2.5.To allow a measure of the accuracy of the approximate solutions, a 1-g static solutionis derived using the proposed approximate method. The 1-g static solution is the limitdynamic solution when the period of the dynamic motion becomes infinitely long. 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 \u2014 1) WmnL\/HThe 1-g static moment acting at the base of the wall is obtained from Eq. 3.29M3 = G\/3gE2ammn sin(bH)(3.35)bnWmnThe total thrust against the wall due to 1-g static horizontal force is determined bydoing a double summation for modes m and n from Eq. (3.34). A normalized thrustratio is introduced and defined asTHRUST RATIOTOTAL . THRUST(3.36)pwhere Amax is the peak ground acceleration in m\/sec2, ft\/sec2 or other consistentunit.Comparison of the accuracy of approximate solutions The approximate solutions are used to obtain the total 1-g static force for two wall-soil systems: one with aChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 26semi-infinite backfill and the other one with a finite backfill. The semi-infinite backfill isapproximated by using L\/H=5.O and the finite backfill is represented by using L\/H=l.5.Results from the different analyses are compared in Figure 3.2 (a) and 3.2 (b) for L\/H=5.Oand L\/H=1.5, respectively. The following observations may be made based on the results.The proposed model gives results that are in very good agreement with Wood\u2019sexact results for both L\/H=5.O and L\/Hz=l.5. The approximation of the proposed model works even better for walls retaining finite backfill (L\/H=1.5).Usually this model gives total force slightly less than the exact total force.The o,=O model yields results that are in very good agreement with the exactresults for L\/H=5.O. For wall-soil systems with L\/H=5.O, the accuracy of theo,=O model is comparable to that of the proposed model. The difference between the two models is that the o-!,=O model overestimates the response butthe proposed model underestimates the response slightly.However for L\/H=1.5, the o,=O model does not give as good results as theproposed model does. The solutions from the proposed model are much closerto the exact solutions than those from the o,=O model. The o,=O model mayoverestimate the total force by about 18%. The proposed model only underestimates the total force by 4%.When the v=O model is applied, the accuracy of the solution is very goodprovided v < 0.3. As v exceeds 0.3, the solutions start to deviate from the exact solutions. For L\/H=5.0, the accuracy of the solution from the v=0 modelbecomes unacceptable as v> 0.4.Recently Veletsos and Younan (1994) made studies on rigid walls with horizontallyChapter 3. Dynamic Thrusts on Thgid Walls with Uniform Elastic Backfill5 272.0(a) L\/H=5.O\u2014WOd exacto 1 .5 proposed model1.0f_Q50.1 0.2 0.3 0.4 0.5Poisson\u2019s ratio20___ ______r(b)o 1 .5 Woods exactproposed modeljl.0 FI I0.Q\u2014 0.1 0.2 0.3 0.4 0.5Poisson\u2019s ratioFigure 3.2: Comparis0 of the accuracy of approm solutjo for d-w1 systems(a) L\/1150 (b) L\/H15Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 28semi-infinite backfills. The accuracy of the o,=O model and v=0 model were examinedin that paper against Wood\u2019s rigorous solution. They reached similar conclusions to theabove regarding the accuracy of these two models. Their conclusions were made for thesemi-infinite backfills only, which is similar to the case with L\/H=5.0.The studies presented conclude that the proposed model gives the best approximationto solutions for the rigid-wall systems with infinite backfills and finite backfills. Therefore, the proposed model will be used for all further studies with 8 = \/3 = 2\/(1 \u2014 v).Static 1-g solution using 8 = 2\/(1 \u2014 v) Additional analyses were carried to studythe accuracy of the solution for the entire range of L\/H ratios. Figure 3.3(a) and Figure3.3(b) show the relationship between the normalized thrust ratios and the L\/H ratios forv = 0.3, 0.4 and 0.5. The results from Wood\u2019s 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 of L\/H ratio. In general, the thrust ratios determinedfrom this study are about 5 % less than that predicted by Wood. Although Wood\u2019s solution that was obtained from a 2-dimensional plane strain analysis is more accurate thanthat from this study, the small amount of error in the thrust ratio is compensated for bythe convenience of using the much simpler expression of the total thrust shown in Eq.(3.34). The simplicity in determination of the total thrust leads to ready application inengineering.The thrust ratios plotted in Figure 3.3 are found to be independent of the shear modulus G of the backfill. The total thrusts increase with the increase of 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 with Uniform Elastic Backfihls 291.2 -__________: (a) Woods1.00 -F\u2014- a \u2022 a. Poisson\u2019s ratio 0.30.6 oooo Poisson\u2019s ratio 0.4D - Poisson\u2019s ratio 0.5F-0.2 -1111111111111111111111 iii iiii liiIiIIIiiIliiiii1 2 3 4 5L\/H1.2 -: (b) author\u2019s1.0 :-0 -0.8.-...-.Pcsson\u2019sratio0.3- oaooo Poisson\u2019s ratio 0.40.6 &P\u00f6iss\u00f6\u00f1\u2019dtio U.5D -F-0.2 -.0.0 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 15L\/HFigure 3.3: Normalized thrust ratios for 1-g static solution(a) Wood\u2019s solution (b) author\u2019s solutionChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backilils 3010.0Poisson\u2019s ratio 0.3E nnnr Poisson\u2019s ratio 0.4o.u Poisson\u2019s ratio 0.56.0 \u2014____ci)-c(\u2022 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 I I I I I I I I1 2 3 4 5L\/HFigure 3.4: Heights of thrusts due to 1-g static horizontal force (wall height H=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 total thrust is calculated to be 1960 kN for per unitwide wall. For values of ii other than 0.3, 0.4 and 0.5, the 1-g static thrusts are easilyobtained using Eq. (3.34).The height of the resultant thrust above the base of the wall due to the 1-g statichorizontal force is plotted in Figure 3.4 against the L\/H ratio for v = 0.3, 0.4 and 0.5.The heights of resultant thrusts are identical for v = 0.3, 0.4 and 0.5. They remainconstant when Lull is greater than 1.0. The heights of the resultant thrusts are about0.611 above the base of the wall.Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backlills 313.4 Dynamic thrusts under sinusoidal motionsDynamic amplification of structural response depends on the ratio of the frequency of theinput excitation to the fundamental frequency of the structure. Resonant response occurswhen the excitation frequency matches the fundamental frequency. For wall-soil systems,a simple approach is to take the fundamental frequency of the backfill w8 to approximatethe frequency of the system. This representation is especially useful for wall-soil systemswith long backfills, where the fundamental frequency of the 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-soil system W is more critical.Therefore, two frequency ratios fRi and fR2 are used to investigate the dynamicamplification of the wall-soil systems. The frequency ratio fRi is defined by the ratiobetween the angular frequency w of the input motion and the natural angular frequencyw3 of the fundamental mode of the infinite horizontal backfill. The frequency ratio fRiis 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 ratio fR2 is defined by the ratio between the angular frequency of theinput motion (w) and the fundamental angular frequency (w11) of the wall-soil system.The frequency ratio fR2 is expressed asfR2 = (3.38)Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 321.2 \u2014 \u2014-- - \u2014\u2014-\u201cr01.0 -\u201c :-\u2014 --D05.0a,0.6 -L, 1.5a) .0.4 -0002 --\u2022\u2022___\u2014\u2014--.1 10 100 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 to fRi =1.0 and fRi = 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 with Uniform Elastic Backfills 33a much smaller number of modes could be used to compute the dynamic response adequately for engineering purpose, 600 modes were used here to get an \u2018exact\u2019 solution forassessing the accuracy of the results. From Eq. 3.28, it can be seen that the solutionconverges very fast as the number of mode n increases. The number of mode n requiredis usually much less than the number of mode m required in the dynamic solution. Studies also showed that a larger number of mode n is necessary for wall-soil systems withsmaller L\/H ratios. Therefore a combination of n=2 and m=300 was used for L\/H=5.0,and another combination of n=6 and m=100 was used for L\/H=1.5.The amplitudes of steady-state dynamic thrusts were determined using Eq. 3.28 andEq. 3.30 for harmonic excitations at different frequencies. The amplitudes of dynamicthrusts are normalized according to Eq. (3.36). The normalized thrust ratios are plottedagainst the frequency ratio fRi in 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 as the excitation frequency approachesthe fundamental frequency of the wall-soil system. At resonant conditions, the peak dynamic thrusts are 2.4pHAmaz for L\/H=5.0 and 3.OpH2Amaa, for L\/H=1.5. Because thestatic thrusts are 1.OpH2Amax for L\/H=5.0 and 0.86pH2Ama,,, for L\/H=1.5, their corresponding dynamic amplification factors are 2.4 for L\/H=5.0 and 3.5 for L\/H=1.5. Theresults suggest that the dynamic amplification for wall-soil systems with finite backfillsis larger than that for wall-soil systems with semi-infinite backfills.The results also show that resonance occurs at a frequency ratio fRi = 1.05 forL\/H=5.O and at fRi = 1.55 for L\/H=1.5. Hence for two identical walls with identicalsoil properties, the fundamental frequency of the wall-soil system with L\/H=1.5 is about1.45 times that with L\/H=5.0.Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 340(I,DZI-4U)DF\u2014i(a) L\/H = 5.0 STEADY STATE RESPONSE3-21Poisson\u2019s ratio = 0.4damping ratio 10%modes: N=2, M=30004111111111 I I I 11111 I I I 111111111111111111111111111110 1 2 3 4FREQUENCY RATIO, fRiE (b) L\/H = 1.5 STEADY STATE RESPONSE550 1(1111 (IjI III 1111111 ( 111111(1111111! juhhhhhi II0 1 2 3 4FREQUENCY RATIO, fRiFigure 3.6: Normalized thrust ratios for sinusoidal motions (a) L\/H=5.O (b) L\/H 1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 353.5 Dynamic thrusts under earthquake motionsDynamic thrusts are computed for wall-soil systems with L\/11=5.0 and L\/H=1.5. Thewalls have a fixed height 11=10 m. The soil backfill has Poisson\u2019s ratio ii = 0.4, unitweight 7 = 19.6 kN\/m3 and a constant damping ratio A=10%.Two earthquake acceleration records were used in the analysis, the SOOE accelerationcomponent of the 1940 El Centro earthquake, and the S9OE 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 scaled down to 0.07g to simulate the linearelastic response of the wall-soil system under a small input motion. The peak acceleration of the Loma Prieta input is 0.067g.The time histories of dynamic thrusts were computed using Eq. (3.28) by mode superposition method. As examples, time histories of dynamic thrusts against rigid wallsare shown in Figure 3.7 for the El Centro input and in Figure 3.8 for the Loma Prietainput. From Figure 3.7 (a) it is interesting to see that the high frequency content of theinput motion has been filtered out when the input motion passes through a relatively softbackfill with G=9810 kPa. For this case, the computed peak dynamic thrust is about182 kN\/m.A typical time history of the height of dynamic thrust is illustrated in Figure 3.9.The most frequently occurring height of dynamic thrust is 0.6211 above the base of thewall. In the region of peak thrust a height of 0.611 would be seem a good value for design.The impact of frequency ratios fRi and fR2 on the dynamic thrusts against rigid wallsI I Jill 1111111111111111111111111111111111111112 4 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 Backfills 36200100-0-\u2014100.\u2014200(a) L\/H= 5.0-H=10 m, Poissons ratio 0.4shear modulus 9810 kPa, damping 10%0zF(1)FC-)z>-z-F(I)zIC-)z>-0102001000\u20141\u2014200- (b) L\/H= 1.5-:00-- Lvj. vvvvv -vv - vvrH=10 m, Poisons ratio 0.4shear modulus 39240 kPa, damping 10%El Centro input (O.07g)1111111111111 I liii I liii 111111111111111111111111110 2 4 6 8 10Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic BackililszF\u2014C\u2019)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\u2014100--\u201420O(a) L\/H=5.0H\u2014lOm, Posson\u2019s ratio 0.4,shear modulus 9810 kPa; damping 10%Loma Prieta input011111111111111111 II I 111111 III5 10 15TIME ( SEC )200100\u2018\u2018III\u201920Ez-Iv)DFC-)z>-0(b) L\/H=1.5I______0.\u2014100-\u2014200V\u2019\\JVV\\J1) RJVVH=lOm; Poissons ratio 0.4;shear modulus 5805 kPa; damping 10%Loma Prieta input111111! 111111 I1IIII1II1IIIIII I 111111110 5 10 15 20Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 38Table 3.1: Peak dynamic thrusts for walls with L\/H=5.0 and 11=10 m,G w8 Dynamic thrust (kN\/m) Dynamic thrust (kN\/m)(kPa) rad\/sec (El Centro input) ( Loma Prieta input)613.0 2.750 48.4 77.51360.0 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 the backfill. The shear moduli of thebackfills vary from 613 kPa to 61313 kPa, which gives a distribution of the fundamentalfrequency of the backfill w8 from 2.75 to 27.503 rad\/sec. The computed peak dynamicthrusts are listed in Table 3.1 for wall-soil systems with L\/H=5.0 and in Table 3.2 forL\/H=1.5. The variation of shear moduli and the angular frequencies w8 are also listedin these Tables. For the El Centro input, the maximum values among the 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\/m for L\/H=5.0and 170.0 kN\/m for L\/H=1.5. These results suggest that the peak dynamic thrusts arelittle dependent on the L\/H ratio under earthquake motions.The frequency ratios fRi and fR2 are determined using Eq. 3.37 and Eq. 3.38, respectively. In these equations the excitation frequency w of the input motion is required.For earthquake motions, the excitation frequency w is taken to be the predominant frequency of the earthquake. The predominant frequency of an earthquake motion is thefrequency at which the response spectral acceleration has the maximum value in theChapter 3. Dynamic Thrusts on Rigid Wails with Uniform Elastic Backfihls 39Table 3.2: Peak dynamic thrusts for walls with L\/H=1.5 and H=10 m,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 3.667 75.0 86.81905.0 4.848 116.0 -2453.0 5.501 133.7 126.44360.0 7.334 194.6 170.05805.0 8.463 179.1 145.49810.0 11.001 130.9 151.039240.0 22.002 134.5 143.912-___- L\/H=5.0; H=10 m; Poissons rato 0.4;- shear modu!us 9810 kPa; damping 10 z;E 10\u2014 fttQJOpt4tyJt+11rr1t1HEIGHT OF THRUST : 6.20 meters0\u2014 1111111 I I 11111 III IllillIjIl III 111111111 liiio 2 4 6 8 10TIME ( SEC )Figure 3.9: A time history of the height of dynamic thrust, L\/H=5.0Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 40acceleration spectrum. Therefore the excitation frequencies of the El Centro input andthe Loma Prieta input are determined to be 11.64 rad\/sec and 10.13 rad\/sec, respectively.The peak dynamic thrusts are normalized using Eq. 3.36. The normalized thrustratios are plotted against the frequency ratio fRi in Figure 3.10 and against the frequency ratio fR2 in Figure 3.11. Based on results from the limited number of analysesfor earthquake motions, it is suggested that the peak dynamic thrusts are 1.30pH2Amafor L\/H=5.0 and 1.38pH2Ama for L\/H=1.5. Because the static thrusts are 1.0pH2Amafor L\/H=5.0 and 0.86pH2Am for L\/H=1.5, their corresponding the dynamic amplification factors are about 1.3 for L\/H=5.0 and 1.6 for L\/H=1.5.A significant observation is made that the dynamic amplification under earthquakemotions due to resonance is much less than that under sinusoidal motions. Under sinusoidal motions, the dynamic amplification factors at resonance are 2.4 for L\/H=5.0 and3.5 for L\/H=1.5. The dynamic amplification factors under earthquake motions are about50% of that under sinusoidal motions.3.6 Accuracy of the response spectrum methodThe response spectrum method is commonly used to determine the responses of structuressubjected to earthquake motions. The response spectrum method which adds modal values without taking the phases of the modes into account is much simpler than the directmode superposition method. Therefore it is of interest to check on the accuracy withwhich the peak dynamic thrust can be determined using the response spectrum approach.Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclcfills 414E (a) L\/H=5.0 Poisson\u2019s ratio = 0.4damping ratio 10%0 3 modes: N=2, M=300sinusoidal motions00000I\u2014Cr)DI0\u2014 1111111111111111 lijI 111111111 11111111 I 1111111O 1 2 3 4 5FREQUENCY RATIO, fRi4Poisson\u2019s ratio = 0.4(b) L\/H1.5 damping ratio 10%\u2022 modes: N=6, M=1000sinusoidal motionsa earthquake motio(I)DFo 1 2 3 4 5FREQUENCY RATIO, fRiFigure 3.10: Normalized thrust ratios versus fRi for earthquake motions (a) L\/H=5.0(b) L\/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills03(1\u2019D\u20181-03I\u2018-2C,,D\u2018-1FREQUENCY RATIO, fR2Figure 3.11: Normalized thrust ratios versus fR2 for earthquake motions (a) L\/H=5.O(b) L\/H1.5424(a) L\/H=5.0 Poisson\u2019s ratio = 0.4damping ratio 10%modes: N=2, M=300sinusoidal motions04\u2014 III(IIIIIlIIIIlIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0 1 2 3 4FREQUENCY RATiO, fR2: Poisson\u2019s ratio = 0.4: (b) L\/H1.5 damping ratio 10%: modes: N=6, M=100E sinusoidal motions-i3ehquoke motions550 1111111111 I III I liii I I III I 1111111 I III I I lill0 1 2 3 4Chapter 3. Dynamic Thrusts on Rigid Wails with Uniform Elastic Baclcfills 43The accuracy of the response spectrum method is measured by a thrust factor Cpdefined asc=(3.39)where the total spectral thrust P31. is evaluated using the response spectrum methodby summation of peak modal thrusts Fmn. Pma is the exact solution which is evaluatedusing the mode superposition method. The values of the thrust factor Cp gives a measureof the accuracy of the response spectrum method.The peak modal thrust Fmn is determined from Eq. 3.32 using the pseudo-spectralvelocity S of the input motion corresponding the modal frequency Wmn. The pseudo-spectral velocities for the two selected acceleration records, the El Centro input and theLoma Prieta input, are shown in Figure 3.12(a) and 3.12(b), respectively.Since the peak modal thrust only represents the peak 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 modal values (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 response spectrum method, thetotal spectral thrust F8,, is determined byP51. = EY2Fmn (3.40)When the root square summation (RSS) is used in the response spectrum method,the total spectral thrust P8, is determined byChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 442.0-_(a)Ua)U)E 1 .5 DAMPING RATIO 10%\u2022 - - - - DAMPING RATIO 5 %15>o10IUa, \u20140. \/ S SU,,05___ ____-o__ _ ___a,U,O.o \u2014 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 I0.0 0.5 1.0 1.5 2.0PERIOD ( SEC )0.5-(b)Ua,0.4E DAMPING RATIO 10%DAMPING RATIO 5 %0.30-I-,U -.O2________I \u2014\u2014 \u2014\u2014V.,0.1 Ia,U,0.0 \u2014 I I I I I I j I I I I I I I I I I0.0 0.5 1.0 1.5 2.0PERIOD ( SEC )Figure 3.12: Pseudo-spectral velocities of (a) the El Centro input and (b) the LomaPrieta inputChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 45= (3A1)Values of the thrust factor C1\u2019 were determined for both sinusoidal motions arid earthquake motions. The earthquake motions are represented by the El Centro input and theLoma Prieta input. The studies were made for L\/H=5.0 and L\/H=1.5. 