T非饱和土力学大师Fredlund的演讲

WhatistheFutureforSlopeStabilityAnalysis?

(AreWeApproachingtheLimitsofLimitEquilibriumAnalyses?)Introduction

LimitEquilibriummethodsofsliceshavebeen“Good”forthegeotechnicalengineeringprofessionsincethemethodshaveproducedfinancialbenefitEngineersareoftensurprisedattheresultstheyareabletoobtainfromLimitEquilibriummethodsSoWhyChange?ThereareFundamentalLimitationswithLimitEquilibriumMethodsofSlices??TheboundariesforaFREEBODYDIAGRAMarenotknown-TheSHAPEfortheslipsurfacemustbeassumedTheLOCATIONofthecriticalslipsurfacemustbe foundbyTRIALandERRORSHAPEandLOCATIONarethedrivingforceforaparadigmshift

ObjectivesofthisPresentation:ToshowthegradualchangethatisemerginginthewaythatslopestabilityanalysescanbeundertakenToillustratethebenefitsassociatedwithimprovedproceduresfortheassessmentofstressesinaslopeOutlineofPresentationProvideabriefSummaryofcommonLimitEquilibriummethodsalongwiththeirlimitations(2-D&3-D)Takethe

FIRST

stepforwardthroughuseofanindependentstressanalysisTaketheSECONDstepforwardthroughuseofOptimizationTechniquesIsaLimitEquilibriumAnalysisanUpperBoundorLowerBoundSolution?LimitEquilibriumMethodsprimarilysatisfytherequirementsofanupperboundtypeofsolutionReason:theshape

ofslipsurfaceisselectedbytheanalystandtherebyadisplacementboundaryconditionisimposedLimitEquilibriumandFiniteElementBasedMethodsofAnalysesWWWWWWWWNLimitEquilibriumMethodofAnalysisSm=tadldlsndlFiniteElementBased

MethodofAnalysisldtadlQUESTION:HowcantheNormalStressatthebaseofaslicebemostaccuratelycomputed?ConsidertheFreeBodyDiagramsusedtocalculatetheNormalStress?AssumptionforallLimitEquilibriumAnalysisSoilsbehaveasMohr-Coulombmaterials

(i.e.,friction,f',andcohesion,c')Factorofsafety,Fs,forthecohesivecomponentisequaltothefactorofsafetyforthefrictionalcomponentFactorofsafetyisthesameforallslices()[]msnsSFuFc=-+bfsb'tan'SummaryofAvailableEquationsAssociatedwithaLimitEquilibriumAnalysisEquations(knowns): QuantityMomentequilibrium nVerticalforceequilibrium nHorizontalforceequilibrium nMohr-Coulombfailurecriterion n

4nUnknowns:

QuantityTotalnormalforceatbaseofslice nShearforceatthebaseofslice,Sm nInterslicenormalforce,E n-1Intersliceshearforce,X n-1Pointofapplicationofintersliceforce,E n-1Pointofapplicationofnormalforce nFactorofsafety,Fs 1

6n-2SummaryofUnknownsAssociatedwithaLimitEquilibriumAnalysisOneFsperslidingmassForcesActingonEachSliceFocusonSmbyxSmXREREXLSlipsurfaceGroundsurfaceWhRN=snbbfNLPhreaticlineMobilizedShearForce,SmforSaturated-UnsaturatedSoils()[]()swasansmFuuFuFcSbfbfsbbtan'tan'-+-+=Onlynewvariablerequiredforsolvingsaturated-unsaturatedsoilsproblemsistheshearforcemobilizedφb=Frictionanglewithrespecttomatricsuctionua=Pore-airpressureuw=Pore-waterpressure[]Momentequilibrium,Fm:Forceequilibrium,Ff:ååå-ïþïýüïîïíì÷÷øöççèæ-+=NfWx'tanR'tantanuNR'cFbwmfffbbååþýüîíì÷÷øöççèæ-+=aafffbabsincos'tan'tantancos'NuNcFbwfPore-airpressuresareassumedtobezerogaugeNormalforceatbaseofslice:LimitEquilibriummethodsdifferintermsofhow(XR-XL)iscomputedandoverallstaticssatisfiedLimitEquilibriumproblemisindeterminate:Canapplyanassumption(Historicalsolution)Canutilizeadditionalphysics(Futuresolution)()FFuFcXXWNbwLR'tansincostansinsin'faafabab++---=sxbaArea=Interslicenormalforce(E)widthofslice,bsxtxysyDistance(m)Elevation

