Experimental Validation of Modified Barton’s Model for Rock Fractures

ORIGINALPAPER

ExperimentalValidationofModifiedBarton’sModelforRockFractures

PooyanAsadollahi•MarcoC.A.InvernizziSimoneAddotto•FulvioTonon

•

Received:13September2009/Accepted:1February2010/Publishedonline:25February2010ÓSpringer-Verlag2010

AbstractAmongtheconstitutivemodelsforrockfrac-turesdevelopedovertheyears,Barton’sempiricalmodelhasbeenwidelyused.AlthoughBarton’sfailurecriterionpredictspeakshearstrengthofrockfractureswithacceptableprecision,ithassomelimitationsinestimatingthepeaksheardisplacement,post-peakshearstrength,dilation,andsurfacedegradation.ThefirstauthormodifiedBarton’soriginalmodelinordertoaddresstheselimita-tions.Inthisstudy,themodifiedBarton’smodel(thepeaksheardisplacement,theshearstress–displacementcurve,andthedilationdisplacement)isvalidatedbyconductingaseriesofdirectsheartests.

KeywordsRockfractureconstitutivemodelÁ

Barton’sempiricalmodelÁShearstrengthÁDilatancyÁShearstiffness

1Introduction

Innear-surfacegeotechnicalworks,themechanicalbehaviorofrockmassesisinfluencedmorebythefracturesthanbytheintactrock.Severalempiricalandtheoretical

P.Asadollahi(&)ÁF.TononDepartmentofCivilEngineering,UniversityofTexas,

Austin,TX78712-0280,USA

e-mail:pasadollahi@gzconsultants.com

M.C.A.InvernizziÁS.Addotto

Land,Environment,andGeo-EngineeringDepartment,TurinPolytechnic,10192Turin,Italy

constitutivemodels(AmadeiandSaeb1990;BartonandChoubey1977;DesaiandFishman1991;Foxetal.1998;Gensetal.1990;Goodman1976;Huangetal.1993;Jing1990;Jingetal.1993;Kanaetal.1996;LadanyiandArchambault1969;Patton1966;Plesha1987;Qiuetal.1993;Saeb1990;Wangetal.2003;Wibowo1994;Wibowoetal.1993)weredevelopedtosimulatethebehaviorofrockfractures.

Patton(1966)proposedbilinearmodelsofsaw-toothfractures.Plesha(1987)idealizedPatton’ssaw-toothtypeasperitiesanddevelopedaconstitutivemodelbasedontheclassicaltheoryofplasticity.Huangetal.(1993)verifiedPlesha’sexponentialdegradationlawthroughaseriesofexperimentsforfractureshavingsaw-toothtypeasperities.Qiuetal.(1993)revisedPlesha’smodelbyidealizingthesinusoidalasperities,butitwaslesspracticalduetothecomplexityofconstitutiveequation.Saeb(1990)modifiedthefailurecriterionofLadanyiandArchambault(1969).Gensetal.(1990)proposedanelastoplasticconstitutivelawfordescribingthethree-dimensionalmechanicalbehaviorofrockfractures.DesaiandFishman(1991)proposedaconstitutivemodelbasedonthetheoryofplasticityforcharacterizingthemechanicalresponseofsimulatedfracturesundermonotonicloading,unloadingandreverseloading.Wangetal.(2003)proposedanellipticyieldfunctionbasedonassociatedflowruletopredictthebehaviorofrockinterfacesandfractures.Usingtheresultsofaseriesofexperimentalworkonsandstone,Leichnitz(1985)developedaconstitutivelawforrockfracturesthatalsoallowsconsiderationforthenon-linear-ityofthematerialbehavior.Kanaetal.(1996)suggestedtheinterlock-frictionmodelfordynamicshearresponse;theimportanceofsecondorderasperitiesonthedynamicshearbehaviorwasexplainedbyFoxetal.(1998).Sam-adhiyaetal.(2008)introducedageneralizedformulation

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598ofathree-dimensionaljoint/interfaceelementtoaccountfordilatancy,roughness,andundulatingsurfaceofdiscontinuities.

