DOC

Upper bound analysis of slope stability with nonlinear failure criterion based on strength reduction technique

By April Flores,2014-02-19 06:17
8 views 0
Upper bound analysis of slope stability with nonlinear failure criterion based on strength reduction techniqueof,Upper,bound,slope,with,upper,Bound,Slope

    Upper bound analysis of slope stability with

    nonlinear failure criterion based on

    strength reduction technique

    J.Cent.SouthUniv.Techno1.(2010)17:836844

    DoI:10.1007/sl1771-010564-7Springer

    Upperboundanalysisofslopestabilitywithnonlinearfailurecriterionbasedon strengthreductiontechnique

    ZHAOLian-heng(赵炼恒),LILiang(李亮),YANGFeng(杨峰),LUOQiang(

    ),LIUXiang(~lJ)

    SchoolofCivilandArchitecturalEngineering,CentralSouthUniversity,Changsha410075,China

    CentralSouthUniversityPressandSpringerVerlagBerlinHeidelberg2010

    Abstract:Basedontheupperbound1imitanalysistheoremandtheshearstrengthreductiontechnique.theequationforexpressing

    critica1limitequilibriumstatewasemployedtodefinethesafetyfactorofagivenslopeanditscorrespondingcritica1failure

    mechanismbymeansofthekinematica1approachoflimitanalysistheory.Thenonlinearshearstrengthparametersweretreatedas

    variableparametersandakinematieallyadmissiblefailuremechanismwasconsideredforcalculationschemes.Theiterative

    optimizationmethodwasadoptedtoobtainthesafetyfactors.Casestudyandcomparativeanalysisshowthatsolutionspresented

    hereagreewithavailablepredictionswhennonlinearcriterionreducesto1inearcriterion,andthevalidityofpresentmethodcouldbe

    illuminated.Fromthenumericalresults,itcanalsobeseenthatnonlinearparameterm,slopefootgradient,heightofslope/4,slope

topgradient?andsoilbulkdensityyhavesignificanteffectsonthesafetyfactoroftheslope.

    Keywords:nonlinearfailurecriterion;strengthreductionmethod;upper-boundtheoremofli

    mitanalysis;slopestabilityanalysis;

    factorofsafety

    1Introduction

    Soilslopestabilityanalysisplaysanincreasingly indispensableroleinthefieldofgeotechnicalaswellas civilengineering.andithasarousedalotofinvestigation Generalmethodsforslopestabilityanalysisareas follows:1imitequilibriummethod.1imitanalysistheorem. slip.1inefieldmethodandnumericalanalysismethod11.

    Limitanalysistheoryhasbeenwidelyusedbecauseofits definitephysicalsignificanceandstrictsolvingrange. However,themainevaluationindexesoflimitanalysis forslopestabilityarecriticalheight(r)andstability factor(Ns1atpresent,whichdifferfromtheuniversal evaluationindexofsafetyfactor(),thuscausinglotsof inconveniencetothesoilslopestabilityanalysis12].

    Theissuethatcombines1imitanalysistheorywiththe strengthreductiontechniquetocomprehensivelyanalyze thestabilityofslopehasbeenseldomconsidered[36].

    Meanwhile.thelinearMol?-Coulomb(MC)failure

    criterionhasbeenwidelyusedintheseeffortsand techniquesmentionedabove.However.nearlyallthe experimentalresultsshowthatthestrengthenvelopesof almostal1thegeomaterialsarecharacterizedasnonlinear, andthatlinearfailurecriterionisaspecialcaseoffailure

    7-91. criteria

    Anumberofresearchershaveemployednonlinear failurecriteriontocalculatecriticalheight(He)and

    stabilityfactor(Ns)ofslopeswithlimitanalysistheory [712andfiniteelementmethod[13];nonetheless,few studies[14]haveobtainedthesafetyfactorsbyusingthe limitequilibriummethod.However,1imitequilibrium methodisusuallytakenasanonstrictsolutionaccording

    totherandomnessinblockdividingandtheassumption oninterforcebetweenblocks[1].

    Forthereasonsmentionedabove,byusingupper bound1imitanalysisandstrengthreductiontechnique. themainpointofthisworkistogettheupperbound solutionofsafetyfactor()undertheassumptionof nonlinearfailurecriterion.Theinfluencesofnonlinear parametermandotherdifferentparametersonslope safetyfactor()andlatentslidesurfacewereexamined byusingtheiterativeoptimizationmethod,andsome chartsofsafetyfactor(),whichvariedwithnonlinear parametermandotherparameters,werepresentedfor practicaluseinengineering.