(m)txybaArea=Intersliceshearforce(X)VerticalsliceDistance(m)Elevation

(m)ò=baxydyXtò=baxdyEsStressesontheBoundaryBetweenSlicesMorgenstern&Price,1965SummaryofLimitEquilibriumMethodsandAssumptionsMethod

EquilibriumSatisfied

Assumptions

Ordinary

Moment,tobase

EandX=0

Bishop’sSimplified

Vertical,Moment

Eishorizontal,X=0

Janbu’sSimplified

Vertical,Horizontal

Eishorizontal,X=0,empiricalcorrectionfactor,f0,accountsforintersliceshearforces

Janbu’sGeneralized

Vertical,Horizontal

Eislocatedbyanassumedlineofthrust

Spencer

Vertical,Horizontal,Moment

ResultantofEandXareofconstantslope

ForcesActingonOneSliceinOrdinaryorConventionalMethodhWbbaN=sbnNSmForcesActingonOneSliceinBishop’’sSimplifiedandJanbu’sSimplifiedMethodshWbbaN=sbnNSmERELSummaryofLimitEquilibriumMethodsandAssumptionsDirectionofXandEistheaverageofthegroundsurfaceslopeandtheslopeatthebaseofasliceVerticalHorizontalLoweandKarafiathDirectionofXandEisparalleltothegroundVerticalHorizontalCorpsofEngineersDirectionofEandXisdefinedbyanarbitraryfunction.PercentofthefunctionrequiredtosatisfymomentandforceequilibriumiscalledλVerticalHorizontalMomentMorgenstern-Price,GLEAssumptionsEquilibriumSatisfiedMethodForcesActingonOneSliceinSpencer’’s,Morgenstern-Price,andGLEMethodshWbbaN=sbnNSmERELXRXLVariousIntersliceForceFunctionsProposedbyMorgenstern&Price(1965)Spencer’’sWilsonandFredlund(1983)Usedafiniteelementstressanalysis(withgravityswitchedon)todetermineashapefortheIntersliceForceFunctionIntersliceForceFunctionforaDeep-seatedSlipSurfaceThrougha1:2SlopeX=Eλf(x)DefinitionofDimensionlessDistancef(x)islargestatmid-pointInflectionpointsnearcrest&toeGeneralizedFunctionalShapewhere:K=magnitudeoffunctionatmid-slopee=baseofnaturallogC=variabletodefineinflectionpointn=variabletodefinesteepnessω=dimensionlessx-position()2/)(nnCKexfw-=WilsonandFredlund,1983X=Eλf(x)DimensionlessDistanceUniquefunctionof““slopeangle””forallslipsurfaces“C”coefficientversusslopeangleUniquefunctionof““slopeangle””forallslipsurfaces“n”coefficientversustangentofslopeangleComparisonofFactorsofSafetyCircularSlipSurface01.801.851.901.952.002.002.25lJanbu’’sGeneralizedSimplifiedBishopSpencerMorgenstern-Pricef(x)=constantOrdinary=1.928FfFmFredlundandKrahn1975FactorofsafetyMomentandForceLimitEquilibriumFactorsofSafetyForaCirculartypeslipsurfaceMomentlimitequilibriumanalysisForcelimitequilibriumanalysisFredlundandKrahn,1975Lambda,lFactorofsafetyForceandMomentLimitequilibriumFactorsofSafetyforaplanartoeslipsurfaceForcelimitequilibriumanalysisMomentlimitequilibriumanalysisLambda,lFactorofsafetyKrahn2003ForceandMomentLimitequilibriumFactorsofSafetyforacompositeslipsurfaceMomentlimitequilibriumanalysisForcelimitequilibriumanalysisLambda,lFactorofsafetyFredlundandKrahn1975ForceandMomentLimitequilibriumFactorsofSafetyfora““SlidingBlock””typeslipsurfaceMomentlimitequilibriumanalysisForcelimitequilibriumanalysisLambda,lFactorofsafetyKrahn2003ExtensionsofMethodsofSlicestoThree-dimensionalMethodsofColumnsHovland(1977)–3-DofOrdinaryChenandChameau(1982)–3-DofSpencerCavounidis(1987)–3-DFs>2-DFsHungr(1987)–3-DofBishopSimplifiedLamandFredlund(1993)–3-Dwithf(x)onall3planes;3-DofGLEShapeandLocationBecomeEvenMoreDifficulttoDefinein3-DTwoPerpendicularSectionsThrougha3-DSlidingMassSectionParalleltoMovementSectionPerpendiculartoMovementFreeBodyDiagramofaColumnwithAllIntersliceForcesParallelPerpendicularBaseIntersliceForceFunctionsforTwooftheDirectionsX/EV/PFirstStepForwardQuestion:IstheNormalStressatthebaseofeachsliceasaccurateascanbeobtained?IstheNormalStressonlydependentupontheforcesonaverticalslice?ImprovementofNormalStressComputationsFredlundandScoular1999LimitequilibriumandfiniteelementnormalstressesforatoeslipsurfaceFromlimitequilibriumanalysisFromfiniteelementanalysisLimitequilibriumandfiniteelementnormalstressesforadeep-seatedslipsurfaceFromfiniteelementanalysisFromlimitequilibriumanalysisLimitequilibriumandfiniteelementnormalstressesforananchoredslopeFromfiniteelementanalysisFromlimitequilibriumanalysisToillustrateproceduresforcombiningafiniteelementstressanalysiswithconceptsoflimitingequilibrium.(i.e.,finiteelementmethodofslopestabilityanalysis)TocompareresultsofafiniteelementslopestabilityanalysisandconventionallimitequilibriummethodsUsingLimitEquilibriumConceptsinaFiniteElementSlopeStabilityAnalysisObjective:Thecompletestressstatefromafiniteelementanalysiscanbe““imported””intoalimitequilibriumframeworkwherethenormalstressandtheactuatingshearstressarecomputedforanyselectedslipsurfaceHypothesisAssumption:Thestressescomputedfrom“switching-on”gravityaremorereasonablethanthestressescomputedonaverticalsliceMannerof““ImportingStresses”fromaFiniteElementAnalysisintoaLimitEquilibriumAnalysissnFinite

