Evaluation of the stability of anchor-reinforced slopes(预应力锚杆边坡稳定性评价)

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Evaluationofthestabilityofanchor-reinforcedslopes

D.Y.Zhu,C.F.Lee,D.H.Chan,andH.D.Jiang

Abstract:Theconventionalmethodsofslicesarecommonlyusedfortheanalysisofslopestability.Whenanchorloadsareinvolved,theyareoftentreatedaspointloads,whichmayleadtoabruptchangesinthenormalstressdistri-butiononthepotentialslipsurface.Assuchabruptchangesarenotreasonableanddonotreflectrealityinthefield,analternativeapproachbasedonthelimitequilibriumprincipleisproposedfortheevaluationofthestabilityof

anchor-reinforcedslopes.Withthisapproach,thenormalstressdistributionovertheslipsurfacebeforetheapplicationoftheanchor(i.e.,σ0)iscomputedbytheconventional,rigorousmethodsofslices,andthenormalstressontheslipsurfacepurelyinducedbytheanchorload(i.e.,λpσp,whereλpistheloadfactor)istakenastheanalyticalelasticstressdistributioninaninfinitewedgeapproximatingtheslopegeometry,withtheanchorloadactingontheapex.Thenthenormalstressontheslipsurfacefortheanchor-reinforcedslopeisassumedtobethelinearcombinationofthesetwonormalstressesinvolvingtwoauxiliaryunknowns,η1andη2;thatis,σ=η1σ0+η2λpσp.Simultaneouslysolvingthehorizontalforce,theverticalforce,andthemomentequilibriumequationsfortheslidingbodyleadstotheexplicitexpressionforthefactorofsafety(Fs)—ortheloadfactor(λp),iftherequiredfactorofsafetyisprescribed.Thereasonablenessandadvantagesofthepresentmethodincomparisonwiththeconventionalproceduresaredemon-stratedwithtwoillustrativeexamples.Theproposedprocedurecanbereadilyappliedtodesignsofexcavatedslopesorremediationoflandslideswithsteelanchorsorprestressedcables,aswellaswithsoilnailsorgeotextilereinforce-ments.

Keywords:slopes,factorofsafety,anchors,limitequilibriummethod.

Résumé:Lesméthodesconventionnellesdestranchessonthabituellementutiliséespourl’analysedelastabilitédesta-lus.Lorsquedeschargesd’ancragesontimpliquées,ellessontsouventtraitéescommedeschargesponctuelles,cequipeutconduireàmedetelschangementsabruptesnesontpasraisonnablesetnereflètentpaslaréalitésurleterrain,onproposeuneapprochealternativebaséesurleprinciped’équilibrelimitepourl’évaluationdelastabilitédestalusarméspardesancrages.Aveccetteapproche,ladistributiondelacontraintenormalesurlasurfacedeglissementavantl’applicationdel’ancrage,i.e.,σ0,estcalculéeparlesméthodesconventionnellesrigoureusesdestranches,alorsquelacontraintenormalesurlasurfacedeglissementpurementinduiteparlacharged’ancrage,i.e.,λpσp(λpétantlefacteurdecharge),estprisecommeladistributiondelacontrainteanalytiqueélastiqueenuncoininfiniquireprésenteap-proximativementlagéométriedelapenteaveclacharged’ancrageagissantsurlesommet.Alorsonsupposequelacontraintenormalesurlasurfacedeglissementpourletalusarméd’ancragesestlacombinaisonlinéairedecesdeuxcontraintesnormalesimpliquantdeuxinconnuesη1etη2,c’est-à-dire,σ=η1σ0+η2λpσsolutionsimultanéedeséquationsdelaforcehorizontale,delaforceverticaleetdumomentd’équilibrepourlecorpsenmouvementconduitàl’expressionexplicitepourlecoefficientdesécuritéFsoupourlefacteurdechargeλpsilecoefficientdesécuritére-quisestprescrit.Lecaractèreraisonnableetl’avantagedelaprésenteméthodeencomparaisonaveclesprocéduresconventionnellessontdémontréprocédureproposéepeutêtreappliquéeaisémentauxconceptionsdepentesexcavéesoudecomportementdeglissementsavecdesancragesd’acieroudescâ-blesprécontraints,demêmequ’avecdesclousdanslesoloudesarmaturesgéotechniques.Motsclés:talus,coefficientdesécurité,ancrages,méthoded’équilibrelimite.[TraduitparlaRédaction]

Received18August2004.Accepted16May2005.PublishedontheNRCResearchPressWebsiteathttp://cgj.nrc.caon4October2005.

