THE ANALYSIS OF THE STABILITY OF GENERAL SLIP SURFACES

《THE ANALYSIS OF THE STABILITY OF GENERAL SLIP SURFACES》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

THEANALYSISOFTHESTABILITYOFGENERALSLIPSURFACESN.R.MORGENSTERN,B.A.Sc.,Ph.D.*,andV.E.PRICE,M.A.,Ph.D.7SYNOPSISWithintheframeworkoflimitequilibriummethodsDanslacadredesmethodesd’equilibrelimiteofstabilityanalysis,norestrictionneedbeplacedatd’analysedestabilite,iln’yapasbesoind’imposertheoutsetupontheshapeofthepossibleslipsurface.derestrictionsaudepartsurlaformedelasurfacedeInmanycases,thecriticalsurfacemaydeviateglissementCventuelle.Dansbiendescas,lasur-facecritiquepeutdevierd’uneman&esignificativesignificantlyfromacircleoraplaneandthereforead’uncercleoud’unplanatparconsequentunemethodthatfacilitatestheanalysisofsurfacesofmethodequifacilitel’analysedessurfacesdeformesarbitraryshapeisofinterest.Amethodfordoingarbitrairespresenteuninter&t.Onpresenteunethisispresented.Theassumptionsnecessarytomethodepouraccomplircela.Ondiscutedesmaketheproblemstaticallydeterminatearedis-hypothesesnecessairespourqueleproblbmesoitcussed.Thesolutionofthegoverningequationsdetermineaupointdevuedelastatique.Laensuresthatallequilibriumandboundarycandi-solutiondesequationsdominantesgarantitquetouteslesconditionsd’equilibreetdelimitesoienttionsaresatisfied.Themethodhasbeenpro-satisfaites.Lamethodeaet6programmeepourunegrammedforadigitalcomputerandsomeexamplescalculatricedigitaleetondonnequelquesexemplesofitsapplicationaregiven.Comparisonsarealsodesesapplications.D’autresmethodesd’analyseymadewithothermethodsofanalysis.soncomparees.INTRODUCTIONThenecessityofconsideringbodyforces,pore-waterpressures,andavarietyofsoiltypesintheanalysisofthestabilityofearthslopesvitiatestheapplicationofmethodsthatarewell-foundedinthemechanicsofcontinuaandemployrepresentativeconstitutiveequations.Asaresultofthis,limitequilibriummethodsofanalysisarecommonlyused.Thesemethodsinvokenokinematicalconsiderationsregardingsoilbehaviourandhencerequirethattheshapeofthepotentialslipsurfacebeassumed.Theselectionoftheshapeofthissurfaceistherefore,toalargedegree,arbitrary.However,itiscommonlyassumedintheanalysisofslopestabilityproblemsthattheshapeoftheslipsurfaceiscircular.Thechoiceofacircularslipsurfaceisusuallyjustifiedonthegroundsthatthecomputationsaremadesimpler.Byappealingtofieldobservations,itbecomesapparentthatslipsurfacesthatareapproxi-matelycircularhavebeenobserved,althoughnon-circularsurfacesaremorenumerous.ThebestdocumentedcaseofacircularslipistheslideatLodalendescribedbySevaldson(1956).Amongthemanyexamplesofnon-circularslipsinnaturalslopesthatcouldbecitedaretheFolkestone-Warrenlandslip(e.g.Legget,1962)andtheslidesatSurte(Jakobson,1952)andFurre(Hutchinson,1961).Photographsofnon-circularslipsurfacesinnaturalslopeshavebeengivenbyVarnes(1958).Inthecaseofearthdamfailures,theslipintheChingfordreservoirisanexampleofnon-circularmovement(CoolingandGolder,1942)asisthemorerecentconstructionslipintheTittesworthDam(Twort,1964)whereaportionofthedown-streamslopeoftheoldbankslidalongaweakclaylayerduringtheconstructionofthenewbank.Thedegreeofsurfacedisturbanceoftheslidingmassprovidesausefulvisualmethodfordistinguishingbetweencircularandnon-circularmovement.Whenthesoilmassdisplacesasarigidbodyrotation,thedisturbanceofthesurfaceiscomparativelysmall.Moresevere*LecturerinCivilEngineering,ImperialCollegeofScienceandTechnology,London,England.tReaderinNumericalAnalysis,NorthamptonCollegeofAdvancedTechnology,London,England.79 