地质岩土英文文献翻译.docx

地质岩土英文文献翻译.docx

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地质岩土英文文献翻译

InternationalJournalofRockMechanicsandMiningSciences

Analysisofgeo-structuraldefectsinflexuraltopplingfailure

AbbasMajdiandMehdiAmini

Abstract

Thein-siturockstructuralweaknesses,referredtohereinasgeo-structuraldefects,suchasnaturallyinducedmicro-cracks,areextremelyresponsivetotensilestresses.Flexuraltopplingfailureoccursbytensilestresscausedbythemomentduetotheweightoftheinclinedsuperimposedcantilever-likerockcolumns.Hence,geo-structuraldefectsthatmaynaturallyexistinrockcolumnsaremodeledbyaseriesofcracksinmaximumtensilestressplane.Themagnitudeandlocationofthemaximumtensilestressinrockcolumnswithpotentialflexuraltopplingfailurearedetermined.Then,theminimumfactorofsafetyforrockcolumnsarecomputedbymeansofprinciplesofsolidandfracturemechanics,independently.Next,anewequationisproposedtodeterminethelengthofcriticalcrackinsuchrockcolumns.Ithasbeenshownthatifthelengthofnaturalcrackissmallerthanthelengthofcriticalcrack,thentheresultbasedonsolidmechanicsapproachismoreappropriate;otherwise,theresultobtainedbasedontheprinciplesoffracturemechanicsismoreacceptable.Subsequently,forstabilizationoftheprescribedrockslopes,somenewanalyticalrelationshipsaresuggestedfordeterminationthelengthanddiameteroftherequiredfullygroutedrockbolts.Finally,forquickdesignofrockslopesagainstflexuraltopplingfailure,agraphicalapproachalongwithsomedesigncurvesarepresentedbywhichanadmissibleinclinationofsuchrockslopesandorlengthofallrequiredfullygroutedrockboltsaredetermined.Inaddition,acasestudyhasbeenusedforpracticalverificationoftheproposedapproaches.

KeywordsGeo-structuraldefects,In-siturockstructuralweaknesses,Criticalcracklength

1.Introduction

Rockmassesarenaturalmaterialsformedinthecourseofmillionsofyears.Sinceduringtheirformationandafterwards,theyhavebeensubjectedtohighvariablepressuresbothverticallyandhorizontally,usually,theyarenotcontinuous,andcontainnumerouscracksandfractures.Theexertedpressures,sometimes,producejointsets.Sincethesepressuressometimesmaynotbesufficientlyhightocreateseparatejointsetsinrockmasses,theycanproducemicrojointsandmicro-cracks.However,theresultscannotbeconsideredasindependentjointsets.Althoughtheeffectsofthesemicro-cracksarenotthatpronouncedcomparedwithlargesizejointsets,yettheymaycauseadrasticchangeofin-situgeomechanicalpropertiesofrockmasses.Also,inmanyinstances,duetodissolutionofin-siturockmasses,minutebubble-likecavities,etc.,areproduced,whichcauseaseverereductionofin-situtensilestrength.Therefore,oneshouldnotreplacethisin-situstrengthbythatobtainedinthelaboratory.Ontheotherhand,measuringthein-siturocktensilestrengthduetotheinteractionofcomplexparametersisimpractical.Hence,anappropriateapproachforestimationofthetensilestrengthshouldbesought.Inthispaper,bymeansofprinciplesofsolidandfracturemechanics,anewapproachfordeterminationoftheeffectofgeo-structuraldefectsonflexuraltopplingfailureisproposed.

2.Effectofgeo-structuraldefectsonflexuraltopplingfailure

2.1.Criticalsectionoftheflexuraltopplingfailure

Asmentionedearlier,MajdiandAmini[10]andAminietal.[11]haveprovedthattheaccuratefactorofsafetyisequaltothatcalculatedforaseriesofinclinedrockcolumns,which,byanalogy,isequivalenttothesuperimposedinclinedcantileverbeamsasshowninFig.3.Accordingtotheequationsoflimitequilibrium,themomentMandtheshearingforceVexistinginvariouscross-sectionalareasinthebeamscanbecalculatedasfollows:

（5）

（6）

Sincethesuperimposedinclinedrockcolumnsaresubjectedtouniformlydistributedloadscausedbytheirownweight,hence,themaximumshearingforceandmomentexistattheveryfixedend,thatis,atx=Ψ:

（7）

（8）

IfthemagnitudeofΨfromEq.

（1）issubstitutedintoEqs.（7）and（8）,thenthemagnitudesofshearingforceandthemaximummomentofequivalentbeamforrockslopesarecomputedasfollows:

（9）

（10）

whereCisadimensionlessgeometricalparameterthatisrelatedtotheinclinationsoftherockslope,thetotalfailureplaneandthedipoftherockdiscontinuitiesthatexistinrockmasses,andcanbedeterminedbymeansofcurvesshowninFig.

