英语翻译原文.docx

英语翻译原文.docx

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英语翻译原文

StabilityofSlopes

1Introduction

Gravitationalandseepageforcestendtocauseinstabilityinnaturalslopes,inslopesformedbyexcavationandintheslopesofembankmentsandearthdams.ThemostimportanttypesofslopefailureareillustratedinFig.9.1.Inrotationalslipstheshapeofthefailuresurfaceinsectionmaybeacirculararcoranon-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

3TheMethodofSlices

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

Theheighthforeachsliceiss