D.Y.Zhu.1InstituteofMountainHazardsandEnvironment,ChineseAcademyofSciences,Chengdu610015,China.C.F.Lee.2DepartmentofCivilEngineering,UniversityofHongKong,PokfulamRoad,HongKong.

D.H.Chan.DepartmentofCivilandEnvironmentalEngineering,UniversityofAlberta,Edmonton,ABT6G2W2,Canada.H.D.Jiang.CollegeofCivilEngineering,HohaiUniversity,Nanjing210098,China.

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Presentaddress:CollegeofCivilEngineering,ChinaThreeGorgesUniversity,Yichan443000,ChinaCorrespondingauthor(e-mail:leecf@hkucc.hku.hk).

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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

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Fig.1.Diagramofananchor-reinforcedslope.(a)Slopewithnormalstressesontheslipsurfaceinducedbyself-weightandanchorloads,respectively.(b)Geometryforcomputingnormalstressesontheslipsurfacepurelyinducedbyananchor

load.

[2a]

β=

β1+β2

2

[2b]

θ=θ′+β2

β1+β2β β2

2=θ′ 1

2[2c]ω=θ′ θ i=

β1 β2

2

iThus[3a]

PH=Pcosω=Pcos β β

2 i

[3b]

PV= Psinω= Psin β 1 β22

i

Substitutingeqs.[3a]and[3b]intoeq.[1a]gives

Can.Geotech.J.Vol.42,

2005

Fig.2.Stressdistributioninawedgewithconcentratedforcesatitsapex.

σ β1 β2 i cos θ′ β1 β2

[4]

r=

2Pcosr 2 2

β1+β2+sin(β1+β2)

sin β β2 β β2

12i sin θ′ 12

β 1+β2 sin(β1+β2)

Thecircumferentialandshearstressesarestillzero.

Thenormalstressσpontheslipsurfacewithaninclina-tionofαtothehorizontalisobtainedfromstaticanalysisas[5]

σp=σrsin2(θ′+α)

Ifmorethanoneanchorloadisactingontheslope,σtheirindividualcontributions.

pistakenasthesumofUsually,theprestressingofananchorisaccomplishedoverashortduration,andsomecohesivesoilsare,tosomedegree,inanundrainedcondition.Thiswillleadtoachangeinpore-waterpressure( u)withintheslidingmass.Accord-ingtoSkempton(1954), uisrelatedtochangesintheprin-cipalstressesinthesoilbythefollowingrelationship:[6]

u=B[ σ3+A( σ1– σ3)]

where σAandBareporepressureparameters;and σmajorandminorprincipalstresses1and3arechangesinre-spectively.

TheporepressureparametersAandBcandeterminedbylaboratorytests.Forsaturatedsoils,Bapproachesunity.ThevalueofAvarieswiththedegreeofoverconsolidationofthesoil,beingpositivefornormallyconsolidatedsoils(intherangeof0.5–1.0),andincontrast,beingnegativeforheavilyconsolidatedsoils(intherangeof–0.5to0.0).

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Zhuetal.Asisshownineqs.[1a]–[1c],ifonlyoneanchorloadacts,thechangesinmajorandminorprincipalstresseswouldbe[7a] σ1=λpσr[7b] σ3=0

Thus[8]

u=λpσrwhere=BA.

However,iftwoormoreanchorloadsact,thesoilintheslopewillnolongerbeunderuniaxialstress.Becauseweareattemptingtoonlyapproximatelyevaluatetheeffectofthedegreeofdrainageonslopestability, σthesimplealgebraicsumofσ1ishereinassumedtobercausedbyindividualan-chorloads.