MmaxandVmaxwillproducethenormal（tensileandcompressive）andtheshearstressesincriticalcross-sectionalarea,respectively.However,thecombinedeffectofthemwillcauserockcolumnstofail.Itiswellunderstoodthattherocksareverysusceptibletotensilestresses,andtheeffectofmaximumshearingforceisalsonegligiblecomparedwiththeeffectoftensilestress.Thus,forthepurposeoftheultimatestability,structuraldefectsreducethecross-sectionalareaofloadbearingcapacityoftherockcolumnsand,consequently,increasethestressconcentrationinneighboringsolidareas.Thus,thein-situtensilestrengthoftherockcolumns,theshearingeffectmightbeneglectedandonlythetensilestresscausedduetomaximumbendingstresscouldbeused.

2.2.Analysisofgeo-structuraldefects

Determinationofthequantitativeeffectofgeo-structuraldefectsinrockmassescanbeinvestigatedonthebasisofthefollowingtwoapproaches.

2.2.1.Solidmechanicsapproach

Inthismethod,whichis,indeed,anoldapproach,theloadsfromtheweakareasareremovedandlikewisewillbetransferredtotheneighboringsolidareas.Therefore,thesolidareasoftherockcolumns,duetooverloadingandhighstressconcentration,willeventuallyencounterwiththeprematurefailure.Inthispaper,foranalysisofthegeo-structuraldefectsinflexuraltopplingfailure,asetofcracksincriticalcross-sectionalareahasbeenmodeledasshowninFig.5.ByemployingEq.（9）andassumingthattheloadsfromweakareasaretransferredtothesolidareaswithhigherloadbearingcapacity（Fig.6）,themaximumstressescouldbecomputedbythefollowingequation（seeAppendixAformoredetails）:

（11）

Hence,withregardtoEq.（11）,fordeterminationofthefactorofsafetyagainstflexuraltopplingfailureinopenexcavationsandundergroundopeningsincludinggeo-structuraldefectsthefollowingequationissuggested:

（12）

FromEq.（12）itcanbeinferredthatthefactorofsafetyagainstflexuraltopplingfailureobtainedonthebasisofprinciplesofsolidmechanicsisirrelevanttothelengthofgeo-structuraldefectsorthecracklength,directly.However,itisrelatedtothedimensionlessparameter“jointpersistence”,k,asitwasdefinedearlierinthispaper.Fig.2representstheeffectofparameterkonthecriticalheightoftherockslope.Thisfigurealsoshowsthelimitingequilibriumoftherockmass（Fs=1）withapotentialofflexuraltopplingfailure.

Fig.2. Determinationofthecriticalheightofrockslopeswithapotentialofflexuraltopplingfailureonthebasisofprinciplesofsolidmechanics.

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2.2.2.Fracturemechanicsapproach

Griffithin1924[13],byperformingcomprehensivelaboratorytestsontheglasses,concludedthatfractureofbrittlematerialsisduetohighstressconcentrationsproducedonthecracktipswhichcausesthecrackstoextend（Fig.3）.Williamsin1952and1957andIrwinin1957hadproposedsomerelationsbywhichthestressaroundthesingleendedcracktipssubjectedtotensileloadingatinfiniteisdetermined[14],[15]and[16].Theyintroducedanewfactorintheirequationscalledthe“stressintensityfactor”whichindicatesthestressconditionatthecracktips.Thereforeifthisfactorcouldbedeterminedquantitativelyinlaboratorial,then,thefactorofsafetycorrespondingtothefailurecriterionbasedonprinciplesoffracturemechanicsmightbecomputed.

Fig.3. Stressconcentrationatthetipofasingleendedcrackundertensileloading

Similarly,thegeo-structuraldefectsexistinrockcolumnswithapotentialofflexuraltopplingfailurecouldbemodeled.Asitwasmentionedearlierinthispaper,crackscouldbemodeledinaconservativeapproachsuchthatthelocationofmaximumtensilestressatpresumedfailureplanetobeconsideredasthecrackslocations（Fig.3）.Iftheexistinggeo-structuraldefectsinarockmass,aremodeledwithaseriescracksinthetotalfailureplane,thenbymeansofprinciplesoffracturemechanics,anequationfordeterminationofthefactorofsafetyagainstflexuraltopplingfailurecouldbeproposedasfollows:

（13）

whereKICisthecriticalstressintensityfactor.Eq.（13）clarifiesthatthefactorofsafetyagainstflexuraltopplingfailurederivedbasedonthemethodoffracturemechanicsisdirectlyrelatedtoboththe“jointpersistence”andthe“lengthofcracks”.Assuchthelengthofcracksexistingintherockcolumnsplaysimportantrolesinstressanalysis.Fig.10showstheinfluenceofthecracklengthonthecriticalheightofrockslopes.Thisfigurerepresentsthelimitingequilibriumoftherockmasswiththepotentialofflexuraltopplingfailure.Asitcanbeseen,anincreaseofthecracklengthcausesadecreaseinthecriticalheightoftherockslopes.Incontrasttotheprinciplesofsolidmechanics,Eq.（13）orFig.4indicateseithertheonsetoffailureoftherockcolumnsortheinceptionoffracturedevelopment.

Fig.4. Determinationofthecriticalheightofrockslopeswithapotentialofflexuraltopplingfailureonthebasisofprincipleoffracturemechanics.

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3.Comparisonoftheresultsofthet