80N.R.MORGENSTERNANDV.E.PRICEsurfacedisturbanceisusuallyobservedwhenfailureoccursalonganon-circularpath.ThistypeofmovementinducesintenseshearstresseswithinthemassandagrabenformationmayresultasillustratedinFig.1.Itisevidentthatbecauseofdeparturesfromhomogeneitythesoilmassmayslipalonganon-circularsurface,andthustheminimumfactorofsafetymaybeassociatedwithsuchasurface.Currentdesignpracticeoftenproducessectionsinwhichthestrengthparametersofthesoilorthepore-pressuresvaryconsiderablywithinadamoritsfoundationand,asnotedbyBishop(1957),ananalysisbasedsolelyuponcircularslipsurfacesmaysignificantlyover-estimatethefactorofsafety.Theseconditionscanarisewhen:(i)thepresenceofasoftlayerinthefoundationdictatesthepathoftheslipsurfacewiththelowestfactorofsafety,(ii)differenttypesofsoilorrockfillareusedandthecurvatureofthepotentialfailuresurfacevarieswiththedifferentstrengthsorpore-pressuresexistingwithinthezones,(iii)drainageblanketsareusedtofacilitatethedissipationofpore-pressures.Fig.1.Grabenformationduetonon-rotationalslipTheinfluenceoftheforegoingconditionsisillustrateddiagrammaticallyinFig.2.Theanalysisofthestabilityofrockslopesalsorequirestheconsiderationofnon-circularslidingsurfaces.Heretheconfigurationofthepotentiallyunstablezoneisusuallydictatedbythepatternsofjointsandfissuresintherockmassandamethodofanalysisisneededthatallowsonetodeterminethefactorofsafetyofaslopewithanysetofobservedplanesofweak-ness.Itisalsoofinteresttoinvestigatewhetherinthecaseofahomogeneoussoilprofile,asurfacecanbefoundwhichgivesalowerfactorofsafetythantheconventionalslipcircle.CONSIDERATIONOFSOMEPREVIOUSMETHODSAnymethoddevelopedtocalculatethefactorofsafetyofnon-circularslipsurfacesmustsatisfyseveralrequirementsifitistohaveotherthanonlyrestrictedusage.Themethodshouldbeabletoconsiderawidevarietyofsurfacesandanyconfigurationcomposedofsoilswithdifferingshearstrengthproperties.Itmustalsobeabletomakeallowanceforcomplexpore-pressuredistributionsandhencetreatthestabilityproblemintermsofeffectivestress.ItisonthebasisoftheserequirementsthatweshallexcludefromdiscussionthemethodofanalysissuggestedbyNonveiller(1957).Thismethodisdirectedtowardstheanalysisofslipsurfacesthatareconcavedownwardsinadamcomposedoftwomaterialsandrequiresanassumptionregardingtheverticalshearforceattheinterfacebetweenthetwomaterials.Thefactorofsafety,asdefinedbyNonveiller,isstronglydependentuponthisassumption.Themethodisnotusedwidely.Theconditionoflimitingequilibriumtogetherwiththemethodofslicesprovidesasatis-factorybasisfromwhichamethodofsufficientgeneralitymaybedeveloped.ThishasbeenrecognizedbyJanbu(1954,1957),Kenney(1956),Janbu,BjerrumandKjaemsli(1956),andSherard(1960),whohavedevelopedanalysesthatconsiderthelimitingequilibriumofa 82N.R.MORGENSTERNANDV.E.PRICEdP,denotesthewaterpressureonthebaseoftheslice,dN’denotestheeffectivenormalpressure,dSdenotestheshearforceactingalongthebaseoftheslice,andCYdenotestheinclinationofthebaseoftheslicewithrespecttothehorizontal.Theconditionthattherebenorotationofthesliceissatisfiedifthesumofthemomentsaboutthecentreofthebaseofthesliceisequaltozero.Bytakingmomentsaboutthemid-pointofthebaseoftheslicewefindthat:E’[(y-y;)-(-$)]-t-P,[(y-h)-(-$)]-(E’+dE’)[y+dy-y;-dy;+(_$)].-x$-(X+dX$%',+d&J(y+dy)-(h+dh)-$-dP,.g=0(1)I(b)Fig.3.(4Potentialslidingmass;(b)ForcesactingonaninfblitesimalAftersimplifyingandproceedingtothelimitasdx-+0itcanbereadilyshownthat:dPX=;(E’.y;)--yg+;(Pw.h)-y-$...(2)ForequilibriumintheNdirection,dN’+dP,=dWcosa-dXcosa-dE’sina-dP,sina...(3)FromequilibriumintheSdirection,dS=dE’coscr+dP,cosa:-dXsincc+dWsina!....(4)TheCoulomb-Bohrfailurecriterionintermsofeffectivestressesmaybeexpressedas:dS=$[c’dxseta+(dN’)tan$1’1.....