英语翻译原文.docx
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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
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