DegradationofjointasperitieswasinvestigatedbyPlesha(1987),Zubelewiczetal.(1987)Leeetal.(2001),Homandetal.(1999,2001),LadanyiandArchambault(1969),Saeb(1990),HutsonandDowding(1990),Hutson(1987),andHuangetal.(1993).Inaddition,thepredictionofthedilatancyphenomenonofregularorirregularfrac-turessubjectedtodirectshearloadinghasbeenaddressedbynumerousresearcherssuchasPatton(1966),LadanyiandArchambault(1969),Jaeger(1971),Barton(1973;1976),Saeb(1990),Homandetal.(1999;2001),Leichnitz(1985),etc.

Amongthesemodels,Barton’sempiricalmodel(Barton1973,1976;BartonandChoubey1977)haswidelybeenused(GrasselliandEgger2003)becauseitiseasytoapplyandincludesseveralimportantfactorsoffractureproperties(Bandisetal.1983).AlthoughBarton’sfail-urecriterionpredictsthepeakshearstrengthofrockfrac-tureswithacceptableprecision,itshowslimitationsinestimatingthepeaksheardisplacement,post-peakshearstrength,dilation,andsurfacedegradation(Asadollahi2009).

Asadollahi(2009)modifiedtheoriginalBarton’smodeltoaddressitslimitations.Hebuiltandanalyzedadatabaseoftheresultsofdirectsheartestsavailableinthelitera-tures.Thedatabasecontained366directsheartestrecordsandwascalledMonotonicDirectShearTest,MDST,database(Asadollahi2009).TheabilityofBarton’smodeltopredictpeaksheardisplacement,dilation,andstress–displacementcurvewasinvestigatedandmodificationswereproposedtoimproveit.

ThepurposeofthispaperistoexperimentallyvalidatethemodifiedBarton’smodelproposedbyAsadollahi(2009).Section2describestheoriginalBarton’smodel,itslimitations,andmodifications(Asadollahi2009).Section3introducesthemethodologyandtestingequipments.Theresultsoftheexperimentalstudyarepresentedandana-lyzedinSect.4followedbysummaryandconclusionsinSect.5.

2Barton’sOriginalModelVersustheModifiedModelBarton(1973)suggestedthefollowingempiricallawfortheshearstrengths¼r󰀂ofrocklog󰀂fractures:

JCS

󰀃󰀃

ntanJRC10rþ/rð1Þ

nwherernisthenormalstressacrossthefracture;/ristheresidualfrictionangle,whichisequaltobasefriction

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angle,/b,forunweatheredfracturesurfaces;JRCisthejointroughnesscoefficient;andJCSisthejointcompres-sivestrength.2.1ShearStiffness

Thesheardisplacement,dpeak,requiredtoreachthepeakshearstrengthdeterminesthesecantstiffnessoffracturesinshear.Thisisextremelyimportantinputdatainthefiniteelement(Barton1972)anddistinctelement(Asadollahi2004)analyses.Secantpeakshearstiffness,Ks,canbeobtainedasKs¼speak=dpeak:

Barton(1982)suggestedthefollowingempiricalequa-tionforthepeaksheardisplacementofrockfractures:

dpeak1󰀂JRC󰀃

0:33

L

¼500L;ð2ÞwhereLislengthofjointsample(inmeters).

AlthoughBandisetal.(1983)foundthataconstantnormalpeakdisplacementmodel(John1970)isnotalwaysrealistic,Eq.2isindependentofnormalstress.Thenon-linearvariationofshearstiffnesswithnormalstressisduetonon-linearvariationofspeakwithrnandthesmallincreaseofdpeakwithrn(Bandisetal.1983).Inaddition,Wibowoetal.(1993)andWibowo(1994)demonstratedthatneithertheconstantstiffnessmodelnortheconstantdisplacementmodel,bythemselves,fittheobservedshearbehaviorofrockfractures.ThepeaksheardisplacementmeasuredintheexperimentsbyWibowoetal.(1993)wasfoundtoincreasewiththenormalloadorstress.

Moreover,Eq.2predictszeropeaksheardisplacementforsawedfractures,whichisnotconsistentwithexperi-mentalobservations.TheMDSTdatabase(Asadollahi2009)contained19datapointswithzeroJRCandpeaksheardisplacementrangingbetween0.05and2.71mm.Inaddition,inthisstudy,45directsheartestswereperformedonsawedfracturesoffourdifferentrocktypes(underdifferentnormalstresses).Thepeaksheardisplacementofsawedfracturerangedbetween0.3and2.2mm(seeSect.4.2fordetailedresults).