600 modes wereused for obtaining both P, and FmDiscussion of results When the RSS method was used, the relationship between thethrust factor Cp and the frequency ratio fRi was obtained and is shown in Figure 3.13.For harmonic loading the thrust factor Cp changes very much with frequency. The response spectrum method could overestimate or underestimate the peak dynamic thrustby 80% to 100%. For earthquake loading the thrust factor CF is usually greater thanone, mostly around 1.5. The response spectrum method usually underestimates the peakdynamic thrust by as much as 50%. However the thrust factor changes for a different soilprofile and a different frequency ratio. The uncertain variation of Cp makes it difficultto apply the response spectrum method for determining the actual peak dynamic thrust-ma2, in practice.When the absolute summation (ABS method) was used, the relationship between thethrust factor Cp and the frequency ratio fRi was determined and is shown in Figure 3.14.For low frequency ratios, such as fRi < 0.8, the total spectral thrust P, obtaining fromthe ABS method agrees very well with the exact solution Fma, under sinusoidal motions.However under earthquake motions, about 20% overestimate of peak dynamic thrust isexpected for the same frequency ratio. For high frequency ratios, such as fRi > 1.5,the peak dynamic thrust may be overestimated by as much as 50% when the responseChapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Baclcfihls 46o \/ ,, \u2014 Poissons ratio = 0.4C1 L\/fT \u2014 damping ratio 10%- : modes: N=2, M=300 (RSS)o : \u00b0 \u00b0 \u00b0 \u00b0 data for earthquake motionsdata for sinusoidal motions.:::zo 1 2 3 4 5FREQUENCY RATIO, fRi0 \/ . \/ Poisson\u2019s ratio 0.4Sjb) L1H 1.5 damping ratio 10%- modes: N=6, M=100 (RSS)o3 2 = \u00b0 \u00b0 a a data for earthquake motionsdata for sinusoidal motions;D1raOOoO,ac\u2014 I I I liii 111111111 Ij ii I Ililtil Ill liii I liii tillo 1 2 3 4 5FREQUENCY RATIO, fRiFigure 3.13: Variations of thrust factor Cp versus frequency ratio fRi (RSS method) (A)L\/H=5.O, (B) L\/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 474Poisson\u2019s ratio = 0.4: (a) L\/H = 5.0 damping ratio 10%modes: N=2, M=300 (ABS)0o data for earthquake motionsdata for sinusoidal motions(I)D1-aO 1 2 3 4 5FREQUENCY RATIO, fRicLo - Poisson\u2019s ratio 0.4(b) L\/H = 1.5 damping ratio 10%modes: N=6, M=100 (ABS)o3 2: a o o data for earthquake motionsdata for sinusoidal motionsC\/)111111111111111111111111 I IIIIIIIIIIIIIiIIIIIt liii0 1 2 3 4 5FREQUENCY RATIO, fRiFigure 3.14: Variations of thrust factor Cp versus frequency ratio fRi (ABS method) (A)L\/H=5.O, (B) L\/H=1.5Chapter 3. Dynamic Thrusts on Rigid Walls with Uniform Elastic Backfills 48spectrum method is used.Therefore it is recommended that the mode superposition method be used in orderto accurately determine the peak dynamic thrusts against rigid walls. The responsespectrum method may be used for approximately estimating the peak dynamic thrustsagainst rigid walls. The use of ABS method is suggested when the response spectrummethod is selected, especially at low frequency ratios.Chapter 4Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles4.1 IntroductionThe first objective of this chapter is to explore the effects of typical backfill non-homogeneityon the magnitude of dynamic thrusts for elastic response. Two types of soil profiles havebeen analyzed. They are profiles with linear and parabolic variations of shear moduluswith depth.The second objective of this chapter is to evaluate the influence of soil non-linearityon the magnitude and point of application of dynamic thrusts. The moduli and dampingof soils are known to be strain dependent (Seed and Idriss, 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 of dynamic thrust, fundamental frequency of the system, and amplification factors of ground acceleration aredetermined for different intensities of acceleration input.Elastic analysis. The undamped forced vibration equation of motion of the backfill iswritten as82u 82u 82u\u2014 (8G\u2014 + G-) = \u2014p\u2019\u00e4o(t) (4.1)49Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 50ly\u00f4u\/\u00f4y=OH u = 0 (plane strain) G, H 8u13x = 0u0____XLFigure 4.1: A composition of non-homogenous soil profilewhere 6 is equal to 2\/(1-v), ii is Poisson\u2019s ratio of soil, and \u00fco(t) is the base acceleration caused by the earthquake.Figure 4.1 shows the type of non-homogeneous soil profiles that will be analyzed inthis chapter. The backfill behind the wall is consisted of layered soils with different properties in each layer. The wall is considered to be rigid, and it does not move relative tothe base. The boundary conditions for this system are also shown in Figure 4.1.Analytical solutions are in general not possible for nonhomogeneous backfills. Therefore, the finite element method is employed to analyze dynamic response of the wall-soilsystem.4.2 Finite element formulation and its validationA finite element developed especially for this study is shown in Figure 4.2. The elementconsists of 6 nodes with 6 horizontal displacement variables. The displacement field has aChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 51yIIU6 U5 U4Ui U2 xii I____________aFigure 4.2: A composition of the finite element used in SPAWlinear variation along the vertical direction and a quadratic variation along the horizontaldirection.Let the displacement u be represented byu=EN2.u i=1,6The shape functions N are given byN1 =N2 x(b\u2014y)N3 = xyN4 =N5 =N6 = xy(a \u2014 x)Galerkin\u2019s general procedure of weighted residuals is used to develop the stiffness andmass matrices of the finite element shown in Figure 4.2. The stiffness matrix [K] isgiven byChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 52[Kj = + (4.2)The diagonal mass matrix of the element is 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. The global stiffness matrix [K] and the mass matrix [M]are assembled accordingly. The equations of motion in matrix form are written[M]{\u2019ii} + [C]{ri} + [K]{u} = \u2014[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 system are determined by analyzing the eigen values of the system. The damping matrix of each finite element is obtained according tothe desired degree of damping of the element. In this manner the damping matrix [C] isevaluated. A finite element program SPAW was developed based on these considerations.Validation of F.E. method To validate the reliability of the finite element analysis, itwas applied first to two uniform soil profiles for which close-form solutions were obtainedin the previous chapter.The geometric mesh used in the finite element analyses is shown in Figure 4.3. Thismesh consists of 3 elements in each layer with 20 layers. The horizontal dimensions ofthese elements are subjected to change proportionally to the L\/H ratio. This mesh hasChapter 4. Dynamic Thrusts on Rigid Wails with Non-homogeneous Soil Profiles 53.\u2018I__ -.LFigure 4.3: A finite element mesh used for dynamic analysesbeen used for all finite element analyses presented in this chapter.The finite element method is applied first to a uniform 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 component of the 1940 ElCentro earthquake scaled to 0.07g. The time history of dynamic thrust against the wallcomputed using the finite element method is shown by the solid line in Figure 4.4(a). Thedashed line represents the closed form solution in Figure 3.7. The agreement betweenthe two solutions is excellent over the entire time histories of dynamic thrusts.The finite element method is next applied to a second uniform soil profile (L\/H=1.5,H=10.Om) with a higher shear modulus of 0=39240 kPa and .\\ = 10%. This wail-soilsystem is stiffer than the previous one. The fundamental frequency of this system is 2.54Hz. The time histories of dynamic thrusts computed using the finite element methodand the close-form solution are shown in Figure 4.4(b). Here again excellent agreementis observed between the dynamic thrusts computed by the two methods of analysis.HChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 542001000\u2014100\u20142002001000zF\u2014C\u2019)DF\u2014C-)z>-0Iz-I\u2014U)DI\u2014C)z>-0: (a) L\/H= 5.0 from F.E. analysis\u2022- - -- from close\u2014form solution- H=lOm, Poissons ratio 0.4;:Shear Modus 9810 kPo; damping 10%.\u2014 11111111111111111111111111111111(11 (liii j(j I I I I0 2 4 6 8 10TIME ( SEC ): (b) L\/H= 1 .5 from F.E. analysis. - - - - from close\u2014form solutionui\u2019: vvvvv\u2018ir\u2019j VH=lOm, Poisson\u2019s ratio 0.4;- Shear Modulus 39240 kPo; damping 10%- El Centro input (0.07g)\u2014200\u20140 2 4 6 8 10TIME ( SEC )Figure 4.4: Comparisons of dynamic thrusts between the F.E. method and the close-formsolution for uniform soils(a) L\/H=5 (b) L\/H=1.5Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 55These comparative studies verify that the finite element formulation constructed forthis problem is a reliable technique for evaluating the dynamic response of the wall-soilsystem and that the mesh employed is an appropriate one.The finite element mesh used for the rigid wall analysis has been embedded in thecomputer program SPAW. There is no need for other users to construct the mesh. Thedynamic analyses can be performed by just inputting the basic soil properties of eachlayer. The time history analyses only take few minutes in a PC486 33MHz computer.4.3 Linear elastic analyses with non-homogeneous soil profilesIn this analysis the shear modulus G and damping ratio ) for a given soil profile retainconstant values throughout the analysis. They are not considered to be functions ofstrains. ? =10% and v =0.4 are used. The horizontal length of the soil layer is constrained at L\/H=5.0. Two types of soil profiles are examined, a profile with a linearvariation of shear moduli with depth and a profile with a parabolic variation of shearmoduli with depth.Two types of motions, the sinusoidal motion and the earthquake motion, are appliedto the two types of soil profiles. For the sinusoidal motion, the amplitudes of the steadystate dynamic thrusts are determined. For earthquake motion, the peak dynamic thrustsdeveloped during shaking are determined. Again the El Centro input is used as the inputof earthquake motion in this study.Analyses were performed to examine the relationship between the thrust ratio andChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 56the frequency ratio. The frequency ratio fR2 defined 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\/sec2 or other consistent unit.Figure 4.5 shows the relationship between the thrust ratio and the frequency ratiofR2 for linear soil profiles. At resonance, the peak dynamic thrusts are 1.56pH2Ama under sinusoidal motions and 1.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 earthquake motions.Figure 4.6 shows the relationship between the thrust ratio and the frequency ratio forparabolic soil profiles. In general, the thrust ratios are greater than those for linear soilprofiles by 20%. At resonance, the peak dynamic thrusts are 1.87pH2Amax under sinusoidal motions and 1.18pH2JLunder earthquake motions. Because the static thrust isabout 0.82pH2Amax, their corresponding dynamic amplification factors are determinedto be 2.3 for sinusoidal motions and 1.4 for earthquake motions.The dynamic response of parabolic soil profiles is compared with dynamic response ofuniform soil profile in Figure 4.7. For sinusoidal motions, the dynamic thrust ratios forparabolic soil profiles are less than those for uniform soil profiles in the frequency rangeof fR2 < 2.0.The limited analyses conducted suggest that the dynamic thrust at resonance forsinusoidal motions are about 60% greater than that for earthquake motions with samepeak acceleration. It should be noted that the earthquake motions are represented bythe El Centro input only.Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil ProfilesLinear soil profile(a) Sinusoidal STEADY STATE RESPONSEmotionsPoisson\u2019s ratio = 0.4L\/H=5.0, damping 10%57Figure 4.5: Relationships between thrust ratio and frequency ratio fR2 for linear soilproffles (a) sinusoidal motions (b) the El Centro input43-2-1\u20140U)DI\u20140IU)DI\u201404\u2014 111111111111111111111111111111111111111 1111110 1 2 3 4FREQUENCY RATIO, fR23210(b) El Centro input PEAK DYNAMIC RESPONSEPoisson\u2019s ratio = 0.4L\/H = 5.0; damping 10%550IIIIIIIIIIIIIItIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1 2 3 4FREQUENCY RATIO, fR2Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 58Parobolc soil profile4(a) Sinusoidal STEADY STATE RESPONSEmotions2 3 Poisson\u2019s ratio = 0.4L\/H = 5.0; damping 10%o 1 2 3 4 5FREQUENCY RATIO, fR24(b) El Centro input PEAK DYNAMIC RESPONSE2 \u2018 Poisson\u2019s ratio = 0.4L\/H = 5.0; damping 10%(I)DO 1 2 3 4 5FREQUENCY RATIO, fR2Figure 4.6: Relationships between thrust ratio and frequency ratio fR2 for parabolic soilproffles (a) sinusoidal motions (b) the El Centro inputChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 594-Poissons ratio = 0.4o 3- L\/H = 5.0; damping 10%Sinusoidal motions\u00b0\u00b0. Parabolic Soil Profilec::_: 111111111111 IIIl,IIIIIIIIIIIIIItIIIII1I1II1IIII0 1 2 3 4 5FREQUENCY RATIO, fR2Figure 4.7: Comparison of dynamic thrust ratios for parabolic soil profiles and uniformsoil profiles under sinusoidal motions (L\/H=5)The other important aspect of dynamic thrust is the location of the resultant thruston the wall. Typical time histories of heights of dynamic thrusts for the three types ofsoil proffles are illustrated in Figure 4.8. The results are obtained using the El Centroinput as input motion. For linear soil profiles dominant height of dynamic thrust is atO.48H above the wall base. For parabolic soil profiles this height becomes 0.5111 above thewall base. The uniform soil profile gives an average height of 0.62H above the wall base.Generally the height of dynamic thrust increases as the soil profile becomes more uniform.Modal frequencies of wall-soil systems with different soil profiles are shown in Table 4.3. Essentially the natural frequencies of the wall-soil systems become more widelyspaced as the soil becomes more uniform.Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles8\u20146\u20144-2-0Figure 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 I I i I i I i\u2018VS\u2019i \u2018nA( r II [J1Tt2 4Parabolic soil profiles6 8 1 0EU)04,EU)z00\u2019VIEU,00\u2019VIITT1 I!LAIA L dLALLL\u2018r\u2019iiy\u2019I I I2 4Uniform soil profilesivv Vf1l yv F6 81086104\u20142\u201401\u2019\u2019\u2019 I\u2018i\u2019 \u2018I\u2019\u2019\u2019\u2018.JIEF0 2 4 6 8 10Time ( sec )Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 61Table 4.3: Patterns of first 10th natural frequencies for three types of soil profiles (wa,rad\/sec)frequencies frequencies frequenciesflth freq. (linear profile) (parabolic profile) (constant profile)1 11.3 11.35 11.702 13.1 14.20 16.323 15.2 18.80 22.904 16.4 20.60 30.195 19.9 26.00 33.246 25.6 29.10 35.137 26.4 29.70 37.798 27.2 30.80 45.539 29.4 33.40 53.3610 33.2 38.50 55.154.4 Equivalent linear simulation of non-linear response under earthquakeloadsA method of analysis for modelling the non-linear response of soil was proposed by Seedand Idriss (Seed & Idriss, 1967), which is designated the equivalent linear method ofanalysis. In this method a set of elastic shear moduli and viscous damping ratios whichare compatible with a measure of the effective shear strains induced by an earthquakeare used to approximate the hysteretic behaviours of soils. The equivalent linear elasticmethod is widely used for dynamic analyses in practice.The equivalent linear method for modelling nonlinear behaviour is used in this study.Since the level of shear strain in each element is not known at the beginning of the analysis, strain compatible moduli and damping ratios are achieved by an iterative process.The effective strain used to determine moduli and damping is chosen to be 0.65 of theChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 62peak dynamic strain for the earthquake type of motions. The damping ratio of the wall-soil system in each iteration is estimated by using an average value of damping ratios forall soil elements in the system.Two wall-soil systems with parabolic variation of shear modulus are analyzed usingthe equivalent linear technique. One wall has a stiff backfill with a shear modulus of132,000 kPa at the base of the backfill. The other wall has a backfill with a shear modulus of 66,000 kPa at the base. The wall height in each case is 11=10 m with L\/H=5.0. Thewall-soil systems are shaken using the El Centro acceleration record as the base motion.The effects of non-linearity on the dynamic responses of the systems are explored usingincreasing levels of input acceleration. The peak accelerations of input motions vary from0.05g to 0.35g in increments of 0.05g.The data on shear strain dependent moduli and damping presented by Seed and Idriss(Seed & Idriss, 1970) were employed. At shear strain levels of 0.0001, 0.0005, 0.001, 0.005,0.01, 0.05, 0.1, 0.5, and 1.0 percent, the values of G\/GmO.a, were selected as 100, 98.3,95.8, 84.3, 74.3, 43.0, 29.6, 10.9, 6.1 percent, respectively. The values of D\/Dmaa, corresponding to the above strain levels are the follows: 0.018, 0.055, 0.073, 0.158, 0.275,0.457, 0.579, 0.891, 1.0. The maximum damping ratio Dmax is chosen to be 30%.Figure 4.9 shows the dynamic response of the stiff site. The intensity of shaking ismeasured by the peak acceleration of the input motion. The amplification of groundacceleration shows its peak at an input acceleration of 0.15g. The dynamic thrust ratio,the frequency ratio and the damping ratio increase steadily with the increase of the levelof shaking. The increasing level of shaking results in reduction of shear moduli and consequently reduction in the fundamental frequency of the wall-soil system. This resultsChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 63Parabolic soil profile G0 = 132,000 kPo= \u2022e.e. Damping ratios\u2022 00000 Frequency ratios- O0DO Amplifaction of ground acceleration\u2022 AAA Thrust ratiosn4zcocoooo.\u2014--\u2014-- \u2018t .0.0 0.1 0.2 0.3 0.4 0.5Peak Bose Acceleration C g )Figure 4.9: Dynamic responses of a stiff, site due to non-linear effect,G0=132,000 kPain an increase in the frequency ratio fR2. 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 of 0.35g 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 of ground accelerations decreases from 3.0 to 2.2 with the increasinglevel of input acceleration.Chapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 64Parabolic soil profile G0 132,000 kPo1.5 -00000 NON\u2014LINEAR ANALYSES- -- LINEAR ELASTIC ANALYSESL\/H = 5.0, H=lOm(I) Poisson\u2019s ratio 0.40 El Centro inputI0.5\u20140.0 0.1 0.2 0.3 0.4 0.5Peak Base Acceleration ( 9 )Figure 4.10: Effect of level of shaking on the dynamic thrust, G0=132,000 kPaParabolic soil profile G0 = 66,000 kPo\u2014 *-*-*-.-* Damping ratios00000 Frequency ratiosD0DC Amplifoction of ground accelerationAA.& Thrust ratios: L\/H=5.0, H=lOmZ : Poisson\u2019s ratio 0.4El Centro input0 0. 