    2Basicprincipleandassumptions

    2,1Strengthreductiontechnique

    Strengthreductiontechniquewasproposedby BISHOPin1955[15].Theshearstrengthparameters(c Foundationitem:Proiect(20063l8802111)supportedbyWestTrafficConstructionScience

    andTechnologyofChina;Pr0ject(2O08yb0O4)supportedby ExcellentDoctorateDissertationsofCentralSouthUniversity,China;Prect(2o08GO32

    3)supportedbyKeyItemofScienceand

    TechnologyResearchofRailwayMinistryofChina Receiveddate:20091125;Accepteddate:2010-0309

    Correspondingauthor:ZHAOLianheng,PhD;Tel:+86

    13755139425;E-mail:zlh8076@163corn

::!:三鱼二竺

    and)aredividedbyslopesafetyfactor(FD,whichare analyticallydefinedasEq.(1),andmaketheslopereach acritica1state.

    .f(1)'t,

    =arctan(tan)

    wheresafetyfactorservesasthereductionfactorof shearstrengthparameters;cisthecohesivestrength;is theintemalfrictionangle;cfdenotesthereduced cohesivestrength;and(pfstandsforthereducedintemal ctionangle.

    837

    2.2Upperboundanalysisbasedonstrengthreduction methodforslopestability

    Accordingtostrengthreductiontechnique,by substitutingthereducedshearstrengthparametersinto theexpressionofvirtualworkprinciple,thelimit analysistheorycanbecombinedwithreductiontheorem followingacertainstabilitycriterion.Foraslope,this upperboundanalyticprocesscanbedescribedasfollows ifakinematicallyadmissiblefailuremechanismis available.thesafetyfactorisequalto1.0whentheslope heightarrivesatcriticalheightr.Thus,undercertain conditionsofanactualslopeheight/4.ctual,theslope staysatstablestatewhentheactualslopeheightctualis justrightgreaterthanorequaltocriticalheight(:) afterdeducingthestrengthparameters(cand).Atthis moment,Hactual=rcanberegardedastheevaluation indexoftheslopestability.andthereductionfactorof

    originalstrengthparametersispreciselythesafetyfactor ()ofslopestability[46].

    2.3Nonlinearyieldcriterionandenergydissipation Theexperimentalresultsshowthatstrength

    envelopesofalmostallgeomaterialscanbe

    characterizedasnonlinearinDrstressspace,while,a

    nonlinearM.Cyieldcriterioncanusuallybeexpressed as[8

    f=c0?(1+/at)V(2)

    whereo-nandfarenormalandshearstressesonfailure envelope(orsurface),respectively;Co,o-tandmaretest parametersthatcanbemeasuredbytest.Whennonlinear parameterm=1,Eq.(2)reducestothewellknownlinear

    MCyieldcriterion.

    Alimitloadcomputedfromapyramidalfailure surface,whichalwayscircumscribestheactualfailure surface.willbeanupperboundontheactuallimitload [3].Thus,thelinearMCfailurecriterionrepresentedby thetangentiallinewillgiveanupperboundontheactual loadforthematerial,whosefailureisgovernedbythe nonlinearfailurecriterion.Byadoptingthisidea,a tangentiallinetothenonlinearyieldcriterion,is employedbyYANGetal[1012]tocalculatetheenergy

    dissipationofgeomaterials,therebyavoidingthe

    calculationdifficultyunderthenonlinearfailurecriterion Amorecomprehensivedescriptionofthismethodcanbe foundinRefs.1012].

    Then,mobilizedinternalfrictionangleis

    introducedasanintermediatevariableintheformof tan(ot=dr/dan,thetangentiallinetothecurveatthe

    locationoftangencypointcanbeexpressedas f=ct+tant'o-n(3)

    wherectistheinterceptofthetangentiallineonthe r-axis.ctisdeterminedbythefollowingexpression: ct=,c0

    m

    '

    f=——''J

    I

    t'tan

    C0

    +o-t-tant(4)

    Asfornonlinearfailurecriterion,theoriginal strengthindexes(cand)aretobealteredintononlinear shearstrengthindexesctandtastangentiallinemethod asfollows:

    fcf=ct/Fs

    =arctan(tan(Pt/Fs)(5)

    2.4Basicassumptions

    Inordertosolvethestabilityproblemofslopes, someassumptionshavebeenmade.

    (1)Theslopeislongenough.Thereforethis problemcanberegardedasaplanestrainproblem. (2)Thefillingisidealizedasaperfectlyplastic material,andfollowstheassociatedflowrule. (31Therateofexternalworkisduetosoilweight, andthecontributiontoenergydissipationisprovided alongthefailuresurface.

    3Calculationforsafetyfactorofslope

    Inthiswork.arotationalfailuremechanism

    followingalogspiralslipsurfaceisshowninFig.1. whereistheanglerelatedlineCCttolineACtisthe anglerelatedhorizontal1inetolineD:andOhiSthe

    anglerelatedhorizontal1inetolineOC.Thismechanism. whichisconsideredbyCHEN[2],isgeometrically definedbyangles.,,',00,Ohandthemobilized

    internalfrictionanglet.

    Calculationsoftherateofworkdissipationandthe workrateofthesoilweightforrotationa1mechanismcan befoundinCHEN[2].Equatingtheworkrateof externalforcestotheinternalenergydissipationrate,the objectivefunctionofsafetyfactorFscanbewrittenas f0l1ows:

    actual

    e[(o0)]1f,.

    tanOf?W1一一f3f4),b2

    (6)

    wherecnla1denotestheactualslopeheight;functions 838J.