*(5) STABILITYOFGENERALSLIPSURFACES83wherec’isthecohesioninterceptintermsofeffectivestresses$’istheangleofshearingresistance>andFdenotesthefactorofsafetyItshouldbenotedthatequation(5)alsoconstitutesadefinitionofthefactorofsafety.Thefactorofsafetywithrespecttoshearstrengthhasbeenadoptedhere.Itisthatvaluebywhichtheshearstrengthparametersmustbereducedinordertobringthepotentialslidingmassintoastateoflimitingequilibrium.Analternativedefinitionofthefactorofsafetyhasbeendiscussedelsewhere(BishopandMorgenstern,1960).Itisclearthatthefactorofsafetywithrespecttomomentratioscannotbeutilizedinnon-circularanalyseswheretheshapeoftheslidingsurfaceisarbitrary.EliminatingdSfromequations(4)and(5)weobtain:$[cdxseca+(dN’)tan+‘]=dE’cosct+dP,cosa-dXsincc+dWsina..(6)ELEMENT(a)Fig.4.(a)Anelementataninterfacebetweentwoslices;(b)EffectivestressesactingonanelementEliminatingdN’fromequations(3)and(6),anddividingbydxcosU.itcanbeshownthat:C’sec2cc+tan$’dWdXdE’dx-z-ztana-2tanct-dssetaFV31=$+~-d$tan~+~tancf....(7) 84N.R.ANDV.PRICEInthespecifiedco-ordinatesystem,tana=-2,andequation(7)becomes:dE’dP,dXdydWdy=-+-dx+~‘~-~.~..*t8)dxwheredP,=r,.dW.seta...(9)andr,,isthepore-pressureratiodefinedbyBishopandMorgenstem(1960).Therefore,thetwogoverningdifferentialequationsare:X=-&Ely;)-yg+$(P&z)-y$$....(2)(10)TREATMENTOFSTATICALINDETERMINACYIfyisspecifiedassomefunctionofx,wehave,ingeneral,astaticallyindeterminateprobleminvolvingtheunknownfunctionsE’,Xandy;andthetwogoverningdifferentialequations.Theindeterminacyarisesfromourlackofknowledgeofthestressesobtaininginthesoilmass.Ifthestressescouldbedetermined,thedisplacementscouldbepredictedusingarepresentativestress-strainrelationship.Thiswouldobviatetheneedofdoinglimitequilibriumanalyses.Itisourinabilitytoobtainanadequatestressanalysisthatjustifiestheapplicationoflimitequilibriummethods.Atthesametime,notknowingthestressesmakesitnecessarytoinvokeanassumptioninordertorendertheproblemstaticallydeter-minate.Therearethreeclassesofassumptionsthatcanbemade:1.Thedistributionofnormalpressurealongtheslidingsurfacecanbeassumed.Thefrictioncircleanalysisisanexampleofamethodthatadoptsthistypeofassumption.Wehavechosentoeliminatethenormalpressurefromtheequilibriumequationsandtherebyplacetheburdenoftheindeterminancyontheinternalforces.2.Anassumptionmaybemaderegardingthepositionofthelineofthrust.Forexample,ify-ypa(y--2).......(11)themomentequilibriumequation,neglectingthepore-pressureterms,becomesX=E&z$[E(y-z)].......02)Equations(12)and(10)nowdefineastaticallydeterminateproblemwherethemagnitudesofaandFmustbefoundbysatisfyingtheappropriateboundaryconditions.Anattempthasbeenmadetoobtainsolutionsusingequations(10)and(12)butill-conditionedfunctions,withensuingnumericaldifficulties,ariseinthesolutionofthedifferentialequationsanditwasnotfoundpossibletoobtainasatisfactorynumericalprocedure.Itshouldbenoticedthatthemagnitudeofainequation(12)mustbedeterminedaspartofthesolution.IfitisspecifiedattheoutsetasinKenney’ssolution(1956)itisnotpossibletosatisfyallequilibriumandboundaryconditions.3.AssumptionsmaybemaderegardingtherelationbetweentheE’andXforces.Ifweisolateanelementattheinterfacebetweentwoslices,asshowninFig.4(a),theeffective STABILITYOFGENERALSLIPSURFACES85stressesactingonthiselementwillbeasgiveninFig.4(b).ForaspecificgeometryandslipsurfacetheinternalforcesaredeterminedbyYE=crz(y)dy.......(13)IzYandx=T&y)dy......(14)s.?WemaythereforeassumethatX=hj(x)E.......(15)Iff(x)isspecifiedtheproblemisstatically,determinateandXandFmaybefoundfromasolutionofthedifferentialequationsthatsatisfiestheappropriateboundaryconditions.Thefunctionf(x)cantakeanyprescribedforminprinciple.However,thebehaviourofsoilimposescertainlimitationssothatonlyacertainrangeoffunctionswillbereasonableinpractice.Thiswillbediscussedinmoredetailinalaterparagraph.Estimatesofthefunc-tioncanbeobtainedfromelastictheory.Morereliablefieldobservationsofinternalstressesindamswillalsobeusefulinestimatingthedistributionoftheinternalforces.Tosimplifytheequationsithasbeenfoundconvenienttodefinehf(x)byusingthetotalhorizontalstressEinsteadoftheeffectivestressE.ThuswedefineE=E+P,.......