structurewallbase

GEOTECHNICALEARTHQUAKEENGINEERINGGEOTECHNICAL,GEOLOGICALANDEARTHQUAKEENGINEERINGVolume9SeriesEditorAtillaAnsal,KandilliObservatoryandEarthquakeResearchInstitute,BogaziciUniversity,Istanbul,TurkeyEditorialAdvisoryBoardJulianBommer,ImperialCollegeLondon,U.
K.
JonathanD.
Bray,UniversityofCalifornia,Berkeley,U.
S.
A.
KyriazisPitilakis,AristotleUniversityofThessaloniki,GreeceSusumuYasuda,TokyoDenkiUniversity,JapanForothertitlespublishedinthisseries,gotowww.
springer.
com/series/6011GeotechnicalEarthquakeEngineeringSimpliedAnalyseswithCaseStudiesandExamplesbyMILUTINSRBULOVUnitedKingdomwithForewordofE.
T.
R.
Dean123Dr.
MilutinSrbulovUnitedKingdomsrbuluv@aol.
comISBN:978-1-4020-8683-0e-ISBN:978-1-4020-8684-7LibraryofCongressControlNumber:2008931592c2008SpringerScience+BusinessMediaB.
V.
Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,microlming,recordingorotherwise,withoutwrittenpermissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecicallyforthepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.
Printedonacid-freepaper987654321springer.
comForewordMeasurableearthquakesoccurveryfrequentlyinmanypartsoftheworld.
Forexample,Shepherd(1992)lists7283earthquakesrecordedintheCaribbeanAntillesinthe22-yearperiod1964to1985,arateofabout1earthquakeperday.
Somewereduetomovementsofhighlystressedrockatmorethan100kmbelowthegroundsurface(ShepherdandAspinall,1983).
Similarhighlevelsofactivityarefoundinallseismicallyactiveregionsoftheworld.
Astheearthquakevibrationstravelfromthesourcetowardsthegroundsurface,theenergyspreadsoutandalsodissipates,sothatenergydensityreduceswithdis-tancefromsource.
Forthemajorityofevents,shakinghasreducedtolevelsthatpeoplecannotfeelbythetimeitreachesthegroundsurface.
Forsomeevents,suf-cientenergyreachesthesurfaceforpeopletofeelminoreffects.
Forafew,theenergyreachingthesurfaceissufcienttocausemajordamage.
Sinceearthquakeshakingistransmittedthroughground,andsincegroundalsosupportsbuildingsandotherstructures,theartandscienceofgeotechnicalengineer-ingisanimportantpartofearthquakeengineering.
Avarietyofconceptsandtech-niquesaredetailedbyKramer(1996),Day(2002),ChenandScawthorne(2003),andothers.
Someoftheimportantgeotechnicalaspectsare:rTheparticlemechanicalnatureofsoil(MitchellandSoga,2005;LambeandWhitman,1979)rTerzaghi'sPrincipleofEffectiveStress(Terzaghietal,1996)rLinear,isotropicelasticmodels(DavisandSelvadurai,1996)rThetheoryofsoilplasticity(Druckeretal.
,1957;DavisandSelvadurai,2002;Loret,1990)rTheMohr-Coulombfailureenvelope(LambeandWhitman,1979;Das,2004)rThecharacterizationofsoilproperties,andtheoriesofcompressibility,owofwaterthroughsoils,uidization,andconsolidationofsoils(FlorinandIvanov,1961;LambeandWhitman,1979;HeidariandJames,1982;WrothandHoulsby,1985;Terzaghietal,1996;Das,2004)rCriticalstatesoilmechanics,whichseekstoincorporatesoilelasticity,plasticity,strength,density,andconsolidationintoasingleunifyingtheoreticalframework(SchoeldandWroth,1968;AtkinsonandBransby,1978;Muir-Wood,1992;Schoeld,2005)vviForewordrAdvancedsiteinvestigationandlaboratorytestingtechniques(Hunt,2005;Head,2006)rAdvancedmethodsforslopestabilityassessment(Abramsonetal,1996;Corn-forth,2005),andbearingcapacityandlateralearthpressure(eg.