Element

Analysis

for

StressesLimit

Equilibrium

AnalysissntmMohrCircletmIMPORT:ActingNormalStressActuatingShearStressLimitEquilibriumAnalysisFiniteElementAnalysisforStressesBishop(1952)-stressesfromLimitEquilibriummethodsdonotagreewithactualsoilstressesCloughandWoodward(1967)-“meaningfulstabilityanalysiscanbemadeonlyifthestressdistributionwithinthestructurecanbepredictedreliably””Kulhawy(1969)-usednormalandshearstressesfromalinearelasticanalysistocomputefactorofsafety“EnhancedLimitStrengthMethod”BackgroundtoUsingStressAnalysesinSlopeStabilityStressLevelRezendiz1972Zienkiewiczetal1975Strength&StressLevelAdikariandCummins1985Enhancedlimitmethods(finiteelementanalysiswithalimitequilibriumFiniteElementSlopeStabilityMethodsDirectmethods(finiteelementanalysisonly)StrengthLevelKulhawy1969F[]--Z=1313墢¢¢åæèççöø÷÷æèççöø÷÷ìíïïïîïïïüýïïïþïïïDDLLfssss{}F=(c+tan)--c+tanA1313¢¢¢å¢¢¢¢¢¢¢åæèççöø÷÷æèççöø÷÷æèçöø÷ìíïïïîïïïüýïïïþïïïsfsssssfDDLLf*F=(c+tan)K墢¢åsftDDLLDefinitionofFactorofSafetyLoadincreasetofailureStrengthdecreasetofailureanalysis)DifferencesandSimilaritiesBetweentheFiniteElementSlopeStabilityandConventionalLimitEquilibriumDifferencesSolutionisdeterminateFactorofsafetyequationislinearSimilaritiesStillnecessarytoassumetheshapeoftheslipsurfaceandsearchbytrialanderrortolocatethecriticalslipsurfaceWhyhasn’’tFiniteElementSlopeStabilityMethodbeenextensivelyused?DifficultiesandperceptionsrelatedtothestressanalysisInabilitytotransferlargeamountsofdataandfindneededinformationNow:MicrocomputerhavedramaticallychangedourabilitytocombineFiniteElementandLimitEquilibriumanalysesDefinitionofFactorofSafetyKulhawy(1969)where:Sr=resistingshearstrengthorSm=mobilizedshearforceåå=mrFEMSSFbfs}'tan)u('c{Swnr-+=ActuatingShearNormalStressAnalysisStudyUndertakenbyFredlundandScoular(1999)AdoptedtheKulhawy(1969)procedureUsedSigma/WandSlope/WPoisson’sratiorange=0.33to0.48Elasticmodulus,E=20,000to200,000kPaCohesion,c'=10to40kPaFriction,'=10to30degreesComparedconventionalLimitEquilibriumresultswithFiniteElementslopestabilityresultsLocationofCenterofaSectionalongtheSlipSurfacewithinaFiniteElementAnalysisxyx-Coordinatey-CoordinateSlipSurfaceFiniteElement(r,s)srFictitiousslicedefinedwiththeLimitEquilibriumanalysisCenterofthebaseofaslice(x,y)PresentationofFiniteElementSlopeStabilityResultsConditionsAnalyzed:DryslopePiezometriclineat3/4height,exitingattoeDryslope,partiallysubmergedPiezometriclineat1/2heightandsubmergedtomid-heightSelected2:1Free-StandingSlopewithaPiezometricLineExitingattheToeoftheSlope20406080100120204060800CrestPiezometricLineToe21x-Coordinate(m)Note:Dryslopewith&withoutpiezometricliney-Coordinate(m)Selected2:1PartiallySubmergedSlopewithaHorizontalPiezometricLineatMid-Slope20406080100120204060800CrestToe21x-Coordinate(m)WaterPiezometricLiney-Coordinate(m)Note:Dryslopewith&withoutpiezometricline050100150200250300203040506070x-Coordinate(m)Actingandrestrictingshearstress(kPa)CrestToeShearStrengthShearForcePoissonRatio,m=0.33ShearStrengthandShearForcefora2:1DrySlopeCalculatedUsingtheFiniteElementSlopeStabilityMethodLocalandGlobalFactorsofSafetyfora2:1DrySlope012345672025303540455055606570x-CoordinateFactorofSafetyCrestToeLocalF(mLocalF(m=0.33)BishopMethod,F=2.360=2.173GlobalFactorsofSafetyBishop2.360Janbu2.173GLE(F.E.function)2.356Fs(m=0.33)2.342Fs(m=0.48)2.339Ordinary2.226sJanbuMethod,FsssFs=2.342Fs=2.339=0.48)FactorsofSafetyVersusStabilityNumberfora2:1DrySlopeasaFunctionofc'0.00.51.01.52.02.50510152025StabilityNumber,[gHtanf¢/c¢]FactorofSafetyc¢=20kPac¢=10kPac=40kPaFs(m=0.33)Fs(m=0.48)Fs(GLE)2:1DrySlope¢FactorofSafetyVersusStabilityCoefficientfora2:1DrySlopeasaFunctionof0.00.51.01.52.02.50.000.020.040.060.080.100.12StabilityCoefficient,[c¢/gH]FactorofSafetyf¢=30°f¢=10°f¢=20°2:1DrySlopesFs(m=0.33)F(m=0.48)Fs(GLE)sFactorofSafetyVersusStabilityCoefficientasaFunctionoffor2:1SlopewithaPiezometricLine0.01.62.00.000.020.040.060.080.100.12StabilityCoefficient,[c'/gH]Factorofsafetyf=30°f=20°f=10°2:1SlopewithpiezometriclineFs(m=0.33)Fs(m=0.48)Fs(GLE)LocationoftheCriticalSlipSurfaceforaSlopewithaPiezometricLinewithSoilPropertiesofc'=40kPaandf'=30°°70102010060504030908070110506040102030x

