英语翻译原文

StabilityofSlopes

1Introduction

Gravitationalandseepageforcestendtocauseinstabilityinnaturalslopes,inslopesformedbyexcavationandintheslopesofembankmentsandearthrotationalslipstheshapeofthefailuresurfaceinsectionmaybeacirculararcoranon-circularcurve．Ingeneral，circularslipsareassociatedwithhomogeneoussoilconditionsandnon-circularslipswithnon-homogeneousconditions．

Translationalandcompoundslipsoccurwheretheformofthefailuresurfaceisinfluencedbythepresenceofanadjacentstratumofsignificantlydifferentstrength．

Translationalslipstendtooccurwheretheadjacentstratumisatarelativelyshallowdepthbelowthesurfaceoftheslope:thefailuresurfacetendstobeplaneandroughlyparalleltotheslope.Compoundslipsusuallyoccurwheretheadjacentstratumisatgreaterdepth，thefailuresurfaceconsistingofcurvedandplanesections．

Inpractice,limitingequilibriummethodsareusedintheanalysisofslopestability.Itisconsideredthatfailureisonthepointofoccurringalonganassumedoraknownfailuresurface．Theshearstrengthrequiredtomaintainaconditionoflimitingequilibriumiscomparedwiththeavailableshearstrengthofthesoil，givingtheaveragefactorofsafetyalongthefailuresurface．Theproblemisconsideredintwodimensions，conditionsofplanestrainbeingassumed．Ithasbeenshownthatatwo-dimensionalanalysisgivesaconservativeresultforafailureonathree-dimensional(dish-shaped)surface．

2AnalysisfortheCaseofφu=0

Thisanalysis,intermsoftotalstress，coversthecaseofafullysaturatedclayunderundrainedconditions,i.e.Fortheconditionimmediatelyafterconstruction．Onlymomentequilibriumisconsideredintheanalysis．Insection,thepotentialfailuresurfaceisassumedtobeacirculararc.Atrialfailuresurface(centreO，radiusrandlengthLa)isshowninFig.9.2.Potentialinstabilityisduetothetotalweightofthesoilmass(WperunitLength)abovethefailuresurface．Forequilibriumtheshearstrengthwhichmustbemobilizedalongthefailuresurfaceisexpressedas

whereFisthefactorofsafetywithrespecttoshearstrength．EquatingmomentsaboutO：

Therefore

(9.1)

Themomentsofanyadditionalforcesmustbetakenintoaccount．Intheeventofatensioncrackdeveloping，asshowninFig.9.2，thearclengthLaisshortenedandahydrostaticforcewillactnormaltothecrackifthecrackfillswithwater．Itisnecessarytoanalyzetheslopeforanumberoftrialfailuresurfacesinorderthattheminimumfactorofsafetycanbedetermined．

Basedontheprincipleofgeometricsimilarity，Taylor[9.9]publishedstabilitycoefficientsfortheanalysisofhomogeneousslopesintermsoftotalstress．ForaslopeofheightHthestabilitycoefficient(Ns)forthefailuresurfacealongwhichthefactorofsafetyisaminimumis

(9.2)

Forthecaseofφu=0，valuesofNscanbeobtainedfromFig.9.3.ThecoefficientNsdependsontheslopeangleβandthedepthfactorD，whereDHisthedepthtoafirmstratum．

GibsonandMorgenstern[9.3]publishedstabilitycoefficientsforslopesinnormallyconsolidatedclaysinwhichtheundrainedstrengthcu(φu=0)varieslinearlywithdepth．

Example1

A45°slopeisexcavatedtoadepthof8minadeeplayerofsaturatedclayofunitweight19kN／m3:therelevantshearstrengthparametersarecu=65kN／m2andφu=0．DeterminethefactorofsafetyforthetrialfailuresurfacespecifiedinFig.9.4.

InFig.9.4,thecross-sectionalareaABCDis70m2.

Weightofsoilmass=70×19=1330kN／m

ThecentroidofABCDis4.5mfromO．TheangleAOCis89.5°andradiusOCis12.1m．ThearclengthABCiscalculatedas18.9m．Thefactorofsafetyisgivenby：

Thisisthefactorofsafetyforthetrialfailuresurfaceselectedandisnotnecessarilytheminimumfactorofsafety．

TheminimumfactorofsafetycanbeestimatedbyusingEquation9.2.