0.5Peak Bose Acceleration C g )Figure 4.11: Dynamic responses of a soft site due to non-linear effects,G0=66,000 kPaChapter 4. Dynamic Thrusts on Rigid Walls with Non-homogeneous Soil Profiles 65The studies from equivalent linear analyses reveal that the dynamic thrust ratiosusually increase with the levels of input accelerations. For the cases investigated, theincrease of dynamic thrust due to non-linear effect is about 25% of the dynamic thrustobtained from a linear elastic analysis.Part IIDynamic Analyses of Pile Foundations66Chapter 5Dynamic analyses of pile foundationsDynamic soil-pile-structure interaction is a challenging area to geotechnical researchersand engineers. A very common example is the 3-D dynamic analysis of a pile foundationfor a bridge abutment. The analysis involves modelling of soil-pile-soil interaction, theeffects of the pile cap, non-linear soil response, and in many cases incorporates seismically induced pore water pressures. There are many approaches to solving the dynamicresponse of pile foundation.Novak (1991) gave an extensive review of the more widely accepted methods of analysis for piles under dynamic loads. His study showed that pile group response can not bededuced from single pile response without taking pile-soil-pile interaction into accountand that the dynamic characteristics of pile groups are strongly frequency 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 divided into two major categories,elastic continuum models coupling the soil and pile in a unified system and the lumpedmass-spring-dashpot models. The elastic continuum models are mostly used for theanalysis of pile foundation subjected to low level excitation such as problems related67Chapter 5. Dynamic analyses of pile foundations 68to machine foundations. The lumped mass models are formulated by separating the response of piles from the soil medium. The contribution of the 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 of pile foundations 69for these types of solutions.In using Novak\u2019s formulation much of attention is focused on the pile head impedancefunctions. The impedances have a great influence on the response of pile supported buildings and structures. The pile head impedances can be defined as the transfer functionsdescribing the ratios between the complex valued displacement response at the pile headand the harmonic forces (or moments) applied at the pile head.Pile head impedances derived using elastic theory are most appropriate for low levelshaking where the dynamic pile head forces induce essentially elastic strains in the soilaround the pile. The pile head impedance is usually expressed in terms of complex shearmoduli, the real part of which represents the secant elastic 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 of soil while the imaginary component indicates energylosses due to wave propagation away from the pile (radiation damping) and hystereticdamping in the soil. For low level of excitation hysteretic damping is small and systemdamping is mostly dominated by radiation energy losses.The real component of pile head impedance derived using Novak\u2019s plane strain approach has been found to diminish as the frequency approaches zero (Novak and AboulElla, 1978a). This result is not realistic. Novak and Nogami have suggested that planestrain soil reactions can be used provided low frequency corrections are applied. Inthe computer program PILAY, a frequency cut-off is applied for determining pile headimpedances of single piles. Nogami and Novak use static stiffness when the dimensionless frequency a = wro\/V5 is less than about 0.3, where w is the excitation frequency, r0Chapter 5. Dynamic analyses of pile foundations 70is the pile radius and V8 is the effective shear wave velocity of soil in the depth rangewhere maximum pile bending occurs. Alternatively at lower frequencies, the pile headimpedance can be taken as constant and equal to that calculated at a suitable dimensionless frequency, such as a 0.3.In PILAY analysis, constant damping coefficients are assumed when the dimensionless frequency a is less than 0.3. That is, the damping coefficients are assumed to beindependent of frequency when a < 0.3. This assumption does not necessary hold especially when the frequency independent hysteretic damping is significant. Other methodsfor determining the damping of pile foundations are given below. It should be noted,however, the Novak\u2019s solutions were primarily intended for for machine foundations forwhich a is usually greater than 0.3.The imaginary component (damping) of pile head impedance, represents theenergy losses along the pile. For an equivalent viscously damped system, the viscousdashpot coefficient c (damping coefficient) is defined as the ratio of the damping andthe frequency cjj = Cj\/w. The equivalent viscous damping coefficients vary withdepth z along the pile because the hysteretic damping varies with pile defiections andstrain in the near field. Gazetas and Dobry (1984) proposed a simple formulation forcomputing the damping coefficients. According to them, the damping coefficients consistof the radiation and hysteretic damping components, or ci., = C,. + The radiationdamping coefficients c,. given by Gazetas and Dobry have been found in good agreementwith those derived by Novak et al. (1978) and Roesset and Angelides (1980). The expressions proposed by Gazetas and Dobry are frequency and depth dependent. For depthsgreater than 2.5 times the pile diameter the radiation dashpot coefficient is given asChapter 5. Dynamic analyses of pile foundations 71cr = 4rop8V{1+[(hl]125}()\u00b075a\u00b02 (5.1)in which a is the dimensionless frequency a = wro\/ 17, w is the frequency of excitation,r0 is the pile radius and V8 is the free field shear wave velocity. For shallower depths, theradiation dashpot coefficient is given asc,. = 8ropsV()075a_025 (5.2)Recently Gazetas et al. (1993) proposed a simpler expression for the viscous dampingcoefficient along the pile. The frequency dependent radiation damping coefficient forvertical motions is expressed asTI .., \u20140.25Cz\u2014PsVs(Laoand for horizontal motionsc =6p8Vda\u00b025 (5.4)in which d is the pile diameter and a0 = wd\/V3. These formulations were used in aBeam-on-Dynamic-Winkler foundation simplified model by Gazetas et al. (1993).The pile head impedances are often used as foundation spring and dashpot parameters in the analysis of superstructures subjected to earthquake loading. This type ofanalysis in which the pile foundations are replaced by springs and dashpots is usuallycalled uncoupled analysis.The assumption generally made in an uncoupled analysis is that one may use the freefield surface accelerations as input into the base of the superstructure. The assumptionOhapter 5. Dynamic analyses of pile foundations 72actually neglects the influence of foundation-ground may have on the motions of the pilecap. The motions of the pile cap may differ significantly from the motions of the freefield surface due to kinematic interaction between the pile and soil.The influence of kinematic interaction on pile head accelerations has been studied byGazetas (1984) and Fan et al. (1991). In the latter publication, comprehensive studieswere made on the kinematic seismic response of single piles and pile groups. The influence of kinematic interaction may become significant if the stiffness ratio between thepile and the soil is high, such as E\/E8 > 10, 000.An uncoupled superstructure analysis that neglects kinematic interaction appears tobe valid provided the free field surface motions are dominated by relatively low frequencywaves. The neglect of kinematic interaction generally results in an overestimate of dynaniic pile cap motions transmitted to the superstructure.The other difficulty in an uncoupled analysis lies in selecting appropriate equivalentelastic moduli of soil compatible with strains occurring during a strong earthquake. Thereduction of soil stiffness and the increase of damping associated with a strong shakingare sometimes modelled crudely in these analyses by making arbitrary reductions in theshear moduli and arbitrarily increasing the viscous damping. For this reason the resultsof these studies have not proved very useful for the response of pile foundations to earthquake loading.The effect of soil non-linearity on pile head impedances of single piles has been investigated for dynamic pile-head loads by Angelides and Roesset (1981). A cylindricalChapter 5. Dynamic analyses of pile foundations 731.21.0------\u2018(lbf\/ft)0.6 -21L__040.2 d=4ft1t=2.5in.)N=lO cycles\u2022 F=horizontol force at top of pile (kips)0.00.0 0.5 1.0 1.5 ao Z5 3.0FREQUENCY f (Hz)Figure 5.12: Variation of pile horizontal stiffness, k with force and frequency due tosoil non-linearity (after Angelides and Roesset, 1981)region of soil surrounding the pile is modelled by using toroidal finite elements. A consistent boundary matrix was placed at the edge of this core region. The equivalent linearmethod (Seed and Idriss, 1967) was used to model the non-linear soil response. Evenneglecting slippage and gapping, they demonstrated a dramatic reduction of horizontalpile head stiffness by applying harmonic horizontal force at the pile head (Figure 5.12).Similar studies using the program PILE3D described in chapter 8 confirm the finding ofAngelides and Roesset (1981). The effect of soil non-linearity on pile head stiffness issignificant and must be taken into account with appropriate accuracy.Lumped mass-spring-dashpot models. A more complex analysis of the seismic response of single piles which incoporates the non-linear response of soil is based on anapproach in which pile foundation and the superstructure are analyzed as a combinedsystem. The interaction between the pile and the near field soil is modelled using a seriesChapter 5. Dynamic analyses of pile foundations 74of non-linear Winkler springs derived from full scale test measurements or non-linearfinite element solutions (Yegian and Wright, 1973; Arnold et al., 1977; Matlock et al.,1978a,1978b; Bea et al.,1984; Nogami and Chen, 1987). The stiffness of one of thesesprings represents the combined stiffness of the strain softened, near field soil and theexterior free field soil whose properties are governed by the intensity of the earthquakeground motions. The method of analysis relating to the use of Winkler springs is usuallyreferred as lumped mass models.At large displacements, the response of pile foundation is controlled by the non-linearcharacteristics of soil at high strain, pile separation (gapping), slippage and friction. Itis difficult to incorporate these factors in a continuum model. Therefore lumped massmodels, such as these employed by Penzien (1970), Matlock et al. (1978a,1980) and others, have been used to model the pile response at large 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 of single piles. However difficulties existin relating the characteristics of the discrete elements to standard geotechnical parameters of soil. Various non-linear resistance-deflection relationships known as p-y curvesand t-z curves have been proposed. The soil stiffness at a particular depth is establishedusing as input a non-linear soil resistance versus lateral pile deflection (p-y) curve where yrepresents the relative deflection between the pile and the moving ground during shakingand p is the net soil resistance to the pile motion.Specification of the mathematical form of the backbone p-y curves for both static andChapter 5. Dynamic analyses of pile foundations 75cyclic pile head loading of piles in sand and clay are available from several sources (American Petroleum Institute,1979; Stevens and Audibert,1986; Gazioglu and O\u2019Neill,1984;Murchison and O\u2019Neill, 1984). These recommendations have come from the results offull scale pile head loading tests. Extensive data on the p-y curves and non-linear pile response were obtained by Yan (1990) using the hydraulic gradient similitude method andby Gohl (1991) using the centrifuge testing. The most commonly used set of specificationfor constructing p-y curves is based on the recommendations of the American PetroleumInstitute (1986). Mostly used in offshore structures, these p-y curves are available forclay and sand, and they make a difference between static loading and cyclic loading.However the validity of their use for earthquake analysis of piles has not been verified.In SPASM analysis, the response of the structure relies on both the accuracy of p-ycurves for representing the soil non-linearity and the accuracy of time-history input offree field displacements. Verification studies of this method by Gohi (1992) using datafrom centrifuge tests showed that the dynamic response of a structure is sensitive to thetime-history input of free field displacements. The SPASM program underpredicts pilefiexural response. A key difficulty in using SPASM is the accurate determination of freefield input motions to be used along the embedded length of the pile. The dampingproperties are determined separately by methods such as the one proposed by Gazetasand Dobry (1984). The so-called coupled method in SPASM is actually a semi-coupledmethod. The method only couples the super-structure with piles, but it does not couplepiles with their surrounding soils directly. Therefore the SPASM analysis is not applicable to analysis of pile groups.Chapter 5. Dynamic analyses of pile foundations 765.2 Dynamic analysis of pile groupsCurrently pile group stiffness and damping coefficients are widely used in dynamic sub-structuring analysis of superstructure-pile foundation. The analysis of dynamic responseof pile group is limited to elastic response using uncoupled multi-step analysis. Themethod of analysis (Gazetas et al., 1992) involves estimation of the dynamic foundation impedance and effective input motions applied to the base of the superstructure.Dynamic sub-structuring analysis is generally carried out using modal analysis incorporating equivalent elastic pile group stiffness and damping coefficients. The pile groupstiffness and damping coefficients necessary for the analysis are evaluated using one ofthe following methods or a combination.A useful solution to the three-dimensional dynamic boundary-value problem has beendeveloped by Kaynia and Kausel (1982). Results from Kaynia and Kausel (1982) showthat dynamic stiffness and damping of pile group are highly frequency dependent andmay significantly differ from that of a single pile. Both stiffness and damping of a pilegroup can be either reduced or increased due to pile-soil-pile interaction. They mayexhibit very sharp peaks or be affected even for very large pile spacings. The dynamiccharacteristics of a pile group may be explained by pile-soil interaction which dependson the ratio of the wave length to pile spacing. At higher frequencies the waves propagating from a loaded pile in the group may be moving out of phase at the location of anadjacent pile. The occurrence of this phase shift may result in negative interaction coefficients which suggests that the stiffness of a dynamically loaded pile group may in factbe higher than the combined stiffness of a single pile multiplied by the number of pilesin the group. However these analytical results are limited to linear elastic response. Thesharp peaks in dynamic stiffness and damping of the elastic solution may be suppressedChapter 5. Dynamic analyses of pile foundations 77due to soil non-linearity.The concept of the dynamic interaction factor has been proposed by Kaynia andKausel (1982) as an extension of the widely used static interaction factor approach (Poulos, 1971, 1975, 1979). The dynamic interaction factor approach is an approximation ofthe more rigorous pile group analysis. The use of dynamic interaction factors avoids theheavy computing effort involved in a rigorous pile group analysis. A set of interactionfactors is available for floating piles, homogeneous soil and a limited selection of parameters in Kaynia and Kausel (1982) and for vertical vibration in linearly nonhomogeneoussoil in Banerjee (1987). El-Marsafawi et al. (1992a, 1992b) presented approximate procedures for estimating dynamic interaction factors based on boundary element analysis,the work of Kaynia (1982), Kaynia and Kausel (1982), Davies et al. (1985) and Gazetas(1991a, 1991b). These dynamic interaction factors are limited to elastic response, andmostly for homogeneous soil.A procedure for estimating dynamic stiffness and damping of a pile group in nonhomogeneous soil was developed and incorporated in a computer program DYNA3 (Novaket al.,1990). In DYNA3 analysis the Novak plane strain pile soil interaction approachis used to determine stiffness and damping of each single pile, which is similar to thatemployed in PILAY analysis. The dynamic impedance of pile group is then determinedby considering the soil pile interaction (or group effect) based on the concept of dynamicinteraction factors. The dynamic interaction factors used in DYNA3 are the combinationof the static interaction factor by Poulos and Davies (1980) for vertical loading and ElSharnouby and Novak (1986) for horizontal loading and the dynamic interaction factorsby Kaynia and Kausel (1982). Although DYNA3 analysis can deal with nonhomogeneoussoil, the analysis is limited to linear elastic response and to the use of elastic dynamicChapter 5. Dynamic analyses of pile foundations 78interaction factors.The methods for direct group analysis of pile foundations based on a continuum modelare limited to linear elastic behaviour using either boundary element or finite elementtechniques. The linear elastic assumption severely limits the applicability of these models in describing response of pile groups to moderate to strong shaking where significantsoil non-linearity develops and changes the extent of interaction between piles. Whilenon-linear 3-D finite element analyses have been carried out for research purposes to examine pile to pile interaction under static lateral loading (Brown and Shie, 1991), thesemethods are rarely used in practice. Dynamic 3-D finite element analyses of pile groupresponse incorporating non-linear soil response have not been carried out to date.5.3 Objectives of this researchIn following chapters a continuum theory for analyzing dynamic response of single pilesand pile groups is presented. The proposed method of analysis models the dynamic pile-soil-pile interaction as a fully coupled system and also possesses ability of modelling soilnon-linear response under strong earthquake loading.A simplified quasi-3D wave equation is proposed to describe the dynamic motion ofsoil under horizontal shaking. The coupled equations of motions between the pile andsoil are solved using the finite element method. A finite element program PlUMP isdeveloped to compute pile head impedances of single pile and pile group by applyingharmonic forces or moments at pile head. Analyses are carried out in the frequencydomain.Chapter 5. Dynamic analyses of pile foundations 79Studies are carried out to validate the applicability of the proposed quasi-3D modelfor simulating the elastic response of pile 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 vibration tests on anexpanded base concrete pile and on a 6-pile group supporting a large transformer.Attention is then focused on extending the proposed model to incorporate non-linearsoil response under strong shaking. Dynamic analysis of pile foundation is carried outin the time domain and the procedure of this analysis is incorporated in a computerprogram PILE3D. The non-linear characteristics of soil is modelled by using a modifiedequivalent linear method of analysis. Also effective routines are incorporated in PILE3Dto model the yielding of the soil and the gapping that may occur in the area near thepile head.The capability of the quasi-3D model for simulating the non-linear dynamic responseof pile foundation subjected to earthquake loading is validated using data from the centrifuge tests on a single pile and a 2x2 pile group. Under strong shaking soil non-linearityis significant and changes with time. The level of soil non-linearity also varies in spaceat a certain time during shaking. Therefore the dynamic stiffness and damping of pilefoundation change with time. The variations of dynamic stiffness and damping of pilefoundations during shaking are demonstrated for the model pile foundations used in thecentrifuge tests.Chapter 6Elastic Response of Single Piles: Theory and Verification6.1 IntroductionIn this chapter, a quasi-3D finite element method of analysis is proposed to determinethe dynamic response of pile foundations subjected to 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 horizontal shaking direction. Displacements inthe vertical direction and in the horizontal cross-shaking direction are neglected. Therefore a quasi-3D wave equation is established.The finite element method is employed to solve the quasi-3D wave equation in the3-D half-space domain. Elastic analyses are conducted in the frequency domain.Since the elastic solutions developed by Kaynia and Kausel (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 obtained by Kaynia and Kausel (1982). Kinematicresponse of single piles is analyzed; and results are compared with those obtained byFan et al. (1991) who used solutions by Kaynia and Kausel. Data from full-scale forcedvibration testing on a single pile are also used to validate the proposed model.80Chapter 