Evaluation of simplified methods for predicting earthquake-induced slope displacements in earth dams

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EngineeringGeology

journalhomepage:www.elsevier.com/locate/enggeo

Evaluationofsimplifiedmethodsforpredictingearthquake-inducedslopedisplacementsinearthdamsandembankments

ChristopherL.Meehana,⁎,FarshidVahedifardbabUniversityofDelaware,Dept.ofCivilandEnvironmentalEngineering,301DuPontHall,Newark,DE19716,USA

MississippiStateUniversity,Dept.ofCivilandEnvironmentalEngineering,235HWalkerHall,MississippiState,MS39762,USA

articleinfoabstract

Thispaperprovidesareviewandcomparisonofexistingsimplifieddisplacement-basedslidingblockmodels.Analyseswereperformedtoevaluatetherelativeaccuracyoffifteenofthesesimplifiedmodelsforpredictingearthquake-induceddisplacements.Toaccomplishthistask,thepredictivecapabilityofthemodelswasassessedbycomparingmodelpredictionswiththeactualdisplacementsthatwereobservedafterearthquakeshakingin122casehistoriesofearthdamsandembankments.Theresultsindicatethatthemodelpredictionsofdisplace-mentwerelessthantheobserveddisplacementforalargemajorityofthecasehistoriesthatwereexamined.Thedifferencebetweentheobservedandpredicteddisplacementswasrelativelylargeforasignificantpercentageofthecases,foreachmodelthatwasexamined.Theshapesandpositionsofthemodels'relativeerrorcumulativedistributionfunctionsdidnotchangesignificantlyifthecasehistorieswerefilteredtoincludeonlythosewithintermediatelevelsofobserveddisplacement(i.e.,0.01mbobserveddisplacementb1m),whichindicatesthatthesimplifiedmodelsmayexhibitthesamebehaviorforcasesofsmalland/orlargedisplacementsastheydoforcasesintheintermediaterange,providedthatapercentage-basedapproachsuchasrelativeerrorisusedtocomparetheresultsfromdifferentmodels.

©2012ElsevierB.V.Allrightsreserved.

Articlehistory:

Received7May2012

Receivedinrevisedform23September2012Accepted13October2012

Availableonline8November2012Keywords:Dams

EmbankmentsEarthquakesDisplacementSlopestabilitySeismichazard

1.Introduction

Earthquakesposeasignificantthreattoawiderangeofgeotechnicalprojects,includingthosethatinvolvenaturalslopes,earthdams,solid-wastelandfills,retainingwalls,tunnels,orfoundations.Tomini-mizeearthquake-inducedlossesinthesestructures,twoessentialques-tionshavetobeconsidered:first,willearthquakeshakingsignificantlydecreasethestrengthofanymaterialinthestructureoritsfoundation(e.g.,liquefaction,strain-softening)?Ifasignificantlossofsoilstrengthoccurs,thereisastrongpossibilityofcatastrophicstructuralfailure,eitherduringtheearthquakeitselforaftercompletionofearthquakeshaking(Bray,2007).Ifsignificantstrengthlossdoesnotoccur,thesecondquestionthatfollowsis:willanearthquakeimposesignificantpermanentdeformationstoastructuresuchthatitspost-earthquakeperformanceisendangered(Bray,2007)?

Forthosecaseswheresignificantstrengthlossdoesnotoccur,avarietyoftechniqueshavehistoricallybeenusedtoevaluateseis-micslopestability.Thesetechniquestypicallyfallintooneofthefollowingcategories,inorderfromlowtohighcomplexity:force

⁎Correspondingauthorat:UniversityofDelaware,Dept.ofCivilandEnvironmentalEngineering,301DuPontHall,Newark,DE19716,USA.Tel.:+13028316074;fax:+13028313640.

E-mailaddresses:cmeehan@udel.edu(C.L.Meehan),farshid@cee.msstate.edu(F.Vahedifard).

0013-7952/$–seefrontmatter©2012ElsevierB.V.Allrightsreserved.http://dx.doi.org/10.1016/j.enggeo.2012.10.016

basedpseudo-staticmethods,displacement-basedmethods(some-timesreferredasNewmark-typeorslidingblockmethods),andstress-deformationanalysesthroughnumericalmethods,suchasfiniteelementordiscreteelementmethods(KramerandSmith,1997).Asanintermediatelycomplicatedandaccurateapproach,displacement-basedmethodsdevelopedbasedonslidingblocktheoryproduceareliableindexofslopeperformanceunderseismicloadingthroughtheirpredictivecalculationofpermanentearthquake-induceddisplacements(e.g.,KramerandSmith,1997).

SinceNewmark's(1965)introductionoftheslidingblockmethod,numerousdisplacement-basedanalyticalmethodshavebeenpro-posedtoimproveupontheaccuracyofNewmark'soriginalmethod(e.g.,MakdisiandSeed,1978;KramerandSmith,1997;RathjeandBray,2000),tosimplifyitsuse(e.g.,FranklinandChang,1977;AmbraseysandMenu,1988;Jibson,2007;HsiehandLee,2011;RathjeandAntonakos,2011),ortoapplythegeneralconceptofthemodeltoapplicationsbeyondthoseoriginallyproposedbyNewmark(e.g.,RichardsandElms,1979;LingandCheng,1997;Lingetal.,1997).MorerecentstudieshavealsobeenperformedtocharacterizeuncertaintiesassociatedwithNewmark-typemodels(e.g.,StrenkandWartman,2011).Thenumberofdisplacement-basedmodelsthathavebeenproposedisquitesignificant,anditisconsequentlydiffi-cultforpracticingengineerstoascertainwhichmodelshouldbeselectedforapplicationtoagivenproblem.Thispaperwillfocusondesignmethodsthathavebeendevelopedtosimplifytheuseofslidingblockmodels,whichwillhereafterbereferredtoassimplified

C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193181

slidingblockmethods.Itshouldbeemphasizedthat,insomecases,theapplicationofsimplifiedslidingblockmodelsisnottrulya“simple”task,asalargenumberofrelativelysophisticatedinputparametersaresometimesrequired.

Thispaperseekstoperformtwotasks:(1)toprovideathoroughreviewofexistingliteraturethatsummarizesandorganizesalargenumberofsimplifiedslidingblockmodels,makingtheseempiricalequationsmoreaccessibleforusebypracticingengineers,and(2)toevaluatetherelativeaccuracyofanumberofexistingsimplifiedslidingblockmodelsforpredictingearthquake-induceddisplacements.Toac-complishthesecondtask,thepredictivecapabilityoffifteensimplifiedslidingblockmodelsisassessedbycomparingmodelpredictionswiththeactualdisplacementsthatwereobservedafterearthquakeshakingin122casehistoriesofearthdamsandembankments.2.Developmentandevolutionofslidingblockmodels

Newmark(1965)isoftencreditedwiththefirstdevelopmentofadisplacement-based“slidingblock”methodforthedynamicanalysisofearthdamsandembankments.AsnotedbyMarcuson(1995),itisprobablyalsoappropriatetocitetheearlydevelopmentofdisplacement-basedapproachesinseismicslopestabilitytocontribu-tionsmadebyWhitmanandTaylor(e.g.,Taylor,1953).EarlypioneersinthisareaalsoincludeGoodmanandSeed(1965),whousedasim-ilardisplacement-basedmethodinsteadoftraditionalpseudo-staticanalysistoevaluateslopeperformanceunderearthquakeshaking.