FromFig.9.3，β=45°andassumingthatDislarge，thevalueofNsis0.18.Then

TheMethodofSlices

Inthismethodthepotentialfailuresurface，insection，isagainassumedtobeacirculararcwithcentreOandradiusr．Thesoilmass(ABCD)aboveatrialfailuresurface(AC)isdividedbyverticalplanesintoaseriesofslicesofwidthb,asshowninFig.9.5.Thebaseofeachsliceisassumedtobeastraightline．Foranyslicetheinclinationofthebasetothehorizontalisαandtheheight,measuredonthecentre-1ine,ish.Thefactorofsafetyisdefinedastheratiooftheavailableshearstrength(τf)totheshearstrength(τm)whichmustbemobilizedtomaintainaconditionoflimitingequilibrium,i.e.

Thefactorofsafetyistakentobethesameforeachslice，implyingthattheremustbemutualsupportbetweenslices，i.e.forcesmustactbetweentheslices．

Theforces(perunitdimensionnormaltothesection)actingonasliceare：

1.Thetotalweightoftheslice，W=γbh(γsatwhereappropriate)．

2.Thetotalnormalforceonthebase，N(equaltoσl)．Ingeneralthis

forcehastwocomponents，theeffectivenormalforceN'(equaltoσ'l)andtheboundarywaterforceU(equaltoul)，whereuistheporewaterpressureatthecentreofthebaseandlisthelengthofthebase．

3.Theshearforceonthebase，T=τml.

4.Thetotalnormalforcesonthesides,E1andE2.

5.Theshearforcesonthesides，X1andX2.

Anyexternalforcesmustalsobeincludedintheanalysis．

TheproblemisstaticallyindeterminateandinordertoobtainasolutionassumptionsmustbemaderegardingtheintersliceforcesEandX：theresultingsolutionforfactorofsafetyisnotexact．

ConsideringmomentsaboutO，thesumofthemomentsoftheshearforcesTonthefailurearcACmustequalthemomentoftheweightofthesoilmassABCD．ForanyslicetheleverarmofWisrsinα,

therefore

∑Tr=∑Wrsinα

Now,

Forananalysisintermsofeffectivestress，

Or

(9.3)

whereLaisthearclengthAC．Equation9.3isexactbutapproximationsareintroducedindeterminingtheforcesN'．ForagivenfailurearcthevalueofFwilldependonthewayinwhichtheforcesN'areestimated．

TheFelleniusSolution

Inthissolutionitisassumedthatforeachslicetheresultantoftheintersliceforcesiszero．Thesolutioninvolvesresolvingtheforcesoneachslicenormaltothebase，i.e.

N'=WCOSα-ul

Hencethefactorofsafetyintermsofeffectivestress(Equation9.3)isgivenby

(9.4)

ThecomponentsWCOSαandWsinαcanbedeterminedgraphicallyforeachslice．Alternatively，thevalueofαcanbemeasuredorcalculated．Again，aseriesoftrialfailuresurfacesmustbechoseninordertoobtaintheminimumfactorofsafety．Thissolutionunderestimatesthefactorofsafety：theerror，comparedwithmoreaccuratemethodsofanalysis，isusuallywithintherange5-2%.

ForananalysisintermsoftotalstresstheparametersCuandφuareusedandthevalueofuinEquation9.4iszero．Ifφu=0,thefactorofsafetyisgivenby

(9.5)

AsN’doesnotappearinEquation9.5anexactvalueofFisobtained．

TheBishopSimplifiedSolution

Inthissolutionitisassumedthattheresultantforcesonthesidesofthe

slicesarehorizontal，i.e.

Xl-X2=0

Forequilibriumtheshearforceonthebaseofanysliceis

Resolvingforcesintheverticaldirection:

(9.6)

Itisconvenienttosubstitute

l=bsecα

FromEquation9.3，aftersomerearrangement，

(9.7)

Theporewaterpressurecanberelatedtothetotal‘fillpressure’atany

pointbymeansofthedimensionlessporepressureratio，definedas

(9.8)

(γsatwhereappropriate)．Foranyslice,

HenceEquation9.7canbewritten:

(9.9)

AsthefactorofsafetyoccursonbothsidesofEquation9.9,aprocessofsuccessiveapproximationmustbeusedtoobtainasolutionbutconvergenceisrapid．