-

Coordinate

(m)80GLE

(F.E.

function)Fs

(m

=

0.33)Fs

(m

=

0.48)MethodXYRFactorofsafetyGLE(F.E.Function)58.556.037.91.741Fs(m=0.33)57.549.534.71.627Fs(m=0.48)57.553.037.81.661Y-Coordinate(m)LocationoftheCriticalSlipSurfaceforaSlopewithaPiezometricLinewheretheFactorofSafetyisClosestto1.070102010060504030908070506040102030110x

-

Coordinate

(m)80Fs

(m

=

0.48)Fs(m=0.33)GLE

(F.E.

function)sMethodXYRFactorofsafetyGLE(F.EFunction.63.559.039.61.102Fs(m=0.33)63.059.041.51.076F(m=0.48)61.559.542.31.100y-Coordinate(m)FactorofSafetyVersusStabilityCoefficientasaFunctionoffor2:1DrySlope,1/2Submerged0.00.51.01.52.02.53.03.50.000.020.040.060.080.100.12StabilityCoefficient,[c¢/gH]FactorofSafetyf¢=20°°f¢=10°°2:1Dryslope,one-halfsubmergedf¢=30°°Fs(m=0.33)Fs(m=0.48)Fs(GLE)FactorofSafetyVersusStabilityCoefficientasaFunctionoffor2:1SlopeHalfSubmergedwithPiezometricLine0.00.51.01.52.02.50.000.020.040.060.080.100.12StabilityCoefficient,[c¢/H]FactorofSafetyf¢=30°°f¢=10°°f¢=20°°2:1Slope,one-halfsubmergedgFs(m=0.33)Fs(m=0.48)Fs(GLE)1020100605040309080701107050604010203080Fs(m=0.33)Fs