Newmark(1965)assumedthatthedominantmechanismforearthquake-induceddisplacementindamsinvolvedslidingshearalongawell-definedfailuresurface.Heproposedthatthedynamicbehaviorofaslidingmasscouldbesimulatedbymodelingthemassasarigidblockslidingonaninclinedbase.Usingthisapproach,athresholdaccelerationisdefinedthatcorrespondstotheinertialforcethatmustbeappliedtoovercometheshearresistancebetweentheblockandthebase.Incurrentpractice,thisaccelerationiscommonlyreferredtoasthe“critical”or“yield”acceleration,anditisusuallyassumedtobetheinertialaccelerationthatyieldsafactorofsafetyofoneinapseudo-staticanalysisoftheslope.UsingNewmark'sapproach,slidingwillcommencewhentheshaking-inducedaccelerationexceedsthecriticalacceleration.Thecumulativeseismicdisplacementscanbecalculatedbyintegrationofeverywheretherelativevelocityoftheslidingblockisgreaterthanzero.

Inpractice,valuesofcriticalaccelerationaretypicallyestimatedusingatrialanderrorapproachinconjunctionwithconventionallimit-equilibriumslopestabilitymethods.Explicitequationshavealsobeendevelopedtodirectlyestimatethecriticalaccelerationforrelativelyuniformslopesandsimplefailuremechanisms(e.g.,Brayetal.,1998;Jibsonetal.,2000)ornon-circularfailuremechanisms(e.g.,Sarma,1973)asafunctionofcriticalinputparameterssuchasslopegeometry,thecohesionandfrictionangleofthesoil,andtheunitweightofthesoil.Forcertainapplicationssuchasrigorousprobabilisticanalysesorlandslidehazardmapping,relativelysimplefunctionalformsthatcanbeusedtodeterminecriticalaccelerationcansignificantlydecreasetherequiredcomputationaleffort,andmaybeappropriateforusegiventherelativeuncertaintyofmodelinputparameters.

Newmark'sslidingblockmodelwasdevelopedandiscommonlyimplementedfollowinganumberofsimplifyingassumptions.Asignificantamountofresearchhasbeenconductedtoexaminethesensitivityofpredictedseismicdisplacementstotheseassumptions,andinmanycases,newmodelsormodificationstoNewmark'soriginalmodelhavebeenproposedtoimprovetheaccuracyofthepredicteddisplacements.ThelimitingassumptionsassociatedwithNewmark'soriginalmodelandsomeofthepertinentstudiesthathavebeenperformedbyotherstostudytheeffectsoftheseassump-tionsareasfollows:(a)thedynamicresponseofthefailuremassdoesnotaffecttheearthquake-induceddisplacement(e.g.,Makdisiand

Seed,1978;LinandWhitman,1983;Hynes-GriffinandFranklin,1984;KramerandSmith,1997;BrayandRathje,1998;RathjeandBray,2000;Wartmanetal.,2003;RathjeandAntonakos,2011);(b)thepotentialfailuremassoftheslopefailsfollowingarigid-perfectlyplastictypeoffailuremechanism(e.g.,KutterandJames,1989;Yanetal.,1996;Mendezetal.,2009);(c)thecriticalaccelera-tionremainsconstantduringshaking,correspondingtonoincreaseorlossofstrengthduetoearthquakeshaking(e.g.,Houstonetal.,1987;KutterandJames,1989;Matasovicetal.,1997);(d)permanentdisplacementoccursjustinthedownwarddirection,and“upslopesliding”doesnotoccur(e.g.,Yan,1991;Matasovicetal.,1998);(e)theverticalcomponentofthegroundmotiondoesnotaffecttheearthquake-induceddisplacement(e.g.,Yanetal.,1996;LingandLeshchinsky,1998;KramerandLindwall,2004;Sawickietal.,2007);(f)thedisplacementsaccumulatealongasingle,welldefinedfailuresurface(e.g.,KutterandJames,1989;Nguyenetal.,2005;WartmanandStrenk,2006);(g)thesoilshearratedoesn'tinfluencethepermanentdisplacementthatoccurs(e.g.,LemosandCoelho,1991;Tika-Vassilikosetal.,1993);and(h)theeffectofporewaterpressureisignored(e.g.,Sarma,1975;KutterandJames,1989;Meehanetal.,2008).

Inadditiontothemodificationsproposedabove,othershavesuggestedextendingtheuseofNewmark'smethodtoearthquakeengineeringapplicationsbeyondearthdamsandembankments.Insomecases,itisnecessarytomodifytheformulationortheframe-workofthemodelinorderforthisextensiontobereasonable.Someofthemorecommonlyencounteredapplicationsareasfollows:conventionalgravityretainingwalls(e.g.,RichardsandElms,1979;WhitmanandLiao,1985),wasteslopesandlandfills(e.g.,KramerandSmith,1997;Matasovicetal.,1997;BrayandRathje,1998),geosynthetic-reinforcedslopesandmechanicallystabilizedearthwalls(e.g.,Lingetal.,1997;PaulsenandKramer,2004;HuangandWu,2006),anchor-reinforcedslopes(e.g.,Trandafiretal.,2009);rockslopes(e.g.,LingandCheng,1997);andearthquake-triggeredlandslidesandhazardmapping(e.g.,WilsonandKeefer,1983;Jibsonetal.,2000;MilesandKeefer,2000;SaygiliandRathje,2009).3.Simplifiedslidingblockmethods

Inordertopredictearthquake-induceddisplacementsusingNewmark'smethod,itisnecessarytohavebothaninputaccelerationtimeseriesthatcorrespondstotheearthquakegroundmotion,andacriticalaccelerationwhichisrepresentativeofthedynamicshearre-sistanceoftheslope.Asdiscussedintheprevioussection,numerousotheranalyticalmethodshavebeenproposedusingthisframework,manyofwhichalsorequiredeterminationofasitespecificaccelera-tiontimehistoryforinputintotheanalysis.Thedeterminationofasitespecificaccelerationtimehistoryiscommonlyperformedusingaselectionprocessthatlooksforsitesthathavebeenshakenbyanearthquakeofsimilarmagnitude,thatarelocatedatasimilardistancefromtheearthquakesource,andthathavesimilargroundconditions.Insomecases,anumberofaccelerationtimehistoriesareusedinconjunctionwithNewmark'smethodforagivensite,andpostulatedaccelerationrecordsarescaledtoachievethedesiredlevelofshaking.

Theselectionofsitespecificgroundmotionsandappropriatescalingfactorsisarathercomplicatedprocessthattypicallyinvolvesacertainlevelofexpertiseandjudgment(e.g.,Watson-LampreyandAbrahamson,2006).Asaresult,anumberofsimplifiedslidingblockmethodshavebeenproposedthatrequireonlycharacteristicgroundmotioninputparameterssuchasthepeakgroundacceleration(amax),peakgroundvelocity(vmax),earthquakemomentmagnitude(M),Ariasintensity(Ia),etc.intheplaceofaccelerationtimehisto-ries.Inordertodevelopthesemethods,researcherstypicallyhaveperformedanalyticalslidingblockanalysesusingarangeofcriticalaccelerationvaluesincombinationwithadatabaseofgroundmotions.Earthquake-induceddisplacementsarepredictedforeach

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criticalaccelerationincombinationwitheachgroundmotioninthedatabase,andtheresultingdisplacementsareplottedversusthecharacteristicgroundmotionparametersofinterest(e.g.,seismicdisplacement(D)vs.criticalaccelerationratio(ac/amax)isacommonformofdatapresentation).Regressionanalysiscanthenbeusedtodeveloparelationshipbetweenoneormoreofthecharacteristicgroundmotioninputparametersandtheresultingearthquake-induceddisplacement.

