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-structuraldefectsinflexuraltoppling

failure

AbbasMajdiandMehdiAmini

Abstract

Thein-siturockstructuralweaknesses,referredtohereinasgeo-structural

defects,suchasnaturallyinducedmicro-cracks,areextremelyresponsivetotensilestresses.Flexuraltopplingfailureoccursbytensilestresscausedbythemomentduetotheweightoftheinclinedsuperimposedcantilever-likerockcolumns.Hence,geo-structuraldefectsthatmaynaturallyexistinrockcolumnsaremodeledbyaseriesofcracksinmaximumtensilestressplane.Themagnitudeandlocationofthemaximumtensilestressinrockcolumnswithpotentialflexuraltopplingfailurearedetermined.Then,theminimumfactorofsafetyforrockcolumnsarecomputedbymeansofprinciplesofsolidandfracturemechanics,independently.Next,anewequationisproposedtodeterminethelengthofcriticalcrackinsuchrockcolumns.Ithasbeenshownthatifthelengthofnaturalcrackissmallerthanthelengthofcriticalcrack,thentheresultbasedonsolidmechanicsapproachismoreappropriate;

otherwise,theresultobtainedbasedontheprinciplesoffracturemechanicsismoreacceptable.Subsequently,forstabilizationoftheprescribedrockslopes,somenewanalyticalrelationshipsaresuggestedfordeterminationthelengthanddiameteroftherequiredfullygroutedrockbolts.Finally,forquickdesignofrockslopes

againstflexuraltopplingfailure,agraphicalapproachalongwithsomedesigncurvesarepresentedbywhichanadmissibleinclinationofsuchrockslopesandor

lengthofallrequiredfullygroutedrockboltsaredetermined.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）

Sincethesuperimposedinclinedrockcolumnsaresubjectedtouniformlydistributedloads

causedbytheirownweight,hence,themaximumshearingforceandmomentexistattheveryfixed

end,thatis,atx=Ψ:

（7）

（8）

IfthemagnitudeofΨfromEq.

（1）issubstituted

intoEqs.（7）and（8）,

thenthemagnitudes

ofshearingforceandthemaximummomentofequivalent

beamforrockslopesare

computedasfollows:

（9）

（10）

whereCisadimensionlessgeometricalparameterthatisrelatedtotheinclinationsoftherockslope,thetotalfailureplaneandthedipoftherockdiscontinuitiesthatexistinrockmasses,andcanbedeterminedbymeansofcurvesshowninFig.

MmaxandVmaxwillproducethenormal（tensileandcompressive）andtheshearstressesin

criticalcross-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,duetooverloadingandhighstress

concentration,willeventuallyencounterwiththeprematurefailure.Inthispaper,

foranalysisofthegeo-structuraldefectsinflexuraltopplingfailure,asetofcracksincriticalcross-sectionalareahasbeenmodeledasshowninFig.5.ByemployingEq.（9）andassumingthattheloadsfromweakareasaretransferredtothesolidareaswithhigherloadbearingcapacity（Fig.6）,themaximumstressescouldbecomputedbythefollowingequation（seeAppendixAformoredetails）:

11）

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

12）

FromEq.（12）itcanbeinferredthatthefactorofsafetyagainstflexural

topplingfailureobtainedonthebasisofprinciplesofsolidmechanicsisirrelevanttothelengthofgeo-structuraldefectsorthecracklength,directly.However,itisrelatedtothedimensionlessparameter“jointpersistence”,k,asitwasdefinedearlierinthispaper.Fig.2representstheeffectofparameterkonthecriticalheightoftherockslope.Thisfigurealsoshowsthelimitingequilibriumofthe

rockmass（Fs=1）withapotentialofflexuraltopplingfailure.

Fig.2.Determinationofthecriticalheightofrockslopeswithapotentialofflexuraltopplingfailureonthebasisofprinciplesofsolidmechanics.

2.2.2.Fracturemechanicsapproach

Griffithin1924[13],byperformingcomprehensivelaboratorytestsontheglasses,concludedthatfractureofbrittlematerialsisduetohighstressconcentrationsproducedonthecracktipswhichcausesthecrackstoextend（Fig.

3）.Williamsin1952and1957andIrwinin1957hadproposedsomerelationsbywhichthestressaroundthesingleendedcracktipssubjectedtotensileloadingatinfiniteisdetermined[14],[15]and[16].Theyintroducedanewfactorintheirequationscalledthe“stressintensityfactor”whichindicatesthestress

conditionatthecracktips.Thereforeifthisfactorcouldbedeterminedquantitativelyinlaboratorial,then,thefactorofsafetycorrespondingtothefailurecriterionbasedonprinciplesoffracturemechanicsmightbecomputed.

Fig.3.Stressconcentrationatthetipofasingleendedcrackundertensileloading

Similarly,thegeo-structuraldefectsexistinrockcolumnswithapotentialofflexuraltopplingfailurecouldbemodeled.Asitwasmentionedearlierinthis

paper,crackscouldbemodeledinaconservativeapproachsuchthatthelocationofmaximumtensilestressatpresumedfailureplanetobeconsideredasthecracks

locations（Fig.3）.Iftheexistinggeo-structuraldefectsinarockmass,aremodeledwithaseriescracksinthetotalfailureplane,thenbymeansofprinciplesoffracturemechanics,anequationfordeterminationofthefactorofsafetyagainstflexuraltopplingfailurecouldbeproposedasfollows:

13）

whereKICisthecriticalstressintensityfactor.Eq.（13）clarifiesthatthefactorofsafetyagainstflexuraltopplingfailurederivedbasedonthemethodoffracturemechanicsisdirectlyrelatedtoboththe“jointpersistence”andthe

“lengthofcracks”.Assuchthelengthofcracksexistingintherockcolumnsplaysimportantrolesinstressanalysis.Fig.10showstheinfluenceofthecracklengthonthecriticalheightofrockslopes.Thisfigurerepresentsthelimitingequilibriumoftherockmasswiththepotentialofflexuraltopplingfailure.Asitcanbeseen,anincreaseofthecracklengthcausesadecreaseinthecritical

heightoftherockslopes.Incontrasttotheprinciplesofsolidmechanics,Eq.（13）orFig.4indicateseithertheonsetoffailureoftherockcolumnsorthe

inceptionoffracturedevelopment.

Fig.4.Determinationofthecriticalheightofrockslopeswithapotentialofflexuraltopplingfailureonthebasisofprincipleoffracturemechanics.

3.Comparisonoftheresultsofthetwoapproaches

ThecurvesshowninFig.representEqs.（12）and（13