Duetotherepetitivenatureofthecalculationsandtheneedtoselectanadequatenumberoftrialfailuresurfaces，themethodofslicesisparticularlysuitableforsolutionbycomputer．Morecomplexslopegeometryanddifferentsoilstratacanbeintroduced．

Inmostproblemsthevalueoftheporepressureratioruisnotconstantoverthewholefailuresurfacebut，unlessthereareisolatedregionsofhighporepressure，anaveragevalue(weightedonanareabasis)isnormallyusedindesign．Again，thefactorofsafetydeterminedbythismethodisanunderestimatebuttheerrorisunlikelytoexceed7％andinmostcasesislessthan2％．

Spencer[9.8]proposedamethodofanalysisinwhichtheresultantIntersliceforcesareparallelandinwhichbothforceandmomentequilibriumaresatisfied．SpencershowedthattheaccuracyoftheBishopsimplifiedmethod，inwhichonlymomentequilibriumissatisfied,isduetotheinsensitivityofthemomentequationtotheslopeoftheintersliceforces．

Dimensionlessstabilitycoefficientsforhomogeneousslopes，basedonEquation9.9，havebeenpublishedbyBishopandMorgenstern[9.2].Itcanbeshownthatforagivenslopeangleandgivensoilpropertiesthefactorofsafetyvarieslinearlywithγuandcanthusbeexpressedas

F=m-nγu(9.10)

where，mandnarethestabilitycoefficients．Thecoefficients，mandnare

functionsofβ,φ’，thedimensionlessnumberc'/γandthedepthfactorD.

Example9.2

UsingtheFelleniusmethodofslices，determinethefactorofsafety，intermsofeffectivestress，oftheslopeshowninFig.9.6forthegivenfailuresurface．Theunitweightofthesoil，bothaboveandbelowthewatertable，is20kN／m3andtherelevantshearstrengthparametersarec’=10kN/m2andφ’=29°.

ThefactorofsafetyisgivenbyEquation9.4.Thesoilmassisdividedintoslicesl.5mwide.Theweight(W)ofeachsliceisgivenby

W=γbh=20×1.5×h=30hkN／m

Theheighthforeachsliceissetoffbelowthecentreofthebaseandthe

normalandtangentialcomponentshcosαandhsinα

Wcosα=30hcosα

Wsinα=30hsinα

Theporewaterpressureatthecentreofthebaseofeachsliceistakentobeγwzw，wherezwistheverticaldistanceofthecentrepointbelowthewatertable(asshowninfigure)．Thisprocedureslightlyoverestimatestheporewaterpressurewhichstrictlyshouldbe)γwze，wherezeistheverticaldistancebelowthepointofintersectionofthewatertableandtheequipotentialthroughthecentreoftheslicebase．Theerrorinvolvedisonthesafeside．

Thearclength(La)iscalculatedas14.35mm．Theresultsaregivenin

Table9.1

∑Wcosα=30×17.50=525kN／m

∑Wsinα=30×8.45=254kN／m

∑(wcosα-ul)=525—132=393kN／m

4AnalysisofaPlaneTranslationalSlip

Itisassumedthatthepotentialfailuresurfaceisparalleltothesurfaceoftheslopeandisatadepththatissmallcomparedwiththelengthoftheslope.Theslopecanthenbeconsideredasbeingofinfinitelength，withendeffectsbeingignored．Theslopeisinclinedatangleβtothehorizontalandthedepthofthefailureplaneisz．asshowninsectioninFig.9.7.Thewatertableistakentobeparalleltotheslopeataheightofmz(0<m<1)abovethefailureplane．Steadyseepageisassumedtobetakingplaceinadirectionparalleltotheslope．Theforcesonthesidesofanyverticalsliceareequalandoppositeandthestressconditionsarethesameateverypointonthefailureplane．

Intermsofeffectivestress，theshearstrengthofthesoilalongthefailureplaneis

andthefactorofsafetyis

Theexpressionsforσ,τandμare:

Thefollowingspecialcasesareofinterest．Ifc’=0andm=0(i.e.thesoilbetweenthesurfaceandthefailureplaneisnotfullysaturated),then

(9.11)

Ifc’=00anddm=11(i.ee.thhewaatertabllecooinciideswithhtheesurrfaceeofthesloppe)，then：

(99.12))

Itshouuldbbenootedthattwheenc’=0ttheffactooroffsaffetyisiindeppendeentoof

thedeppthzz．Ifcc’isgreaaterthannzerro，thefacttoroofsaafetyyisafuunctiionoofz,,anddβmayexceeedφ’provvideddziisleesstthanacrriticcalvvaluee．

Forattotallstrressanallysisstheesheearsstrenngthparaameteersccuandφuareeuseedwiithaazerrovaalueofuu.

