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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-2SummaryofUnknownsAssociatedwithaLimitEquilibriumAnalysisOneFsperslidingmassForcesActingonEachSliceFocusonSmbyxSmXREREXLSlipsurfaceGround
surfaceWhRN
=snbbfNLPhreaticlineMobilizedShearForce,Sm
for
Saturated-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=Interslice
normalforce(E)widthofslice,bsxtxysyDistance(m)Elevation
(m)txybaArea=Interslice
shearforce(X)Vertical
sliceDistance(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=sb
nNSmForcesActingonOneSliceinBishop’sSimplifiedandJanbu’sSimplifiedMethods
hWbbaN=sb
nNSmERELSummaryofLimitEquilibriumMethodsandAssumptionsDirectionofXandEistheaverageofthegroundsurfaceslopeandtheslopeatthebaseofasliceVerticalHorizontalLoweandKarafiathDirectionofXandEisparalleltothegroundVerticalHorizontalCorpsofEngineersDirectionofEandXisdefinedbyanarbitraryfunction.PercentofthefunctionrequiredtosatisfymomentandforceequilibriumiscalledλVerticalHorizontalMomentMorgenstern-Price,GLEAssumptionsEquilibriumSatisfiedMethod
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Element
Analysis
for
StressesLimit
Equilibrium
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(m)80GLE
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=
0.33)Fs
(m
=
0.48)Me咱th泼odXYRFa帜ct办or解o娃f叫sa而fe请tyGLE(F.E.Fu找nc葡ti扶on脱)58.556.037.91.741Fs(m=0.33)57.549.534.71.627Fs(m=0.48)57.553.037.81.661Y-貌C礼oo培rd本in胖at芽e剖(m蛙)Lo喂ca保ti铜on众o序f钳th唐e碑Cr侍it佩ic贤al探S塑li举p火Su誉rf锣ac径e骄fo聚r绘a抱Sl姑op覆e载w扣it乖h赌aPi通ez样om找et剑ri荒cLi妇newh停er邮e帜th书e让Fa甩ct由or泰o悟f勺Sa京fe侦ty复i脚s她Cl剧os冈es搅t像to绩1喷.070102010060504030908070506040102030110x
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