Alargenumberofsimplifiedslidingblockmethodshavebeenproposedbyothersforapplicationtoavarietyofgeotechnicalearth-quakeengineeringproblems.AsummaryofcommonlyreferencedmethodsisprovidedinTable1.Thecriticalelementsofeachmodelthatareshownineachofthecolumnsinthistablearediscussedinmoredetail,asfollows.3.1.Functionalform

Thiscolumnpresentsthefunctionalformofthesimplifiedslidingblockmethodthatwasprovidedbytheoriginalauthors;thisfunctionalformistheproductoftheregressionanalysesthatwereperformedforthecharacteristicgroundmotionparametersofinterest.Inanumberofthecases(Eqs.1–4),theoriginalauthorsdidnotpresentanequationfortheirsimplifiedmethodintheformthatisprovidedhere,insteadpresentingtheirresultssothatachart-basedsolutionapproachcouldbefollowed.Forthesemodels,theequationsthatarepresentedinTable1aretheresultsofregressionanalysesperformedbyCaiandBathurst(1996)upontheoriginalauthors'chart-basedsolutions.Usingasimilarapproach,thefunctionalformspresentedfortheHynes-GriffinandFranklin(1984)models(Eqs.7and8)weredeter-minedbytheauthorsofthismanuscriptusingregressionanalyses.3.2.Value

Whenusingagivensimplifiedmodel,itiscriticaltounderstandtheintrinsicnatureofthedisplacementvaluethatispredicted.Typically,theregressionanalysisperformedbytheoriginalauthorswascalibratedtoprovideeithera“mean”oran“upperbound”predictionofearthquake-induceddisplacement.MakdisiandSeed(1978)providearangeofdisplacementpredictions,andconsequent-lythechartprovidedbytheoriginalauthorsneedstobeusedtodetermineeithera“mean”(mid-rangevalue)or“upperbound”(topoftherangevalue)ofearthquake-induceddisplacement.

Inordertoconsidertherelativelyunpredictablenatureofstronggroundmotionsandtheuncertaintiesassociatedwithpredictingseismicdisplacementsusingslidingblockmodels,anumberofstatis-ticalandprobabilisticapproacheshavebeenusedtoquantifytheuncertaintyofthedisplacementspredictedusingtheslidingblockmodelequations(e.g.,LinandWhitman,1986;AmbraseysandMenu,1988;Yegianetal.,1991;BrayandTravasarou,2007;RathjeandSaygili,2008;RathjeandSaygili,2011).Ingeneral,thesemethodsareformulatedtoprovidetheprobabilityofexceedanceofagivenseismicdisplacement.Theyareappropriateforprobabilisticseismichazardanalysis.Ingeneral,forthosemodelswhichweredevelopedinaprobabilisticframework,the“median”valuepredictionequationsthatarepresentedinTable1aretheresultofprobabilisticpredictionsofdisplacementwitha50%probabilityofexceedance(e.g.,AmbraseysandMenu,1988;Yegianetal.,1991).3.3.Designatedapplication(s)

Asnotedpreviously,anumberofmodificationshavebeenmadetoNewmark'sanalyticalmethodtoextenditsusetoearthquakeengi-neeringapplicationsbeyondearthdamsandembankments.Inasimilarfashion,simplifiedslidingblockmethodshavealsobeendevelopedorcalibratedforusewithavarietyofearthquakeengineeringapplications.

Thiscolumnprovidestheapplicationforwhichthesimplifiedslidingblockmodelwassuggestedand/orcalibratedforbytheoriginalauthors.3.4.Model

Eachsimplifiedmodelwasdevelopedbaseduponpredictionsofdisplacementthatweremadeusinganunderlyinganalyticalslidingblockmodel.Awidevarietyofanalyticalmodelshavebeenusedtodevelopthesimplifiedmodels,eachwithitsowninherentsetofassumptions.Typically,themostsignificantdifferencebetweentheunderlyinganalyticalmodelsistheassumptionsthattheymakewithrespecttothedynamicresponseoftheslidingmass.Theunderlyinganalyticalmodelscanbebroadlyclassifiedaseither“rigid”(e.g.,Newmark,1965),“decoupled”(e.g.,MakdisiandSeed,1978),or“coupled”(e.g.,KramerandSmith,1997;RathjeandBray,2000).Itisimportanttounderstandthedifferencesintheunderlyinganalyticalmodelswheninterpretingtheresultsfromvarioussimpli-fiedmodels,hencetheinclusionofthiscolumninthetable.3.5.Numberofrecords

Sincesimplifiedmodelsaredevelopedfromregressionanalysis,thenumberofaccelerationtimehistoriesthatareusedintheregres-sionmayaffectthequalityofthemodelprediction.Morerecentsim-plifiedmodelsusemoretimehistoriesforregressionandmodeldevelopmentthansomeoftheearlysimplifiedmodels,andmaycon-sequentlyyieldmorereliablepredictionsofearthquake-induceddisplacement.

4.Casehistoryanalysis

Inordertoassessthepredictivecapabilitiesofanumberofthesimplifiedmodelsdescribedintheprevioussection,itisusefultocomparepredictionsofearthquake-induceddisplacementwiththeactualvaluesthatwereobservedduringaseriesofseismicevents.Adatabaseof122casehistoriesdescribingtheperformanceofearthdamsandembankmentsduringpastearthquakesispresentedindetailinSinghetal.(2007).Thiscasehistorydatabasewascompiledbaseduponanextensiveliteraturereview,andwasusedinthisstudytoevaluatetheperformanceofthesimplifiedmodels.ThecompletelistofcasehistoriesalongwiththepertinentparametersforeachcasehistorycanbefoundinVahedifard(2011).

Distributionsofpertinentearthquakegroundmotionparameters,siteandslopecharacteristicsofinterest,andtheassociatedearthquake-inducedslopedisplacementsintheSinghetal.(2007)databaseareshownareFig.1.Forthesehistograms,datafallingatthebreakbetweenbinsisassignedtothelowerbin,i.e.,aMagnitude8.0earthquakeisassignedtothe“7.5–8.0”binratherthanthe“8.0–8.5”bin.Forbrevity,itisnotpossibletoincludethesourceinformationorindividualdetailsofeachcasehistoryhere;interestedreadersarereferredtoSinghetal.(2007)formoredetailedcasehistoryinformation.

AnumberofthedatabaseparametersshowninFig.1weretakendirectlyfromSinghetal.(2007).Thetechniquesthattheyusedtodeterminethesedatabaseparametersareasfollows.

4.1.Earthquakemomentmagnitude(M),heightofstructure(H),andepicentraldistance(Distance)

Takenfromtheoriginalcasehistoryreference,whichisprovidedinSinghetal.(2007).NotethatDistanceinformationforoneofthecasehistoriesisnotavailableintheSinghetal.(2007)database;consequently,thereareonly121valuesintheDistanceandvmaxdistributions,whichmeansthatthiscasehistorycouldnotbeusedforallofthesimplifiedmodelsthatwereexamined.

C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193

Table1Summaryofsimplifiedslidingblockmodels.Eq.ModelFunctionalform󰀁󰀃v2ac−1maxD¼3amaxamax󰀁󰀃v2ac−2maxD¼0:5amaxamax󰀁󰀃󰀁󰀃v2ac−2ac−0:49maxD¼9:2exp−5:87amaxamaxamax !󰀁󰀃Daclog¼0:85−3:91amaxCamaxT2p󰀁󰀃󰀁󰀃v2acac−0:38maxD¼35exp−6:91amaxamaxamax󰀁󰀃Dac¼f[chart-basedmethod]amaxTDamax󰀁󰀃v2ac−4D¼0:087maxamaxamax󰀁󰀃󰀁󰀃ac4ac3þ0:193−logðDðcmÞÞ¼0:078amaxamax󰀁󰀃2󰀁󰀃acac0:285−1:847þ0:804amaxamax󰀁󰀃󰀁󰀃ac4ac3logðDðcmÞÞ¼−0:116−0:702−aamax󰀁󰀃2󰀁max󰀃acac1:733−2:854−0:287amaxamaxWhitmanandLiao(1985)D¼37󰀁󰀃v2acmaxexp−9:4amaxamax\"󰀁10AmbraseysandMenu(1988)logðDÞ¼0:90þlog1−acamax󰀃2:53󰀁ac󰀃−1:09#MedianGroundandslopesRigid50ValueUpperboundUpperboundMeanMeanUpperboundRangeUpperboundUpperboundEarthdamsandembankmentsEarthembankmentsEarthdamsandembankmentsGravityretainingwallsEarthdamsRigidRigidDecoupledRigidDecoupled91799179354Designatedapplication(s)EarthdamsandembankmentsModelRigid183

No.ofrec.41aaNewmark(1965)1ba2a3a4a567bSarma(1975)FranklinandChang(1977)MakdisiandSeed(1978)RichardsandElms(1979)Hynes-GriffinandFranklin(1984)8bMean9MeanGravityretainingwallsRigid17911Yegianetal.(1991)12Brayetal.(1998)amax !󰀁󰀃Daclog¼0:22−10:12þamaxNeqamaxT2D󰀁󰀃󰀁󰀃ac2ac3−11:4816:38amaxamax󰀁󰀃aclogðD=ðkmaxD5−95ÞÞ¼1:87−3:477amaxmedianEarthdams,embankments,andslopesRigid86Mean13Watson-LampreyandAbrahamson(2006)MeanlnðDðcmÞÞ¼ð5:470þ0:451ðlnðSaðT¼1sÞÞ−0:45Þþ20:0186ðlnðSaðT¼1sÞÞ−0:45Þþ0:596ðlnðARMSÞ−1:0Þþ20:656ðlnðSaðT¼1sÞ=amaxÞÞþð−0:0716ÞðlnðSaðT¼1sÞ=amaxÞÞþ0:802ðlnðDuracÞ−0:74Þþ0:0763ðlnðDuracÞ−0:74Þþ󰀄1ð−0:581Þðlnðamax=acÞþ0:193Þ2Rock-founded,geosynthetic-linedsolid-wastelandfillsEarthslopesDecoupled33Rigid615814BrayandTravasarou(2007)lnðDðcmÞÞ¼−0:22−2:83lnðkcÞ−0:333ðlnðkcÞÞþ20:566lnðkcÞlnðamaxÞþ3:04lnðamaxÞ−0:244ðlnðamaxÞÞþ0:278ðM−7ÞlnðDðcmÞÞ¼−1:10−2:83lnðkcÞ−0:333ðlnðkcÞÞþ0:566lnðkcÞlnðSað1:5TDÞÞþ3:04lnðSað1:5TDÞÞ−20:244ðlnðSað1:5TDÞÞÞþ1:50TDþ0:278ðM−7Þ\"󰀁#󰀃󰀁󰀃ac2:341ac−1:438logðDðcmÞÞ¼0:215þlog1−amaxamax\"󰀁logðDðcmÞÞ¼−2:710þlog0:424M1−acamax󰀃2:335󰀁acamax󰀃−1:478#22MeanEarthandwasteslopesCoupled(rigid)Coupled(non-rigid)137615Mean16Jibson(2007)MeanNaturalslopesRigid227017þMeanMeanMeanMeanNaturalslopesRigid(scalar)2383181920SaygiliandRathje(2008)log(D(cm))=2.401logIa−3.481logac−3.230log(D(cm))=0.561logIa−3.833log(ac/amax)−1.474󰀁󰀃󰀁󰀃acac2þ−20:93lnðDðcmÞÞ¼5:52−4:43amaxamax󰀁󰀃󰀁󰀃ac3ac442:61−28:74þ0:72lnðamaxÞamaxamax󰀁󰀃󰀁󰀃acac2þ−20:84lnðDðcmÞÞ¼−1:56−4:58amaxamax󰀁󰀃3󰀁󰀃4acac44:75−30:5−0:64lnðamaxÞþ1:55lnðvmaxÞamaxamax󰀁󰀃v2acD¼245:4maxexp−8:86amaxamax21MeanRigid(vector)22Ebelingetal.(2009)UpperboundRock-foundedstructuresRigid122(continuedonnextpage)184

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Table1(continued)Eq.ModelFunctionalform

23

D¼65:44

v2󰀁󰀃max

aexp−8:86acmaxamax

aFunctionalformpresentedbyCaiandBathurst(1996).bFunctionalformpresentedbytheauthors.

4.2.Peakhorizontalgroundacceleration(amax)

Takenfromtheoriginalcasehistoryreference,whichisprovidedinSinghetal.(2007).Typically,thesevalueswereestimatedintheirorig-inalreferenceusingavarietyofattenuationrelationships.Inafewofthecases,thepeakgroundaccelerationvaluesweremeasureddirectlyonsite.Thepeakseismiccoefficient,kmax,istheunitlessequivalentofamaxandisdefinedaskmax=amax/g,wheregisthegravitationalacceleration.

4.3.Predominantperiodofearthquakegroundmotion,(Tp)

Takendirectlyfromtheoriginalcasehistoryreference,ifavailable.Otherwise,estimatedfromavailabledatausingthemethodpresentedbyIdriss(1991).

4.4.Fundamental(elastic)periodofdam(TD)

Takendirectlyfromtheoriginalcasehistoryreference,ifavailable.Otherwise,determinedusingtheapproachdescribedbyGazetasandDakoulas(1991)andAndrusandStokoe(2000).4.5.Critical(yield)acceleration(ac)

Calculatedusingapseudo-staticslopestabilityanalysisprocedurethatutilizedBishop'smodifiedmethodforanalyzingcircularfailuresur-faces.Soilparametersnecessaryfortheseanalyses(e.g.,unitweight,co-hesion,andfrictionangle)weretakenfromlaboratoryorfieldtestsresultsiftheywereavailableintheoriginalreference.Otherwise,gener-icsoilpropertieswereused,whichweredeterminedbaseduponmate-rialdescriptionsthatwereprovidedintheoriginalreference(atableofassumedparametersbasedonsoiltypeisprovidedinSinghetal.,2007).Thecriticalcoefficient,kc,istheunitlessequivalentofacandisdefinedaskc=ac/g.

4.6.Resultantobserveddisplacement(D)

Calculatedbytakingthedotproductoftheverticalandhorizontalcomponentsoftheobserveddisplacementvectorandaunitvectoralignedalongtheaverageinclinationofthebaseoftheslidingsurface.Thebaseinclinationanglewasdeterminedfromthecriticalfailurecirclefromthepseudo-staticanalysesthatwereperformed.

Anumberofthesimplifiedslidingblockmodelsthatwereexam-inedinthisstudyrequiretheuseofinputparametersthatarebeyondwhatwasprovidedintheoriginalSinghetal.(2007)database.Theseparametersweredeterminedbytheauthorsofthispaperusingthefollowingmethodology.

4.7.Peakgroundvelocity(vmax),spectralaccelerationwith5%dampingataperiodof1s(Sa(T=1s)),spectralaccelerationwith5%dampingatadegradedperiodof1.5TD(Sa(1.5TD))

Determinedfromearthquakemagnitude(M),thedistancefromtheearthquakesourcetothesite(Distance),andthetime-averagedshearwavevelocityforthetop30minthefoundation(Vs30)usingtheBooreandAtkinson(2008)ground-motionpredictionequations.Vs30valueswereestimatedusingsiteconditiondataprovidedby

ValueDesignatedapplication(s)Model

No.of

rec.

Mean

Singh(2009)inconjunctionwiththegeneralrangeofVs30valuespro-videdbyBooreetal.(1997).

4.8.Rootmeansquareofacceleration(ARMS)andthedurationforwhichtheaccelerationisgreaterthanthecriticalacceleration(Durac)DeterminedusingtheequationsprovidedbyWatson-LampreyandAbrahamson(2006).