Examplee9.33

Alongnatuuralsloppeinnaffissuuredoverrconssoliddateddclaayissincclineedatt12°°tothhehoorizoontall．Thewateertaableisaatthhesuurfacceanndseeepaggeissrouughlyyparralleeltootheesloope．Aslliphhasddevellopeddonapllaneparaallelltothesurffaceataadeppthoof5m．Thesatuurateedunnitwweighhtofftheeclaayiss20kN／m3．Thepeakkstrrengtthpaarameeterssareec’=10kN/mm2andφ’=26°；theresiiduallstrrengtthpaarameeterssareecr’=0aandφr’=18°.Dettermiinettheffactooroffsaffetyalonngthhesllippplanee(a)iinteermsofttheppeakstreengthhparrametters(b)iinteermsofttherresiddualstreengthhparrametters．

Withthhewaatertablleatttheesurrfacee(m=11)，ataanyppointtontheslippplaane，

Usingttheppeakstreengthhparrametters，

Thenthhefaactorrofsafeetyiisgiivenby

Usingttherresiddualstreengthhparrametters，thefacttoroofsaafetyycannbe

obtaineedfrromEEquattion9.122:

GeneraalMeethoddsoffAnaalysiis

MorgenssternnanddPriice[99.4]ddevellopeddaggenerralaanalyysisinwwhichhalllbouundarryanndeqquiliibriuumcoondittionssareesattisfiiedaandiinwhhichthefailluresurffacemaybeaanysshapee，circcularr，non--circcularrorcomppoundd．ThesoillmasssabbovethefaillureplanneissdivvideddinttoseectioonsbbyanumbberoofveerticcalplannesaandtthepprobllemiisreenderredsstatiicalllydeetermminattebyyasssuminngarelaationnshippbettweenntheeforrcesEanndXontthevvertiicalbounndariiesbbetweeeneeachsecttion．Thissasssumpttionisoofthhefoorm

X=λf(x))E(99.13))

whereff(x)iisannarbbitraaryffuncttiondesccribiingttheppatteerniinwhhichtheratiioX//Evaariessacrrossthesoillmasssanndλisaascaaleffactoor．ThevaluueoffλisoobtaiinedasppartoftthessoluttionalonngwiithttheffactooroffsaffetyF．ThevaluuesoofthhefoorcessEaandXXanddtheepoiintoofapppliccatioonoffEccanbbedeetermmineddateachhverrticaalbooundaary．Foranyassuumedfuncctionnf(xx)ittisneceessarrytooexaamineetheesollutiooninndettailtoeensurrethhatiitissphyysicaallyreassonabble((i.e..nosheaarfaailurreorrtennsionnmusstbeeimpplieddwitthinthesoillmasssabbovethefailluresurfface)).Thhechhoiceeofthefuncctionnf(xx)dooesnnotaappeaartooinffluenncettheccompuutedvaluueoffFbbymooretthanabouut5%%anddf(xx)=lisaacommmonassuumptiion．

TheeanaalysiisinnvolvvesaacommplexxproocesssofiterratioonfoorthhevaaluessofλanddF，desccribeedbyyMorrgenssternnanddPriice[99.5]，andtheuseofaacommputeerissesssentiial.

Bell[99.1]propposeddammethoodoffanaalysiisinnwhiichaallttheccondiitionnsoffequuilibbriummareesattisfiiedaandttheaassummedffailuuressurfaacemmaybbeoffanyyshaape．Thesoillmasssissdivvideddinttoanumbberoofveerticcalssliceesanndsttaticcalddeterrminaacyiisobttaineedbyymeaansoofannasssumedddisstribbutioonoffnorrmalstreessaalonggtheefaiilureesurrfacee．

Sarma[[9.6]]devveloppedaametthod，baseedonntheemetthodofsslicees，inwhhichthecritticallearrthquuakeacceeleraationnreqquireedtooprooduceeaccondiitionnoflimiitinggequuilibbriummisdeteerminned．Anaassummedddistrributtionofvvertiicalinteerslliceforccesiisussediinthheannalyssis．Agaiin，allthecondditi