6. Elastic Response of Single Piles: Theory and Verification 816.2 Dynamic analyses of pile foundations: formulationUnder vertically propagating shear waves (Figure 6.1) the soils mainly undergo sheardeformations in XOY plane except in the area near the pile where extensive compressiondeformations in the direction of shaking develop. The compression deformations alsogenerate shear deformations in YOZ plane, seeing Figure 6.1. Under the light of theseobservations assumptions are made that dynamic motions of soils are governed by theshear waves in XOY plane and YOZ plane, and the compression waves in the shakingdirection, Y. Deformations in the vertical direction and normal to the direction of shakingare neglected. Comparisons with full 3-D elastic solutions confirm that these deformationsare relatively unimportant for horizontal shaking.Let v represent the displacement of soil in the shaking direction, Y. The compressionforce is 8G1. The shear force in XOY plane is G1, and the shear force in YOZplane is The two shear waves propagate in Z direction and X direction, respectively. The inertial force is p$; Applying dynamic force equilibrium in Y-direction,the dynamic governing equation under free vibration of the soil continuum is written as82v 82v 82vG\u2019 + + Ps (6.1)where G* is the complex shear modulus, Ps is the mass density, and = 2\/(1\u2014v) fora Poisson\u2019s ratio v. Since soil is a hysteretic material, the complex shear modulus G* isexpressed as = G(1 + i 2A), in which G is the shear modulus of soil, and A is thehysteretic damping ratio of soil. The radiation damping will be included later.The displacement field at any point in each element is modelled by the nodal displacements and appropriate shape functions. A linear displacement field is assumed inChapter 6. Elastic Response of Single Piles: Theory and Verification 82\/\/\/\/\/ \/\/\/Y\/\/\/\/\/7\/\/\/\/\/\/\/\/\/\/\/\/, \/\/\/\/\/, 7\/\/,. \/\/\/\/\/\/\/_ \/_40.Direction of shakingFigure 6.1: The principle of quasi-3D dynamic pile-soil interaction in the horizontaldirectionstructural masszSoil7,,\/\u2018shear\u2018shearI_____4CompressionYPile3-D finite elementsChapter 6. Elastic Response of Single Piles: Theory and Verification 85156 221 54 \u2014131221 412 131 .3l254 131 156 \u2014221\u2014131 _3l2 \u201422l 412The radiation damping is modelled using velocity proportional damping. Theing force Fd per unit length along the pile is given byFd=c,,-- (6.7)where ca, is the radiation dashpot coefficient for horizontal motion.A simple expression for the radiation dashpot coefficients c, which was proposed byGazetas et al. (1993) and is given in Eq. 5.4, is used in the analysis. Applying the sameprocedure as that used to obtain mass matrix, the radiation damping matrix [C\u2019] for apile element is156 221 54 \u2014131221 412 131 _3l254 131 156 \u2014221\u2014131 _3j2 \u2014221 4j2The global dynamic equilibrium equation in 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 loads applied.pEAl[M ]ji\u2014 420 (6.6)damp-* Cr1pi1e (6.8)Chapter 6. Elastic Response of Single Piles: Theory and Verification 86__Kv4..0e=o =Figure 6.3: Pile head impedances6.3 Pile head impedancesThe impedances are defined as the complex amplitudes of harmonic forces (or moments) that have to be applied at the pile head in order to generate a harmonic motionwith a unit amplitude in the specified direction (Novak,1991).The concept of translational and rotational impedances is illustrated in Figure 6.3.The translational, the cross-coupling, and the rotational impedances of pile foundationsused in this analysis are defined as\u2022 K: the complex-valued pile head shear force required to generate unit lateraldisplacement (v=1.O) at the pile head while the pile head rotation is fixed (S = 0).\u2022 K8: the complex-valued pile head moment generated by the unit lateral displacement (v=1.0) at the pile head while the pile head rotation is fixed (S = 0)Chapter 6. Elastic Response of Single Piles: Theory and Verification 87\u2022 K88: the complex-valued pile head moment required to generate the unit pile headrotation ( 1.0) while pile head lateral displacement is fixed (v=0)Since the pile head impedances K,,,,, K,,8K88 are complex valued, they are usuallyexpressed by their real and imaginary parts as= k1 + i C (6.10)= +i (6.11)in which and are the real and imaginary parts of the complex impedances,respectively, and i = cj = Cj\/w = coefficient of equivalent viscous damping; andw is the circular frequency of the applied load. and C1 are usually referred as thestiffness and damping at the pile head. All the parameters in Eq. 6.10 are dependent onfrequency w.Determination of impedances K,,,,, K,,8 and K88 Pile head impedances will beevaluated as functions of frequency by subjecting the system to a series of 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*] \u2014w2[M]}{vo} = {P0} (6.12)or[K]global{VO} = {P0} (6.13)Chapter 6. Elastic Response of Single Piles: Theory and Verification 88where[K1gzabo1 [K*j + i w[C] \u2014w2[.M] (6.14)According to the definition, impedances K,,,, and Kue can be found by applying a unithorizontal displacement at the pile head under the condition of a fixed pile head rotation.Eq.6.13 becomes0[K]gia&ai 1.0 = K,,,, (6.15)0.0 K,,8where v are the displacements of the nodes other than pile head.Dividing Eq. 6.15 by Ku,, and eliminating the row of zero rotation, one obtainsIv\/K,,l 1 0 1[K]9z0b = (6.16)I v J (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 pile head impedances is toapply a unit horizontal force at the pile head and calculate the complex displacement atthe pile head v. Therefore the pile head impedances K,,,, and K,,9 are determinedK,,,, = (6.17)K,,9 = (6.18)Chapter 6. Elastic Response of Single Piles: Theory and Verification 89Using the same principle, the rotational impedance K88 can be determined by applyinga unit moment at the pile head under the condition of a fixed pile head horizontal displacement. The rotational impedance K88 is determined asK88 = (6.19)where 8 is the rotation at the pile head caused by the unit moment at the pile head.Because of reciprocity principle, the cross-coupling impedance K8 and K8 are identical.6.4 Verification of the proposed model: pile head impedancesIn order to assess the accuracy of the proposed quasi-3D finite element approach, theimpedance functions for single piles are determined and compared with the analyticalresults by Kaynia and Kausel (1982). Analyses were performed in the frequency domainfor elastic conditions. Impedances K8,K88 will be presented as functions of thedimensionless frequency a0, where a0 is defineda0 = (6.20)in which w is the angular frequency of the exciting loads (force and moment) at thepile head, d is the diameter of the pile, and V1 is the shear wave velocity of the soilmedium. For a uniform soil profile with a shear modulus G and a mass density p, V8 iscomputed by V8 =It is found that for given values of E\/E8 and a0, the ratios K\/(E8d),K8\/(E3d2),and K98 \/(E8d3)hold unchanged for any soil modulus E5 of a uniform soil profile. Therefore the normalized impedances K\/(E5d),K8\/(Ed2),andK88\/(E3), are presentedChapter 6. Elastic Response of Single Piles: Theory and Verification 90single piles:LdEp\/Es = 1,000L\/d > 15 (floating pile)soil damping 5 %Poisson\u2019s ratio 0.4Figure 6.4: A pile-soil system used for computing impedances of single pilesas a function of the dimensionless frequency a0 when other parameters are given.Figure 6.4 shows a pile-soil system and its relative parameters. A ratio of Er\/ES =1, 000, which was used by Kaynia and Kausel, is adopted for the analysis, where E andare the Young\u2019s moduli of the pile and the soil, respectively. The soil medium has aPoisson\u2019s 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 half of the full mesh is required to model the response ofpile-soil interaction. The half mesh, shown in Figure 6.5, consists of 1463 nodes and 1089elements. The use of half mesh reduces the size of the global matrix by a factor of 4.0with a corresponding large reduction in computational time. It took about 300 secondsto determine the dynamic impedances for each frequency using a 486 PC computer.Chapter 6. Elastic Response of Single Piles: Theory and Verification 91VI,\/Figure 6.5: Finite element modeffing of single pile for computing impedancesWhen the symmetric condition is applied, the Young\u2019s modulus E and the massdensity Pp of the pile should be reduced by a factor of two in 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 the other half of the full mesh. For the same reason, theapplied loads (force and moment) at the pile head should be reduced by a factor of 2.0.These adjustments are automatically included in the program PlUMP.Discussion of results The normalized quantities K\/(E3d),K6\/(E9d2),K98\/(E3d)are presented as functions of dimensionless frequency a0. Since the impedances are complex quantities, their values are expressed in term of their real partsk1(stifFness) andimaginary partsC1(damping) according to Eq.6.l0. Hence the normalized stiffness anddamping are compared with those obtained by Kaynia and Kausel (1982), and they areshown in Figure 6.6, 6.7, and 6.8, respectively.Chapter 6. Elastic Response of Single Piles: Theory and Verification 9212-10..- - -- Kaynia and Kausel (1982)8 = \u2022 \u2022 Proposed model6-4-.\u201dEf2-I I I I I I I I I I I I I I I I I I I I I I I0.00 0.10 0.20 0.30 0.40dimensionless frequency, a012: Kaynia and Kausel (1982)10.: \u2022 Proposed model-0\u2014 IIIIliIIIIII1IIIIIIIllIiillIlIIIIIIIIJ0.00 0.10 0.20 0.30 0.40dimensionless 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 of Single Piles: Theory and Verification 9312-10- ---- Kaynia and Kausel (1982)a a Proposed modelN8-Cl) -a)C -4-2-0\u2014 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 I0.00 0.10 0.20 0.30 0.1dimensionless frequency, a012-Koynia and Kausel (1982)10 - \u2022-\u00b0-- Proposed model0\u2014 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 I0.00 0.10 0.20 0.30 0.40dimensionless frequency, a0Figure 6.7: Normalized stiffness k9 and damping Cue versus a0 for single piles(E\/E5 = 1000, \u2018 =0.4, A=5%)Chapter 6. Elastic Response of Single Piles: Theory and Verification 9440------ Koynia and Kausel (1982)Proposed model20-(I)1 :\u2014 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 I0.00 0.10 0.20 0.30 0.40dimensionless frequency, a020-- Kaynia and Kausel (1982)- Proposed modelcjio0\u2014 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0.00 0.10 0.20 0.30 0.40dimensionless frequency, a.0Figure 6.8: Normalized stiffness k99 and damping C versus a0 for single piles(E\/E6 = 1000, z\u2019 =0.4, )=5%)Chapter 6. Elastic Response of Single Piles: Theory and Verification 95The computed impedances K8, and K88 agree well with those obtained by Kaynia and Kausel. Both solutions show that dynamic stiffnesses increase slightly as thedimensionless frequency a0 increases. The computed quantities are slightly smaller thanthose by Kaynia and Kausel, which implies their modelling of the pile-soil systems resultsin stiffer response. The translational stiffness computed by Kaynia and Kausel areabout 10% larger than those from current study. The other two stiffnesses k8 and k98show less sensitivity to the method of computation. The differences of k8 and k98 fromthe more exact solutions are about 5%.However, the difference of impedance between the two solutions is insignificant inpractice when soil non-linearity is an important factor. In many cases reduction of soilshear moduli with the increase of shear strain is significant. Quantitatively modelling ofreduction of shear moduli, especially in the near field of the pile, is important for thedetermination of pile head impedances. It will be shown later that the proposed methodhas the ability to determine the non-linear dynamic impedances as the soil moduli decrease with the increase of shear strain under strong shaking.Comparison with solutions by Novak et al. Using the mesh shown in Figure 6.5,stiffness k,and damping C were computed for a Er\/ES ratio of 295. The results arethen compared to solutions presented by Novak and Nogami (1977), and Novak\u2019s approximate (Novak,1974). Figure 6.9(a) and 6.9(b) show comparisons on stiffness anddamping, respectively. The values of dynamic stiffness computed by Novak and Nogamiare about 25% larger than those computed by author. For a0 > 0.3, values of dynamicstiffness from Novak\u2019s approximate solution are about 10% larger than those computedby author. For dimensionless frequency a0 less than 0.3, dynamic stiffness computed byChapter 6. Elastic Response of Single Piles: Theory and Verification 968-Novak and Nogami (1977)6 \u2014 \u2014\u2014 Novak\u2019s Approximate (1974)42:0.00 0.10 0.20 0.30 0.40dimensionless frequency a08-Novak and Nogami (1977)6- \u2014 \u2014\u2014 Novak\u2019s Approximate (1974)\u2022 Proposed modelIo0\u2014 i i i III I I I I I I I I I I I I I II I I I I I I I I I0.00 0.10 0.20 0.30 0.40dimensionless frequency a0Figure 6.9: Comparison of stiffness and damping C with solutions by Novak andNogami (1977), Novak (1974)Chapter 6. Elastic Response of Single Piles: Theory and Verification 978-6-___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 I I I I I I I I I I I I I I I I I I I I0.00 0.10 0.20 0.30 0.40dimensionless frequency, a0Figure 6.10: Comparison of stiffness for different mesh sizeNovak\u2019s approximate method diminishes with the decrease of frequency. The dampingfrom Novak and Nogaini or Novak\u2019s approximate method is normally larger than damping computed in the present study (Figure 6.9(b)).Effect of the number of finite elements The number of finite elements used in theanalysis has some influence on the impedances computed using the proposed method.Theoretically accuracy of the 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 computed by two differentmeshes. It is clear that some differences exist between results from the two meshes aroundChapter 6. Elastic Response of Single Piles: Theory and Verification 98______pile head motion free field motionsingle free-head piles:L Ep\/Es = 1,000 or 10,000dLid = 201 .5Lsoil damping 5%Poisson\u2019s ratio 0.4-4base motionFigure 6.11: Pile foundation for analysis of kinematic responsea0=O.3. A finer mesh with more finite elements is better to represent the dynamic response accurately.6.5 Verification of the proposed model: kinematic interactionA pile-soil system shown in Figure 6.11 is subjected to a harmonic displacement vbei1tat its rigid base. The dynamic response at the pile head may be same as or very closeto the dynamic response at the free field surface if the pile is very flexible. However inmany cases the dynamic response at the pile head differs significantly from the responseat the free field surface because piles are generally much stiffer than soil and thus modifysoil deformations. This type of interaction between piles and soils is called kinematicinteractionChapter 6. Elastic Response of Single Piles: Theory and Verification 996.5.1 Kinematic interaction factorsThe dynamic motions at the free field surface and at the pile head are different due to thekinematic interaction between the pile and the soil. Let the harmonic displacements atthe free field surface be represented by v11 eif1t , and at the pile head by vet and &peiat,in which v, and 8, are the complex amplitudes of the translational displacement and therotational displacement, respectively.Absolute values of complex amplitudes of harmonic displacements are used for determining the kinematic interaction factors. The kinematic interaction factors I,. and I, aredefined after Gazetas (1984) as= (6.21)= d(6.22)in which Ui,, U1I and are the absolute values of the complex amplitudes v,, V11and , respectively; and d is the diameter of the pile.6.5.2 Computed kinematic interaction factorsThe kinematic interaction factors are obtained for pile-soil systems with a flexible pile(E\/E8 = 1, 000), and with a stiff pile (E\/E, = 10, 000). The other parameters of thesystems are shown in Figure 6.11.The accuracy of the quasi-3D finite element method is checked against the boundaryintegral method developed by Kaynia and Kausel (1982) and used by Fan et al. (1991).Chapter 6. Elastic Response of Single Piles: Theory and Verification 100The computed kinematic interaction factors I, and \u2018\u00e71 as functions of the dimensionless frequency a0 are plotted in Figure 6.12 for E\/E8=1,000 and in Figure 6.13 forE\/E=10,000 together with the interaction factors obtained by Fan et al. (1991). Acomparison of the two sets of factors shows that there is very good agreement betweenthe quasi-3D solutions and the boundary element solutions.The kinematic interaction becomes more significant when the stiff pile is placed inthe soil. ForE\/E8=10,000, the response of the pile head is significantly reduced whenthe dimensionless frequency a0 is greater than 0.25. At a0 = 0.35, the amplitude ofthe translational displacement at the pile head is 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 expanded base concrete pile was conducted and reportedby Sy and Siu (1992). The vibration test was carried out by applying very low harmonicloads at the structural mass, which generated elastic response in the system. This provides an opportunity to validate the quasi-3D model for elastic response.6.6.1 Description of site condition and test resultsThe testing site is located in the Fraser river delta south of Vancouver. The soil profileat the testing site consists of 4 m of sand and gravel fill overlying a 1 m thick silt layerover fine grained sand to 40 m depth. A seismic cone penetration test (SCPT 88-6)was conducted 0.9 m from the test pile location. In addition a mud-rotary drill hole (Chapter 6. Elastic Response of Single Piles: Theory and Verification 1012.0\u2022 --Fan et al. (1991)(a) - -- Proposed model\u2018palo\u2022 Ep\/Es = 1000_D 1.5 -0o 10-CDC -0 -0.5 -I.1).90.0 \u2014 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 I0.0 0.1 0.2 0.3 0.4 0.5dimensionless frequency0.5 -(b) Fan etal.(1991)0.4 - ..01.Proposed modelEp\/Es = 10000.3 -____4-\u2019C.)CD0.2 -C.)CD0.14-\u2019.90.0 \u2014 1Ti I I I I I I I I I I I I I I I I I I I I I I I I0.0 0.1 0.2 0.3 0.4 0.5dimensionless frequency a0Figure 6.12: Kinematic interaction factors versus a0 for Er\/ES = 1, 000Chapter 6. Elastic Response of Single Piles: Theory and Verification 1022.0 -(a)----- Fanetal. (1991)....Proposed model1.5 - Ep\/Es = 10,000o1.0 -\u20184-C00.5 -0.0 - 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 I0.0 0.1 0.2 0.3 0.4 0.5dimensionless frequency a00.5 -(b) Fan etal. (1991).Proposed model- Ep\/Es = 10,0000tO.3-\u2018IC.2 0 2 -C.)C\u2019,00.1 -0.0\u2014 IiIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII0.0 0.1 0.2 0.3 0.4 0.5dimensionless frequency a0Figure 6.13: Kinematic interaction factors versus a0 for Er\/ES 10, 000Chapter 6. Elastic Response of Single Piles: Theory and Verification 103CPT Oc (bar) SPT N (blows\/O.3m) Vs (m\/s)0 125 250 0 10 20 30 40 0 100 200 3000\u2014 iiIi \u2014 iiiiIiiiitiiIii, \u2014 iiIiitiIiiit:\u2014,A\u2019.. . 0\u2022\u2022 4..-.--- 0\u2022IC p- - \u2022-. 