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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

ForcesActingonOneSliceinSpencer’s,Morgenstern-Price,and

GLE

Methods

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Element

Analysis

for

StressesLimit

Equilibrium

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ac散tor价ofSaf银etyf¢=乐30弓°f¢=酿10犹°f¢=2龙0°2:粮1化Dr爷y迈Sl剂op波esFs(m=育0.访33舰)F(m=币0.都48峡)Fs(GL扎E)sFa傻ct增or声o昌f肿Sa衔fe甲ty端V淹er舒su自s挂St听ab茶il里it秘y板Co览ef哨fi觉ci香en博t规as述aFun咬cti铺on扛offor隆2:蒜1S智lop众ew针ith井aPi赵ez住om萌et阶ri心cLin股e0.00.40.旱81.王21.炼62.矿00.0要00.川020.0皮40.0与60.0端80.肚100.桥12St将ab颤il揉it题y蜻Co片ef骑fi励ci艳en如t,衫[c指'/gH]Fac跳tor抹of很sa充fet访yf=3恒0°f=子20欢°f=转10输°2:1摇Sl婚ope烘wi掏thpi滩ez奶om谱et圈ri卵cli奸neFs(m=0.33)Fs(m=0.48)Fs(GLE)Loc培ati响on炼of臣the棕Cr淘iti议cal担Sl必ip吼Sur彼fac猫ef栽or劳aS本lop裹ew呆ith打aPi谈ez达om欲et傍ri曾cLin殃ewi蒜th喷S蓬oi番l桃Pr析op桶er工ti婚es乒o朋fc'群=某4亿0kPaandf'胳=口30马°70102010060504030908070110506040102030x

-

Coordinate

(m)80GLE

(F.E.

function)Fs

(m

=

0.33)Fs

(m

=

0.48)Me论th部odXYRFac禾tor袭of锈sa汇fet肉yGLE(F.E.Fu划nc瞧ti珍on育)58.556.037.91.741Fs(m=0.33)57.549.534.71.627Fs(m=0.48)57.553.037.81.661Y-暮C晨oo葱rd凡in丝式at哑e饮(m捞)Lo旅ca臂ti歌on梳o箱f炊th欺e愁Cr紫it铁ic狗al劲S亿li仿p布Su湿rf盘ac逮e源fo耻r揉a筹Sl港op膊e扔w捕it顾h渴aPi寨ez丙om悲et税ri么cLi嚼newh旬er宽e拉th饲e肿Fa愁ct佩or梦o篇f猛Sa慈fe口ty刘i客s册Cl沈os纠es岸t陷to具1历.070102010060504030908070506040102030110x

-

Coordinate

(m)80Fs

(m

=

0.48)Fs(m=0.33)GLE

(F.E.

function)sMet开hodXYRFac女tor碎of仙sa杂fet裙yGLE(F.E短F涨un吐ct芦io坊n.63.559.039.61.102Fs(m=0.33)63.059.041.51.076F(m=0.48)61.559.542.31.100y-浪C盒oo荒rd高in临at弃e兆(m免)Fac咱tor差of鞠Sa洒fet苹yV老ers醋us衰Sta毙bil危ity火Co蜂eff咽ici前ent夹as坑aFun率cti球on灶offo愧r代2:搜1Dr恨y所Sl手op杠e,与1渔/2描S扩ub蜂me膏rg率ed0.00.室51.溪01.52.暑02.祸53.03.50.0累00.叹020.0浸40.纯060.举080.1滑00.丘12St贱ab风il按it舍yCoe日ffi东cie鱼nt,涨[聋c¢/gH]Fac盾tor划of费Sa惜fet迎yf¢=屋20爪°f¢=约10铅°2:鼠1粱Dr将y馒sl暖op新e,距o煮ne贸-h短al烤f卫su砍bm匪er购ge咱df¢=鞭30独°Fs(m=臣0.窜33妈)Fs(m=绒0.戒48涨)Fs(GL歇E)Fac昌tor掌of涂Sa警fet箱yV争ers贷us待Sta恋bil拌ity显Co育eff锡ici愈ent向as裂aFun怎cti征on灭offo先r剪2:盐1愿Sl瞒op秋eHa粥lf放S猫ub姿me务rg掉ed室w抱it常hPi辱ez艳om猴et敢ri冲cLin斗e0.00.51.01.52.02.50.松000.辩020.0济40.咐060.葛080.1当00.锦12Sta轨bil誓ityCo秤ef骨fi笼ci笨en凯t,抹[c¢/H]Fac普tor停of轮Sa碗fet沿yf¢=3草0°f¢=已10榨°f¢=距20轻°2:末1朵Sl垂op化e,涉o谢ne宾-h淘al航f贴su谢bm匪er绸ge谎dgFs(m=扣0.染33谊)Fs(m=0香.48树)Fs(GL躲E)1020100605040309080701107050604010203080Fs(m=0.温33剧)Fs

(m

=

0.48)GL磁E部(F访.E激.倾Fu蝴nc爪ti糖on掘)sMet敬hodXYRFa军ct茂or会o热f兆sa亭fe谢tyGLE(F.E.Fun眼cti真on58.058.540.22.303Fs(m=0.33)52.550.531.82.259F(m=0.48)51.551.531.02.273Loc库ati跳on梢of屿the毅Cr尽iti拜cal该Sl罗ip揪Sur伪fac鼻ef得or足aHa咱lf抛S壶ub公me菌rg琴edSlo域pe马whe僻re妖the神So删il盆Pro佳per酷tie幼sa须rec'宜=4雕0kP卡aandf'=蛙30遗°x始-Coo号rdi竞nat纤e(m价)y-恒Co善ord挤ina队te樱(m)Con从clu被sio奴nsfro贵mSte鹅p1For炎war色dNor嘉malan鼓dAc设tu落at唐in床g仔Sh科ea酿rst假re品ss恩es励f傍ro仙m抗a松fi客ni乓te悦e钞le霜me破nt交a助na誉ly求si圾s局ap轰pe熄ar柿t导o温pr兰ov沿id斩e羞a末mo粒re钩r蚕ea梢so杯na术bl配e冠re御pr守es督en伏ta的ti词on旺o蚀f泊th验e羡st夜re简ss犹s誉ta醉te时i棵n夫a沿sl单op第eTh练eEnh纱anc最ed胖Lim涛itme姻th旅od支b原yKu突lh甩aw苹y(19浩69)ap搬pe伏ar严s鄙to棚o梯pe灭n奔th功e挡wa译y轿to夸s否im奔ul以at可e攀mo丙re博c铅om略pl略ex桨s碧lo叼pe乎s漂ta晕bi值li扔ty谢p茎ro雁bl学em回sEnh废anc胸ed招Lim崭itmet匹hod轮sc复an兔rea洲dil梳yb逼eu叮sed睛inro定ut族in测e颠en恨gi傅ne架er科in千g梯pr目ac沿ti漫ceGlo皂bal拢fa健cto洁rs销of威saf疫ety念ap么pea跌rt鱼ob坦ee车sse面nti捷all鞭yt颂he

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