(16)andthepointofapplicationytofthetotalstressbyEy,=Ely;+P,h.....,(17)Theninsteadof(15)weassumeX=Af(x)E.......(18)THESOLUTIONOFTHEEQUATIONSInordertobeabletoinvestigatethestabilityofasoilmasswithanyslopeandpropertiesithasbeenassumedthatthepotentialslidingbodymaybedividedintoanumberoffiniteslicesbyverticallineswithco-ordinatesx0,x1...xn.Thisdivisioniscarriedoutsothatwithineachslicetheportionoftheslipsurfaceislinear,theinterfacebetweendifferentsoiltypesandporepressurezonesarelinearandthefunctionfdefinedbyequation(18)dependslinearlyonx.Hencewithineachslicewehavey=Ax+B.......(19)$.px+q..*....andf=kx+m.......(21)Equation(19)and(20)allowasectionwithanyarbitraryshapetobeapproximatedintheanalysis.Equation(21)assumesthattheratiooftheinternalforceschangeslinearlyoverasegmentoftheslidingbody.Thisassumptionisnotundulyrestrictivebecausekandwzmaybechosentovaryfromsegmenttosegmentandanycontinuousdistributionofinternalforcescanbeapproximatedinthisway.Usingequations(16)to(21),equation(2)becomesX=g(EyJ-yg.......(22)andequation(10)becomes(Kn+L)g+KE=Nx+P......(23) 86N.R.MORGENSTERNANDV.E.PRICEwhereK=A,&f+,)......L4mfy+4)+1-Ay.,...N=p!k?$+A-r,(l+A2)y[1...tan4’andP=$(1+Aa)+q7+&r,(1+P)Y..IEquation(23)canbeintegratedacrosseachsliceinturnstartingwiththevalueE=0atthebeginningoftheslipsurface.If,foreachslice,xismeasuredfromthebeginningofthatslice,thenthesolutionwhichsatisfiesE=E,whenx=0.......(24)isE=-!.-....L+KxThevalueofEattheendofthesliceisdeterminedandthisgivesthestartingvalueofEforthenextsliceunlesstheendoftheslipsurfacehasbeenreached.TheboundaryconditiontobesatisfiedattheendoftheslipsurfaceisE=E,whenx=x,.......(26)whereE,isusuallyzeroSatisfyingequation(25)anditsboundaryconditionsisaloneinsufficienttoensurecom-pleteequilibriumsinceequation(22)mustalsobesatisfied.ThiscanbedonebydeterminingytfromthevaluesofEandfoundfromequations(25)and(18)providedtheysatisfythefollowingnecessarycondition.Byintegratingequation(22)wehave:M=E(y,-y)=j(X-E$)dx.....(27)20SinceingeneralM=0whenX=X,,,then:M,=j(x-Efg)d%=0......10Ifequation(28)issatisfied,valuesofytcanbefoundfromequation(27)therebyensuringthateachsliceisinmomentequilibrium.HenceinordertofindvaluesofhandFsuchthatalltheequationsofequilibriumaresatisfied,westartwithguessedvaluesofhandFandthenintegrateacrossalltheslicestoobtainthevaluesofE,andM,whichingeneralwillnotbothbezero.Then,byasystematiciterativemethodofmodifyingXandF,valuesarefinallyobtainedforwhichE,andM,arezero.Thishasbeenprogrammedforanelectroniccomputerandthenumericaltechniquesusedwillbereportedelsewhere.Thereisonecomplication,however,whichwillbedescribedhere.InsomecasesthereismorethanonepairofvaluesofXandFwhichsatisfytheequationsforagivensurfaceandonehastodecidewhichisthephysicallymeaningfulsolution.TherulewhichhasbeenadoptedisthatthevaluesofXandFshouldmakethevaluesof(L+Kx)positiveforthecompleterangeofxoftheslipsurface.Withthisrule,inallthecasesexamined,uniquevaluesofhandFhavebeenfound.Fromequation(25)itcanbeseenthatif(L+Kx)iszeroforanyvalueofx,Ebecomesinfiniteatthatpoint.Thisisnotphysicallypossible.Althoughthevaluesof(L+Kx)arediscontinuousacrossanyoftheboundariesoftheslicesx,,x2,...,x,_~,onewouldnotexpect(L+Kx)tochangesignacrosstheseboundaries.Ifweconsideraverythin STABILITYOFGENERALSLIPSURFACES87sliceroundeachx,inwhichsomesoilpropertiesareveryquicklyvaryingbutcontinuous,thenif(L+Kx)changessignacrossadiscontinuity,itwouldbezeroatsomepointinthecorres-pondingthinhypotheticalregion.Thisindicatesthat(L+Kx)shouldbeofthesamesignforallx,andfromexperienceitappearsthatitshouldbepositiveduetothedominanceofthecontributionofunitytoLinequation(23b).SomespecialassumptionsrelatingtheXandEforcesareofinterest.Ifitisassumedthatx=2......,.itcanbeshownthatyt=y.........