0\u2022- 0 \u2022r115-- -d 0 \u20220I\u2014 00\u2014- I \u2014 0\u2022w 20- - -0025- -\u2022 . :30 -:SCPT 88\u2014B\u2022. \u2022....DH 88\u20142- \u2014 -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 of Single Piles: Theory and Verification 104FFI\u2014ELECTROMAGNETIC SNAJ 15 (floating pile)soil damping 5 %Poisson\u2019s ratio 0.4S \u2022IFigure 7.3: A pile-soil system used for computing impedances of pile groups7.5 Elastic response of pile group: results and comparisonsThe dynamic impedances of a 4-pile (2x2) group with s\/d=5.0 are presented, in which sis the centre to centre distance of two adjacent piles and d is the pile diameter. A rigidpile cap is rigidly connected to the 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 ratio Pa\/Pp 0.7 isapplied.The dynamic impedances of 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 base is assumed at a depth of 5L (L=length of pile) beyond thetip of the pile. The dynamic impedances are evaluated at the bottom of the pile cap.Chapter 7. Elastic Response of Pile Groups: Theory and Verification 117Discussion of results In order to show the pile group effect, dynamic impedances ofthe pile group are normalized to the static stiffness of the pile group expressed as thestiffness of a single pile times the number of piles in the group. The normalized dynamicimpedances of the pile group, which are called the dynamic interaction factors, are therefore defined asa,,,,N.kV\u00b0V(7.13)a,,9N.k,,\u00b09(7.14)K88a99=(7.15)LV 96in which k,,,, k,,\u00b09,k906 are static stiffnesses of a single pile identical to those in the pilegroup that is placed in the same soil medium, and N is the number of piles in the pilegroup ( N=4 for a 4-pile group). K99 in Eq. 7.15 is the individual rotational impedanceat the head of each pile due to the geometrical and material properties of the pile.In order to present the results graphically, the complex-valued dynamic interaction factors a23 are separated by their real parts a(stiffness) and imaginary partsa(damping). The computed dynamic interaction factor a(stiffness) is compared inFigure 7.4(a) with that by Kaynia and Kausel (1982). Very good agreement is observedbetween the two solutions for dimensionless frequency a0 < 0.28. For a0 > 0.28, thecomputed values are about 25% higher than those by Kaynia and Kausel. The computedinteraction factors a(damping) are in good agreement with those by Kaynia and Kausel(Figure 7.4(b)).The dynamic interaction factors a,,,,, and a99 of the pile group are shown in Figure 7.5(a) for stiffness component and in Figure 7.5(b) for damping component. TheChapter 7. Elastic Response of Pile Groups: Theory and Verification 118results show that the translational stiffness (or cx) shows the greatest effect of grouppile-soil interaction; whereas the rotational stiffness k88 (or aee) shows the least effect(Figure 7.5(a)). However their corresponding damping components show the reversetrend (Figure 7.5(b)). For a0 < 0.3 the stiffness interaction factors ave, and cxee arein the range of 0.6, 0.7 and 0.9, respectively.Because the piles are rigidly connected to the pile cap at the pile heads, the totalrotational impedance of the pile group consists of both the rocking impedance Krof the pile group and the rotational impedance K98 at the head of each pile= K. + K89 (7.16)Following the notation used by Kaynia and Kausel, the total rotational impedance of thepile group K7 is normalized as K\/(N. r? in which k\u00b0 is the static verticalstiffness of a single pile placed in the same soil medium.The normalized quantity K\u2019\/(N k) is compared in Figure 7.6 with thatby Kaynia and Kausel. Very good agreement between the two solutions is seen for thestiffness component (Figure 7.6(a)). Good agreement between the two solutions for thedamping components also exists for a0 < 0.3 (Figure 7.6(b)). For a0> 0.3, the computeddamping component is about 25% less than that by Kaynia and Kausel.It has been shown that the proposed model can well simulate the dynamic characteristics of pile groups. The conclusion is drawn from comparisons with analytical resultsby Kaynia and Kausel. To further verify the applicability of the proposed model forsimulating the elastic response of pile group, field vibration tests of a 6-pile group areChapter 7. Elastic Response of Pile Groups: Theory and Verification 1193-(a) stiffness component2-Kaynia and Kausel (1982)- Proposed model.2 2x2 pile group; s\/d=5C 1-I.9 0-0.00 0.10 0.20 0.30 0.40dimensionless frequency a03-: (b) damping component2-04-,C)0\u20184\u2014\u2014 0\u2014 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 I0.00 0.10 0.20 0.30 0.40dimensionless frequency, a0Figure 7.4: Comparison of dynamic interaction factor a.,, with solution by Kaynia andKausel for 2x2 pile groups (E\/E3 = 1000,s\/d 5.0)Chapter 7. Elastic Response of Pile Groups: Theory and Verification 1203-(a) stiffness component- TranslationCross\u2014coupling2 - Rotation2x2 pile group; s\/d=50 -C)0oz:0a)-9 0\u20140.00 0.10 0.20 0.30 0.40dimensionless frequency, a03-(b) damping component2-U\u20190-I-,C)0c 1-0C-)00- . . .. ::.-90.00 0.10 0.20 0.30dimensionless frequency, a0Figure 7.5: Dynamic interaction factors a, a66 versus a0 for 2x2 pile groups(E\/E, = 1000,s\/d = 5.0)Chapter 7. Elastic Response of Pile Groups: Theory and Verification 1214-(a) stiffness componentKaynia and Kausel (1982)3 Proposed model- 2x2 pile group; s\/d=52--.-1\u20140\u2014 iiiiiiiiiitiiii,iiiiiiiiiiiiiiiiiiiiiii0.00 0.10 0.20 0.30 0.40dimensionless frequency, a04-- (b) dompng component3-2-1-__________________-0- iiiiiijtIIlIIIIIIIJ t0.00 0.10 0.20 0.30 0.40dimensionless frequency, a0Figure 7.6: Comparison of normalized total rotational impedance K7\/A with solutionby Kaynia and Kausel for 2x2 pile groups (E\/E3 = 1000, s\/d=5, A N * Th\u2019?k\u00b0 )Ohapter 7. Elastic Response of Pile Groups: Theory and Verification 122analyzed 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 group foundation of a large transformer bank (Bank 79) located at the Duwamish substation, Seattle,Washington. Test data and analytical results were reported by Crouse and Cheang(1987). The foundation of the transformer consists of a pile cap with 6 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 finite element method of analysis. Results of this analysis are used to verify the applicability ofthe proposed model for pile group.7.6.1 Description of vibration and its testing resultsThe transformer and foundation system has been described in detail in Crouse andCheang (1987) and are briefly summarized below. The soil profile at the location oftransformer bank consists of mostly loose to medium dense sand to silty sand, with somedense sand or gravelly sand layers, overlying very dense gravelly sand glacial till at 12.2m depth. The ground water table was at a depth of 3.7 m. The in-situ shear wave yelocities (V3) measured from a downhole seismic survey in the sand to silty sand depositsare 125 m\/sec in the upper 3.7 rn and 165 rn\/sec below 3.7 m depth. Figure 7.7 showsthe idealized soil proffle at Duwaniish Station according to Crouse and Cheang (1987).Response of Pile Groups:LAYER DEPTH0THICKNESS II,) fpI)4\u2019 2 2204\u2019 6 66010 4 10 11102\u2019 13 13682\u2019 15 14634\u2019 18 160620\u2014 5 22.5 1820S3\u2019 26.5 201030 4\u2019 30 21774\u2019 34 23676\u2019 39 26054050Theory and Verification\u2018m. \u20227(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 30937.8 119 30Figure 7.8 shows the transformer bank and the pile foundation. The transformer,weighing 326 kip is anchored to concrete pedestals which is a continuous part of the pilecap. The pile cap has a dimension of 13.4 ft by 8.00 ft. The pile cap is embedded beneaththe ground surface as shown in Figure 7.8. The pile foundation consists of 6 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 at 40 ft depth. The compositecompressional rigidity (EA) and the flexural rigidity (El) of the each concrete filled pipepile are 5.1 x 108b and 4.24 x 107b.ft2,respectively, where E= Young\u2019s modulus, A:=cross section area, and 1= bending moment of inertia.123Chapter 7. ElasticSANDToSILTYSAND iGRAVELLY ::\u2018.SANDSANDToSILTYSANDGRAVELLYSANDGLACIALTILLIFigure 7.7: Idealized1987)YM80LS\u2022 VERTICAL CONFINING STRESSSHEAR STRENGTHLOWSTRAIN SHEAR MODULUS-TOTAL DENSITY\u2022 - 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 of Pile Groups: Theory and VerificationPLAN- ELEVATIONPILE CP0SS.SECTION1\u2014CONCRETEt 0.172\u201d12 0.0.rutFOUNDATION PLAN124Figure 7.8: Setup of a full-scale free vibration test on a 6-pile group (after Crouse andCheang, 1987)\u2018CI?4T10 0 ola. si\u2014Io 0I \u2018I II\u2014\u2014Izi .\u2014I 2!.\u2014Chapter 7. Elastic Response of Pile Groups: Theory and Verification 125A sling was attached to the transformer, and then pulled and quickly released to let thestructure vibrate freely. The motions were recorded at various locations on the foundation. Tests were performed in both principal horizontal directions, NS or Y-axis and EWor X-axis, of the transformer foundation.The resonant frequencies and damping ratios of the transformer foundation systemwere determined from the recorded time histories of transient vibrations by Crouse andCheang (1987). The measured fundamental frequencies in NS and EW directions are 3.8Hz and 4.6 Hz, respectively. The measured damping ratios in the NS and EW directionsare 6% and 5%, respectively.7.6.2 Computed results using the proposed modelIn present analysis the soil profile shown in Figure 7.7 is used except the shear modulusdistribution modified by Sy (1992) is adopted. According to Sy (1992), a correction onmeasured shear wave velocity was made to account for the soil densification due to pileinstallation. An increase of 4% in low strain shear modulus Gma values was applied tothe measured free field values for this correction. Poisson\u2019s 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) show the finite element models used in the analysis for obtaming dynamic impedances in NS direction and EW direction, respectively. Dynamicimpedances of the pile foundation are computed covering an excitation frequency rangefrom 0 Hz to 6 Hz. At this stage of the analysis, the effect of pile cap embedment isnot taken into account. The computed impedances of the pile group corresponding tothe excitation frequencies at 3.74 Hz in NS direction and at 4.63 Hz in EW direction areChapter 7. Elastic Response of Pile Groups: Theory and Verification 126listed in Table 7.6. The stiffness and damping values shown in this Table are referencedto the bottom of the pile cap.In order to determine the resonant frequencies of the transformer-pile cap system,dynamic response of the system is computed by subjecting the system to horizontal harmonic forced excitation at different frequencies. Figure 6.18 shows the structural massand its supporting dynamic impedances (springs and dashpots). Eq. 6.23 is used againfor obtaining dynamic response of the structural mass at a particular excitation frequency.Harmonic force and moment were applied at the centre of gravity of the transformer-pilecap system. According to Crouse and Cheang, the transformer and pile cap have a totalmass of 1.13 x i0 slugs. The height of centre of gravity of the system is 10.9 ft above thepile head. The moments of inertia are 4.38 x lO5slug.ft2 and 5.46 x lO5slug.ft2 aboutaxes in the NS direction and in the EW direction, respectively.The response curves of the transformer-pile cap system are shown in Figure 7.10 indashed lines. The resonant frequency of the system is the excitation frequency at whichthe peak dynamic response of the system occurs. The damping ratios are determinedusing the the bandwidth method from the response curves. The computed resonant frequencies and damping ratios are shown in Table 7.5. Results are presented together withthe measured values.Effect of pile cap embedment In the previous analysis the effect of pile cap-soil interaction was not included. According to Crouse and Cheang, gaps between the pile capand the soil underneath it may exist due to settlement of soil. Also conventionally theeffect of pile cap-soil interaction has been included by considering the soil reaction actingChapter 7. Elastic Response of Pile Groups: Theory and Verification 127\/\/ \/ \/ \/\/\/\/\/N\/\/\/[\/ \/\/\/\/\/\/\/\/\/\/ \/ \/ ,\/\/\/jY\/\/\/\/\/\/A\/ I A\u20184,\/ 1 \/\/A\/Figure 7.9: 3-D finite element models of the 6-pile foundation (a) NS direction, (b) EWdirection(a)\/ \/ \/ \/\/\/\/\/E\/\/\/ \/\/h\/\/ 00I\u2019ll\u2019__I(b)\/\/\/ \/\/\/\/A\/\/4\/ \/ 1\/\/\/__t1IIChapter 7. Elastic Response of Pile Groups: Theory and Verification 128Table 7.5: Computed resonant frequencies and damping ratios without the effect of pilecap embedmentResonant frequencies (Hz) Damping ratiosComputed Measured Computed MeasuredNS EW NS EW NS EW NS EW3.50 4.37 3.8 4.60 0.08 0.06 0.06 0.05on the vertical sides of the pile cap (Prakash and Sharma, 1990 & Novak et al.,1990).Therefore in this analysis only the soil reactions acting on the vertical sides of the pilecap are included and the soil reactions acting on the base of the pile cap are not considered. The side reaction due to pile cap embedment usually result in increased foundationstiffness and damping. According to Beredugo and Novak (1972), the foundation stiffnessand damping due to pile cap embedment can be determined using the plain strain soilmodel.The rectangular pile cap had an area of 106.67 ft2, an equivalent radius of 5.83 ft.The embedment depth of the pile cap was 1.75 ft, and the shear modulus of soil at thatdepth was 6.0 x iO psf. Based on these data, using Beredugo and Novak\u2019s solution, thestiffness and damping due to pile cap embedment are determined at the bottom of thepile cap asTranslationk = 4.20 x 106 lb\/ftC = 1.33 x iO . lb\/ftCross-couplingChapter 7. Elastic Response of Pile Groups: Theory and Verification 129Table 7.6: Computed stiffness and damping of the transformer pile foundationN-S (f=S. 74 Hz) E- W (f=4. 6S Hz)with without with withoutembed. effect embed. effect embed. effect embed. effectk(lb\/ft) 2.46e-b7 2.04e+7 2.55e+7 2.13e+78(lb\/rad) -2.21e+7 -2.57e+7 -2.43e+7 -2.79e+7k\u2019(lb.ft\/rad) 1.34e+9 1.25e-b9 2.59e-l-9 2.50e+9C(lb\/ft) 10.Oe+6 7.05e+6 8.81e-l-6 5.O5eH-6Ce(lb\/rad) -5.lOe+6 -7.62eH-6 -1.97e-f-6 -5.25e--6C\u201d(1b.ft\/rad) 9.20e-l-7 6.91e+7 1.74e+8 1.44e+8k8 = 3.68 x 106 lb\/rad= 1.16 x iO . ib\/radRotation= 9.34 x 1W lb.ft\/rad= 1.03 x 106 w lb.ft\/radAfter adding the stiffness and damping due to pile cap embedment to these of the pilegroup, the combined stiffness and damping of the transformer foundation are determinedand they are shown in Table 7.6 at the resonant frequencies.The dynamic response of the transformer-pile cap system is obtained including theeffect of pile cap embedment. Figure 7.10 shows the response curves of the transformerpile cap system when the effect of pile cap embedment is included. They are comparedwith those when the embedment effect is not included. It can be seen that the pile capembedment results in increased stiffness and damping of the transformer foundation.Chapter 7. Elastic Response of Pile Groups: Theory and VerificationC,)C0aCr,a)+w&+w&&N\u2014S direction(6\u2014pile group)130without embedment effect_____with embedment effectc\\JS.pI\u2019\u2018II II IJ,\u2022 I\u2022 II I\u2022 I,,Iimeasuredfrequency\u2014 11111111111111111110 2 4 6 8frequency, HzE\u2014W direction(6\u2014pile group): without embedment effectwith embedment effect- measured\u2022\u2018 frequencyEJI I I I I I I I I I I II I I I0 2 4 6 8frequency, HzFigure 7.10: Response curves of the of transformer-pile cap system (a) NS direction (b)EW directionChapter 7. Elastic Response of Pile Groups: Theory and Verification 131Table 7.7: Measured and computed resonant frequencies and 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 0.09 0.09 0.06 0.05The computed and measured resonant frequencies and damping ratios of the transformer-pile cap system are given in Table 7.7. The computed resonant frequencies including theeffect of pile cap embedment are 3.74 Hz and 4.63 Hz in the NS direction and in the EWdirection, respectively. The corresponding measured resonant frequencies are 3.8 Hz and4.6 Hz. The computed resonant frequencies match very well with the measured frequencies in both principal directions. However, the computed damping ratios are about 50%to 80% higher than the measured damping ratios.Chapter 8Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction8.1 IntroductionIn this chapter the quasi-3D finite element method of analysis described in chapter 6and 7 is used to model dynamic response of pile foundations subjected to earthquakeloading. Since earthquake excitation is a random process, the non-linear finite elementanalysis is conducted in the time domain. The use of time-domain analysis makes itpossible to model the variations of soil properties with time under earthquake loading.Therefore, adjustments in the proposed model are made first to accommodate the time-domain analysis. Then studies are focused on modelling non-linear response of the soilunder earthquake loading.A finite element program PILE3D has been developed for dynamic analysis of pilefoundation under earthquake loading. In PILE3D, the shear stress-strain relationship ofsoil is simulated to be either linear elastic or non-linear incrementally elastic. When thenon-linear option is used, the shear modulus and the hysteretic damping are determinedusing a modified equivalent linear approach based on the levels of dynamic shear strains.Features such as shear yielding and tension cut-off are incorporated in PILE3D also. Thedynamic response of pile groups can be effectively modelled using PILE3D.132Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 1338.2 Quasi-3D finite element analysis in the time domainThe basis of the quasi-3D finite element method of analysis for pile foundations has beengiven in Chapter 6 and Chapter 7. In these chapters, dynamic analyses were performed inthe frequency domain. To accommodate the time-domain analysis presented herein, someadjustments are required in the formulation of the global dynamic equilibrium equationgiven 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 diagonal mass matrix can save both computational time and space. Thus the diagonal mass matrix formulations are used in PILE3Dto construct [M]eiem for both the soil element and the beam element. The diagonal massmatrices for the soil element and the beam element are 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 and damping 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 damping ratio .\\ of soil is included usingequivalent viscous damping and its formulation is given below.A procedure for estimating damping coefficients for each individual element proposedby Idriss et al. (1974) is employed in PILE3D. The main advantage of this procedureis that a different degree of damping can be applied in each finite element according itsshear strain level. The damping is essentially a Rayleigh-type damping, assuming theChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 134damping is contributed one half by mass and the other half by stiffness. In each timeperiod LT the global stiffness matrix [K] is computed based on the individual shearmodulus in each element, and the global mass matrix [M} is always constant through thetime domain. The fundamental natural frequency w1 of the pile-soil system is obtainedby solving the corresponding eigenvalue problem.The fundamental frequency of the pile-soil system w1 is then applied to every soilelement in the system. The damping matrix for a soil element is given by[C]eiem [M]eiem -+- \/3. [K]eiem (8.3)in whicha = \u2018e1em= elem\/LL1and )telem is the hysteretic damping ratio of soil in the element and 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 from each individual finite element. Therefore under earthquakeloading, the global dynamic force equilibrium equation in matrix form is given by[M]{\u2019\u00fc} + [C]{\u2019\u00f4} + [K]{v} = \u2014[M]{I} i,(t) (8.4)in which ib(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 Seismic Soil-Pile-Structure Interaction 1358.3 Solution scheme for dynamic equationThere are two methods for solving a dynamic equation, the mode-superposition methodand the direct step-by-step integration method (dough and Penzien,1975; Newmark,1959;Wilson et al.,1973).The mode-superposition method is very useful when a linear system is to be analyzed. This method requires the evaluation of the vibration modal frequencies and theircorresponding modal vectors. It basically uncouples the response of the system, andevaluates the response of each mode independently of others. The main advantage ofthis approach is that the dynamic response of a system can be evaluated by consideringonly some vibration modes even in systems that may have many degrees of freedom; thusthe computational efforts may be significantly reduced. However the mode-superpositionapproach is not applicable to non-linear systems.The direct step-by-step integration method is applicable to both linear and non-linearsystems. The non-linear analysis is approximated by analyses of a succession of differentlinear systems. In other words, the responses of the system are computed for a short timeinterval assuming a linear system having the same properties determined at the start ofthe interval. Before proceeding to the next time interval, the properties are determinedso that they are consistent with the state of displacement and stress at that time.The direct step-by-step integration procedure developed by Wilson et al. (1973) wasemployed in PILE3D to solve the dynamic equation Eq.8.4.Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 136The equation Eq.8.4 is solved by an incremental form[Mj{} + [Cj{ii} + [K]{v} = \u2014[M]{I} z..i,(t) (8.5)Since Eq. 8.5 is used for solving the incremental values of dynamic response, thedynamic equilibrium should be ensured by checking equation Eq. 8.4 after each step ofintegration. During the dynamic analysis unbalanced force { L.P}unbal is computed aftereach integration step{J.P}unbal = \u2014 [C]{)} \u2014 [K]{v} \u2014 [M]{I} . (t) (8.6)This unbalanced force is then added to the right hand side of equation Eq. 8.5 in thenext step of integration to satisfy dynamic equilibrium.8.4 Non-linear analysisTypical relationships between shear stress and strain at different strain amplitudes underdynamic cyclic loading is shown in Figure 8.1. Firstly the shear stress increases with theshear strain non-linearly. Secondly the loading-unloading curve forms a hysteresis ioop.The shape of the T\u2014 7 curve determines the degree of reduction of shear modulus withshear strain, and the area of the hysteresis ioop represents the amount of strain energydissipated during the cycle. The dissipated energy implies the degree of material damping at this strain level.In a dynamic analysis involving hysteretic non-linearity, a rigorous method of analysisin modelling the shear stress-strain behaviour is to follow the actual loading-unloadingreloading curve. This method has been successfully applied in 1-D ground motion analyses (Finn et al., 1977, Lee and Finn, 1978) and 2-D plane strain analyses (Finn et aL,1986). However this method requires updating the tangential shear modulus for all soilChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Stru ct tire Interaction 137stressstrainFigure 8.1: Hysteretic stress - strain relationships at different strain amplitudeselements and building up stiffness matrix in every time step of integration. This procedure is too time-consuming for 3-D analysis.The equivalent linear method is employed in PILE3D to model the soil non-linearhysteretic behaviour. The equivalent linear method was initially proposed by Seed andIdriss (1967), and it has been widely accepted and used in soil dynamic analyses. Thespirit of this method is that the hysteretic behaviour of soil can be approximated by aset of effective shear moduli and viscous damping which are compatible with the levelsof shear strains. Figure 8.2(a) shows typical relationships between the ratio G\/Gmaxeffective shear modulus G over the shear modulus at very low strain Gmax, 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 Seismic Soil-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 \u2014\u2014 0\u2022 Hardin (1965) o24 \u2014 o Drnevich. Hall and Richort 0966 -_______________0 Matnishito. Kishido and Ky00967)\u2022 Silver and Seed (1969) \u2014Donovan (1969)20\u2014 V Macdin and Drnevch (1970) \u2014 7V Kishido and Takono 0970) 7\/1\u20ac,_\u2014.\u2018.\u2014i.I? \u2018k \/6 .\u2018,-v \u2018cito- - K,-. to-aShea: Swain-percentChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 139The equivalent linear method has been applied in the computer code SHAKE (Schnable et al., 1972) for 1-D ground motion analyses and QUAD-4 (Idriss, 1974) for 2-Dplane strain analyses. Here the procedures of this method are incorporated in the quasi-3D dynamic pile-soil interaction analysis. However they are used with some modifications.In SHAKE, analysis is performed using initial shear moduli and dampings for theentire input acceleration record; then a second analysis is performed for the same acceleration record using constant shear moduli and dampings which are compatible tothe effective shear strains obtained from the first analysis. In other words, the shearmodulus and damping are determined according to the shear strain using curves such asthose shown in Figure 8.2. Iteration process is used to achieve the compatibility betweenmodulus, damping and shear strain. Analyses cease when the differences between themoduli and dampings in the two subsequent analyses are with the desired given values.There is a disadvantage of the equivalent linear method used in SHAKE. A set of constant values of shear modulus and damping is usually not appropriate to represent thenon-linear behaviour of soil within the whole time domain. Especially under earthquaketype of loading the level of shear strain usually changes very much from the beginning,through the middle to the end of the time domain. The common-used criterion of takingthe effective shear strain equal to 65% of the maximum shear strain may not applicablein many cases.A procedure of applying the equivalent method based on the periodical level of shearstrain is proposed to overcome this problem. This procedure requires no iteration process. Figure 8.3 shows the idea of this method. In this procedure an assumption is madethat the effective shear moduli and damping in a period of time \/.T, remain constantChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 140responses, Atimeti ti+1 tjFigure 8.3: The principle of modified equivalent linear methodand are determined by the peak shear strain in the previous period T_1 based on relationships such as those shown in Figure 8.2. The shear moduli and damping ratios inall soil elements are then determined for the period of time. PILE3D has the capabilityallowing the use of different curves input by users.The selection of the length of time period is based on the fundamental frequency ofthe input earthquake motion, and this length can be selected by users. The length oftime period is selected neither too short to take too much computational time nor toolong to lose the accuracy of the analysis. A length between 0.4 second and 1.0 second isgood for the earthquake records used in the analysis presented in the thesis.AA,AAiAtj+1Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 141shear stress r normal stress uTf___________arstrain rna1 strainTf 7shear yielding tension 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, shear yielding of soil materialoccurs when the shear stress exceeds the shear strength of the material. When yielding occurs the soil element may develop significantly large shear deformation under verysmall increment of shear stress. In other words the shear modulus of the soil is significantly reduced due to shear yielding. The shear yielding 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 the principle ofthis procedure. The shear stress used for this purpose is the maximum shear stress inthe vertical plane; while the shear strength r1 is input by users according to the type ofsoil. The shear strength r1 may be determined by c+o0 . tanq5 ( c is the cohesion, q isthe friction angle, and is the effective overburden stress).Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 142On the other hand, cracking or tensile failure may occur when the total lateral normal stress of a soil element exceeds the tensile strength of soil. For sands any tensilestress developing may lead to a tensile failure. When a tensile failure occurs soil hasno resistance. A no-tension model shown in Figure 8.4(b) is used to accommodate thisphenomenon. One of the criteria built in PILE3D for checking tension is0dynamc > O + 0j (8.7)where o is the static lateral stress of the soil element, and 0j is the tensile strengthof soil, and0dynamic is the dynamic lateral stress computed during the analysis.The options of checking shear yielding and allowing no-tension to develop in soils areavailable only when a dynamic step-by-step integration procedure is used. These featuresovercome the difficulties that may be encountered in a pure elastic analysis. In a pureelastic analysis no controls on the shear yielding or the tensile failure can be enforced.The application of these features has been found very effective 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 parameters are used.Shear modulus of soil The key parameter required for this analysis is the low-strainshear modulus of soil. The low strain shear modulus is also called the maximumshear modulus. The Gmax can be determined accurately by measuring the shear waveChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 143velocity V3 of soil, thenGmax = p V2= 7\/g V2 (8.8)where p is the 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 equations may beapplied to estimate the low strain shear modulus Gmax. A useful empirical equationproposed by Hardin and Drnevich (1972) is of the formGm = 32O.8(2.9\u00f7 (\u00b0(OCR\u2019 (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 and clays. For claysit is recommended that the equation be used when the void ratio is in the range of 0.6to 1.5. The variation of constant k with plasticity index PT is given in Table 8.8.For most practical purposes, Seed and Idriss (1970) proposed another useful expression for estimating Gmax of granular soils (sands and gravels)Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 144Table 8.8: Relationship between Hardin and Drnevich constant 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 0.48 0.50Gma, = 1000(k2)maw(\u00f8)\u00b05 inpsf 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 )6o values was proposed by Seedet al. (1986)(k2)ma = 20(N1) (8.11)For clays the maximum shear modulus may be calculated based on the undrainedshear strength, S,, using the equationGmcut = Kctay Su (8.12)in which Kday is a constant for a given clay. From Seed and Idriss (1970) the typicalvalues of Kciay vary between 1000 to 3000.Damping ratio and other parameters The second input required in the analysis isthe relationship between damping ratio and the shear strain of soil. Although variationChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 145j0-4 3- 3 ioCyclic Shear Strain t rFigure 8.5: Comparison of damping ratios for sands and gravelly soils (after Seed et al.,1986)may be expected for specific soil material, 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 Figure 8.5. Other parameters required in the analysis are Poisson\u2019s ratio andshear strength of soil.8.7 Aspects relative to analysis of pile groupThe program PILE3D has the capability of simulating the dynamic response of a pilegroup supporting a rigid pile cap. Analysis is done in a fully coupled manner. The rigidpile cap is represented by a concentrated mass at the centre of gravity of the pile cap,and the mass is rigidly connected to the piles by a massless rigid bar.Figure 8.6 shows the principle for modelling a pile cap in PILE3D. The pile cap ismodelled as a rigid body in the analysis. The motions of the pile cap are represented20\u2018.. I I I0 Oato lot Gravels and Gravelly So\u00e1lSINs invhqahon\u2014 Averoq. \u2018Vlue for Sands .\u2014;-A \u2014j\u2014 --Uir and .ower eouncs Idriss \/p ,\u2018\u2014 ,EEE\u00b0E-- .\u2014\u2014\u201410_IChapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 146rocking impedanceFigure 8.6: A diagram showing the representation of pile group supporting structureby the motions of the pile nodes which are connected to the bottom of the pile cap.These nodes are called the pile head nodes. The pile head nodes have identical defiections (translations and rotations in the vertical plane) as the pile cap does. Therefore aprincipal restraining pile head node can be used to represent the motions of all pile headnodes or of the pile cap. In the finite element analysis the translation and rotation ofall pile head nodes in the group are restrained to the principal pile head node. In thisway, identical defiections of all pile head nodes are achieved. The mass of the pile capis connected to the principal pile head node so that all pile head nodes share the samemass, stiffness and defiections.On the other hand, under seismic loading 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 because the vertical and horizontal motions are uncoupled.Chapter 8. Non-Linear Analysis of Seismic Soil-Pile-Structure Interaction 147The rocking impedance of the the pile group is included in the analysis by using thefollowing procedures.The rocking stiffness and damping is updated at selected times during the horizontalmode analysis by using PILE3D in the vertical mode. The procedure for computingrocking impedance Krr has been given in chapter 7. The rocking impedance is computedusing the current values of strain dependent shear moduli and damping ratio of soil. Thecurrent rocking impedance (stiffness and damping) is then transferred to the pile cap asrotational stiffness and damping.The rocking impedance of pile group is important to the dynamic response under horizontal shaking, and it has been properly treated in the dynamic analysis of pile groupusing PILE3D.Chapter 9Analyses of Centrifuge Tests of Pile Foundations9.1 IntroductionIn this chapter the proposed quasi-3D finite element method of analysis is used to analyze the non-linear response of model pile foundations subjecting to horizontal loadingin centrifuge tests. Centrifuge tests on a single pile and a 2x2 pile group are analyzedusing the computer program PILE3D. The computed results are compared with thosemeasured in the centrifuge tests. The ability of the proposed model for simulating thenon-linear response of pile foundations is evaluated.During strong shaking, the shear modulus and damping ratio of soil medium changewith time, which causes corresponding changes in the dynamic impedances of pile foundations. These variations in dynamic impedances with time during strong shaking areevaluated for the model pile foundations. This is the first time that the time-histories ofdynamic impedances have been calculated.9.2 Dynamic analysis of centrifuge test of a single pile9.2.1 Description of centrifuge test on a single pileA centrifuge test on a single pile was carried out at the California Institute of Technology (Caltech) by B. Gohi (1991). Detailed data on the centrifuge test are given by Gohi148Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 149(1991). Details of the test may also be found in a paper by Finn and Gohi (1987). Acentrifuge acceleration of 60g was used for the test.Figure 9.1 shows the soil-pile-structure system used for the single pile test. The effectof the super-structure was simulated by clamping a rigid mass at the pile head. Thepile head mass was instrumented using a non-contact photovoltaic displacement transducer and an Entran miniature accelerometer. The locations of the accelerometer andlight emitting diode (L.E.D.) used by the displacement sensor are shown in Figure 9.1.The pile head displacements were measured with respect to the moving base of the soilcontainer. The prototype parameters of the single pile test is shown in Figure 9.2. Themodel pile has a unit weight of 74.7 kN\/m3. The prototype 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 loose sand with a void ratio co = 0.78and a mass density p = 1.50Mg\/rn3 The friction angle of the sand was determined tobe 300. Gohi (1991) has showed that the low strain shear moduli of the sand foundationvary as the square root of the depth, and that they can be quantitatively evaluated byusing the Hardin and Black (1968) equation()\u2014 e01 I NO.5V\u2018 + e0where e0 is the in-situ void ratio of the sand and o is the mean normal effective confiningstress in kPa. The mean normal effective confining stress is computed from the effectivevertical stress oChapter 9. Analyses of Centrifuge Tests of Pile FoundationsLED.\u2014_c1 __[216.5 L_......b... J-_\u2014_ AccelerometerSoil surface \u2014.. +No.2 iNo.4Location ofStrain gaugesNo. 6No. 3Base of centrifuge \u2014\u2014 Pile tipbucket_________________________________4seismic motion150No. I+ No.5\u2022 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\/mIcg 53.11 kN.se.rhcg= 0.99 m0.99 m1\u201912 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 of Pile Foundations 1511-1-2K00m O\u2022j (9.2)Using e0 0.78 and a lateral stress coefficient K0 0.4 for the loose sand, Gohi showedthat measured shear wave velocities and those computed by using the Hardin and Blackequation are in good agreement.A horizontal acceleration motion is input at the base of 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 at the top of the super-structure wererecorded. Dynamic moments at the selected locations along the pile shown in Figure 9.1were also recorded during the test.The computed Fourier amplitude ratios of the pile head response and the free fieldmotion with respect to the input motion are given in Figure 9.3(a) and Figure 9.3(b),respectively. The natural frequency of the free field 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 prototype scale. Figure 9.4 showsthe finite element model used in the analysis. The sand deposit is divided into 11 layers. A decreasing thickness of layer is used toward the soil surface. This arrangementwould allow more detailed modelling of the stress and strain field where lateral soil-pileChapter 9. Analyses of Centrifuge Tests of Pile Foundations 152A - -__A AV - ..--. T!f-w\u2019\u2019? \u2018. .\u2018\u2018jA-0 2 4 6 8Frequency (Hz) -Frequency (Hz)Figure 9.3: Computed Fourier amplitude ratios (a) pile amplitude 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\u2018--00 2 4 6 8 10Chapter 9. Analyses of Centrifuge Tests of Pile FoundationsGEO .SCRL 0I I15 30153\/II I RA.SYMMETRY I I\u2019HSHAKING DIRECTIONFigure 9.4: The finite element modeffing of centrifuge testinteraction is strongest. The pile is modelled using 15 beam elements including 5 elements above the soil surface. The super-structure mass is treated as a rigid body, andits motion is represented by a concentrated mass at the centre of gravity. A very stiffbeam element with fiexural rigidity 1000 times that of the pile was used to connect themass and the pile head. The motion of any point in the rigid super-structure can bedetermined according to its geometric relationship to the reference point, the pile head.The present finite element mesh consists of 666 nodes and 456 elements.The finite element analysis was carried out in the time domain. The non-linear analysis was performed to account for the changes of shear moduli and damping ratios dueto dynamic shear strains. According to Gohi (1991), the shear-strain dependency of theshear modulus and damping ratio used in the analysis is shown in Figure 9.5. The maid-mum shear modulus Gm was calculated according to Eq. 9.1. The maximum dampingratio of the loose sand was taken as 25% following Gohi (1991).\/\/AXIS OF \/\/\/\/W\/ \/Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 154LllLEiZIfl:* DAPI11GI I I IIllI \u2014i\u2014i. \u2022\u2014\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2014\u2022\u2014\u2022i\u2022\u2022\u2022\u2022\u2022\u2022\u2014\u2014t\u2014\u2022\u2014\u2014i\u2022 \u2014 I I 111110.001 0.01 0.1 1SHEAR STRAIN (%)Figure 9.5: The relationships between shear modulus, damping and the shear strain forthe loose sandResults of analysis The computed and measured accelerations in the free field andat the pile head are shown in Figure 9.6 and Figure 9.7, respectively. There is goodagreement between the measured and the computed accelerations.The computed and measured time histories of displacements at the top of the structureare plotted in Figure 9.8. The computed displacements are smaller than the measureddisplacements in the first 10 sec of motion. 