Inordertoovercometheselimitations,Asadollahi(2009)performedacorrelationanalysesonMDSTdata-basetofindthebestempiricalequationforthepeaksheardisplacementofrockfractures.AlthoughBartonfoundthatthepeaksheardisplacementincreaseswithJRC,theoppositewasfoundincorrelationanalysesofMDSTdatabase.TheregressionanalysisfoundthatcosinefunctiondescribestherelationshipbetweenJRCandpeaksheardisplacementthebest.Consequently,thefollowingempiricalequationwasintroduced(Asadollahi2009):

ExperimentalValidationofModifiedBarton’sModelforRockFractures599

d¼0:0077ÂL0:45󰀄rn

󰀅0:34󰀂󰀂JCS󰀃󰀃peak

JCS

cosJRClog10

rnð3Þ

2.2Stress–DisplacementCurve

Barton(1982)showedthatthemobilized(pre-orpost-peak)shearstrengthcanbeexpressedusingtheconceptof

roughnessmobilization,JRCmobilized,inEq.1.TheratioJRCmobilized=JRCpeakcanbeestimatedfromtheratiod/dpeakusingthevaluesgiveninTable1.Bartonassumedthat,atasheardisplacementequalto100dpeak,themobilizedJRCbecomeszero.Itseemstobejustanapproximationfortheendofthecurve(Asadollahi2009).Moreover,evenafterthisamountofdisplacement,thefracturesurfaceisnotthesameasinsawedfractures(JRCmobilized=0).

Inordertopredictthepre-peakstress–displacementcurve,Asadollahi(2009)performedcorrelationanalysesofMDSTdatabaseandintroducedTable2,whichgivestheratioofJRCmobilized=JRCpeakand/mobilized=/basefordif-ferentmagnitudesofd/dpeak.Inaddition,heproposedthefollowingequationtopredictthepost-peakmagnitudeofJRCmobilized=JRCpeakfromd=dpeak:

JRCmobilized󰀂dpeak󰀃0:381JRC¼d;ð4Þ

peakthisempiricalequationwasobtainedbycorrelationanal-ysesandfittedtheMDSTdatabasethebest.2.3Dilatancy

Barton(1982)indicatedthatdilationbeginswhenJRCmobilized=0andmobilizedtangentdilationangle,dt,canbeobtainedfromthefollowingdð1=MÞJRC󰀂relationship:

JCS

󰀃t¼mobilizedlog10r;ð5Þ

n

Table1Recommendedmodelforshearstress–displacement(Barton1982)

Non-planarfractures

Planarfractures(JRCB5)

JRCmobilizedJRCmobilizeddd

peakJRCpeak

dd

peakJRCpeak

0-/r/i0-/r/i0.300.300.60.750.60.751.01.01.00.952.00.852.01.04.00.704.00.910.00.5010.00.725.00.4025.00.5100

0

100

0

Table2Pre-peakmobilizationofthebasefrictionangleandJRC

/mobilizedJRCmobilizedddpeak

/base

JRCpeak

0.00.00.00.250.750.00.500.900.670.600.920.831.0

1.0

1.0

whereMisadamagecoefficientthattakesvaluesof1or2forshearingunderloworhighnormalstress,respectively(OlssonandBarton2001).

Almosthalfofthedirectsheartestsfoundintheliter-aturebyAsadollahi(2009)displayednegativedilation(contraction),whichwasneglectedbyBarton’smodel.Mostofthetime(notalways),theinitialcontractionisduetothemismatching.Sincethenegativedilationcanbeseeninmostofthedirectsheartestsandconsideringthatisconservativeinthestabilityanalysesofrockblocks(e.g.,intunneling),itisrecommendedthatcontractionbeconsidered.