5.Relativeaccuracyofsimplifiedslidingblockmodels

Asdescribedpreviously,inordertoassessthepredictivecapabilitiesofthedifferentsimplifiedmodels,itisusefultocomparepredictionsofearthquake-induceddisplacementwiththeactualvaluesthatwereobservedduringseismicevents.Asthecasehistorydatabasethatwasusedforassessmentiscomposedprimarilyofearthdamsandembank-ments,itisnotappropriatetotesttheperformanceofsimplifiedslidingblockmodelsthathavebeendevelopedforotherapplications.Conse-quently,forthisreason,thefollowingmethodsthatarelistedinTable1werenotincludedinthecasehistoryanalysis:RichardsandElms(1979),WhitmanandLiao(1985),Brayetal.(1998),andEbelingetal.(2009)(bothmodels).

AsshowninTable1,BrayandTravasarou(2007)haveproposedtwodifferentsimplifiedslidingblockmodels,oneforapplicationto“rigid”or“nearlyrigid”potentialslidingmasses(thosehavinganinitialfundamentalperiod,TDb0.05s),andtheotherforapplicationtonon-rigidslidingmasses(thosewithTD≥0.05s).Inthedatabaseofcasehistoriesthatwasexamined,allofthecasehistoriescorre-spondtocaseswhereTDisgreaterthan0.05seconds;consequently,BrayandTravasarou's“rigid”slidingmassapproach(Eq.14)wasalsonotincludedinthecasehistoryanalysis.

IntheSinghetal.(2007)database,notenoughinformationisavailabletodetermineAriasintensity(Ia)values(Arias,1970)withanydegreeofconfidenceusingempiricalcorrelations(e.g.,theapproachproposedbyTravasarouetal.,2003).Consequently,thethirdandfourthmodelsproposedbyJibson(2007),Eqs.18and19,cannotbereliablytestedinthiscasehistoryanalysis.Inanycase,asnotedbyJibson(2007),thesemodelshavethetendencytopredictvaluesthatarerelativelyclosetoeachotherforagivengroundmotion,sotheomissionofthesemodelsisnotbelievedtosignificantlyaffecttheconclusionsthatweremadeinthisstudy.

SaygiliandRathje(2008)havealsoproposedtwodifferentsimpli-fiedmethodsforpredictingearthquake-induceddisplacement,onethatusesascalarapproachinthecalculationprocessandonethatusesavectorapproach(i.e.,Eqs.20and21).Followingtheirvectorap-proach,itispossibletouseuptofivedifferentequationswithdifferentcombinationsofinputgroundmotionparameters.Fouroftheseequa-tionsrelyoninputvaluesofeitherAriasintensity(Ia)and/orthemeanperiodofearthquakeacceleration(Tm),whichrequireadditionalassumptionsandcalculationsfromthedatathatisavailableintheSinghetal.(2007)database.Consequently,onlyoneofthevector-basedequa-tionsprovidedbySaygiliandRathje(2008)ispresentedinTable1andisusedinthecasehistoryanalysis(Eq.21).

FifteenofthesimplifiedslidingblockmodelspresentedinTable1areappropriateforusewiththeSinghetal.(2007)database,andconsequentlyweretestedinthecasehistoryanalysis:Eqs.1–5,7,8,10,11,13,15–17,20,and21(Eq.numbersshowninTable1).Eachof

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Fig.1.Distributionsofearthquakegroundmotionparameters,siteandslopecharacteristicsofinterest,andearthquake-inducedslopedisplacementsthatwereusedinthecasehistoryanalysis.

thesemodelswasdevelopedbaseduponcalibrationorregressionanaly-sisthatwasperformedbytheoriginalauthorsoverarangeofinputpa-rameters.Asaresult,forthemodelsthatweretested,itisnotreasonabletoexpectthattheywouldbeabletomakeaccuratepredic-tionsofslopebehaviorforinputparametersoutsideoftheircalibrationrange.Consequently,forpurposesofthecasehistoryanalysisthatwasperformed,eachmodelwasonlytestedforcasehistoriesthatfallintotheappropriaterangeofmodelinputparameters.Thismeansthatonlyasubsetoftheoriginal122casehistoriesinthedatabasecouldbeusedtoassessthepredictivecapabilitiesofeachmodel.TherangefilteringcriteriathatwasusedandtheassociatednumberofcasehistoriesthatweretestedforeachmodelareprovidedinTable2.

Forthefifteensimplifiedslidingblockmodelsthatweretested,predictionsofearthquake-induceddisplacementweremadeforeachcasehistorywhichsatisfiedtherangefilteringcriteriashowninTable2,usingtheappropriatecasehistoryinputparametersfromthedatabase.Fig.2showstheobservedversuspredicteddisplace-mentsforeachofthesimplifiedslidingblockmodelsthatwasexam-ined.Asshown,withveryfewexceptions,almostallofthecasehistoriesthatwereanalyzedfallabovetheunitylineforeachofthemodels.Notethatallofthepointswhichfallabovetheunityline

correspondtoanunconservativemodelprediction.Thepredictedseis-micdisplacementsforeachcasehistorywerethencomparedwiththeobservedseismicdisplacements,andtherelativeerrorinthemodelpredictionwascalculatedusingEq.(24).Thedifferencebetweentheobservedandpredicteddisplacementsandtherelativeerrorforeachmodeloverallofthecasehistorieswerethenvisualizedusinghisto-gramsandcumulativedistributionfunctions.Inordertoperformside-by-sidecomparisonsoftheresultsfromanumberofmodels(saymorethanthreeorfour),cumulativedistributionfunctionswerefoundtobemoreuseful,astheplotstendtoappearsignificantlylesscluttered.Fig.3showsthecumulativedistributionofthedifferencevalues(Dobserved−Dpredicted)forthesimplifiedslidingblockmodelsthatwereexaminedinthecasehistoryanalysis.Fig.4showsthecumu-lativedistributionofrelativeerrorforthesimplifiedslidingblockmodelsthatwereexamined.RelativeError¼

Dobserved−Dpredicted

x100%

Dobserved

ð24Þ

IntheSinghetal.(2007)database,anumberofthecasehistoriesthatarepresentedcorrespondtoeitherverysmallorverylarge

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Table2

Casehistoryutilizationforeachsimplifiedslidingblockmethod.Eq(s).

Model(s)

RangeselectionNo.ofcriteriaapplied

casesuseda1,2Newmark(1965)0b(ac/amax)≤1

78,373Sarma(1975)

0.05≤(ac/amax)≤0.9560,354FranklinandChang(1977)0.02≤(ac/amax)≤1.065,375MakdisiandSeed(1978)0.05≤(ac/amax)≤0.957,

32

6.5≤M≤8.25

7,8Hynes-GriffinandFranklin0.01b(ac/amax)b0.643,19(1984)

M≤8

10AmbraseysandMenu(1988)0.05≤(ac/amax)≤0.9527,206.6≤M≤7.2

11Yegianetal.(1991)0.05≤(ac/amax)≤0.946,275.6≤M≤7.713Watson-Lampreyand0.1≤ac≤0.3(g)66,33Abrahamson(2006)

4.5≤M≤7.9

15

BrayandTravasarou(2007)

0.01≤ac≤0.5(g)22,7

4.5≤M≤9.00≤TD≤2(s)