Beforetheapplicationofanchorloads,wecancalculatethefactorofsafetyfortheslopewithexistingmethodsofslicesandobtainthenormalstressontheslipsurfaceσ0.Af-tertheanchorloadsareapplied,thefactorofsafetyfortheslopewouldchangeandwouldneedtoberecalculated.Ratherthanusingtheconventionalmethodsofslices,whichmakeassumptionsabouttheintersliceforces,weapplytheprincipleofthenewlyproposedprocedure(Zhuetal.2003)bymodifyingthenormalstressontheslipsurfaceandusingittocomputethestabilityoftheanchor-reinforcedslopes.Becausetherearethreeequilibriumconditionsforthewholeslidingbody,andoneunknown(i.e.,thefactorofsafety,Fs)istobedetermined,wecanassumenormalstress(σ)ontheslipsurface,withtwoauxiliaryunknowns.Natu-rally,theσnormalstress(σ)iscontributedbytwoparts:σandλweassume0thatpp.Torendertheproblemdeterminate,[9]

σ=η1σ0+η2λpσp

whereη1andη2aretheauxiliaryunknowns.

Aconstantfactorofsafety(Fs)isassignedtothewholeslipsurface.TheshearresistancealongtheslipsurfaceisdeterminedbytheMohr–Coulombfailurecriterionandtheprincipleofeffectivestress:[10]

τ=

1

F[(σ u u)tanφ′+c′]s

whereφ′andc′aretheeffectiveinternalfrictionangleand

cohesion,respectively.

Forsimplicity,supposethat[11]

ψ=tanφ′;c=c′

Fromeqs.[10]and[11],itfollowsthat[12]

τ=

1F[(σ u)ψ+c] 1λp σ1ψsFs

Fromthehorizontalandverticalforceequilibriumandthemomentequilibriumwithrespecttoanarbitrarilyspecifiedpoint(xc,yc),oneobtains[13a]

∫b

a( σs′+τ kcw)dx= λp∑Px

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[13b]∫b

a(σ+τs′ w)dx=

λp∑Py

[13c]

∫b

a( σs′+τ)(yc s)+(σ+τs′ w)(x xc)

kcw(yc 0.5∑s 0.5g)]dx

=λpPx(yp yc)+λp∑Py(xp xc)

wherePx(positivetotheright)andPverticalcomponentsy(positivedownwards)arehorizontalandofanchorloadP(thesuffixidentificationisomittedforsimplicity);s(x)andg(x)denotethecurvesoftheslipsurfaceandtheground,re-spectively;w(x)denotestheself-weightofasliceofunitwidth;ands′(x)istheinclinationoftheslipsurface(i.e.,s′=tanα).

Assumingthat[14a]Fb

x=∫akcwdx

[14b]

Fb

y=∫awdx

[14c]Mbc=∫a

[kcw(yc 0.5s 0.5g)+w(x xc)]dx[14d]

∑Mp=∑Px(yp yc)+∑Py(xp xc)

[14e]rσ(x)= s′(yc s)+x xc[14f]

rτ(x)=yc s+s′(x xc)

andconsideringeqs.[9]and[12],eqs.[13a]–[13c]arere-writtenas[15a]

∫b

a(η1σ0+η 2λpσp) s′+ψ1

F dxs

=Fx λp∑Px+

1b

F(uψ c)dxs∫a

+λp

b

F σ1ψdxs

∫a[15b]

∫b

a(η1σ0+η2λpσp) 1

1+s′ψF

dxs=Fy+λp∑Py+

1b

Fs′(uψ c)dxs∫a

+λpb

Fs

∫as′ σ1ψdx

[15c]

∫b

a(η1σ0+η2λ pσp) 1

rσ+rτψF

dxs

=Mc+λp∑Mp+

1b

Frτ(uψ c)dxs∫a

+λpb

Frτ σ1ψdx

s

∫aSolvingeqs.[15a]–[15c]simultaneouslywillyieldsolutionstothefactorofsafety(Fs)—ortheloadfactor(λp)ifFauxiliaryunknowns(λandλsisprescribed—andthe12)aswell.

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1346Solutiontothefactorofsafety

Ifthemagnitudeoftheanchorloadsisgiven,thesolutiontothefactorofsafetyofthereinforcedslopeisderivedinthissection.Assuming[16]

ωc=uψ c+λp σ1ψ

eqs.[15a]–[15c]arerewrittenas

[17a]ηb 1∫aσ0

s′+ψ1 b1

F dx+η2s ∫aλpσp s′+ψF dx

s =Fx λp∑Px+

1∫b

Fa

ωcdxs[17b]ηb 1∫aσ0

1+s′ψ1 F dx+η∫bλ 2pσp 1

1+s′ψF dx

s as =Fy+λp∑Py+

1b

Fa

s′ωcdxs∫[17c]