Choudharyetal,2004;KumarandGhosh,2006)rLiquefactionandthesteadystateconcept(Castro,1969;SeedandIdriss,1971;Poulos,1981;VaidandChern,1985;Seed,1988;Ishihara,1995;JefferiesandBeen,2006)rShakingtableandcentrifugemodeltesting(Schoeld,1980;ArulanandanandScott,1994;Taylor,1994)rThedevelopingtheoriesofunsaturatedsoilmechanics(FredlundandRahardjo,1993)rTheuseofadvanceconstitutivemodels(Loret,1990;YamamuroandKaliakin,2005)withniteelementmethods(ZienkiewiczandTaylor,1989,1991;BrittoandGunn,1987;Finn,1999;Potts,2003)rTheglobalgathering,processing,anduseofcollectiveexperience(YoudandIdriss,2001)Basedontheseandotherfactors,advancesinunderstandinghavebeenincor-poratedindesigncodesincludingtheUniformBuildingCode(UBC,1997),theInternationalBuildingCode(IBC,2006),Eurocode8(2004),APIRP2A(2005),ISO19901(2004),andmanyothers.
Tosupportthesedevelopments,itcanbehighlydesirabletodocumentsomesimpliedmodelsthatareeasiertounderstand,retainandexplainthefundamentalphysicsinvolved,andprovidewaysofassessingtherelevance,reliability,andap-plicabilityofmoresophisticatedapproaches.
Itisalsoratherusefultobeabletoidentifythemostsignicantpublicationsinatechnicalliteraturethatisnowveryextensiveindeed.
ThemonographpresentssomeoftheAuthor'sdescriptions,casehistories,experiencesandcommentsonavarietyofsimpliedmodelsforengineer-ingdesignandanalysis.
Thisisvaluablebothforpersonsnewtothesubjectwhowilllearnofthewide-rangingconsiderationsinvolved,andtootherexperiencedpractitionerswhowillbeabletocompareexperienceswiththosesharedhere.
SeniorLecturerinGeotechnicalEngineering,E.
T.
R.
DeanUniversityoftheWestIndiesPrefaceThismonographcontainsdescriptionsofnumerousmethodsaimedateaseandspeedofuseformajorproblemsingeotechnicalearthquakeengineering.
Commentsonassumptions,limitations,andfactorsaffectingtheresultsaregiven.
Casestudiesandexamplesareincludedtoillustratetheaccuracyandusefulnessofsimpliedmethods.
Alistofreferencesisprovidedforfurtherconsiderations,ifdesired.
Mi-crosoftExcelworkbooksreferredtoinAppendicesandprovidedonanaccompany-ingCDareforthecasestudiesandexamplesconsideredinthemonograph.
Someofthereasonsforusingthismonographarementionedbelow.
Manycodesandstandardscontainrecommendationsonbestpracticebutcompli-ancewiththemdoesnotnecessarilyconferimmunityfromrelevantstatutoryandle-galrequirements(asstatedinBritishStandards).
Someseismiccodesandstandardswererevisedaftermajoreventssuchasthe1995Hyogo-kenNambuandthe1994Northridgeearthquakes.
Codescontainclauseswithoutreferencestotheoriginalsourcesformoredetailedconsiderationswhencasesthatrequiresuchconsiderationappearinpractice.
Codesdonotcontainexplanationsofthestatementsexpressedinthem.
Codesarebriefregardinggroundpropertiesandgroundresponse.
Forexample,Eurocode8–Part5requiresassessmentoftheeffectsofsoil-structureinteractionincertaincircumstancesbutdoesnotspecifythedetailsoftheanalyses.
Therefore,theuseofcodesandstandardsalonemaynotbesufcientinengineeringpractice.
Inengineeringpractice,thereisoftenratherlittleinteractionbetweenstructuralandfoundationdisciplines.
Structuralengineersoftenconsidergroundinasim-pliedwayusingequivalentsprings.
Geotechnicalengineersconsideroftenonlyloadingfromstructuresonfoundations.
Dynamicsoil-structureinteractionisverycomplexandanalyzedmainlybyspecialistingeotechnicalearthquakeengineering.
Thismonographshouldhelpgeotechnicalandstructuralengineerstocommunicateeffectivelytobetterunderstandsolutionsofmanyproblemsingeotechnicalearth-quakeengineering.