Ontheotherhand,Barton(1982)proposedEq.6forthetangentdilationangleateachsheardisplacementandTable1formobilizedJRC.BasedonTable1,JRCmobilizedisnegativeuptod/dpeak=0.3.Therefore,thetangentdilationangleshouldbenegativeuptod/dpeak=0.3.Inaddition,JRCmobilizediszeroatd/dpeak=0.3andthenithasapositivevalue.Asaresult,dilationdisplacementshoulddecreaseuptod/dpeak=0.3andthenincrease.Thus,d/dpeak=0.3shouldcorrespondtotheminimumdilationdisplacement,notthepointatwhichthefracturestartstodilate.Consequently,Eq.6isinconsistentwithTable1.Inordertoaddresstheabove-mentionedweaknessandinconsistency,Asadollahi(2009)analyzedMDSTdatabaseandfoundthat:•Atdh¼0:5dpeak;theaveragemagnitudesofdvisclosetozero.•

Atdh¼d󰀄peak;dilation

d󰀄can󰀅󰀅bedescribedthebestasv¼13tanJRClog10JCS

rn

:Thus,heproposedthefollowingempiricalquadraticequationforthedilationdisplacement,dv,ateachpre-peaksheardisplacement,dv1󰀂dhB󰀂dpeak:

JCS󰀃󰀃󰀂dh󰀃󰀂

2d󰀃d¼tanJRClogh

peak310rdÀ1:

npeakdpeak

ð6Þ

Inaddition,fromananalyticalpointofview,thetangent

dilationangle󰀄ateach󰀅

sheardisplacement(Eq.5)canbe

describedasdddv󰀄peak󰀅¼tan󰀄JRC󰀄󰀅󰀅mobilizedlogd

d10JCSrn

:dvpeak

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600UsingEq.4topredictJRCmobilized,thefollowinginte-grationwasintroducedforthepost-peakdilationdis-placementatd(dh

peak

¼l(Asadollahi2009):dZl 󰀂dv¼dpeak

peak󰀃0:381tanJRC󰀂JCS

󰀃!

peakdlogh

10rnÂd󰀂1dh

󰀃)dþðdpeakvÞpeak

ð7Þ2.4ValidityDomainsoftheModifiedModel

MDSTdatabasecontained366datapointsofdirectsheartestsinwhichthemagnitudesofJRCrangedbetween0and20withanormaldistribution.Therefore,themodi-fiedmodel(Eqs.3,4,6,and7)isvalidforallvaluesofJRC.

Inaddition,inMDSTdatabase,thern/JCSratiorangesbetween0.001and0.1.Consequently,althoughthisrangecoversalmostallstressesthatonemaybefacedinapracticalrockengineeringproblem,itcanbeconsideredasavaliditydomainofthemodifiedmodel.

Finally,MDSTdatabasecontainednocasewithd[15dpeak.AlthoughAsadollahi(2009)demonstratedthatEq.4worksatleastaswellasBarton’stable(Table1)ford[15dpeak,itisrecommendedtouseEqs.4and7withcautionatlargesheardisplacements.

3MethodologyandTestingEquipment

Thepurposeofthisexperimentalstudywastovalidatethe

newlydevelopedmodeltopredicttheshearbehaviorofrockfractures(Eqs.3,4,6,and7).Inordertovalidatethemodelforallrocktypesandfracturecharacteristics,areasonablerangeofallparametersthatmayaffecttheshearbehaviorofthefracturesshouldbecoveredintheexperi-mentalstudy.However,coveringallrangesofallparam-etersisnotfeasibleduetothelimitationsintime,funding,andavailableequipment.

Inordertovalidatethemodelindependentofrocktypeandrockhardness,theexperimentalstudywasperformedonfourdifferentrocktypes:twoweakrocks,includingweaklimestone(calledLimestone1)andredsandstone,andtwohardrocks,includinggraniteandmetamorphiclimestone(calledLimestone2).3.1UniaxialCompressiveStrength

ToevaluatetheUniaxialCompressiveStrength(UCS)oftheintactrocks,threedifferentkindsoftestswerecarriedout:theSchmidtHammertest,thepointloadtest(PLT)andtheUCStestwithstress–straincurve.

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TheSchmidthammermeasuresthereboundofaspringloadedmassimpactingagainstthesurfaceoftherock(orconcrete),AS5873andC805.TheL-hammerusedinthisexperimentalstudy(impactenergy=0.075mkg)whichissuitableformeasuringUCSvaluesdowntoabout20MPaanduptoatleast300MPa.TheSchmidthammerreboundnumberisanarbitraryscalerangingfrom10to100.Thehigherreboundgivesthehighercompressivestrengthoftherock.Inthisstudy,thefollowingempiricalequationoriginallysuggestedbyMiller(1965)andadoptedlaterbyBartonandChoubey(1977)andISRM(1978)isusedtocorrelateUCSandreboundnumber,R:log10ðUCSÞ¼0:0088cRþ1:01;

ð8Þ

wherecisthedrydensityofrock(kN/m3)andtheunitofUCSisMPa.

