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Zhuetal.Introduction

Anchorsandsoilnailsarecommonlyusedtostabilizepo-tentiallyunstableslopes.Theanchorloadsnotonlydirectlyprovidetheforcesand(or)momentscounteractingthoseforcestendingtodestabilizetheslopebutalsoimprovetheshearresistancealongtheslipsurfacebyincreasingthenor-malstressonthatsurface(HobstandZajic1983;Bromhead1994).Evaluationofslopestability,includingtheanchorloads,isimportantforthedesignofstabilizationmeasuresinvolvinganchors.

Limitequilibriummethodsofsliceshavebeenwidelyusedforcalculatingfactorsofsafetyfornaturalandconstructedslopes(Duncan1996).ThecommonlyusedmethodsincludethoseproposedbyFellenius(1936),Bishop(1955),Morgen-sternandPrice(1965),Spencer(1967),andJanbu(1973).Inprinciple,alltheseconventionalmethodsofslicescouldac-commodateanchorloadsorothertypesofconcentratedforcesactingupontheslope.Themoststraightforwardtreat-mentofconcentratedforcesistoincludethemasexternalforcesactingoncorrespondingslices(Hutchinson1977;FredlundandKrahn1977;Zhuetal.2001).However,suchatreatmentwillleadtoanunreasonablyabruptincreaseinnormalstressonthebaseoftheassociatedslices(Krahn2003).Thismeansthatthecontributionofanchorloadstotheincreaseinshearresistanceissolelyrelatedtotheshearstrengthofthatassociatedsegmentontheslipsurface,aswillbeshownlaterinthispaper.Thisisevidentlyunreason-ablefromboththeoreticalandpracticalpointsofview,asthenormalstressesontheslipsurfaceinducedbyananchorwouldnotbeconcentratedonanarrowsegment.Thus,ques-tionsareraisedonthereasonablenessofdirectlyusingcon-ventionalmethodsofslicesforanalysingthestabilityofanchor-reinforcedslopes.

Notwithstandingtheabovelimitation,themethodsofslicesaregenerallyacceptedasareliableanalyticaltoolforslopestability,astheyhavebeenfoundtogiveapproxi-matelyequalfactorsofsafety(within15%tolerance)aslongastheysatisfythecompleteequilibriumconditionsforthewholeslidingbody.Thecommonlyusedrigorousmethodsofslicesgenerallyassumecontinuous(andoftenrathersmooth)distributionoftheinclinationsofintersliceforces(MorgensternandPrice1965;Spencer1967)orcontinuouslocationofthelineofthrustacrosstheslidingmass(Janbu1973),therebyresultingincontinuousdistributionofnormalstressesalongtheslipsurface.Suchassumptionsofcontinu-ityapproximatelyreflecttherealcharacteristicsofthoseslopessubjecttogravity,pore-waterpressures,andseismicforces.However,whentheslopeisacteduponbyaconcen-tratedloadatthegroundsurface,boththeinclinations(andthemagnitude)oftheintersliceforcesandthelocationofthelineofthrustarenolongercontinuousacrosstheslidingbody,butthenormalstressdistributionalongtheslipsurfaceshouldstillremaincontinuous.Thus,iftheconventionalas-sumptionsaremadeinthiscase,theresultantcharacteristicsoftheintersliceforcesandthenormalstressesontheslipsurfacewouldbereversedandcontrarytoreality.Toover-comethisinherentshortcomingoftheconventionalmethods,weproposeanalternativebasedontheassumptionofcon-tinuousnormalstressdistributionalongtheslipsurface.Be-foretheapplicationofanchorloads,thenormalstressesontheslipsurfaceareassumedtobethosecalculatedbythe

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conventionalrigorousmethodsofslices(e.g.,theMorgenstern–PricemethodortheSpencermethod).Thenormalstressesinducedbytheanchorloadareapproximatelyobtainedfromanelasticsolution.Thelinearcombinationofthesetwopartsconstitutesthedistributionofnormalstressesontheslipsur-faceoftheanchor-reinforcedslope.Solvingthecompleteequilibriumequationsfortheslidingbodyyieldsthefactorofsafetyfortheslopewithgivenanchorloadsorthemagni-tudeoftheanchorloadrequiredtostabilizetheslopewithaspecifiedvalueforthefactorofsafety.

Basicformulation

Atypicalslopewithanchorloads(λpP3,λpP2,withλastheloadfactor)isshowninFig.1a.Forgeneralpurposes,ptheslipsurfaceisofarbitraryshape.Inadditiontothean-chorloads,theslopebodyissubjecttoself-weight(γ),hori-zontalseismicforce(kshowninthefigure).Withoutcγ)andthepore-wateractionofanchorpressureloads,u(notthefactorofsafetycanbecalculatedbyusinganymethodofslicesaccommodatingthegeneral-shapedslipsurface.TheMorgenstern–Pricemethod(MorgensternandPrice1965),withanintersliceforceofconstantinclination,issuggestedforthispurpose.Thedistributionofnormalstresses(σtermsoftotalstress)canbeobtainedasaby-productof0,inthecomputationprocess.

Inresponsetotheactionofanchorloads,anadditionalnormalstressdistribution(λpσp)isinducedalongtheslipsurface.Considerasingleanchorload,P,actingatpoint(xp,yp)ontheslopeatanangleofitothehorizontal,asshowninFig.1b.Theinducednormalstressontheslipsur-faceisdenotedbyσp.Becausetheanalysisiswithintheframeworkoflimitequilibrium,thenormalstressontheslipsurfaceisnotrequiredtobetheoreticallyexact.Thus,forpracticalpurposes,σpisassumedtobetheelasticstressas-sociatedwithaninfinitewedgewithitstwoedgesconnect-ingthepointofactionofPandthetwoendsoftheslipsurface.Fortunately,theanalyticalsolutiontoσpisavailablefromthemechanicsofelasticity.AsshowninFig.2,apairofforces,PH(horizontal)andPinfinitewedgewithitssymmetricalV(vertical)actattheapexofanaxisinthehorizontaldirectionanditsedgeslyingatangleofβtothehorizontal.Accordingtothemechanicsofelasticity(TimoshenkoandGoodier1970),thestressesatapointwithpolarcoordinates(r,θ)inthewedgeare[1a]σPθPr=

HcosVsinθr(β+0.5sin2β)

+

r(β 0.5sin2β)

[1b]σθ=0[1c]

τrθ=0

whereσristheradialstress;σθisthecircumferentialstress;andτrθistheshearstress.

NowconsiderthecorrespondingwedgeshowninFig.1b,βwithitslowerandupperedgesextendingatanglesofβhorizontal,respectively.Theconcentratedforce1and2tothePliesatanangleofωwiththesymmetricalaxisMM′.Thepolarcoordinatesofthepointconsideredare(r,θ′),corre-spondingto(r,θ)inthecoordinatesysteminFig.2.Fromthegeometricalrelation,wecanseethat