(m

=

0.48)GLE(F.E.Function)sMethodXYRFactorofsafetyGLE(F.E.Function58.058.540.22.303Fs(m=0.33)52.550.531.82.259F(m=0.48)51.551.531.02.273LocationoftheCriticalSlipSurfaceforaHalfSubmergedSlopewheretheSoilPropertiesarec'=40kPaandf'=30°x-Coordinate(m)y-Coordinate(m)ConclusionsfromStep1ForwardNormalandActuatingShearstressesfromafiniteelementanalysisappeartoprovideamorereasonablerepresentationofthestressstateinaslopeTheEnhancedLimitmethodbyKulhawy(1969)appearstoopenthewaytosimulatemorecomplexslopestabilityproblemsEnhancedLimitmethodscanreadilybeusedinroutineengineeringpracticeGlobalfactorsofsafetyappeartobeessentiallythesameformostsimpleslopesSelectionofPoisson’sratiohassomeeffectontheEnhancedLimitfactorofsafetyFactorsofSafetyappeartodifferslightlyfor:LowcohesionvaluesHighanglesofinternalfrictionHowdotheResultsfromEnhancedLimitMethodsComparetoLimitEquilibriumMethods?LocalFactorsofSafetycanalsobecomputedbytheEnhancedLimitMethodSecondStepForwardQuestion:IsitpossibleforthecomputertodeterminetheShapeofthecriticalslipsurface?IsitpossibleforthecomputertodeterminetheLocationofthecriticalslipsurface?ImprovementonShapeandLocationHaandFredlund2002OptimizationTechniques(i.e.,DynamicProgramming)canbeusedtofindthepathwaywhichminimizesafunctionoftheshearstrengthavailabletotheactuatingshearstresswithinasoilmassHypothesisAssumption:Thestressescomputedfrom“switching-on”gravitycanbeusedtorepresentthestressstateinthesoilmassSlopeStabilityAnalysisUsingDynamicProgrammingCombinedwithaFiniteElementStressAnalysisDynamicProgramming(DP)optimizationtechniquesforslopestabilityanalysis(Spencer‘‘sMethod)wasintroducedbyBaker(1980)Yamagami&Ueta(1988)andZouetal.(1995)improvedontheBaker(1980)solutionbycouplingDynamicProgrammingwithaFiniteElementstressanalysisDefinitionofFactorofSafetySmoothcurveDiscreteform(1)(2)"Stage"B"Statepoint""n+1"AY"i""1"Riii+1kjSijk...ii+1...Fs=(ShearStrength)/(ActuatingShearStress)òò=BABAfsdLdLFttåå==DD=niiiniiifsLLF11ttDefinitionof“ReturnFunction““;Gstage"i+1"stage"i"lijlijfttfijijjsijtijqkijsijtElement(ij)Element(ij)R=ResistingShearStrength:S=ActuatingShearStressFs=(ShearStrength)/(ActuatingShearStress)Difficulttominimize!å=D-=niiisfiLFG1)(ttdLFGsBAf)(tt-=òå=-=niisiSFRG1)(ActuatingShearForcesandResistingShearS=ActuatingShearStressR=ResistingShearStrengthåå====D=neijijijneijijiiilSLS11ttåå====D=neijijfneijijifilRLRiji11ttijbijwaijaijneijijiluuucR}tan)(tan)({'1'ffs-+-+=å=Definitionof“OptimalFunction“:MinimumValueof“ReturnFunction““=theoptimalfunctionobtainedatpoint{k}ofstage[i+1],=theoptimalfunctionobtainedatpoint{j}instage[i],and=thereturnfunctioncalculatedwhenpassingfromthestatepoint{j}instage[i]tothestatepoint{k}instage[i+1].where:Introducean““optimalfunction””,H=OptimalFunctionG=ReturnFunctionå=-==niisiSFRGG1min)(minmin)(jHi),()()(1kjGjHkHiii+=+)(1kHi+)(jHi),(kjGiBoundaryConditionsof“OptimalFunction““Attheinitialstage,(i=1):Atthefinalstage,(i=n+1):where:=thenumberofstatepointsinthefinalstageH=OptimalFunction0)(1=jH1...1NPj=),()()(1kjGjHkHnnn+=+å=+-==niisimnSFRGkH11).()(...1=n+1NPk1+nNPTheMinimum(orOptimal)TravellingTimeProblemDYNAMICPROGRAMMINGSOLUTION116487511114121H1(1)=092747H1(1)=13AH(2)=812310B5674STAGENUMBER1234567d=(4,2)3G(1,2)=33105243252827224415532BATHEMINIMUMTRAVELLINGTIMEPROBLEMAnalyticalSchemeoftheDynamicProgrammingMethodEntrypoint"1""InitialABpoint"Y"Statepoint"...ii+1...XBB"n+1"X...StageNo."Exitpoint"Si"Gridelement"boundary""Searching