(30)andthefactorofsafetycanbefoundfromamodifiedformofequation(26)alone.Theassumptionstatedbyequation(29)issimilartothatadoptedintheconventionalcircularanalysiswhichhasbeendiscussedbyBishop(1955).Althoughthisassumptionsimplifiesthenumericalcalculationsitisnotacceptablebecausetheimpliedinternalforcesarelikelytobephysicallyinadmissible.Thereisalsonoreasonfortherelationbetweentheinternalforcestodependsolelyupontheinclinationofanarbitrarilychosenslipsurface.AnotherassumptionhasbeensuggestedbyJanbu(1954)andbySherard(1960)intheform:X=a,hE.........(31)where#issomespecifiedconstant.Itisclearthatoverallmomentequilibriumcannotbeensuredif#isspecified.Itis,how-ever,possibletomaketheproblemstaticallydeterminatebyassuming:X=hE.........(32)wherehmustbecomputedtogetherwithFfromequations(26)and(28).Beforedescribingsomeexamplesoftheapplicationoftheprecedinganalysis,itisim-portanttodiscussseveralaspectsofsoilbehaviourthathavenotbeenconsideredinthemethodandhowthisomissioninfluencestheinterpretationoftheresults.Fortheanalysistobephysicallyacceptable,notonlymusttheequilibriumandboundaryconditionsandfailurecriterionalongtheslipsurfacebesatisfiedbuttheimpliedstateofstresswithinthesoilmassmustalsobepossible.Inparticular,thefailurecriterionwithinthesoilmassabovetheslipsurfacemustnotbeviolatedand,sinceitiscommonlyacceptedthatsoilsdonottaketension,nostateoftensionmustbeimpliedtoexistabovetheslipsurface.Foranarbitrarilychosenslipsurfaceitisnotpossibletoensurethatalltheseconditionsaresatisfied.Itisthereforenecessaryineachparticularcasetocalculatetheinternalforcesandthepositionofthelineofthrustinordertoinspectwhetherthefailurecriterionisexceededinternallyandwhetherastateoftensionisimplied.Intheformercasethisiseasilydonesincetheallowableratiobetweentheshearforceandthelateralthrustmaybereadilycomputedfromthefailurecriterion.Toinvestigatewhethertensionisimplied,thepositionofthelineofthrustmaybecalculatedfromequation(22)andifitfallsoutsideofthepotentialslidingmasstensionmustexistwithin.Inthediscussionoftheexamplesthatfollows,itwillbeseenthatthefactorofsafetyandtheinternalforcesarerelativelyinsensitivetovariationintheassumedfunctionrelatingtheinternalforcesprovidedthatthefunctionisreasonable.Hence,foranarbi-trarilychosenslipsurfaceitmaynotbepossibletoobtainphysicallyadmissibleinternalforcesandinsuchacaseitmustbeconcludedthatthesurfaceitselfisunlikely.Theselectionofareasonablefunctionalsorequiressomecomment.Asmentionedearlier,functionscanbeestimatesfromavailableelasticitysolutionsusingequations(13)and(14).Alternatively,theymaybespecifiedonthebasisoftheintuitiveassumptionthatformostcross-sectionsthehighertherateofcurvatureoftheslipsurface,thegreatertheratiobetween 88N.R.MORGENSTERNANDV.E.PRICEtheshearandhorizontalforcesatthesliceinterface.Theporepressuresinthesoilmassconstituteanotherfactortobeconsideredwhenestimatingthefunction.Inparticular,ifthesliceinterfaceisinazoneofhighporepressuretheamountofshearthatcouldbemobilizedwouldbereducedaccordinglyandthefunctionshouldthereforetakealowervalueinthisregion.Ultimately,reliablefieldmeasurementsofinternalstresseswillprovidethebestguidetoestimatingthisfunction.Inthefollowingexamples,comparisonsaremadeforparticularcross-sectionsusingfunctionsthatareconsideredreasonableandsomethathavebeenchosenarbitrarily.0)Fig.5.