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 of moments in the pile at the soil surface and at a depthChapter 9. Analyses of Centrifuge Tests of Pile Foundations 1550.3MEASURED\u2022 COMPUTEDii01z \u2022 \u2022.4 4\u2022I\u2014$1 .I \u2022 a \u2022\u2022 \u20224ck . \u2018 St I aLi_i \u2018aw :() .C)\u20140.3 i i i i i i i i i i0.00 10.00 20.00 30.00TIME ( SEC )Figure 9.6: The computed versus measured acceleration response at the free fieldof 3 m (near point of maximum moment) are plotted against the measured time historiesin Figure 9.9 and Figure 9.10, respectively. Satisfactory agreement between the computed and the measured moments is observed in the range of larger moments.The distributions of the computed and measured bending moments along the pile atthe instant of peak pile deflection are shown in Figure 9.11. The computed moments agreevery well with the measured moments. The bending moments increase to a maximum ata depth of 3.5 diameters, and then decrease to zero at a depth around 12.5 diameters.The moments along the pile have same signs at any instant time, suggesting that theinertial interaction caused by the pile head mass was dominant and the pile was vibratingin its first mode. The measured peak moment is 325 kN.m, whereas the computed peakmoment is 344 kN.m with an error of 6% overestimate.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 1560.10z\u20140.00\u20140.100.300.200.10z00.00\u20140.10< \u20140.20\u20140.300.00 10.00 20.00TIME ( SEC )Figure 9.7: The computed versus measured acceleration response at the pile head30.00TIME ( SEC )Figure 9.8: The computed versus measured displacement response at the top of thestructureChapter 9. Analyses of Centrifuge Tests of Pile FoundationsFigure 9.9: The computed versus measured moment response at the soil surfacezFz0zzaD400157z\u2018 200Fzw0z\u201420CzbJaD\u201440C0.00 10.00 20.00TIME ( SEC )MEASURED:------- COMPUTED.I I I I I I30.005002500\u2014250\u20145000: MEASURED- COMPUTEDI I I I I I I I I I I 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 depth D=3m: SOIL SURFACA\u2022qboeoeo COMPUTEDo\u00b0\u00b0\u00b0 MEASUREDI I I I I I IBENDING MOMENT (KN.M)Figure 9.11: The computed versus measured moment distribution of the pile at peak piledeflectionUnder earthquake loads the shear moduli and damping ratios of soil vary with boththe time and the location. The proposed non-linear analysis is capable of tracing thevariation of soil properties at any instant during shaking. As examples of this feature,distributions of soil shear moduli at depths of 0.25 m and 2.10 in are plotted in Figure9.12 at an instant t=12.58 sec, and in Figure 9.13 at another instant t=17.11 sec. It isseen that at a certain depth such as 2.10 m the soil shear moduli in the near field of thepile are much less than the shear moduli in the far field.At any instant during shaking, a set of soil properties are determined for each soilelement. Therefore the dynamic impedances of the pile foundation can be evaluatedcorresponding to soil properties at this instant. The variation of dynamic impedances ofpile foundations with time during shaking is evaluated. These non-linear pile impedancesChapter 9. Analyses of Centrifuge Tests of Pile Foundations 158200\u20142-\u20144-\u20146--\u20148-\u201410-\u201412\u2014400 1009.2.3 Non-linear pile impedancesChapter 9. Analyses of Centrifuge Tests of Pile Foundations 159single pile at 12.58 secinitial shear modulus 12945 kPa(a) at depth 0.25 minitial shear modulus 36610 kPa(b) at depth 2.10 mFigure 9.12: 3-D plots of the distribution of shear moduli at t=12.58 secChaPt& 9. AnalYses oftfuge Tests of Pile Foundations 160sirg pi\u2019e at 17.ll secjnitia shear mod0l 1 294 kPa(a) at depth 0.25 minitial shear modU5 36610 kpaC.2(J)(b) atdepth2.l\u00b0mjgute 9.13: 3-D plots of the distnbutbon of sheat oduli at t1711 SecChapter 9. Analyses of Centrifuge Tests of Pile Foundations 161include effect of pile-soil kinematic interaction and inertial interaction from the superstructure.Dynamic impedances under an earthquake motion The concept of dynamicimpedance is formulated to reflect the complex force-deflection relationship of a pile foundation under harmonic pile head loading at a specific excitation frequency. This conceptis well suited for the dynamic analysis of a machine foundation which is normally excitedat specific frequencies. However under earthquake excitation, the pile foundations donot usually vibrate at a constant frequency. Strictly speaking, the concept of dynamicimpedance does not apply directly in dynamic analyses of pile foundations involvingearthquake motions.Conventionally dynamic impedances are used in sub-structuring analysis of pile supporting structures. Pile head impedances are usually computed at the dominant frequency of excitation appropriate to an earthquake motion. In normal design prior to anearthquake event, it would appear logical to select the fundamental frequency of the pileas the frequency of interest, provided the earthquake motions are expected to contain significant frequency contents around the fundamental frequency of the pile. An alternativetechnique is to explore variations of dynamic stiffness and damping of the pile foundationin the frequency range of an earthquake motion, such as from 0 Hz to 10 Hz. The latterapproach is used in this study to look into the characteristics of dynamic stiffness anddamping as functions of frequency.The dynamic impedances (stiffness and damping Ci,) of the single pile are computed using program PlUMP. These impedances are determined at the ground surface.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 162Excitation frequencies of 1.91 Hz, 6.0 Hz and 10 Hz are selected to explore the effect ofexcitation frequency on dynamic impedances of pile foundations. Frequency is limited tothe 0-10 Hz range considered appropriate for seismic loading.Stiffnesses of the pile foundation during shaking At the excitation frequencyf=1.91 Hz, the dynamic stiffnesses of the pile decrease dramatically as the level of shaking increases (Figure 9.14). The dynamic stiffnesses experienced their lowest values inthe 10 to 14 seconds range when the maximum accelerations occurred at the pile head.It can be seen that the translational stiffness ku,, decreased more than the rotationalstiffness kee or 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 of 125,000 kN\/rad. ke showed theleast effect of shear strain. It decreased to 138,000 kN.m\/rad which is 63.6% of its initialstiffness of 217,000 kN.m\/rad. The stiffnesses rebounded when the level of shaking decreased with time. Representative values of the pile stiffnesses k,,,, ,k,,9 and k66 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 in Figure 9.14. Thesestiffnesses are 27.6%, 52% and 73.7% of their original stiffnesses.The variations of translational stiffness k,,,, and rotational stiffness k66 with time atdifferent excitation frequencies are shown in Figure 9.15. It can be seen that the excitation frequency has little influence on the dynamic stiffness of the pile foundation forfrequency less than 10 Hz. The dynamic stiffnesses of the pile foundation may be considered independent of frequency under seismic loading.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 163250000 -200000 kee (kN.m\/rd)150000 -:100000 k8 (kN\/rczct50000 -k (kN\/m)o\u2014 1111111111 1111111 I 111111110 5 10 15 20 25 30TIME ( SEC )Figure 9.14: Variation of stiffnesses k9,k99 of the single pile at f=1.91 Hz250000 -DDD frequency 1.91 Hza frequency 6.00 Hz200000frequency 10.0 Hz150000-s5I I I I I I I I I I I I II I I I I I I I I I0 5 10 15 20 25 30TIME ( SEC )Figure 9.15: Variation of stiffnesses ku,., and k with time under different excitationfrequencyChapter 9. Analyses of Centrifuge Tests of Pile Foundations 16450000frequency 1.91 Hzfrequency 6.00 HzI frequency 10.0 Hz40000 -30000 A A\/20000-100001.1111111111111111 I 111111111 I I I0 5 10 15 20 25 30TIME(SEC)Figure 9.16: Variation of translational damping C,,, versus time under different frequencyDampings of the pile foundation during shaking The variatin of damping withthe excitation frequency is different from that of the stiffness. The values of dampingsusually increase with the excitation frequency due to the frequency-dependent radiationdamping. Figure 9.16 shows typical variations of the translational damping C,,,, versustime as the excitation frequency increases. It is seen that the amount of incrementaldamping due to the change of frequency is roughly proportional to the amount of incremental frequency. Same patterns are observed for the cross-coupling damping C,,9 andthe rotational damping C98. Under these observations and due to the fact that hystereticdamping is independent of frequency w, the following expression is proposed to representdampings C3 of pile foundations under seismic loadingC, \u2014 C2 + Rw\u00b075 (9.3)\/ AqJf\/where C represents the frequency-independent hysteretic damping, and R.,, is theChapter 9. Analyses of Centrifuge Tests of Pile Foundations 16525000 -_________________________\u00b0-\u00b0\u00b0 translationcross\u2014couplingrotation20000C\u2022g 15000-Q.. IIIIIIIIIIIIIIIIIIIIIIIII;III0 5 10 15 20 25 30TIME(SEC)Figure 9.17: Variation of hysteretic dampings C, C9 and C of the single pileradiation damping constant, w is the excitation frequency w = 2irfThe hysteretic damping C were determined by using very low excitation frequency(such as f=0.01 Hz) when computing impedances. Figure 9.17 shows the variations ofhysteretic damping contributions C, (translation), C (cross-coupling) and C (rotation) with time. The rotational hysteretic damping is less sensitive to the change ofshaking level than the other two hysteretic dampings. Representative values of 12,000kN\/m, 11,000 kN\/rad, and 12,000 kN.m\/rad may be selected to represent the hystereticdampings of Crn,, C,,9 and C99, respectively.The radiation damping constants are computed using Eq. 9.3. As expected theradiation damping constants H1, fail into a relatively narrow zone in the time domainwhen the frequency changes from 1.91 Hz to 10 Hz. Figure 9.18 shows the translationalradiation damping constant R,,,, with time for the three frequencies analyzed. A value of>>-2000.g 1000C0.900850 of R, is typical for representing the translational radiation damping. Applying thesame concept, a value of 600 is found typical for both R1,6 and R60.The results of studies on pile impedances are summarized in Table 9.1. Since thedamping C3 are functions of excitation frequency the resulting damping coefficients(c,j = Cq\/w) are different at different excitation frequencies. 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 the corresponding damping coefficients change to 741 kN.sec\/m, 604 kN.sec\/rad and 635 kN.m.sec\/rad if theexcitation frequency changes to 31.41 rad\/sec (f = 5 Hz).Effect of structural mass on pile impedances In the analysis described early theeffect of the structural mass on the dynamic impedances has been included. The heavystructural mass would significantly increase the level of non-linearity of soil in the nearChapter 9. Analyses of Centrifuge Tests of Pile Foundations3000166: frequency 1.91 Hz: -- frequency 6.00 Hz: 000D frequency 10.0 Hz.I\u2014I I I I I I I I I I I I I I I I I I I III I I I I0 5 10 15 20 25 30TIME ( SEC )Figure 9.18: Variation of radiation damping constant R. at different frequenciesChapter 9. Analyses of Centrifuge Tests of Pile Foundations 167Table 9.9: Parameters of dynamic impedances of single pileAverage Minimum Hysteretic Radiation dampingstiffness stiffness damping C constant R3k(kN\/m) 40,000 20,000C(kN\/m) 12,000 850ke(kN\/rad) 65,000 45,000C9(kN\/rad) 11,000 600kee(kN.m\/\u2019rad) 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 behaviour ofsoil can be adequately modelled.Studies were conducted to explore the effect of structural 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 mass on 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 of the pile foundation with the full structural mass are muchless than those without structural mass. This is because soils in the near field of the pileare much more strongly mobilized when the structural mass is present during shaking.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 168translational stiffness k,,%kN\/m)200000o o o o o full structural massno structural mass150000 -(I)U)w100000UU(I)50000,IIIlIIIIIIIIIIIIIIIIIIIIIIII0 5 10 15 20 25 30TIME ( SEC )rotational stiffness k00(kN.m\/rad)250000oOo 0 full structural massno structural mass200000ci)UzUU\u201c 150000100000 I I I I I I I I I I I I I I I ( I I I I I I0 5 10 15 20 25 30TIME ( SEC )Figure 9.19: Comparison of dynamic stiffnesses of pile foundations with full structuralmass and without structural massChapter 9. Analyses of Centrifuge Tests of Pile Foundations 1699.2.4 Computational timesThe non-linear analysis was carried out in the time domain. The average CPU time usinga PC-486(33MHz) computer needed to complete one step integration is 7.0 sec for thefinite element grid shown in Figure 9.4, and 3 hours of CPU time are required for an inputrecord of 1550 steps. The computation time is much shorter for a linear elastic analysis when the shear moduli of soil foundation remain constant through the tMime domain.The average computational time for computing the dynamic impedances using PlUMPis 50 seconds for one set of soil properties. The total computational time required to generate curves as shown in Figures 9.14 is about 30 minutes.9.3 Dynamic analysis of centrifuge test of a pile group9.3.1 Description of centrifuge test on 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 setup is shown in Figure 9.20. The pileswere set in a 2x2 arrangement at a centre to centre spacing of 2 pile diameters or 1.14m. The properties of piles in the group are same as those of the single pile. Two of thefour piles were instrumented. The piles in the group were rigidly clamped to a stiff pilecap, and four cylindrical masses were bolted to the cap at locations shown in the Figure9.20. The top of the pile cap was the location where the four pile heads were clamped tothe structural mass. A pile cap accelerometer and a displacement L.E.D. were placed atlocations shown in the figure. The displacement L.E.D. was located 46 mm (2.76 m inprototype scale) above the four pile heads.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 170After being converted to the prototype scale, the pile cap has a mass of 220.64kN.sec2\/mand a mass moment of inertia about centre of gravity I = 715.39 kN.sec2.m.The centre of gravity 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 dry dense sand with a void ratioe0=0.57and a mass density p = 1.70Mg\/rn3 The friction angle of the dense sand was 45\u00b0. Gohl(1991) showed that the small strain shear modulus Gmcza, 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 earthquake acceleration motion. Peakaccelerations of up to 0.14g were applied to the base of the foundation and were dominated by frequencies in the range of 0 to 5 Hz. The free field accelerations were stronglyamplified through the sand deposit to values of up to 0.26g at the surface. Pile capaccelerations of up to 0.24g and displacements up to 60 mm were recorded during thetest. Residual displacements of up to 10 mm remained at the end of earthquake motion.9.3.2 Dynamic analysis of the pile groupIn the analysis, the pile cap was treated as a rigid body. Hence all pile heads, which wererigidly connected to the bottom of the pile cap, bad identical translational and rotationaldefiections as the pile cap. In the finite element analysis, restrained nodes are used toimpose the identical motions on the nodes. This technique of restraining nodes has beendescribed in the earlier chapter and was applied in the analysis of pile groups.Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 171PILE CAPCAP MASSESHEAD CLAMPACCELEROMETERSURFACEc PILE CAP MASS NO. 2 (DISPLACEMENTLED LOCATED ON TOP OF MASS)CYLINDRICALMASS BOLTEDTO PILE CAP(14.3mm 0, 38.1mm HIGH)CAP MASS NO.3ACCELEROMETERSLOTS TO ALLOWVARIABLE PILESPACINGS0 20mmI -ISCALESIDE VIEWFigure 9.20: The layout of centrifuge test for 4-pile group (after Gohi, 1991)Chapter 9. Analyses of Centrifuge Tests of Pile Foundations 172Another important issue in the analysis of pile groups is to incorporate rocking stiffness and damping into the analysis of horizontal motions. In the proposed quasi-3Danalysis, the horizontal motions are uncoupled with the vertical motions. The rockingstiffness induced by the vertical resistance of piles can not included directly in the quasi-3D analysis of horizontal motions. However the rocking impedances (resistance) maysignificantly restrain the rotational motions of the pile heads, and the effect of rocking impedances on the horizontal motions must be taken into account. The rockingimpedances may be even larger than the rotational impedances from the pile heads, especially when the pile spacing is large.The rocking impedances of the pile cap are included in the analysis of pile groups.They are computed by performing a separate quasi-3D analysis in the vertical mode during the analysis in the horizontal mode. This impedance calculation is made using thecurrent values of strain dependent shear moduli and damping ratios of soil. The theoryand procedures for computing such rocking impedances have been given in chapter 7.After the rocking impedances were computed for each time period, they are transferredto the pile cap as rotational stiffness and damping.The finite element mesh used for the pile group analysis is shown in Figure 9.21. Themesh used in the present analysis has 947 nodes and 691 elements. The sand foundationwas modelled by 11 horizontal layers with a smaller thickness toward the sand surface.Each pile was modelled using 15 beam elements including 5 elements for the part abovesoil surface. A very stiff massless beam element was used to connect the structural massto the pile heads.The analysis was carried out using the non-linear option to simulate the changes ofChapter 9. Analyses of Centrifuge Tests of Pile Foundations 173GEO.SCRLEd\u2014lLrA,L JH-4SHAKING DIRECTIONFigure 9.21: Finite element modelling of the 2x2 pile groupshear moduli and damping of soil with the shear strain. According to Gohi (1991), theshear-strain dependency of current shear moduli G and damping ratio D of the densesand are shown in Figure 9.22. The small strain shear moduli Gm are functions of theoverburden stresses and are estimated according to Eq. 9.1. The maximum hystereticdamping ratios Dm of the sand foundation are taken as 25 % following Gohi (1991).Results of analysis 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 of the 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 of Centrifuge Tests of Pile Foundations 174075 -MODULUS DAMPING050- -x -025 -c.cc ; 10.0001 0.001 0.01 0.1 1SHEAR STRAIN (%)Figure 9.22: The relationships between shear modulus, damping and the shear strain forthe dense sanddisplacement response showed a permanent residual displacement of about 10 mm at theend of earthquake motion. This is because the analysis is carried out using the equivalentlinear elastic approach.The computed moment time history in the instrumented pile at a depth of 2.63 min the area of maximum moment is plotted against the measured moment time 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 of Pile Foundations\u20140.0750 5 10 15 20 25 30TIME ( SEC )1750.40.2z.00.0w-JwC-)-0.2\u20140.4- MEASUREDCOMPUTED4Mb\u00f8I I I I I I I I I I I I I I I I I I I I I I I I0 5 10 15 20 25TIME ( SEC )Figure 9.23: The computed versus measured acceleration responses at pile cap300.0500.025\u20140.000\u20140.025\u20140.050 -MEASUREDCOMPUTEDi\u2014\u2014-t----- ---\u2014iI IFigure 9.24: The computed versus measured displacement at top of structural massChapter 9. Analyses of Centrifuge Tests of Pile Foundations 176MEASUREDCOMPUTED\u2014. 200z Uiu0 U0 - :\u2014100\u2018. .,.\u2018 \u2018. \u2018.