地质岩土英文文献翻译Word格式.docx
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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