0.002≤Sa(1.5TD)≤2.7(g)Dpredicted>1cm16,17Jibson(2007)(ac/amax)b1.042,29

0.05≤ac≤0.4(g)5.3≤M≤7.6

20,21SaygiliandRathje(2008)0.05≤(ac/amax)≤146,30

0.05≤ac≤0.3(g)4.5≤M≤7.9amax≤1(g)

aNote:Thefirstnumbershowninthiscolumncorrespondstothenumberofcasesthatwereanalyzedafterapplyingtherangefilteringcriteriaonly(e.g.,Figures3and4).Thesecondnumbercorrespondstothenumberofcasesthatwereanalyzedafterapplyingboththerangefilteringcriteriaandthecriteria0.01mbDobservedb1m(e.g.,Figures5and6).

observeddisplacementsafterearthquakeshaking.Itcanbereason-ablyarguedthatthesesmallandlargedisplacementvaluesdonotrepresentrobustcasehistoriesfortestingslidingblockmodels.Inthecaseofsmallobserveddisplacements,say1cmorless(e.g.,theselectioncriteriapresentedbyBrayandTravasarou,2007),theobserveddisplacementsaresmallenoughthattheycanbereasonablyconsideredtobezeroforgeotechnicalapplications.Inthecaseoflargeobserveddisplacements,say1mormore(e.g.,themaximumallowabledisplacementfordamsproposedbyFranklinandChang,1977),thereisanincreasedpossibilitythatthesoilhaslostsignificantstrength,asituationwherenearlyallslidingblockmodelsarenotrecommendedforuse(e.g.,MakdisiandSeed,1978).Additionally,atlargedisplacements,kinematicfactorsrelatedtotheslidinggeometrywillplayanincreasedroleinthedisplacementthatoccurs,furtherincreasingthedifferencebetweenpredictedandobservedresults.

Thevaluesthathavebeenproposedfortheseupperandlowerdisplacementboundariesaresomewhatarbitrary;however,fortherea-sonspreviouslymentioned,theauthorshypothesizethatslidingblockmodelsmayyieldmoreaccuratepredictionsatintermediaterangesofearthquake-induceddisplacement.Toexplorethishypothesis,analysisofthecasehistorieswasrepeatedusingboththerangefilteringcriteriashowninTable2andthecriteria:0.01mbDobservedb1m.ThenumberofcasesthatwereexaminedforeachmodelusingthismorestringentselectioncriterionisshowninTable2.Figs.5and6showthecumula-tivedistributionofthedifferencevaluesandtherelativeerrorforthosecaseswhere0.01mbDobservedb1m,respectively.6.Discussionofresults

Themethodthatwasusedforassessmentofthesimplifiedslidingblockmodelsisacomparisonofpredictedvalueswiththeactualvaluesthatwereobservedduringseismicevents.Consequently,sep-aratecomparisonsshouldbemadeformodelsthatweredeveloped

basedupon“mean”(or“median”)valuepredictionswiththosethatweredevelopedbasedupon“upperbound”predictions.Intheanaly-sesthatwereperformed,thefollowingmodelswerecharacterizedas“upperbound”predictionmodels:Newmark(1965)—Eqs.1aand1b,FranklinandChang(1977)—Eq.4,andHynes-GriffinandFranklin(1984)—Eq.7.Theremainingmodelsthatwereassessedwerechar-acterizedas“mean”(or“median”)valuepredictionmodels.

Byexaminingthecumulativedistributionfunctionsofthediffer-encevaluesandtherelativeerrorforthe“mean”predictionmodels,thefollowingsignificantconclusionscanbedrawn.

Forallofthemodelsthatwereexamined,therelativeerrordistri-butionsaredominatedbypositivepredictionsofrelativeerror.AsshowninFig.4,67%ormoreofthecasesthatwereexaminedfellonthepositivesideofthespectrumforeverymodel.Ifjustoneofthemeanpredictionmodelsisexcluded(Watson-LampreyandAbrahamson,2006),thenumberofcasehistorieswherepositiverelativeerrorwascalculatedincreasesto81%ormoreforallofthemodelsthatwereexamined(Figure4).Ifonlythecaseswhereinter-mediatedisplacementswereobservedareincludedintheanalysis(0.01mbDobservedb1m),74%ormoreofthecasesthatwereexam-inedyieldedpositivevaluesofrelativeerror(Figure6).Theseobser-vationsareofsignificantconcern,aspositivevaluesofrelativeerrorcorrespondtocasehistorieswheretheobservedseismicdisplace-mentwasgreaterthanthepredictedseismicdisplacement—anunconservativemodelprediction.Thereasonforthisunconservativetrendinthemodels'predictionsisnotclear—theauthorshypothe-sizethatitmaybeattributedtooneormoreofthefollowingfactors:(1)somesortofsystematicassumptionsthataremadebytheunder-lyingslidingblockmodelsthatareusedinthesimplifiedmodels'formulations;(2)theinherentinabilityofslidingblockmodelstocapturetheearthquake-inducedvolumetriccompressionthatoccursinanembankmentduringshaking,asnotedbyBrayandTravasarou(2007);and(3)asystematicoccurrenceofsoilstrengthdegradationduringshaking,whichwasnotaccountedforinthesimplifiedslidingblockmodelsthatwereexamined(asthesimplifiedslidingblockmodelsallutilizedaconstantcriticalacceleration).

Forthosecasehistorieswheresignificantmovementoccurred,thedifferencebetweenthepredictedandtheobserveddisplacementswasquitelargeformanyofthemodelsthatweretested.AsshowninFig.3,thepercentageofcasehistorieswhereDobserved−Dpredicted>1mrangedbetween11%(forWatson-LampreyandAbrahamson,2006)to64%(forBrayandTravasarou,2007).Thisobservationissomewhatdis-concerting,asthepossibilityofunderpredictingactualdisplacementsbymorethan1mcouldhavenegativeimplicationsforthedetermina-tionofovertoppingfreeboardforearthdams.Astheoccurrenceoflargedisplacementswasnotwell-predictedformanyofthecaseswhereitoccurred,designersusingthesemethodsshouldbeawareoftherelativepotentialofthedifferentsimplifiedmodelsforsignificantunderpredic-tionofdisplacements.

Asnotedpreviously,itcanbereasonablyarguedthatslidingblockmodels(particularlysimplifiedslidingblockmodels)shouldnotbeusedforcaseswheresignificantsoilstrengthlossorlargeearthquake-induceddisplacementsoccur.Unfortunately,asthedifferencebetweentheobservedandpredicteddisplacementsislargeformanyofthecases,thedesignerwouldnotknowinadvanceofearthquakeshakingifamoresophisticatedanalysismethodwaswarranted.Nevertheless,itisalsoinstructivetolookatthemagnitudeofthedifferencesbetweentheobservedandpredicteddisplacementsforcaseswheretheobserveddisplacementsareinthe“intermediate”range(0.01mbDobservedb1m).AsshowninFig.5,themagnitudeofthedifferencevaluesinthe“inter-mediate”rangeismuchsmallerthanifallofthecasesareconsidered,astheverylargeobserveddisplacementcasesareremovedfromconsider-ation.AsshowninFig.5,the“intermediate”modeldifferencedistribu-tionshavethesamegeneralshapeandtendtofallinthesameapproximaterangeforeachofthemeanvaluepredictionmodelsthatweretested.

C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193187

Fig.2.Observedversuspredicteddisplacementsforthesimplifiedslidingblockmethodswhichwereexaminedinthecasehistoryanalysis.

Ingeneral,bycomparingFigs.3and5,itcanbeobservedthattheshapeofthe“difference”distributionfunctionschangessignificantlyifonlythe“intermediate”rangecasesareconsidered.Thisisnotsurprising,asthelargedisplacementcasesarefilteredoutusingthisap-proach.Incontrast,bycomparingFig.4withFig.6,itcanbeobservedthat,withtheexceptionofWatson-LampreyandAbrahamson(2006),mostoftherelativeerrordistributions'shapesandlocationsdonotchangesignificantlywhenthe“intermediate”displacementfilteringcriteriaisapplied.Thisindicatesthat,forfuturecomparisonsbetweenmodels,itmaynotbenecessarytofiltertheresultsbyobserveddisplacementifthefinalcriterionforcomparisonistherelativeerror.

