搜索
1InternationalJournalofRockMechanicsandMiningSciencesAnalysisofgeo-structuraldefectsinflexuraltopplingfailureAbbasMajdiandMehdiAminiAbstractThein-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,Criticalcracklength21.IntroductionRockmassesarenaturalmaterialsformedinthecourseofmillionsofyears.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-structuraldefectsonflexuraltopplingfailure2.1.CriticalsectionoftheflexuraltopplingfailureAsmentionedearlier,MajdiandAmini[10]andAminietal.[11]haveprovedthattheaccuratefactorofsafetyisequaltothatcalculatedforaseriesofinclinedrockcolumns,which,byanalogy,isequivalenttothesuperimposedinclinedcantileverbeamsasshowninFig.3.Accordingtotheequationsoflimitequilibrium,themomentMandtheshearingforceVexistinginvariouscross-sectionalareasinthebeamscanbecalculatedasfollows:(5)(6)Sincethesuperimposedinclinedrockcolumnsaresubjectedtouniformlydistributedloads3causedbytheirownweight,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-structuraldefectsDeterminationofthequantitativeeffectofgeo-structuraldefectsinrockmassescanbeinvestigatedonthebasisofthefollowingtwoapproaches.2.2.1.SolidmechanicsapproachInthismethod,whichis,indeed,anoldapproach,theloadsfromtheweakareasareremovedandlikewisewillbetransferredtotheneighboringsolidareas.Therefore,thesolidareasoftherockcolumns,duetooverloadingandhighstressconcentration,willeventuallyencounterwiththeprematurefailure.Inthis