    Cent.SouthUniv.Techno1.(2010)17:836844

    gradient55.,soi1bulkdensityy=l8.6kN/mcohesion forcec=l6.7kN/m~andinterna1frictionangle=12.. Comparisonsweremadewithdifferentmethods (simplifiedbishopmethod,Swedishslicemethod.Janbu methodandfinitedifferencemethod)onsafetyfactor andlatentslipsurface.whichareoutlinedinFig.2. Fig.1Rotationalfailuremechanismforslopestabilityanalysis for_=dependonanglesOh,00,o,,',and,andcan

    befoundinCHEN[2].Accordingtotheupperbound theorem,thesolutionofEq.(6)fallsintoaclassical

    optimizationproblem.Theleastupperboundforsafety factorcanbefoundbysolvingthefollowingsetof equations:

    jo,-o,-o,0Fs7{aa(tan)()

    00<,?,tan(Or?0,Hc=Hat1

    InEq.(7),theunknownquantitiesareoh,0o,and t;ineffect,safetyfactorisanimplicitfunctionatthe sametime.So,iterativeoptimizationcalculationis adoptedtoobtaintheleastupperboundforsafetyfactor byreducingnonlinearshearstrengthindexes(cfand t).

    OnceOh,00,,ctandtarefound,thegeometryof thecriticalfailuresurfaceiscompletelydefined.LandD aretheparametersusedtodrawthepositionofthe potentialslidingsurfaceinFig.1.Thereare :

    Il

    sin(0h+)?(Hactual/r0)

    and

    .=sin(fl-fl')sinsin?.mal'.

    "

    sin(Ohsin+)?l

    (8)

    whereLexpressesthedistancebetweenthefailure surfaceatcrestandedgeoftheslope;Drepresentsthe distancebetweenthefailuresurfaceatbottomofthe slopeandslopetoe,andD=0meanstheslippingsurfaces passingthroughtheslopetoe.

    4Comparisonsandanalysis

    4.1Comparisonwithcalculationonlinearfailure

criterion

    Anembankmentslopeexamplebasedon1inear failurecriterionwascitedtoillustratethevalidityofthis method.Theparametersinthisexampleareasfollows: slopeheightH=6m.slopetopgradient=0..slopefoot 1.80

    1.174

    l179

    :..

    Upperbound1imitanalysisF

    SwedishslicemethodF

    Janbumethod

    Simplybishopmethod

    Finitedifferencemethod

    82

    80

    79

    74

    50

    Fig.2Comparisonofdifferentmethodsoncriticalsliding surfacesandsafetyfactorbasedonlinearfailurecriterion AsseenfromFig.2,slopesafetyfactorobtainedby presentmethodisapproximatelythesameasthatby othermethodsandabsoluteerrorisnomorethan2.1%: the1atentslipsurfacesobtainedbytraditional limitequilibriummethodsareadjacenttoeachother, whichconfirmthevalidityofthismethod.

    4.2Comparisonwithcalculationonnonlinearfailure criterion

    Toshowthevalidityofthepresentapproachunder

    theassumptionofnonlinearfailurecriterion,theexample, inwhichr0=90kN/m.at=247.3kN/mwereusedbv ZHANGandCHEN71,waschosenlikeotherauthors

    includingDRESCHERandCHRISTOPOULOS8],

    C0LLINSetal[9J,YANGandYIN[10andLI[13].

    11able1showsacomparisonofstabilityfactor?c

    obtainedbyZHANGandCHEN71withtheresultsof

    stabilityfactorNs,criticalheightHandcorresponding safetyfactorcomputedbythepresentsolution.Forthe casesanalyzed,sevenvaluesofnonlinearparameterm aretakenintoaccountfrom1.Ot02.5andfourslopesare consideredwithslopefootgradientvaryingfrom45.to 90..Inaddition.slopetopgradienta=0.andsoilbulk densityg=20.0kN/mareconsideredatthesametime. AsshowninTable1.forfl=90..75o60.and45. with=1.02.5.themostabsolutedifierencebetween thepresentcomputationalstabilityfactorandthat obtainedbyZHANGandCHEN7]is0.72%,1.51%,

    2.94%and7.43%.respectively.Andallofthepresent stabilitVfactorJvsolutionsarelessthanthoseobtained \

    J.Cent.SouthUniv.T_echno1.(2010117:836844839

    byZHANGandCHEN71.Intermsoftheupperbound

    limitanalysismethod.thesmallerthestabilityfactorNs, thebettertheupperboundsolution.

    Theoreticallyspeaking,givenaslopewithcertain parameters,itiStruethatsafetyfactor)isI.0when slopesreachcriticalheight()andstabilityfactor() 2-6].Also,whenslopesreachcriticalheight()and stabmtyfactor(),safetyfactors()forallofthecases

Report this document

For any questions or suggestions please email
cust-service@docsford.com