Foreachrocktype,severalspecimenswerepreparedbyeitherofthefollowingmethods:(1)cutting59592cmpiecesusingalapidaryslabsaw;(2)coring(5-cmdiame-ter)specimensusingcoredrillpress‘‘SupermaxHRD-700H’’.AftertakingSchmidthammertests,PLTswereperformedoneachspecimenemployingPLTmachineGCTS8LT100.

BasedonISRMSuggestedMethodsforDeterminingPointLoadStrength(ISRM1985),theUCScanbecal-culatedfromthePLT.Thepointloadindexisdefinedasfollows:IPðsÞ¼

D2;ð9Þ

e

whereDeistheequivalentcorediameter.Forthenon-circularcrosssectionitisequaltoqffiffiffifficaseof

4pA

;inwhichAistheminimumcross-sectionalareaofaplanetroughthespecimenandtheplatencontactpoints.

Thevaluesofthepointloadindex,Is,shouldbemodi-fiedfordiametercorrections:IðsÞ50¼FÂIðsÞ;ð10ÞF¼ðDe=50Þ0:45;

ð11Þ

inwhichDehastheunitofmm.Twolowestandtwohighestvaluesofpointloadindiceswereremovedfromthedatasetandtheremainingvalueswereaveraged.TheISRMsuggestedmethodfordeterminingpointloadstrengthproposesthattheUCSis20–25timespointloadindex.

Theuniaxialcompressiontestwithstress–straincurvemeasurestheuniaxialorunconfinedcompressivestrength,Young’smodulus,andPoissonratiooftherockmaterial(BradyandBrown2004).Foreachrocktype(exceptforLimestone1),threesampleswerecoredusingdrillpress‘‘SupermaxHRD-700H’’,trimmed,andgroundemployingspecimengrinder.Aservo-hydraulictestingmachine,

ExperimentalValidationofModifiedBarton’sModelforRockFractures601

designedforuniaxial/triaxialtests,wasusedforperforminguniaxialcompressiontests.UCStestsonLimestone1werenotperformedbecauseoflackofrockmaterial.AttemptstofindexactlythesameLimestonewereunsuccessful.3.2JointCompressiveStrength(JCS)

TheJCSatlowstresslevelsisequaltotheunconfinedcompressionstrength,rc,oftheintactrockifthefractureisunweathered,butmayreducetoapproximatelyrc/4forweatheredfractures(Barton1971).TheSchmidthammercanbeemployedtomeasuretheJCSvaluesofweatheredrockfractures[Miller’s(1965)method].

Forthecaseofartificialsawedfractures,thefractureisunweatheredandundamagedandthusJCSshouldbeequaltoUCS.However,theprocessofmakingartificialroughfracture(shearingtheintactrockorbreakingbyhammer)mayinducemicro-fractures,whichreducetheJCS.InordertoobtainanestimationoftheratioofJCStoUCSforthecaseofroughfractures,tenSchmidthammertestswereperformedonbothsawedandroughfracturesofeachrocktype.TheSchmidthammertestsonroughfracturesweredoneafterperformingdirectsheartestandopeningthespecimenring.TheUCSsestimatedusingthereboundvaluesobtainedon(sheared)roughfracturesareequalto60%ofthosepredictedusingthereboundvaluesmeasuredon(intact)sawedfractures.Sincetheprocessofshearingtheroughfracturescausessomeadditionaldamagestothefractureandthusdecreasesitscompressivestrength,itisestimatedthattheratioofJCStoUCSshouldbearound0.8.Therefore,inthefollowing,JCSofroughfracturesobtainedaccordingtotheabove-mentionedprocedureisassumedtobeabout0.8timesUCSofthecorrespondingtointactrock.

3.3DirectShearTest

Severaldirectsheartestswereperformedonartificialsawedandroughfracturesofeachrocktype.Thepurposeofthedirectsheartestsperformedonsawedfractureswastoobtainthebasefrictionanglesandtovalidatethepro-posedmodificationinthecaseofsawed(orplanar)frac-tures(Eq.3forJRC=0).Ontheotherhand,thedirectsheartestsperformedonroughfractureswerecarriedouttovalidatethemodificationmadeonBarton’smodelregard-ingthepeaksheardisplacement(Eq.4forJRC=0),stress–displacementcurve(Table2;Eq.4),anddilationdisplacement(Eqs.6and7).

