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

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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

hereagreewithavailablepredictionswhennonlinearcriterionreducesto1inearcriterion,andthevalidityofpresentmethodcouldbe

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

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.

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

(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

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

optimizationproblem.Theleastupperboundforsafety factorcanbefoundbysolvingthefollowingsetof equations:

jo,-o,-o,0Fs7{aa(tan)()

00&lt;,?,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

1.174

l179

:..

Upperbound1imitanalysisF

SwedishslicemethodF

Janbumethod

Simplybishopmethod

Finitedifferencemethod

82

80

79

74

50

4.2Comparisonwithcalculationonnonlinearfailure criterion

Toshowthevalidityofthepresentapproachunder

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

includingDRESCHERandCHRISTOPOULOS8],

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

11able1showsacomparisonofstabilityfactor?c

obtainedbyZHANGandCHEN71withtheresultsof

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

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