(a)Exampleofacircularslipsurface;(b)ThreeassumedfunctionsSOMEAPPLICATIOKSOFTHEAKALYSISOFGENERALSLIPSURFACES1.AcircularslipsurfaceIthasbeenshown,thattoallpracticalpurposes,thefrictioncirclemethodofanalysisandslipcircleanalysis,asdevelopedbyBishop(1955),givethesamefactorsofsafetyforagivencirclewhentheproblemisconsideredintermsofeffectivestresses(BishopandMorgenstern,1960).HowevertheroutineanalysispresentedbyBishopbeingthefirststageofamorecompleteiterativeprocessdoesnotsatisfystaticalequilibrium.Inparticular,theinfluenceoftheXforcesisneglected.Nevertheless,thefactthatthetwomethodsgivethesamefactorofsafetywhileimplyingdifferentinternalforcedistributionssuggeststhatforcircularslipsurfaces,thefactorofsafetyisrelativelyinsensitivetothedistributionoftheinternalforces,asdiscussedindetailbyBishop(1955).ByassumingdifferentrelationshipsbetweentheXandEforcesandcomputingthefactorofsafetyforapotentialcircularslipsurfaceitispossibletodemonstratethisinsensitivity.Thecalculationshavebeencarriedoutusingthegeneralslipsurfaceanalysisandhencethepropertythattheslipsurfaceiscircularhasnotbeenusedexplicitly.Aslip-circleanalysishasfirstbeencarriedoutonanelectroniccomputerforthecircleshowninFig.5(a)usingtheprogrammedescribedbyLittleandPrice(1958).Thefactorof STABILITYOFGENERALSLIPSURFACES89safetywasfoundtobe2.098.ThreedifferentassumptionsregardingtherelationshipbetweentheinternalforcesareshowninFig.5(b).Allthreeassumptionshavebeenselectedarbi-trarily.ThevaluesofFandhfoundfromthethreenon-circularanalysescalculatedusingtheserelationshipsaresummarizedinTable1togetherwiththefactorofsafetyobtainedfromtheslip-circleanalysis.Table1.AnalysisofacircularslipsurfaceCaseIFhSlip-circleanalysis..(2.098j-Generalizedmethod(AssumptionI)..2.045-0.256General&dmethod(AssumptionII)..2,136-0.208General&dmethod(AssumptionIII)..2.134-0.393iIAlthoughthefactthatthevaluesofFwithassumptionsIIandIIIaresosimilarisacoincidencetheresultsillustratethat,atleastforacircularsurfaceinahomogeneoussection,thefactorofsafetyisnotstronglydependentupontheinternaldistributionandthattheassumptionsinBishop’scirculararcanalysisprovidereasonableresults.2.Anon-circularsurfaceimahomogeneoussectionFig.6(a)illustratesanassumednon-circularslipsurfaceinasectioncomposedofasinglesoiltypehavingonlyonepore-pressureratio.ThissectionhasalsobeenstudiedbyKenney(1956)whofoundthatthefactorofsafetywasapproximately1.7usinghismethod.*UsingKenney’sfiguresforhisfirstiteration,thefactorofsafetygivenbyJanbu’sanalysisisreadilyshowntobe1.73(Janbu,BjerrumandKjaemsli,1956).TwodistributiveassumptionshavebeenconsideredinthecomputeranalysisandtheyaregiveninFig.6(b).Thesecondassump-tionisconsideredreasonable.Thefactorofsafetyforbothcaseswasfoundtobealmostthesame.TheresultsaretabulatedinTable2andthecomputedinternalforcesforbothcasesareplottedinFig.6(c).Table2.Analysisofanon-circularsurfaceinahomogeneoussectionACase!FiKenney’sanalysis.11.66-1.721Janbu’sanalysis..1.73Generalizedmethod(AssumptionI)..1.592-0.328Generalizedmethod(AssumptionII)../1.609/-1.190ItisclearthatbothKenney’sandJanbu’smethodsoverestimatethefactorofsafetyontheunsafeside,butbylessthan8%.*InhisanalysisKenneyassumedseveralpositionsforthelineofthrustandfoundthatthefactorofsafetyvariedbetween1.66and1.72. 90N.R.MORGENSTERNANDV.E.PRICEAsinthecaseoftheslip-circleanalysis,thefactorofsafetyforthissurfaceisnotsensitivetotheassumptionoftherelationshipbetweentheinternalforces.Thiswillnotalwaysbethecase.Theinfluenceofthisassumptiondependstoalargedegreeupontheshapeoftheslipsurfaceandthevariationofstrengthparametersandpore-pressuresalongit.Thelineofthrustcompatiblewiththecalculationsbaseduponthesecondassumptionhasalsobeencomputed.ItisshownbythedottedlineinFig.6(a).Thenumbersinparenthesesgivetheratioofthedistancebetweentheslipsurfaceandthelineofthrusttotheheightofthesliceateachinterface.ItisapparentthatnointernaltensionalstressesareimpliedandFig.6.