Byexaminingthecumulativedistributionfunctionsofthediffer-encevaluesandtherelativeerrorforthe“upperbound”predictionmodels,thefollowingsignificantconclusionscanbedrawn.Foreachofthe“upperbound”modelsthatwastested,alargepercentageofthecasesthatwereanalyzeddonotinfactyieldupperboundmodelpredictionsofdisplacement(Figures3–6).ForNewmark's(1965)method,99%ofthecasesanalyzedhavediffer-encevaluesandrelativeerrorvaluesgreaterthanzero,whichmeansthattheobservedvaluesaregreaterthanthepredictedvalues—notatrue“upperbound”prediction.Inasimilarfashion,largenumbersarealsoobservedforFranklinandChang's(1977)method(98%)andHynes-GriffinandFranklin's(1984)method(66%).Theseobservationsareconsistentwiththetendencyformodelunderpredictionthatwasobservedduringanalysisofthe“mean”valuemodelresults.

Ingeneral,withthepossibleexceptionofHynes-GriffinandFranklin's(1984)upperboundmethod,theupperboundmodels

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Fig.3.Cumulativedistributionofdifferencebetweenpredictedandobserveddisplacementsforeachofthesimplifiedslidingblockmodels(sampledfromallobserveddisplacementcases).

yieldcumulativedistributionfunctionsthatareinthesamerangeasthe“meanvalue”predictionmodels(Figures3–6).

Undoubtedly,theinputparametersofsomeofthecasehistoriesinthedatabasemayhavesomeerrorsduetouncertaintyorassumptionsthatweremadebySinghetal.(2007)ortheoriginalcasehistoryauthors.Theuseofcasehistoriesthatarenotwelldocumentedcanbeconsideredsomewhatundesirable,astherearemanyuncertaintiesintheinputparametersthatcanhaveasignificanteffectonthemodelpredictionresults.Foranumberofthecasehistoriesthatareutilizedherein,theaccelerationatthesitelocationisunknown,andneededtobeestimatedusingattenuationrelationships.Forsomeofthecases,thematerialparametersareunknown,andneededtobeestimatedusingavailabledata.Clearly,thereisanuncertaintyassociatedwiththeuseofattenuationrelationshipsforestimatingthesite-specificgroundmotionparameters,whichcanleadtosignificantuncertaintyinthepredicteddisplacements.Inasimilarfashion,theuseofgenericsoilpropertiesforsomecasehistoriesisalsoundesirable,asitaffectsthecriticalaccelerationthatisusedinthemodels.Forbothoftheseinputparameters(groundmotionsandsoilstrength),smalluncer-taintiesintheinputscanyieldsignificanterrorsontheoutputsideoftheequation.Consequently,theuseofalargecasehistorydatabasethathassomeuncertaintywithitsinputparametersforsomeofitscasehistoriescanbeaneasilycriticizedapproachtosimplifiedslidingblockmodelassessment.Unfortunately,thedatathatwehaveavailableatthistimedoesnotallowforindependentassessmentoftherelativecontributionsof“modelerror”and“inputparametererror”.Takinganotherviewpointhowever,onecanobservethatmanyofthemodelsthatwereusedhereinweredevelopedusingaslidingblockapproachthatnecessitatedasignificantnumberofsimplifyingassumptions.Theseslidingblockmodelswerethensubjectedtoalargedatabaseofbothrecordedandsyntheticgroundmotions,whichyieldedawiderangeandsignificantscatterinthemodelpredic-tionsthatwereusedtodevelopthesimplifiedmodels.Thisentireap-proachtosimplifiedslidingblockmodeldevelopmentissyntheticinnature,whichnecessitatescomparisonwithrealcasehistoriestoassessmodeleffectiveness.Anumberofauthorshavetestedtheirsimplifiedmodelsusingafewwell-documentedcasehistories,andtheresultshavebeeningoodagreement.However,asthewell-documentedcasehistorydatabaseissosmall,testingasimplifiedmodelusingonlyafewdatapointsplacesextensiveemphasisontoofewpointsofobserva-tion.Moreover,evenforwell-documentedcasehistories,therecanbealargeuncertaintyinthemodelinputparameters(e.g.,shearstrength,groundmotions,watertableposition,degradationofsoilstrengthduringshaking,etc.),allofwhichcanhaveasignificanteffectonthemodelresults.

Basically,giventhenatureofthecasehistorydatathatiscurrentlyavailable,therearetwoapproachestosimplifiedslidingblockmodelassessment:(1)useasmallnumberofwell-documentedcasehisto-ries,or(2)useamuchlargerdatabaseofcasehistoriesthatcomprisearangeofuncertaintyintheinputparameters.Areviewofexistingliteratureindicatesthatthefirstapproachhasbeenmuchmorecommonthanthesecond.However,thisapproachtocasehistory

C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193189

Fig.4.Cumulativedistributionofrelativeerrorforeachofthesimplifiedslidingblockmodels(sampledfromallobserveddisplacementcases).

assessmentdisregardsmanyobservationsoffieldbehaviorthatmaybeuseful,simplybecausewedon'thaveasmuchinformationaswewouldlikeformanysites.Italsoplacestoomuchemphasisononlyafewcasehistorieswhereextensivedataisavailable,whichisstatis-ticallyproblematic.

Theapproachutilizedhereinfocusesontheuseofalargercasehistorydatabase.Forassessmentofsimplifiedslidingblockmodels,thismayactuallybeamoreappropriateapproach,asthesemodelswilltypi-callybeusedbypractitionersforsitesthatcomprisearangeofuncertain-tyinmodelinputparameters.Inpractice,formanysites,itisrelativelyuncommontohavesitespecificgroundmotionsorathoroughunder-standingofthesoilshearstrength.Inanycase,ifthesefactorsarewellknown,theuseofmoresophisticatedmodelsthanthoseutilizedhereiniswarranted.Giventhestatisticalnatureofinputparametervariationforsimplifiedmodels,whereparameterswillbeover-estimatedforsomecasehistoriesandunderestimatedinothers,onewouldexpectthatanaccuratesimplifiedslidingblockmodelwouldyieldpredictionstrendingaroundthemeanofobservedfielddisplacementsifalargeenoughcasehistorydatabasewasusedformodelassessment.

Theoverarchingtrendsindistributionbehaviorthatwereobservedforallofthesimplifiedslidingblockmodelsoverallofthecasehistoriesinthedatabasepresentaclearandconsistentpictureofmodelbehaviorinabroadsense.Surprisingly,allofthemodelstendedtounderpredicttheobservedearthquake-induceddisplacements.Thisisanissueofconcernforpracticingengineersthatusethesesimplifiedmodels,especiallyconsideringthattheapproachesthatwerefollowedinthisstudytoestimatemodelinputparametersarebelievedtobegenerallyconsistentwithwhatmightbedonebyengineersinpracticeonsimilarprojectswheremodelinputparametersarenotwellknown.

