地质岩土英文文献翻译.docx
《地质岩土英文文献翻译.docx》由会员分享,可在线阅读,更多相关《地质岩土英文文献翻译.docx(25页珍藏版)》请在冰豆网上搜索。
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.
ViewWithinArticle
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.
ViewWithinArticle
3.Comparisonoftheresultsofthet