Fs=

ηbbb

1aσ0ψrτdx+η2a

λpσpψrτdx a

rτωcdx

η1∫a

σ0rσdx η2∫a

λpσprσdx+Mc+λp∑Mp

Theaboveequationsarerearrangedas

[18a]η 1 A1+1FA′ 1 +η2 A2+

1A′ 2 =A3+1

A′3s Fs Fs[18b]η 1 B1+1FB′ 1 +η 2 B2+1 1

s FB′2 =B3+

B′3s Fs[18c]

FDs=η+Dη+DE1η1+E2η2+E3

inwhich[19a]Ab

b

1= ∫as′σ0dx;

A′1=∫aψσ0dx

[19b]Ab

b

2= ∫a

s′λpσpdx;

A′2=∫a

ψλpσpdx

[19c]

A3=Fx λp∑Px;

A′b

3=∫a

ωcdx

[19d]Bb1=∫a

σ0dx;B′b

1=∫a

s′ψσ0dx

[19e]Bb

b

2=∫aλpσpdx;

B′2=∫a

s′ψλpσpdx

[19f]

B3=Fy+λp∑Py;

B′b

3=∫a

s′ωcdx

[19g]Dbb

1=∫a

σ0ψrτdx;D2=∫a

λpσpψrτdx;

Db3= ∫a

rτωcdx

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[19h]

Eb

b

1= ∫a

σ0rσdx;

E2= ∫a

λpσprσdx;

E3=Mc+λp∑Mp

Equations[18a]–[18c]canbeanalyticallyresolved,resultinginanexplicitsolutiontothefactorofsafety(Fs)asfollows:

[20]

Fts=

3

+

+

wherep,q,andtcanbecomputedwiththeparameters

shownineqs.[19a]–[19h].Thebriefderivationofeq.[20]ispresentedinAppendixA;fordetails,seeZhuetal.(2003).

Solutiontorequiredanchorloads

Inthedesignofmeasuresforstabilizingfailedslopesorslopeshavingunacceptablestabilityconditions,themagni-tudeoftherequiredanchorloadsisoftenneeded.Inthiscase,themagnitudeoftherequiredanchorloadscanbecal-culatedbytrialanderrorusingeq.[20]untiltheslopeat-tainsthespecifiedfactorofsafety.Itcanalsobedirectlycomputedusinganotherexplicitexpression,thederivationofwhichisgivenbelow.Assuming[21a]ωx= s′+ψ

1F;ωy=1+s′ψ

1s

F;s

ωr=rσ+rτψ

1Fs

[21b]ωu=

1

F(uψ c);ωb=

1

F σ1ψs

s

eqs.[15a]–[15c]arewritteninmatrixformas

a12

a13 [22]

a11

η1 a21a22a23

λpη2 c1

= a31

a32

a 33

λ c2 p c3

inwhich

[23a]a11=∫b

b

aσ0ωxdx,

a12=∫a

σpωxdx,

ab

13= ∫a

ωbdx+∑Px

[23b]ab

b

21=∫a

σ0ωydx,

a22=∫a

σpωydx,

ab

23= ∫a

s′ωbdx ∑Py

[23c]ab

31=∫a

σ0ωrdx,

ab

32=∫a

σpωrdx,

a33= ∫b

a

rcωbdx ∑Mp

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Zhuetal.

[23d]cb

b

1=Fx+∫aωudx,

c2=Fy+∫a

s′ωudx,

c3=Mc+∫b

a

rcωudx

Thesolutiontoeq.[22]followstheCramerrule,with[24a]λp= 3 [24b]η1= 1 [24c]η2= 2 3

inwhich

a11

a12a13[25a] =a21

a22a23a31a32a33c1

a12a13[25b] 1=c2

a22a23c3a32a33a11

c1a13[25c] 2=a21

c2a23a31c3a33a11

a12c1[25d] 3=a21

a22c2a31

a32

c3

Illustrativeexamples

Example1

Aslopewithaheightof15mandaninclinationof45°isshowninFig.3a.Theslopemassconsistsoftwotypesofsoils,whoseparametersarepresentedinFig.3a.Theanchoristobeappliedatthehalfheightoftheslopewithanincli-nationof30°tothehorizontal.