(a)Non-circularslipsurfaceinahomogeneoussection;(b)Twoassumedfunctions:(c)Internalforcedistributionsthatthecomputedinternalforcesarephysicallyacceptableonthebasisofthiscriterion.Theyarenot,ofcourse,theonlyadmissiblesetofinternalforces. STABILITYOFGENERALSLIPSURFACES91Theyieldcriterionwithintheslidingsoilmasswouldbeviolatedifthecalculatedshearforceatasliceinterfacenecessaryforequilibriumexceededtheshearingresistancethatcouldbemobilizedalongtheinterface.Sincethetotalnormalforceactingontheinterfacehasbeencalculatedandsincetheporepressures,andstrengthparametersobtainingalongtheinterfacearealsoknowntheavailableshearingresistanceforthesecondassumptionmayalsobecomputed.TheyarecomparedwiththecalculatedXforcesinFig.6(a)wheretheratioR,oftheavailableshearingresistancetotheshearforcerequiredforequilibriumisgiven.Allvalueslieaboveunityandthereforetheyieldcriterionissatisfiedwithintheslidingmass.Itshouldbenotedthattheshearandnormalstressesthatmustbeactingonacriticalsurfacearenotthesameasthoseactuallyinthesoilmass.However,itisnotpossibleingeneraltomakecomparisonsbecauseofthelackofknowledgeofstressdistributionscomputedusingrepresentativestress-strainrelations.3.Aty+icalearthdamsectionThenon-circularsurfacespecifiedinFig.7(a)istypicalofthetypeofsurfacethatshouldbeanalysedduringthedesignofanearthdamonafoundationwithlowshearstrengthproperties.*_EXE55POREPRE55URE(RBOVEDRRWDOWNLEVEL)ATE:I)r,-TOTRL5~~~55(USINGWBIERCEDDENSTiEBELOWDRRWDOWNLEVEL)2)THEPOREiRE55URE;RRiThOSELChi;!DE;E3TO3aTAINRFTERRAPIDDRAWDOWN)3)RVERRCEBULKDE?&IlYOFMFITERIFIL152.1T3NNE5h3I*34(a)R55uMPTIONnbIDII0I23456789POINT5ONSURFFICEINXDlRaCTlON0))Fig.7.(a)Typicalnon-circularslipsurfaceinadamwithaweakfoundation;(b)ThreeassumedfunctionsInthisexamplethepore-pressuresarethosethatmightobtainafterarapiddrawdownofthereservoir.AnanalysisofthissurfaceusingJanbu’smethodgaveafactorofsafetyof1.28(Janbu,BjerrumandKjaernsli,1956).Threedistributionshavebeenassumedforthe 92N.R.MORGENSTERNANDV.E.PRICEinternalforcesforanalysisonthecomputer.ThesedistributionsareshowninFig.7(b).TheresultsofthecalculationsarepresentedinTable3.Table3.Analysisofanon-circularsurfaceintypicalearthdamCaseFhJanbu’sanalysis--1.28-Generalizedmethod(AssumptionI)--1.346-0.132Generalizedmethod(AssumptionII)--1.2821-0.022GeneralizedMethod(AssumptionIII)--1.279-0,046IIForthiscaseweseethatJanbu’smethodgivesafactorofsafetythatisthesameasthatobtainedusingasensibledistribution.Thefactorofsafetyfromthecomputeranalysisisalsoreasonablyinsensitivetothedistributiveassumption.Notenoughcomputationshavebeencarriedoutsofartoenableonetomakegeneralstatementsabouttheinfluenceofthisassumptiononthefactorofsafety.However,itissignificantthatinthisexamplewhichisrepresentativeofactualdesignconsiderationstheinfluenceappearstobeslightforreasonableassumptions.CONCLUSIONSAmethodhasbeendevelopedforthedeterminationofthefactorofsafetyofaslidingbodyofanyshapecontainingmaterialswithvaryingshearstrengthparametersandpore-pressures.Itisbasedsolelyupontheprinciplesoflimitingequilibrium.Notonlymusttheshapeofthepotentialslipsurfacesbechosenbutanassumptionmustalsobemaderegardingthedistributionofinternalforces.Thefactorofsafetydoesnotappeartobeverysensitivetothisassumption.Reasonableassumptionscanbeinferredeitherfromaknowledgeoftheapproximateinternalstressdistributionorfromfieldobservationsofinternalstresses.Themethodisusefulinthedesignofearthdamsortheanalysisofthestabilityofnaturalslopesandcuttings.Thesolutionensuresthatallequilibriumandboundaryconditionsaresatisfied.Comparisonintwocaseswithotherexistingsolutionsrevealsthattheycanbeinerrorbyasmuchas8%ontheunsafeside.Furthercomputationsarerequiredtoinvestigatetheoccur-renceofthesedifferencesinmoredetail.Theuseofinternalstressmeasurementstoaidinthedeterminationoftherelationshipbetweentheinternalforcesplaysaroleanalogoustocarryingouteffectivestresscalculationswheretheinfluenceofthepore-pressureisconsideredintheanalysis.Oneofthemainadvantagesofaneffectivestressanalysisisthatthefactorofsafetydependsuponthemagni-tudeofthepore-pressures.Ifthesehavebeenpredicted,itispossibletoobservethemandhencecorroborate,inpart,thedesigncalculations.Theuseofinternalstressmeasurementstohelpdefinethepossiblerelationshipsbetweentheinternalforcesmeansthatcomputationsarebaseduponanothermeasurablequantityand,tothatdegree,maybefurthersubstantiatedbyfieldobservations.Inthecaseofacircularslipsurfacetreatedasacaseofanon-circularanalysis,itwasshownthatboththemethodpresentedhereandthatsuggestedbyBishop(1955)gaveapproximatelythesameresult.Furthermore,thefactorofsafetyinthiscase,aswellasinthetwootherexamplesdiscussed,isinsensitivetovaryingtherelationshipbetweentheinternalforces. STABILITYOFGENERALSLIPSURFACES93Bothearthpressureandbearingcapacityproblemscanalsobeconsideredintermsoflimitequilibriumanalyses.ItwouldbepossibletoextendthesolutiondevelopedheretothecomputationofearthpressurecoefficientsandbearingcapacityfactorsinamannersimilartothatdescribedbyJanbu(1957).Theinfluenceofearthquakescouldalsobeintroducedintotheanalysis.Thisiscommonlydoneinlimitequilibriummethodsbyconsideringaninclinedbodyforce,somefractionofgravity,actingoneachslice.Thisforcecanbeincorporatedintotheequationsofequilibrium.Howevertheshearstressdistributionwithinasoilmasssubjecttoanearthquakewillalsochangeanditwouldbenecessarytouseanassumptionfortheinternalforcesinthiscasewhichdiffersfromthatusedinthestaticcase.ACKNOWLEDGEMENTTheAuthorsgratefullyacknowledgetheassistanceofMrsJ.SkinnerandothermembersofthestaffoftheEnglishElectric-LeoLondonComputingServiceinprogrammingthesolu-tionandcarryingoutthecomputations.TheAuthorsarealsogratefulformanyhelpfuldiscussionswithDrA.W.Bishop.ThisstudywassupportedinitsearlystagesbyaresearchgrantawardedbytheDepart-mentofScientificandIndustrialResearch.REFERENCESBISHOP,A.W.,1955.“Theuseoftheslipcircleinthestabilityanalysisofearthslopes.”Gdotechnique,5:1:7-17.BISHOP,A.W.,1957.Contributiontodiscussionon“TheUskschemeforthewatersupplyofSwansea”.G.A.R.SheppardandL.B.Aylen,PYOC.Instnciu.Engrs,7:281.BISHOP,A.W.,andN.MORGENSTERN,1960.“Stabilitycoefficientsforearthslopes.”Gdotechnique,10:4:129-150.COOLING,L.F.,andH.Q.GOLDER,1942.“Theanalysisofthefailureofanearthdamduringconstruc-tion.”J.Instnciv.Engu,19:38-55.HUTCHINSON,J.N.,1961.“AlandslideonathinlayerofquickclayatFurre,CentralNorway.”GCotech-nique,11:2:69-94.JAKOBSON,B.,1952.“ThelandslideatSurteontheGotaRiver.”PYOC.Roy.SwedishGeotech.Inst.No.5,120pp.JANBU,N.,1954.“Applicationofcompositeslipsurfacesforstabilityanalysis.”Proc.EurqbeanConf.onStabilityofEarthSlopes,Stockholm,3:43-49.JANBU,N.,1957.”Earthpressureandbearingcapacitybygeneralizedprocedureofslices.”Proc.FourthInt.Conf.SoilMech.,2:207-212.JANBU,N.,L.BJERRUM,andB.KJAERNSLI,1956.“Soilmechanicsappliedtosomeengineeringproblems.”NorwegianGeotech.Inst.,Pub.No.16.KENNEY.T.C.,1956.“Anexaminationofthemethodsofcalculatingthestabilityofslopes.”MA.Thesis,UniversityofLondon.LEGGETT,R.F.,1962.“Geologyandengineering.”(2ndEd.)p.426,McGrawHill,NewYork.LITTLE,A.L.,andV.E.PRICE,1958.“Theuseofanelectroniccomputerforstabilityanalysis.”GLotech-nique,8:3:113-120.NONVEILLER,E.,1957.”Stabilnostnehomogenihnasipa.”HidroteknickiInstitut,Belgrade.SEVALDSON,R.A.,1956.“TheslideinLodalen,October6,1954.”Gdotechnique,6:4:167.SHERARD,J.L.,1960.“Aninvestigationoftheinfluenceofsideforcesinsliceandwedgemethodsofstabilityanalysisforearthdams.”Unpublishedreport.TWORT,A.C.,1964.“ThenewTittesworthdam.”J.InstnWaterEng.,18:125-179.VARNES,D.J.,1958.“Landslidetypesandprocesses”in”Landslidesandengineeringpractice“.Ed.byE.B.Eckel,HighwayResearchBoard,SpecialReport29,Washington,D.C.