Specialistsinnon-lineardynamicsanalysesneedtorecognizethatthemotionofanon-linearsystemcanbechaoticandtheoutcomescanbeunrepeatableandunpredictable.
BakerandGollub(1992),forexample,showthattwoconditionsaresufcienttogiverisetothepossibilityofchaoticmotion:thesystemhasatleastthreeindependentvariables,andthevariablesarecoupledbynon-linearviiviiiPrefacerelations.
Equivalentlinearandsimpliednon-lineardynamicanalysisdescribedinthismonographcanbeusedtoavoidpossiblechaoticoutcomesofacomplexnon-lineardynamicanalysis.
Groundmotioncausedbyearthquakesischaoticandthereforegreateraccuracyofsophisticatedmethodslosesitsadvantage.
Expectedgroundmotioncanbepredictedonlyapproximately,andsimpliedanalysesarefasterandeasiertoolsforparametricstudiescomparedtosophisticatedmethods.
UnitedKingdomMilutinSrbulovAcknowledgementsProfessorMaksimovicpersuadedmetoswitchprofessionfromconcretestructurestogeotechnicsrightaftermygraduation.
HepioneeredstudiesofsoilmechanicspaidbyEnergoprojektCo.
atImperialCollegeintheU.
K.
TheMScsoilmechanicsstudyin1984/85enabledmetoobtainthepositionofaresearchassistantlater.
IwashonoredandprivilegedtoworkwithProfessorAmbraseysonanumberofresearchprojectssupportedbytheEngineeringandPhysicalScienceResearchCounciloftheUnitedKingdomandbytheEPOCHprogramoftheCommunityofEuropeanCountriesatImperialCollegeinLondonduringtheperiod1991–1997.
Thesimpliedapproachusedinourresearchisdirectlyapplicabletoroutineengi-neeringpractice.
DrE.
T.
R.
Deanreviewedseveralofmypapersandwasofgreathelpwithhisdetailedandprecisecommentsfortheimprovementoftheinitialversionsofthepapers.
Hekindlyreviewedthemonographandmadeasignicantcontributiontowardstheimprovementoftheclarityandreadabilityofthetext.
ElsevierpublisherskindlygrantedpermissiontoreproduceFig.
5B,Fig.
10,Fig.
11,2/3ofDiscussion,andAppendixAofthepaperbyAmbraseysandSr-bulov(1995)inprintandelectronicformatinalllanguagesandeditions.
Elsevierpublisherskindlygrantedpermissiontoreproducepages255to268ofthepaperbySrbulov(2001)inprintandEnglishversion.
PatronEditorepublisherskindlygrantedpermissiontoreproducepartsofmypaperspublishedinthejournalEuropeanEarthquakeEngineering.
TheAmericanSocietyofCivilEngineerskindlygrantedpermissiontoreproduceinprintandelectronicversionTable2fromZhangetal.
(2005)paper.
ixContents1WellKnownSimpliedModels11.
1Introduction11.
2SourceModelsofEnergyReleasebyTectonicFault11.
2.
1ASimpliedPoint-SourceModel11.
2.
2AnAlternative,PlanarSourceModel41.
2.
3CaseStudyComparisonsofthePointandPlanarSourceModels51.
3SlidingBlockModelofCo-SeismicPermanentSlopeDisplacement61.
3.
1Newmark's(1965)SlidingBlockModel61.
3.
2CommentsonNewmarks's(1965)SlidingBlockModel.
.
.
71.
4SingleDegreeofFreedomOscillatorforVibrationofaStructureonRigidBase101.
4.
1DescriptionoftheModel101.
4.
2CommentsontheModel111.
5Summary122SoilProperties132.
1Introduction132.
2CyclicShearStiffnessandMaterialDamping142.
2.
1ShearStiffnessandDampingRatioDependenceonShearStrain162.
3StaticShearStrengthsofSoils182.
4CyclicShearStrengthsofSoils202.
5TheEquivalentNumberofCyclesConcept232.
5.
1AnExampleofEquivalentHarmonicTimeHistories252.
6WaterPermeabilityandVolumetricCompressibility262.