Beforetheanchorisapplied,thefactorofsafetyforthisslopeis0.998,calculatedwiththeSpencermethod.Whenananchorloadof300kNperunitlengthisappliedtotheslopeandadrainedconditionisassumed(i.e.,=0),thefactorofsafetyoftheslopeisincreasedto1.286inthepresentap-proach.ThenormalstressdistributionovertheslipsurfaceaftertheapplicationoftheanchorloadisshowninFig.3b.Itcanbeseenthatundertheactionoftheanchorload,thenormalstressontheslipsurfaceiscontinuousandfairlysmoothinshape,withamaximumvalueof103kPaoccur-ringincloseproximitytothepointofactionoftheanchorload.Ifaminimumfactorofsafetyisrequiredfortheslope,thentheminimumanchorloadcanbedirectlycomputedbyusingeq.[24a]withavalueof485kN/m.

Forcomparisonpurposes,theSpencermethod,withcon-ventionaltreatmentofanchorloads,isalsousedinthisex-ample,andthecorrespondingresultsareshowninFig.3a.Inthiscase,thefactorofsafetyfortheslopewiththeanchor

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Fig.3.Slopeprofileandnormalstressesontheslipsurfaceforexample1.(a)Slopeprofileandsoilparameters.(b)Normalstressesonslipsurfacecomputedbyconventionalandpresentmethods.GWL,groundwaterlevel.

loadof300kN/mis1.357,whichis6%largerthanthatpro-videdintheabovesolution.Fromthepracticalpointofview,suchadifferenceisrathersmall.Theassociatednor-malstressdistributionontheslipsurfaceisalsoshowninFig.3b.Itcanbeseenthatthenormalstressontheslipsur-faceincreasesabruptlyatthepointimmediatelyunderthepointofactionoftheanchorload.Thisisquiteunreasonablefromthestaticpointofview,andthusonecannotensurethattheconventionalprocedureisvalidforanchorloadsinallcases(Krahn2003).

Example2

Theslopeprofileofanotherexampleandthesoilparame-tersareshowninFig.4.Threeanchorsaretobeappliedtostabilizethisslope.Foraslopewithoutapredefinedfailuresurface,thestabilizationmeasureshouldensurethatallpo-tentialslipsurfaceshavefactorsofsafetygreaterthanaspecifiedvalue,say1.2forthisexample.Alllocalcriticalslipsurfaceswithfactorsofsafetyof<1.2arelocatedbyus-ingthecriticalslipfieldmethod(Zhu2001).Atotalof11criticalslipsurfacesareplottedinFig.4.Thevaluesoffac-torsofsafety(Floadss0)correspondingtotheseslipsurfaceswith-outanchorarepresentedinthesecondcolumnofTable1.Toevaluatetheeffectonslopestabilityofpossibleexcesspore-waterpressureinducedbyabruptapplicationoftheanchorload,weassumethattheporepressureparameter()variesbetween0and1.0.

Thefactorsofsafetywithanchorloads(P1000kN/m)andtheloadfactorsrequiredby1=Pthe2=Pspecified3=factorofsafetyof1.2arepresentedinTable1for=0.00,0.25,0.50,0.75,and1.00.ItcanbeseenfromTable1that

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Fig.4.Slopeprofileandsoilparametersforexample2.

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2005

Table1.Valuesoffactorsofsafetyandrequiredloadfactors.

themostcriticalslipsurfaceisashallowsurface(No.11)passingthroughthetoeoftheslope.However,inthecaseof=0.00(i.e.,drainedcondition)forthisshallowslipsur-face,theincreasedfactorofsafetyisthelargest,andthere-quiredloadfactoristheleast.Inotherwords,thismostcriticalslipsurfacewithoutanchorloadistheleastcriticalaftertheapplicationofanchorloads.Iftheslopeistomeettheprescribedstabilityconditions,theanchorloadsshouldbedesignedwithdueconsiderationtothesecondmostcriti-calslipsurface(No.7),whichpassesbelowthetoeoftheslope:itisassociatedwiththelowestfactorofsafetyforthegivenanchorloads,anditalsorequiresthelargestanchorloadstoattainthespecifiedfactorofsafety.ItisevidentfromTable1thatwithanincreaseinporepressureparame-ters(thefactorofsafetydecreasesandtherequiredloadfactorincreases.Itshouldbenotedthatthelocationsandin-clinationsoftheanchorsshowninFig.3areselectedonlyforthepurposesofillustration.Inpracticalapplication,itisrecommendedthatanoptimizationprocessbeperformedtodetermineanoptimumcombinationofanchors.Theproce-dureproposedherewouldserveasausefultoolforthispur-pose.