InthecaseofLimestone1,alapidaryslabsawwasusedtocut8cmsamples.However,thedrillpresswasemployedtocore5-cmdiametersamplesfromthesand-stone,thegranite,andLimestone2.Wheneverasawedfracturewasrequired,thesampleswerecutintohalfwiththeslabsaw.

Inthecaseofweakrocks(Limestone1andthesand-stone),artificialroughfracturesweremadebyshearingintactrocksampleunder1MPanormalstressuptofailureandreturningtheshearactuatortotheoriginalpositionafterremovingthenormalstress.However,inthecaseofhardrocks(graniteandLimestone2),artificialroughfracturesweremadebybreakingthesamplesinhalfbyahammer.Foreachdirectsheartest,therocksamplewasinsertedinthespecimenringandfixedusingAnchoringandPatchingCementmanufacturedbyRockiteasencapsulat-ingmaterial.

GeotechnicalConsultingandTestingSystems(GCTS)servo-hydraulictestingmachine(RDS-300)wasemployedforthedirectsheartests(Fig.1).Themachineissuppliedwithoneshearboxmadeupofanupperandalowerpart.Theupperpartcanbemovedverticallyandthelowerpartcanbemovedhorizontally.Twoactuators,oneactingverticallyandoneactinghorizontally,areusedtoapplytheforcesinthetwodirections(degreesoffreedoms).Twolinearrailbearingsareusedforguidanceofthelowerboxinordertohaveacontrolledlinearmovement.

Theservo-hydraulictestingmachineiscomposedofa500kNcompressionframe,adirectshearapparatus,andelectro-hydraulicshearandnormalloadactuatorswith300and500kNloadcapacity,respectively.Themaximumstrokeis100mmintheverticaldirectionand±50mminthesheardirection.

Fourplatesaroundthesphericalseatingforthenormalactionpreventrotationoftheupperhalfoftheshearbox(Fig.1a,d).ItshouldbementionedthatBoulon(1995)Jafarietal.2003;Jafarietal.2004)alsopreventedthisrotationusingtwobrushlessservo-motors.Thetwowallsofajointcanmovesymmetrically,sonorelativerotationoccursduringtheshearingdisplacementandthenormalforceremainscenteredontheactivepartofthejointatanygiventime.

Inthesheartest,thenormalandsheardisplacementsweremeasuredbymeansoflinearvariabledifferentialtransducers(LVDTs).TheverticaldisplacementbetweentheshearboxismeasuredbyfourLVDTs,positionedinasquarepatternaroundthespecimen,oneineachcorner.EachoftheLVDTshasameasurementrangeof12mm.TheaveragevalueofthesefourLVDTsisusedtorepresentthevertical(normal)displacementpresentedinthe‘‘Results’’.Therelativedisplacementbetweentheshearboxinthehorizontal(shear)directionismeasuredbyoneLVDT,whichhasa100mmrange.ThesensitivitiesoftheLVDTsare0.025mmforsheardisplacementand0.0025mmfornormaldisplacement.

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4ResultsandDiscussion4.1UCSandJCS

InordertoobtaintheUCSofintactrocks,threedifferentkindsoftestswereperformed:Schmidthammertests,PLTs,andUCStestswithstress–straincurve.TheresultsofthesetestsaresummarizedinTable3.TheUCSvaluesevaluatedusingSchmidthammertests,PLTtests,andUCStestswithstress–straincurveareconsistentwitheachother.TheadoptedmagnitudesofUCSandJCSfordifferentrocktypesarealsopresentedinTable3.4.2DirectShearTestsonSawedFractures

Directsheartestswereperformedontwotofoursamplesofeachrocktypeunderdifferentnormalstressesrangingbetween0.2and6MPa.Table4presentsthepeakshearstrengthsandpeaksheardisplacementsobtainedinthese

Fig.1Servo-hydraulictestingmachine(GCTSdirectsheartestsystem,RDS-300)

teststogetherwiththeappliednormalstressesandlengthofthesamples.