7Summary283SeismicExcitation293.
1Introduction293.
2SeismicHazard293.
2.
1TypesofEarthquakeMagnitudes303.
2.
2TypesofSource-to-SiteDistances31xixiiContents3.
2.
3TypesofEarthquakeRecurrenceRates313.
2.
4RepresentationsofSeismicHazard323.
2.
5SourcesofEarthquakeData393.
3FactorsAffectingSeismicHazard.
413.
3.
1EarthquakeSourceandWavePathEffects413.
3.
2SedimentBasinEdgeandDepthEffects453.
3.
3LocalSoilLayersEffect543.
3.
4TopographicEffect573.
3.
5SpaceandTimeClustering(andSeismicGaps)583.
4ShortTermSeismicHazardAssessment603.
4.
1HistoricandInstrumentalSeismicDataBased.
603.
4.
2ObservationalMethod623.
5LongTermSeismicHazardAssessment653.
5.
1TectonicDataBased653.
5.
2PaleoseismicDataBased673.
6Summary704SlopeStabilityandDisplacement.
734.
1Introduction734.
2SlopeStability734.
2.
1LimitEquilibriumMethodforTwo-DimensionalAnalysisbyPrismaticWedges744.
2.
2SingleTetrahedralWedgeforThree-DimensionalAnalysisofTranslationalStability844.
3ShearBeamModelforReversibleDisplacementAnalysis864.
3.
1Two-DimensionalAnalysis.
864.
3.
2Three-DimensionalEffect.
884.
4SlidingBlockModelsforPermanentDisplacementAnalysis894.
4.
1Co-SeismicStage.
894.
4.
2Post-SeismicStage944.
5BouncingBallModelofRockFall994.
5.
1CaseStudyofBedrina1RockFallinSwitzerland1034.
5.
2CaseStudyofShimaRockFallinJapan.
1054.
5.
3CaseStudyofFutamataRockFallinJapan1064.
6SimpliedModelforSoilandRockAvalanches,DebrisRun-OutandFastSpreadsAnalysis1074.
6.
1EquationofMotion1084.
6.
2MassBalance1104.
6.
3EnergyBalance1114.
7Summary1175SandLiquefactionandFlow1195.
1Introduction1195.
2ConventionalEmpiricalMethods1205.
2.
1LiquefactionPotentialAssessment120Contentsxiii5.
2.
2FlowConsideration1225.
3RotatingCylinderModelforLiquefactionPotentialAnalysisofSlopes.
1235.
3.
1ModelforCleanSand1235.
3.
2ModelforSandwithFines1265.
4RollingCylinderModelforAnalysisofFlowFailures.
1355.
4.
1ModelforCleanSand1355.
4.
2ModelforSandwithFines1365.
5Summary1396DynamicSoil–FoundationInteraction1416.
1Introduction1416.
2AdvancedandEmpiricalMethods1426.
2.
1NumericalMethods,CentrifugeandShakingTableTesting.
1426.
2.
2SystemIdenticationProcedure.
1426.
3DiscreteElementModels1436.
3.
1LumpedMassModelFormula1436.
3.
2ClosedFormSolutioninTime1506.
3.
3TimeSteppingProcedure1566.
4SingleDegreeofFreedomOscillatoronFlexibleBaseforPiledFoundationsandFlexuralRetainingWalls1686.
4.
1GroundMotionAveragingforKinematicInteractionEffectConsideration1706.
4.
2AccelerationResponseSpectraRatiosforInertialInteractionEffectConsideration1726.
5Summary1857BearingCapacityAndAdditionalSettlementofShallowFoundation.
.
1877.
1Introduction1877.
2BearingCapacity:Pseudo-StaticApproaches1877.
3BearingCapacity:EffectsofSub-SurfaceLiquefaction1887.
4BearingCapacity:EffectsofStructuralInertiaandEccentricityofLoad1897.
4.
1AnExampleofCalculationofBearingCapacityofShallowFoundationinSeismicCondition1907.
5AdditionalSettlementinGranularsoils1917.
5.
1ExamplesofEstimationofAdditionalSettlementCausedbySandLiquefaction1927.
6Summary1938SeismicWavePropagationEffectonTunnelsandShafts1958.