\u2014200\u2014300 i i0TIME ( SEC )Figure 9.25: The computed versus measured moment at depth D=2.63 m111110 200 400MOMENT (KN.M)30011111111 I LIII 11111111111 I III5 10 15 20 25 300w20\u20142-\u20144--\u20146\u2014\u20148\u2014\u201410_\u2014400 \u2014200BENDING:__-. MEASUREDQ o.\u20ac COMPUTEDFigure 9.26: Distribution of moments at peak pile cap displacementChapter 9. Analyses of Centrifuge Tests of Pile Foundations 1776.OE+5 \u20144.OE+5 \u20142.OE+50.OE\u2014t\u20140 \u20140 5 10 15 20 25 30TIME ( SEC )Figure 9.27: Variation of stiffnesses k8 of the 4-pile group at ft=1.91 Hz9.3.3 Non-linear impedances of the 4-pile groupApplying the same technique as used in the analysis of single pile, dynamic impedances ofthe 4-pile group are evaluated. To illustrate the results of this analysis, the translationalstiffness k and the cross-coupling stiffness k06 of the 4-pile group are presented.The variations of stiffnesses and k9 of 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, and k9 were reduced dramatically at times when strong ground motions occurred. decreased to 80,000 kN\/m from 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 about 17% and 38% of their initial stiffnesses.Part IIISummary and Suggestions for Future Work178Chapter 10Summary and Suggestions for Future Work10.1 Dynamic thrusts on rigid wallsApproximate methods for determining the dynamic thrusts against rigid walls subjectedto horizontal dynamic loads were presented. Analytical solutions were obtained for thedynamic thrusts against rigid walls.Calibration was made by comparing the approximate 1-g static solutions with Wood\u2019sexact l-g static solution. Results showed that the proposed model using the shear beamanalogy produces best approximation to solution of the rigid-wall problem. The computed total static thrusts are about 5% less than those from Wood\u2019s solution for anyLull ratio, in which H is the height of the wall and L is the half length of the soil backfillconfined by two vertical rigid walls.Dynamic analyses have been performed for wall-soil systems with uniform backfillsand subjected to both sinusoidal motion and earthquake motion. The wall-soil systemswith semi-infinite backfihls are approximated by 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 the peak dynamic thrusts are 2.4pHAmaa,for L\/H=5.O and 3.OpH2Am for L\/H=1.5. The static thrusts are 1.OpH2Ama, for179Chapter 10. Summary and Suggestions for Future Work 180L\/H=5.O and 0.86pH2Amaa, for L\/H=1.5. Therefore, their corresponding dynamic amplification factors are 2.4 for L\/H=5.0 and 3.5 for L\/H=1.5. These results stronglysuggest that the use of a static force for dynamic loading may result in serious underestimate of the dynamic thrusts against rigid walls.The earthquake motions are represented by the scaled records of the El Centro andthe Loma Prieta ground motions. The results of analyses show that the peak dynamicthrusts are 1.3OpH2Amaa, for L\/H=5.0 and 1.38pH2Ama for L\/H=1.5. Their corresponding dynamic amplification factors are about 1.3 for L\/H=5.0 and 1.6 for L\/H=1.5. Thedynamic amplification factors under earthquake motions are about 50% of that undersinusoidal motions.Finite element formulations for evaluating the dynamic thrusts against rigid wallswith non-homogeneous soil profiles have been presented. Comparisons with close-formsolutions showed that the dynamic thrusts can be predicted accurately using the proposed finite element method of analysis.An efficient computer program SPAW has been developed for determining the dynamic thrusts and moments against rigid walls. The program was designed for dynamicanalysis of rigid walls with arbitrary non-homogeneous soil layers under sinusoidal motions and earthquake motions. A finite element mesh necessary for the analysis has beenembedded in SPAW so that only the properties of soil layers are required for input. Thecomputational time for a dynamic time-history analysis is only few minutes in a PC48633MHz computer.Chapter 10. Summary and Suggestions for Future Work 181For soil profiles with linear variation of shear modulus with depth, the analyses revealed that the peak dynamic thrust at resonance is L56pH2Amaa, for sinusoidal motions.The peak dynamic thrust reduces to 1.OpH2Aa,, for the scaled El Centro input. For soilprofiles with parabolic variation of shear modulus with depth, the peak dynamic thrustat resonance is 1.87pH2Ama for sinusoidal motions. The peak dynamic thrust reducesto 1.l8pH2Amax for the El Centro input.Because the static force is about O.7lpH2Amax for linear soil profiles and 0.82pH2Amaxfor parabolic soil profiles. The corresponding dynamic amplification factors are 2.3 forsinusoidal motions and 1.4 for earthquake motions. The dynamic thrust at resonance forsinusoidal motions are about 60% greater than that for earthquake motions with samepeak acceleration.Results from the equivalent linear elastic analyses show that the non-linearity of soilshas significant effects on the dynamic thrusts. For parabolic soil proffles the peak dynamic thrust of a wall-soil system under strong shaking (0.35g) due to non-linear effectis about 25% higher than that obtained using the linear elastic analysis. This result wasobtained using the scaled El Centro input as the base motion. For wall-soil systems thathave different soil profiles or that are subjected to other earthquake motions, the finiteelement program SPAW can be readily applied to evaluate the dynamic response of thesystems.The conclusions made on the study of dynamic thrusts on rigid walls are presentedas follows:Chapter 10. Summary and Suggestions for Future Work 1821. The proposed simplified wave equation has been successfully applied to determinethe dynamic thrusts and moments acting on rigid walls. It implies that the dynamicmotions of soil structure under horizontal motions are mainly governed by the wavesin the horizontal directions.2. It is recommended that the mode superposition method be used to determine thepeak dynamic thrusts against rigid walls. The use of the absolute summation ofmodal thrust is suggested when the response spectrum method 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 be used to represent the peak dynamicthrust. The use of static solution may cause significant error of dynamic thrust especially at resonance. A dynamic analysis is required under dynamic loading.4. The height of the resultant dynamic thrust is generally suggested to be at 0.611above the base of the wall for a wall height of H. However for non-homogeneoussoil profiles, such as linear soil profiles or parabolic soil profiles, the height maydecrease to about 0.511.5. The peak dynamic thrusts become large when the soil profile becomes more uniform.Under sinusoidal motions, the peak dynamic thrusts at resonance are 1.56pH2Am,1.87pH2Amacj, and 2.4OpHAmax for linear, parabolic and uniform soil profiles, respectively. For scaled El Centro input, the peak dynamic thrusts are 1.00pH2Amam,1.l8pH2Amczx and 1.3OpH2Amaz for linear, parabolic and uniform soil profiles, respectively. These results are obtained for wall-soil systems with L\/H=5.0 andA=10%.6. The effect of soil non-linearity on the dynamic thrust seems to be significant forChapter 10. Summary and Suggestions for Future Work t83rigid walls. A 25% increase of peak dynamic thrust may be expected for wall-soilsystems subjected to earthquake motions with peak accelerations in the order ofO.35g, compared to results from a linear elastic analysis.Suggestions For future research, studies should be focused on including the effectsof seismic pore water pressures in a saturated backfill on the dynamic thrusts. Thedevelopment of pore water pressure may significantly reduce the shear moduli of soils.Also one should be aware of that the mechanism of liquefaction in soil layers confined byrigid walls is different from that in a free field. The shear strains of soils are much lessespecially in the area near the wall.10.2 Dynamic analyses of pile foundationsA quasi-3D finite element method of analysis has been proposed to determine dynamicresponse of pile foundations subjected to horizontal loading. The proposed model isbased on a simplified 3-D wave equation. The 3-dimensional dynamic response of soilis simulated by displacements in the horizontal shaking direction. Displacements in thevertical direction and in the horizontal cross-shaking direction are neglected. The use ofthe proposed simplified wave equation greatly saves the computing space and computing time for the finite element analysis. However it maintains adequate accuracy in themodelling of the dynamic response of pile foundations.The proposed quasi-3D theory is first incorporated using the finite element methodin the frequency domain. This formulation is used for the analysis of elastic responseof pile foundations. Elastic solutions of Kaynia and Kausel (1982) have been used tovalidate the proposed model for elastic response. Dynamic impedances of single pilesand 2x2 pile groups have been computed and compared with those obtained by KayniaChapter 10. Summary and Suggestions for Future Work 184and Kausel (1982). Also kinematic interaction factors of single piles have been computedand compared with those obtained by Fan et al. (1991) who used solutions by Kayniaand Kausel. The computed results from the proposed quasi-3D model agreed well withthe results by Kaynia and Kausel.Full scale vibration tests of a single pile and a 6-pile group have been analyzed using the proposed quasi-3D model. For the single expanded base pile, the computedfundamental frequency of the pile cap system was 6.67 Hz, which agreed well with themeasured fundamental frequency of 6.5 Hz. The computed damping ratio is 6% compared to a measured damping ratio of 4%. For the 6-pile group supporting a transformerbank, the computed fundamental frequencies of the transformer-pile cap were 3.74 Hz and4.63 Hz in the NS and EW directions, respectively, which agreed well with the measuredfundamental frequencies of 3.8 Hz and 4.6 Hz in the corresponding two directions. Thecomputed damping ratios of the system in the two directions were 9% and 9%, comparedto measured damping ratios of 6% and 5%. The damping ratios were overestimated usingthe proposed model.The proposed quasi-3D method is also formulated in the time domain using the finiteelement method. This formulation is targeted for the analyses of non-linear response ofpile foundations under earthquake loading. The time-domain analysis allows modellingthe variations of soil properties with time under earthquake loading.The procedures of non-linear time-domain analysis are incorporated in the computerprogram PILE3D. In PILE3D, the shear stress-strain relationship of soil is simulated tobe either linear elastic or non-linear elastic. When the non-linear option is used, theshear modulus and the hysteretic damping are determined using a modified equivalentChapter 10. Summary and Suggestions for Future Work 185linear approach based on the computed levels of dynamic shear strains. Features suchas shear yielding and tension cut-off are incorporated in PILE3D as well. The dynamicresponse of pile groups can also be effectively modelled using PJLE3D.Centrifuge tests of a single pile and a 2x2 pile group have been analyzed using theproposed quasi-3D finite element method of analysis. The ability of the program PILE3Dto model the dynamic response of the pile supported structures under moderately strongshaking has been proven adequate for engineering purpose. These studies suggest that theshear-strain dependent shear moduli and damping ratios of soil is modelled adequatelyin the analysis by using the modified equivalent linear approach. Also the approximatemethod for modelling the shear yielding and the tension cut-off seems to yield satisfactory results.The other important feature of the proposed method is that the time histories ofdynamic impedances of single piles and pile groups can be computed. This is the firsttime that this has been done.The results of analyses showed that stiffnesses of the pile foundations decrease withthe level of shaking; the dampings of the pile foundations increase with the level of shaking. In a seismic event, the translational stiffness kr,, 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 of the centrifuge tests, the translationalstiffness would decrease to as much as 15% of its initial stiffness; however the rotationalstiffness decreases to about 60% of its initial stiffness. The variation of the cross-couplingstiffness k,,6 is around 50% 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, f 10Hz. However the damping usuallyincreases with excitation frequency. The damping can be represented by 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 of analysis 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 of results 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 of elastic response of single 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 Future Work 187Kaynia and Kausel (1982) and data from full scale vibration tests of a single pileand a 6-pile group.3. A quasi-3D finite element program PILE3D has been developed for the analysesof non-linear response of single piles and pile groups. Analyses are performed inthe time domain. The characteristics of soil non-linearity with strains is modelledusing the modified equivalent linear approach. Other features such as the modellingof shear yielding and tension cut-off are incorporated in PILE3D. The program isapplicable for analyses of single piles as well as pile groups.4. The proposed model for modelling non-linear response of pile foundations underearthquake loading has been validated using data from centrifuge tests of a singlepile and 2x2 pile group. Satisfactory results have been obtained.5. Time-dependent variations of dynamic impedances of pile foundations during shaking can be evaluated and have been demonstrated for the model pile foundationsused in the centrifuge tests. The results of analyses showed that stiffnesses of pilefoundations decrease with the level of shaking. The translational stiffness k,,1, decreases the most at high strain level; the rotational stiffness k98 decreases the least.However, the damping of pile foundations increases with the level of shaking.Suggestions for future work For future research on dynamic response of pile foundations, studies should be focused on including the effects of seismic pore water pressureson the response of pile foundations. This next step is very important because pile foundations are often used at potentially liquefiable sites.Bibliography[1] American Petroleum Institute (1986). \u201c Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms,\u201d 16th edition, Dallas, TX.[2] Angelides, D.C., and Roesset J.M., (1981). \u201cNon-linear Lateral Dynamic Stiffnessof Piles,\u201d ASCE, Jour. of the Geotech. Engineering Division, 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,\u201d OTC 2749, 9thOffshore Technology Conference, Houston, Texas, pp. 189-198.[4] Bea, R.G., Litton, R., Nour-Omid, S. and Chang, J.Y. (1984). \u201cA Specialized Design and Research Tool for the Modelling of Near-Field Soil Interactions,\u201d OTC4806, 16th Offshore Technology Conference, Houston, Texas, pp. 249-252.[5] Brown, D.A. and Shie, C.F. 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Englewood Cliffs, New Jersey, pp.349-381.[51] Poulos, H.G. (1971). \u201cBehaviour of Laterally Loaded Piles. II- Pile Groups,\u201d Journal of the Soil Mechanics Division, ASCE, Vol. 97, No. SM5, pp. 733-751.[52] Poulos, H.G. (1975). \u201cLateral Load Deflection Prediction for Pile Groups,\u201d Journalof the Geotechnical Engineering Division, ASCE, No. OT1, pp. 19-34.[53] Poulos, HG. (1979). \u201cGroups Factor for Pile-Deflection Estimation,\u201d Journal ofthe Geotechnical Engineering Division, ASCE, No. GT12, pp. 1489-1509.Bibliography 194[54] Poulos, H.G. and Davis, E.H., (1980). Pile Foundation Analysis and Design. JohnWiley & Sons.[55] Prakash, S. and Sharma, H.D., 1990. Pile Foundation in Engineering Practice. JohnWiley & Sons, Inc.[56] Rayleigh, L., Theory of Sound, Vol. I, Dover Publications Inc., New York, N.Y.,1945.[57] Roesset, J.M. and Angelides, D. (1980). \u201c Dynamic Stiffness of Piles,\u201d from \u201cNumerical Methods in Offshore Piling,\u201d Institution of Civil Engineering, pp.75-81[58] Schnabel, P.B., Lysmer, J., and Seed, H.B., (1972). \u201cSHAKE: A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites,\u201d Report No.EERC 72-12, Earthquake Engineering Research Centre, University of California,Berkeley, Calif., Feb..[59] Scott, R.F. (1973) \u201cEarthquake-Induced Earth Pressures on Retaining Walls,\u201d Proceedings, Fifth World Conference on Earthquake Engineering, Rome, Italy.[60] Seed, H. B., and Whitman, R. V. (1970) \u201c Design of Earth Retaining Structuresfor Dynamic Loads,\u201d Lateral Stresses in the Ground and Design of Earth-RetainingStructures, ASCE.[61] Seed, H.B., Wong, R.T., Idriss, I.M., and Tokimatsu, K., (1986). \u201c Moduli andDamping Factors for Dynamic Analyses of Cohesionless Soils,\u201d Jour. of Geotech.Engineering Division, ASCE, 112(11), 1016-1032.[62] Seed, H.B., and Idriss, I.M. (1967) \u201c Analysis of Soil Liquefaction: Nigatta Earthquake,\u201d Jour. Soil Mechanics and Foundation Division, ASCE, Vol. 93, No. SM3,pp. 83-108.[63] Seed, H.B., and Idriss, I.M. (1970)\u201d Soil Moduli and Damping Factors for DynamicResponse Analyses,\u201d Report No. ERRC 70-10, Earthquake Engineering ResearchBibliography 195Center, University of California, Berkeley, California, December[64] Stevens, J.B. and Audibert, J.M.E. (1979). \u201c Re-examination of P-Y Curve Formulations,\u201d 11th Offshore Technology Conference, Houston, Texas, pp. 397-401.[65] Sy, A., (1992). \u201c An Alternative Analysis of Vibration Tests on Two Pile GroupFoundations,\u201d Piles under Dynamic Loads, ASCE Geotech. Special Publication No.34, 136-152.[66] Sy, A. and Siu, D., (1992). \u201c Forced Vibration Testing of An Expanded Base Concrete Pile,\u201d Piles under Dynamic Loads, ASCE Geotech. Special Publication No.34, 170-186.[67] Tajimi, H. (1966;1969). \u201c Earthquake Response of Foundation Structures\u201d, Rep. ofFac. Sci. Eng., Nihon University, 1966.3, 1.1-3.5 (in Japanese). See also Tajimi, H.Dynamic Analysis of a Structure Embedded in an Elastic Stratum\u201d, Proc. 4thWCEE, Chile.[68] Veletsos, A.S. and Younan, A.H. (1994). \u201cDynamic Soil Pressures on Rigid VerticalWalls,\u201d International Journal of Earthquake Engineering and Structural Dynamics,Vol. 23, No.3, pp275-301[69] Wilson, E.L., Farhoomand, L., and Bathe, K.J., (1973). \u201cNonlinear Dynamic Analysis of Complex Structures,\u201d International Journal of Earthquake Engineering andStructural Dynamics, Vol. 1, No.3, Jan.-March.[70] Wood, J. H. (1973) \u201c Earthquake-Induced Soil Pressures on Structures ,\u201c Ph.Dthesis submitted to the California Institute of Technology, Pasadena, Calif., USA.[71] Yan, L. (1990). \u201cHydraulic Gradient Similitude Method for Geotechnical ModellingTests with Emphasis on Laterally Loaded Piles.\u201d Ph.D. Thesis, Faculty of Graduate Studies, Univ. of British Columbia, Vancouver, Canada[72] Yegian, Y. and Wright, S. (1973). \u201cLateral Soil Resistance-Displacement Relationships for Pile Foundations in Soft Clays,\u201d OTC 1893, 5th Offshore TechnologyBibliography 196Conference, vol. 2, pp.663-671.Appendix iDerivation 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 \u00f4vTy_G(\u00e3_+_) (.1)In the shear beam = 08u= (.2)The normal stress o in the direction of shaking is determined using the assumption=02G \u00f4v uo.y=l_2\u20191\u201d8yth0ThereforeEly vEluEly 1\u2014vElxThe normal stress o in the direction of shaking is given by2G Elu Ely1 2VE(hL)Elx+MElyl(.5)Substituting for2 Elu= (.6)1\u2014ti O3197198Substituting Eq. .2 and Eq. .6 into the equilibrium equation(7)ox oy POt2one finds that2 02u 82u 02u1 \u2014+G = (.8)Therefore by comparison with Eq. (3.6) and (3.7)(.9)Eq. .8 satisfies equilibrium but does not satisfy second order compatibility.","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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