ofconcentratedforces.Althoughtheextensionoftheconventionalmethodstoincludeanchorloadsisstraightfor-ward,anunreasonablenormalstressdistributionontheslipsurfacewouldariseasaresult.Analternativeprocedureisproposedinthispaperforamorerationalanalysisofanchor-reinforcedslopes.Withthisprocedure,thenormalstressontheslipsurfaceisassumedtobealinearcombina-tionoftwopartsinvolvingtwoauxiliaryunknowns:onepartcorrespondstotheunreinforcedslopeobtainedusingcon-ventionalmethods;theotherpartisinducedsolelybytheanchorloads,withanapproximateclosed-formsolution.Solvingthethreeequilibriumequationsyieldsexplicitsolu-tionstothefactorofsafetywithgivenanchorloadsandtotherequiredanchorloadswithaspecifiedfactorofsafety.Thedisadvantagesofconventionalproceduresindealingwithanchorloadscanthusbeovercome.Thismethodcanserveasapromisingtoolforthedesignofstabilizationmea-suresinvolvinganchorsorsoilnailandgeotextilereinforce-mentsforfailedslopesandforthosehavingunacceptablestabilityconditions.

Acknowledgements

ThestudywasfinanciallysupportedbytheResearchGrantsCouncilofHongKongandtheJockeyClubResearchandInformationCenterforLandslipPreventionandLandDevelopment,UniversityofHongKong.Theirsupportis

Conclusions

Thelimitequilibriummethodsofsliceshavebeenwidelyusedforanalysingthestabilityofslopeswithouttheaction

预应力锚杆边坡稳定性评价 原文 另附译文 请自行下载

Zhuetal.gratefullyacknowledged.PartofthisworkwascarriedoutattheUniversityofAlberta,Canada.Theauthorsaregrate-fultoProf.N.R.Morgensternforgivinginvaluableguidanceonthisstudy.DuringD.Y.Zhu’sstudyleaveinCanada,supportedbytheUniversityofHongKong,Dr.J.H.Chenprovidedusefulhelp,aswellasdiscussionsonthework.TherevisionofthisworkwaspartiallysupportedbytheNa-tionalNaturalScienceFoundationofChina(grantNo.40472138)andtheLaboratoryofHazardPreventionandMitigationatChinaThreeGorgesUniversity.

References

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1349

AppendixA.Solutionforthefactorofsafety

Solvingeqs.[18a]and[18b]forη1andη2,oneobtainsT0+

1

T1+1

[A1a]ηFF2T21=

s

G0+sFG1+

s

FG2sS0+

1

S1+1[A1b]ηFFS22=

G0+FG1+

s

F2G2s

where

[A2a]T0=A3B2 A2B3;

T1=A3B′2+A′3B2 A2B′3 A′2B3;

T2=A′3B′2 A′2B′3

[A2b]S0=A1B3 A3B1;

S1=A1B′3+A′1B3 A3B′1 A′3B1;

S2=A′1B′3 A′3B′1

[A2c]G0=A1B2 A2B1;

G1=A1B′2+A′1B2 A2B′1 A′2B1;

G2=A′1B′2 A′2B′1

Substitutingeqs.[A1a]and[A1b]intoeq.[18c]andrear-rangingyieldsacubicfunctionofFs,asfollows:[A3]F3s+t2F2

s+t1Fs+t0=0

where[A4a]tD0= 1T2+D2S2+D3G2E1T0+E2S0+E3G0

[A4b]tE1=1T2+E2S2+E3G2 D1T1 D2S1 D3G1

E1T0+E2S0+E3G0[A4c]tE2=

1T1+E2S1+E3G1 D1T0 D2S0 D3G0

E1T0+E2S0+E3G0

Equation[A3]isrewrittenas: 3

[A5] Fts 2 3 +p Fts 2

3

+q=0where[A6a]p= t223

+t

1[A6b]q=

127t32 1

3t1t2+t0Solvingeq.[A5]givestheexpressionforthefactorofsafety

Fsasineq.[20].

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