1Introduction1958.
2WavePropagationEffectonCutandCoverTunnelsandShafts.
.
.
.
1958.
2.
1CaseStudyoftheDaikaiStationFailurein1995.
1968.
2.
2CaseStudyofaTenStoryBuildinginMexicoCity199xivContents8.
3WaveRefractionEffectonDeepTunnelsandShafts2018.
4Summary2029CommentsonSomeFrequentLiquefactionPotentialMitigationMeasures2039.
1Introduction2039.
2StoneColumns2039.
3SoilMixing2049.
4ExcessWaterPressureReliefWells2059.
4.
1AnExampleforPressureReliefWells2089.
5Summary208Appendices–MicrosoftExcelWorkbooksonCompactDisk211A.
1CoordinatesofEarthquakeHypocentreandSite-to-EpicentreDistance211A.
2LimitEquilibriumMethodforNortholtSlopeStability212A.
3SingleWedgeforThree-DimensionalSlopeStability214A.
4Co-SeismicSlidingBlock215A.
5aPost-SeismicSlidingBlocksforMaidipoSlipinFrictionalSoil.
.
.
.
215A.
5bPost-SeismicSlidingBlocksforCatakSlipinCohesiveSoil216A.
6BouncingBlockModelofRockFalls216A.
7SimpliedModelforSoilandRockAvalanches,DebrisRun-OutandFastSpreads216A.
8Closed-FormSolutionforGravityWalls219A.
9aTimeSteppingProcedureforKobeWall219A.
9bTimeSteppingProcedureforKalamataWall.
219A.
10AccelerogramAveragingandAccelerationResponseSpectra.
219A.
11BearingCapacityofShallowFoundation223A.
12ExcessPoreWaterPressureDissipation.
223References225Index241ListofSymbolsSymbolDescriptionσh/hhorizontalaxialstressgradientinhorizontaldirectionτhn/ngradientofshearstressinverticalplaneindirectionnormaltotheplaneτhv/vgradientofshearstressinverticalplaneinverticaldirection2u(1)/t2secondgradientofhorizontaldisplacementintime(1-downslope)u/vhorizontaldisplacementgradientinverticaldirectioncapparentcohesionofreinforcedsoilφequivalentfrictionanglealongslidingblockbaseσaveragecompressivestressonslidingblockbaseθinclinationtothehorizontalofslidingblockbase.
.
θrotationalaccelerationofacylinderaroundapoint.
.
uhorizontalacceleration.
θ1nrotationalvelocityofagravitywall.
.
θon,.
.
uonrotationalandhorizontalaccelerationsofagravitywall/αexponentoftheratioγγ1rαangleofslidingblockinclinationtohorizontal/kexponentoftheratioσmP1a(N1)60normalizedblowcounttoanoverburdenpressureof100kPaandcorrectedtoanenergyratioof60%aanexponenta(i)acceleration(initial)a,b,ccoefcientscalculatedfrommeasuredincrementaldisplacementsu,v,wa1rateofgroundaccelerationincrementduringatimeintervalA1,2seismicwaveamplitudes1and2Abareaofthemasscontactwiththebaseandsidesac(h,r)criticalhorizontalaccelerationinsliding(h)orrocking(r)acrcriticalaccelerationxvAffoundationareaaf,phorizontalpeakfoundationaccelerationAfaulttectonicfaultareaAgamplitudeofgrounddisplacementag,thorizontalgroundaccelerationahhorizontalacceleration(foraharmonicload)aipeakinputaccelerationofaSDOFOalgroundaccelerationatdepthlalongthepile/wallattimetAlooptheareaofthehystereticloopaogroundaccelerationatthebeginningofatimeintervalapeak,depthpeakhorizontalgroundaccelerationatdepthaphpeakhorizontalgroundsurfaceaccelerationapeak,surfaceapvpeakverticalgroundsurfaceaccelerationarrockfallaccelerationjustbeforetheimpactAsareaofslopeslidingsurfaceAu(d)upstream(downstream)verticalcrosssectionareabhorizontaldistancebetweenthebackofawallandthewallcentroidb(i)breadthofwedgebase(interfacei)BbwidthofanequivalentballofrockfallbcbreadthofarectangularpilecapBfdiameterofanequivalentcircularfoundationbjbreadthofjointjBsnumberof(sub)basementsinabuildingBwwallbasewidthcsoilshearstrength(cohesion)atzerocompressivestressCtranslationaldashpotcoefcientc(j)soilcohesionindrainedcondition(atjointj)C0,1,2constantschhorizontalcoefcientofinertiaforceinducedbygroundmotioncnamplitudeofthenthharmonicoftheFourierseriescpgroundlongitudinalwavevelocityCssoilconstantintheshearstrengthandshearstrainrelationshipcssoilcharacteristicwavevelocityctgroundtransversalwavevelocitycuundrainedshearstrengthofliqueedsandlayercu(1)undrainedcohesion(inonecycle)curresidualundrainedshearstrengthofliqueedsandcv(r)coefcientofconsolidation(inradialdirection)cvmverticalcoefcientofinertiaforceinducedbygroundmotionCθrotationalsoildashpotcoefcientxviListofSymbolsdminimaldistancefromthelocationofinteresttothesurfaceprojectionofafaultD50anaveragediameterofsoilparticlesdcdepthfactordedistancebetweenwellscentretocentreDffoundationdepthbelowgroundleveldg,thorizontalgrounddisplacementintimedhhorizontaldistancebetweenthelocationwheretheloadFisactingandthelocationwherethestressiscalculatedDldepthofliqueedsoillayerdppilediameterdphpeakhorizontalgroundsurfacedisplacementdrradialdistancemeasuredfromcentreofthewelldsstraight-line(slant)distancebetweentheearthquakehypocenterandarecordingsiteDsmaximumsurfacedisplacementoftectonicfaultdtchangeofthicknessofwedgejointdtj,ejoint(j)thicknesschangeedistancebetweenwallcentroidanditsbaseEYoungmodulusEdenergydensityatahypocentraldistanceEfftheoreticalfree-fallenergyofhammerElossenergylossduetoplasticdeformationofimpactedsurfaceEmactualenergydeliveredbyhammerEototalenergyreleasedattheearthquakesourceEpYoungmodulusofpileEsanaveragelateralearthforceEttotalenergyreleasedattheearthquakesourceperunitareaofthesourceffrequencyofshearstressreversalFavraveragefactorofsafetyofagroupofwedgesFggroundresistingforcetorockfallpenetrationonimpactFi,jlocalfactorsofsafetyalongwedgejointsi,jFmmodicationfactorofsedimentstransversalwavevelocitiesFNnormalandstrike-slipfaultindicatorFOunspeciedfaultindicatorFppointloadFrsoilreactionforceatwallbaseFSfactorofsafetyofslopestabilityFTreverse(thrust)faultindicatorFvverticalfoundationcapacityGshearmodulusggravitationalaccelerationGbaveragetransversalwavevelocityrange3601]probabilityofatleastoneexceedanceofaparticularearthquakemagnitudeinaperiodoftyearsPaatmosphericpressurePbsoilresistingforceactingatthebasePfaxialcomponentofrockfallimpactforcePIsoilplasticityindexpncharacteristicaxialstressListofSymbolsxixpoeffectiveoverburdenstressatthefoundationdepthPrsoilreinforcementforcePsimprovementinshearingresistancefromsoilreinforcementforcePrRradiusofanequivalentballofrockfallrcylinderradiusr1radiusofthenesmodelRbratiobetweenthehorizontaldistancesfromastationtosedimentbasinedgeandthedepthofsedimentsatthelocationofthestationrccorrelationcoefcientrdstressfactorwithdepthreahalfofthedistancederfsourceslantdistancerhradiusofanequivalentdisksforthehorizontalmotionrMCradiusofMohr–CoulombcircledenedbyEquation(9.