Foreachrocktype,shearstrengthversusnormalstresscurvewasdrawnforalldirectsheartestsperformedonsawedfractures.SinceJRCisequaltozero,theinclinationofthetrendlinepassedthroughtheoriginwouldbeequaltotan(/b).ThebasefrictionangleofeachrocktypeandR2ofthetrendlinepassedthroughdatapointsaregiveninTable4.Barton’sempiricalequation(Eq.2)suggestszeropeaksheardisplacementforsawedfractures.However,As-adollahi(2009)introducedEq.3forpeaksheardisplace-mentwhichworksforallrangesofJRC,evensawedfractures.

Figure2aandTable5showtheabilityofEq.3topredictthepeaksheardisplacementofsawedfractures.ExceptforLimestone2(veryhardrock),Eq.3worksverywell.Inallcases,predictionsobtainedwithEq.3aremuchbetterthanthezerovaluegivenbyBarton’sequation(Eq.2)anddepictedinFig.2b.

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ExperimentalValidationofModifiedBarton’sModelforRockFractures603

Table3ResultsofSchmidthammer,PointLoadTests(PLT),andUCStestsaswellasfinaladoptedvaluesforUCSandJCSfordifferentrocktypes

Rocktype

SchmidthammertestNumberoftests

Limestone1

15

Averagereboundvalue18.6

Unitweight,c(kN/m3)25.0

UCS(MPa)26

PointLoadTest(PLT)TestNo.12345678Is(50),Height(mm)18.1418.4418.1120.4318.20.7920.7220.17

P(N)1,5702,4101,9201,3101,7602,1401,9101,300

De(mm)30.4233.1533.4635.5230.7634.8134.9435.07

Is(50)(MPa)1.361.821.430.1.491.501.330.901.3430

3,04,0204,3904,1503,1502,9303,4103,7802,750

34.8835.7639.0335.9932.3537.2734.6037.34.07

2.722.702.582.762.471.852.412.321.992.4253

7,1706,2907,04,1005,8807,0305,04,4204,6207,370

42.4033.5035.1936.1834.5835.5633.9736.4933.90.98

3.704.685.442.714.174.774.292.883.374.014.0088

3,8404,6603,7702,9205,8006,9303,5605,2405,840

39.0243.0829.6935.4337.3145.4438.0733.7135.54

2.262.353.381.993.653.222.173.863.962.8663

FinaladoptedmagnitudesforUCSandJCSUCS(MPa):155JCSsawedJCSroughfracturesfracturesUCStestTestno.

Height(mm)

Diameter(mm)

UCS(MPa)

FinaladoptedmagnitudesforUCSandJCSUCS(MPa):28JCSsawedJCSroughfracturesfractures(MPa):28(MPa):22.5

Average.(MPa)

UCS(MPa)

Sandstone

10

27.0

25.5

41

1234567Is(50),18.7019.7323.5520.0716.1421.4518.4722.1117.95

123

110.06105.01111.53

51.0251.0051.91

43.944.234.841

AverageUCS

FinaladoptedmagnitudesforUCSandJCSUCS(MPa):41JCSsawedJCSroughfracturesfractures(MPa):41(MPa):33

Average.(MPa)

UCS(MPa)

Granite

20

48.4

26.5

138

123456710Is(50),27.6217.2719.0820.1718.4119.4917.7320.517.7825.99

123

105.5294.4996.26

51.2351.1951.34

108.1130.2141.5127

AverageUCS

FinaladoptedmagnitudesforUCSandJCSUCS(MPa):127JCSsawedJCSroughfracturesfractures(MPa):127(MPa):101

Average.(MPa)

UCS(MPa)

Limestone2

10

49.6

27.0

155

1234567Is(50),23.6928.8113.5319.3121.8631.6522.4117.4419.42

123

104.7595.16102.20

50.0650.5850.53

173.0188.5157.8173

AverageUCS

(MPa):155(MPa):124

Average.(MPa)

UCS(MPa)

123

604

Table4Resultsofdirectsheartestsperformedonsawedjoints

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P.Asadollahietal.