Asanalternativetodevelopmentofadditionalsimplifiedslidingblockmodels,perhapsitiswarrantedtouseshearstrengthsthathavebeenreducedtoaccountforsoilsofteningduringearthquakeshakingastheinputstosimplifiedslidingblockmodels.Thereduc-tionofshearstrengthtoaccountforstrengthdegradationduringaseismiceventcanbeincorporatedinslidingblockmodelsintwodifferentways:1)usingreducedshearstrengthparametersfortheslopestabilityanalysiswhichisperformedtodeterminethecriticalaccelerationinasimplifiedslidingblockmodelanalysis.Forexample,theuseofresidualratherthanpeakshearstrengthhasbeenrecommendedandadvocatedforseismicstabilityanalysisanddesignofearthstructures(e.g.,Bolton,1981;Jewell,1996;Leshchinsky,2001;LiuandLing,2012);2)usingvariablecriticalaccelerationinaslidingblockmodel,ratherthana“simplified”slidingblockmodel.Since,ingeneral,thecriticalaccelerationistheonlyparameterthatrepresentstheshearstrengthofsoilinsimplifiedslidingblockmodels,utilizingacriticalaccelerationwhichvariesasafunctionofmobilizedseismicdisplacementcanarguablyreflectthestrainsoft-eningphenomenon(e.g.,Matasovicetal.,1997;JibsonandJibson,2003).Forthecasehistoriesthatwereexaminedinthisstudy,eitherofthesesuggestedapproacheswouldyieldhigherpredictionsofdis-placementthatwouldbeinbetteragreementwithobservedfielddisplacements.

Analternativetosimplifiedslidingblockmodelsareregressionequa-tionswhicharedevelopedusingcasehistoriesofobservedearthquake‐

190C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193

Fig.5.Cumulativedistributionofdifferencebetweenpredictedandobserveddisplacementsforeachofthesimplifiedslidingblockmodels(sampledfromthecaseswhere0.01mbDobservedb1m).

induceddisplacements,ratherthansimulatedearthquakedisplacementsthataregeneratedusingsomesortofslidingblockmodel.Usingthisapproach,regressionanalysisisperformedtodeveloparelationshipbetweencharacteristicgroundmotioninputparametersandtheearth-quake‐induceddisplacementsthatareobservedforagiventypeofearthstructure(i.e.,earthdams,naturalslopes,retainingwalls,etc.)overarangeofcasehistories(e.g.,Singhetal.,2007,VahedifardandMeehan,2011).Themainobstacletoeffectivelyutilizethisapproachistherelativelylownumberofwell‐documentedcasehistories.

Asanadditionalsidenote,itisworthmentioningthatslidingblockanalysesprovideco-seismicdisplacementpredictions,whiletheobserveddisplacementsfromcasehistoriescombinedatathatisbothco-seismicandpost-seismic.Thisobservationbringstolighttheimpor-tanceofpost-seismicdisplacementsindesign;forexample,rainfallafteranearthquakemayincreasetheporepressuresontheslipsurface,whichcancauseadditionaldisplacementsduringsmallerafter-shocks.Thisphenomenonisclearlydifferentfromlossofstrengthduringco-seismicmovements(e.g.,SarmaandChlimintzas,2000).Un-fortunately,goodearthquake-inducedembankmentdisplacementdataisdifficulttoobtain,anddatathatdifferentiatesco-seismicandpost-seismicdisplacementsinaclearfashionisquiterare.Totrulyunderstandthesephenomena,moresophisticatedanalysismethodsarewarrantedwithasmallerdatasetofwell-instrumentedcasehisto-ries,priortoextrapolationouttoalarger,more“uncertain”datasetofthetypethatisusedinthecurrentstudy.

7.Summaryandconclusions

SinceNewmark's(1965)introductionoftheslidingblockmethod,numeroussimplifiedslidingblockmodelshavebeenproposed.Thispaperprovidesathoroughreviewofexistingliteraturethatsumma-rizesandorganizesalargenumberofsimplifiedslidingblockmodels,inanattempttomaketheseempiricalequationsmoreaccessibleforusebypracticingengineers.Analyseswerealsoperformedtoevalu-atetherelativeaccuracyofanumberofexistingsimplifiedslidingblockmodelsforpredictingearthquake-induceddisplacementsinearthdamsandembankments.Toaccomplishthistask,thepredictivecapabilityoffifteensimplifiedslidingblockmodelswasassessedbycomparingmodelpredictionswiththeactualdisplacementsthatwereobservedafterearthquakeshakingin122casehistoriesofearthdamsandembankments.Theresultsfromthesecomparisonsindicatethatforallofthesimplifiedslidingblockmodels,themodelpredictionsofdisplacementwerelessthantheobserveddisplace-mentforalargemajorityofthecasehistoriesthatwereexamined.Thisobservationwastrueforboththe“mean”and“upperbound”predictionmodels.Itwasalsotruewhetherornotthecasehistoriesusedintheanalysiswerefilteredtoincludeonlythosecaseswheretheobserveddisplacementswereinthe“intermediate”range(0.01mbDobservedb1m).

Thedifferencebetweentheobservedandpredicteddisplacementswasrelativelylarge(>1m)forasignificantpercentageofthecases,

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Fig.6.Cumulativedistributionofrelativeerrorforeachofthesimplifiedslidingblockmodels(sampledfromthecaseswhere0.01mbDobservedb1m).

foreachmodelthatwasexamined.Predictionerrorsofthismagnitudecanhaveasignificantconsequencewithrespecttothepost-shakingperformanceofearthdams,anddesignersusingthesemethodsshouldbeawareoftherelativepotentialofthedifferentsimplifiedmodelsforsignificantunderpredictionofdisplacements.

Theshapesandpositionsoftherelativeerrordistributionsforeachmodel,forallofthecasehistoriesthatwereanalyzed,weregenerallysimilartotheshapesandpositionsoftherelativeerrordis-tributionsforthosecaseswheretheobserveddisplacementswereinthe“intermediate”range(0.01mbDobservedb1m).Consequently,themodelsmayexhibitthesamebehaviorforcasesofsmalland/orlargedisplacementastheydoforcasesintheintermediaterange,providedthatapercentage-basedapproachsuchasrelativeerrorisusedtocomparetheresultsfromdifferentmodels.Notationacamaxac/amaxARMSβCDD5-95criticaloryieldacceleration(g)

peakhorizontalgroundacceleration(g)criticalaccelerationratio(unitless)rootmeansquareofacceleration(g)

inclinationofslidingplanetohorizontal(degrees)

aconstantforSarma'smethod(1975)whichisequaltocos(β−θ−ϕ')/cosϕ′(unitless)

resultantobservedseismicdisplacement(m)

timebetween5%and95%oftheAriasintensityofearthquake(s)

durationforwhichtheaccelerationisgreaterthanthecriticalacceleration(s)

ϕ′effectiveshearstrengthparameterofthesoil(°)Hheightofearthdamorembankment(m)IaAriasintensity(m/s)kccriticalcoefficient(kc=ac/g,wheregisthegravitational

acceleration)

kmaxpeakseismiccoefficient(kmax=amax/g)MearthquakemomentmagnitudeNeqequivalentnumberofuniformcycles(Yegianetal.,1991)θinclinationofinertiaforcetohorizontal(degrees)

Sa(1.5TD)spectralaccelerationwith5%dampingatdegradedperiod

equalto1.5TD(g)

Sa(T=1s)spectralaccelerationwith5%dampingat1sec(g)TDinitialfundamentalperiodoftheslope(s)Tmmeanperiodofearthquakeacceleration(s)Tppredominantperiodofearthquakeaccelerationrecord(s)vmaxpeakgroundvelocity(cm/s)Vs30time-averagedshearwavevelocityforthetop30minthe

foundation(m/s)

DuracAcknowledgments

ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationundergrantno.CMMI-0844836.Theauthorswouldliketo

192C.L.Meehan,F.Vahedifard/EngineeringGeology152(2013)180–193

thankDr.DebasisRoyandMr.RaghvendraSinghfromIndianInstituteofTechnology,Kharagpurforprovidingthecasehistorydatabasethatwasusedinthemodelevaluation.ThefirstauthorwouldalsoliketoacknowledgethesupportoftheFulbrightCenterinFinlandandthe2012–2013Fulbright-TampereUniversityofTechnologyScholarAward,whichprovidedsupportforworkonthismanuscript.

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