1)rpileahalfofpilediameterrrradiusofanequivalentdisksfortherotationalmotionru(,j)excessporewaterpressureratio(atjointj)rvradiusofanequivalentdisksfortheverticalmotionrwradiusofawellSslidingforceatthebaseofarigidretainingwallsaxistoaxisspacingbetweensoil-cementmixturewallsSAstiffsoilsiteindicatorscshapefactorSfaveragesliponthefaultduringanearthquakeSSsoftsoilsiteindicatorStnumberofstoreysabovegroundlevelSuminimaluniaxialcompressivestrengthofsamplestakenfrommixedsoilTperiodofvibrationttimeTi(j)forceactinginthedirectionthatisparalleltothesurfaceofawedgebasei(interfacej)t1timewhencylinderwillstartrotationtachtimecorrespondingtoachTdperiodoftherstmodeoffreevibrationofadamTeqvperiodofequivalentharmoniccycleTftransversalcomponentofrockfallimpactforceTishearforceatwedgejointiTMreturnperiodofearthquakesexceedingmagnitudeMTpageoftectonicplatesubductionTrearthquakerecurrenceperiodTsthetime(inseconds)necessaryforaseismicwavetopassalongLsxxListofSymbolsTvtimefactortwthicknessofsoil-cementmixturewallsT()transversalforceatthetopofthecolumnduetothehorizontaldisplacementandrotationθuhorizontaldisplacementU(z,r)overalldegreeofconsolidation(atdepthz,radiusr)u1one-waypermanenthorizontalcomponentofdisplacementsonslopinggroundu2two-waypermanentdisplacementsoflevel(horizontal)groundufowdistanceuf(ω)surfaceamplitudeofthefreeeldgroundmotionuohorizontalwalldisplacementutexcessporewaterpressureattimetvverticaldirectionVvolumeofmovingmassalongtravelpathv1lowersoilwavepropagationvelocityvhhorizontalbasevelocityvinincomingvelocityofrockfallvlvelocityofpropagationofthelongitudinalwavesvmmovingmassvelocityvoinitialvelocityvoutvelocityofbouncedrockfallVpvelocityofaparticlevphpeakhorizontalgroundsurfacevelocityVrrateoftectonicplatesubductionvtvelocityofpropagationofthetransversalwavesvtpgroundvelocitybelowthepile/walltipattimetvtTsgroundvelocitybelowthepile/walltipattimetTsWweightW1weightofthenesmodelWDdissipatedenergybymaterial(hysteretic)dampingWftectonicfaultwidthWsstrainenergyxshortestdistancebetweentheforceNandpointAinFigure5.
5yshortestdistance(levelarm)betweentheforceNtanφandpointAinFigure5.
5ypileshortestdistancebetweenpilecentroidandtheneutralaxisofrotationzdepthzmdatumabovemovingmassatrestpositionτa(,i)availablesoilshearstrength(atjointi)τeshearstressnecessarytomaintainlimitequilibriumBConstantofproportionalitybetweenγi(j),eandi(j),eListofSymbolsxxi(i(j),e)relativehorizontaldisplacementofabeamend(magnitudesofkinematicallypossibletangentialdisplacementsalongjointsi,jofwedges)Etransientpartoflateralearthforcei(j),ekinematicallypossibleshearstrainalongjointiorjMθmassmomentofinertiaofthetrappedsoilbeneathwallforPoisson'sratiogreaterthan1/3sfoundationsettlementttimesteptwtimelagbetweenarrivaloflongitudinalandtransversalwavesuincrementofgroundsurfacedisplacementinxdirectionvincrementofgroundsurfacedisplacementinydirectionwincrementofgroundsurfacedisplacementinzdirectionxincrementalhorizontaldistancealongrockfalltrajectoryjustbeforetheimpactxhorizontallengthoverwhichchangeofthicknessofmovingmasshasbeenachievedyincrementalverticaldistancealongrockfalltrajectoryjustbeforetheimpactzchangeofthicknessofmovingmassεincrementalaxialstrainφdifferencebetweenangleofsoilfrictionatzeroeffectivestressandbasicangleofsoilfrictionγincrementalshearstrainσvadditionalverticalstressatadepthz>0causedbypointloadFatthegroundsurfacesumofenergylossoveratravelpathofmovingmassNaxialcomponentoftheresultantofallforcesactingontheslipsurfaceTshearcomponentoftheresultantofallforcesactingontheslipsurfaceαangleinFigure5.
4and5.
11α1(2)anglebetweennormaltotheinterfaceanddirectionofpropagationofwavepathsontwosidesofaninterfaceαjangleofinclinationoftangentialdisplacementvectorwithrespecttojointdirectionαllocalangleofinclinationtothehorizontalattheimpactplaceofrockfallβinclinationtothehorizontalβllargerinclinationofthegroundsurfaceslopeortheslopeofthelowerboundaryoftheliqueedzoneinpercentβrfangle(positiveupwards)withthehorizontalatthebeginningofrockfallβttuningratioxxiiListofSymbolsδbfrictionanglebetweensoilandwallbackδi(j),asheardisplacementindirectshearapparatuscorrespondingtoavailableshearstressτaatajointi(i.
e.
j)δi(j),esheardisplacementindirectshearapparatuscorrespondingtomobilizedshearstressτeatajointi(i.
e.
j)δpplasticdeformationindirectionperpendiculartotheimpactsurfaceδrresidualangleofsoilfrictionεi(j),aaxialstrainintriaxialapparatuscorrespondingtoavailableshearstressτaatajointi(i.
e.
j)εi(j),eaxialstrainintriaxialapparatuscorrespondingtomobilizedshearstressτeatajointi(i.
e.
j)φfrictionangleincyclicconditionφ(j)soilfrictionangle(atjointj)indrainedconditionφ1peakfrictionalangleinstaticconditionφbbasicangleofsoilfricitionφnphaseangleespectivelyofthenthharmonicoftheFourierseriesγshearstrainγsubmergedunitweightofnon-liqueedsoilγhvshearstraininverticalplaneγi(j),ashearstraincorrespondingtoavailableshearstressτaatajointi(i.
e.
j)γi(j),eshearstraincorrespondingtomobilizedshearstressτeatajointi(i.
e.
j)γrreferentshearstrainγsunitweightofsoilparticleγsoilunitweightofsoilγwunitweightofwaterηviscosityofsoilηawabsoluteviscosityofwaterηwangleofinclinationtothehorizontalofbackllbehindaretainingwallκ,κ1exponenttoshearstrainintheshearstrengthandshearstrainrelationshipλaveragerateofoccurrenceoftheeventwithconsideredearthquakemagnitudeμshearmodulusoftheEarth'scrustνPoisson'sratiooangleofinclinationtotheverticalofthebackofawallθrotationangleθ1anadditionalinternalrotationaldegreeoffreedomθbrelativerotationofabeamendθoangleofwallrotationListofSymbolsxxiiiθranglebetweenthereinforcementdirectionandanormaltowedgejointθαdifferencebetweenanglesα1andα2ρsoilunitdensityρ1lowersoilunitdensityρwwaterunitdensityσmmeaneffectiveconningstressσvverticaleffectivestress(fromoverburden)σ3lateralconningeffectivepressureσdhorizontalcompressivestressesactingonthedownstreamverticalcrosssectionsofmovingmassσhaxialstress,positivewhentensileσuhorizontalcompressivestressactingontheupstreamverticalcrosssectionsofmovingmassσvtotaloverburdenpressure(atdepthvbelowwalltop)σ()axial(effective)stress,positivewhencompressiveτshearstressτbshearstressatthebaseandsidesτdverticalshearstressactingonthedownstreamverticalcrosssectionsofthemassτhnshearstressintheplaneperpendiculartotheplanewithinwhichhorizontaldisplacementoccursτhvshearstressintheverticalplane(behindwallatdepthv)τppeakshearstrengthτuverticalshearstressactingontheupstreamverticalcrosssectionsofthemassω(n)circularfrequency(ofnthharmonicoftheFourierseries)ωdcircularfrequencyofaninputmotionωefundamentalcircularfrequencyofundampedcoupledlinearelasticSDOFOωggroundcircularfrequencyωhcircularfrequencyofhorizontalmotionωocircularfrequencyoftheoutputmotionωrnaturalfrequencycorrespondingtotherotationalmotionofadynamicmodelωsnaturalcircularfrequencyofpile(s)/wallinxedbaseconditionξdampingratioξeequivalenthystereticdampingratioξgsoilhystereticdampingratioξhradiationdampingratioofapilegroupinhorizontaldirectionξminminimumdampingratioξrradiationdampingratioofapilegroupinrotationξsstructuralhystereticdampingratioxxivListofSymbols