Rocktype

Specimennumber

Lrns

dPR2lineartrendline/b(°)

(mm)

(MPa)

(MPa)

(mm)

passedthroughshearstrengthversusnormalstresscurve(%)Limestone1195.031.00.7490.9699.4

35.7

5.03.5182.202

80.96

0.50.3260.421.00.7261.102.01.5401.704.0

3.1001.253100.05.03.5001.803.02.1001.203.02.1201.103.0

2.0981.35486.700.40.3600.550.60.4800.400.8

0.6500.55Sandstone150.900.30.3580.4298.331.4

0.50.3920.500.80.6560.581.00.7260.631.50.8840.72251.10

0.20.2240.360.40.3670.460.60.4200.531.20.7850.672.01.4000.794.02.5031.016.0

3.4811.15Granite151.150.50.1520.3094.724.9

1.00.2990.601.50.4790.902.00.7930.524.0

1.7920.80251.150.80.2460.651.80.6350.552.51.3500.773.51.5750.5.5

2.3600.90Limestone2151.030.20.3800.5765.339.2

0.40.4450.600.60.5820.950.80.7120.651.2

0.10.69250.900.50.5680.920.70.5920.780.90.7220.951.10.8320.851.3

0.985

1.06

ExperimentalValidationofModifiedBarton’sModelforRockFracturesFig.2Predictedversusmeasuredpeaksheardisplacementforsawedfractures605

(a) Modified Barton’s model: Peak shear displacement (b) Barton’s original model: Peak shear displacement predicted using Equation (3) predicted using Equation (2) Table5AbilityofEq.3inpredicatingthepeaksheardisplacementofsawedfracturesRocktypeandadoptedmagnitudeofJCSLimestone1(JCS=28MPa)Sandstone(JCS=41MPa)Granite(JCS=127MPa)

Limestone2(JCS=155MPa)Fig.3ComparisonbetweenmeasuredpeaksheardisplacementandtheirpredictedvaluesusingEqs.2and3forthecaseofsawedfractures(JRC=0)predicted

ÞAverageÆSTDðdmeasured

dpredicted

ðdmeasuredÞmax

dpredicted

ðdmeasuredÞmin

d1.05

0.790.780.41±±±±0.330.250.190.081.731.461.060.560.600.560.500.31

(a)(b)(c)(d)(e)1.51.00.50.0(f) Specimen; Specimen 2 peak (mm)n (kPa)0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 (g)(h)(i)(j)123

606P.Asadollahietal.

Figure3showstheabilityofEq.3inconsideringtheeffectofnormalstressonthepeaksheardisplacement(comparedtoEq.2).

AlthoughforsawedfracturesBarton’soriginalmodeldoesnotsuggestastress–displacementcurve,themodifiedmodelsuggestedTable2toquantifythemobilizationofbasefrictionangle.Figure4comparesthemeasuredratio/mobilized//baseateachsheardisplacementwiththepre-dictedvaluesusingTable2.Themeasuredvaluesof/baseand/mobilizedareobtainedusingthefollowingequations:

󰀂󰀃speak

/base¼arctan;

rn

󰀂󰀃s;/mobilized¼arctanrn

ð12Þð13Þ

wherernisthenormalstress;speakisthepeakshearstrength;andsistheshearstressatagivensheardis-placement.ItcanbeseenthatTable2worksbetterthanBarton’soriginalmodel.

Fig.4Comparisonbetweenmeasuredratioof/mobilizedateachsheardisplacement/baseandpredictedonesusingTable1andBarton’smodel(sawedfractures)123

2(a)2(c)2(b)2(d)ExperimentalValidationofModifiedBarton’sModelforRockFractures607

4.3DirectShearTestsonRoughFractures

Directsheartestswereperformedonthreesamplesofeachrocktypeunderdifferentnormalstressesrangingbetween0.5and2MPa.Table6summarizestheresultsofdirectsheartestsconductedontheroughfracturesofdifferentrocktypes.

4.3.1PredictionofShearDisplacementatFailureForeachsample,JRCwasback-calculatedbasedonthemeasuredvaluesofotherparameters(JCS,basefrictionangle,normalstress,andshearstrength).Table7summa-rizesthecalculationsofJRCandpeaksheardisplacements

bothusingBarton’sequation(Eq.2)andthemodifiedempiricalequation(Eq.3).

Figure5andTable8compareEqs.3and4witheachotherbygivingtheirratioofpredictedtothemeasured

dpredicted

peaksheardisplacement,dmeasured;forroughfracturesofdifferentrocktypes.Fordifferentrocktypes,thefol-lowingconclusionscanbedrawnfromTables7and8andFig.5:

•Equation3worksbetterthanEq.2inpredictingpeaksheardisplacementofrockfracturesofLimestone1,because:–

predicted

AlthoughthevalueofðdmeasuredÞAveragecalculatedusingEq.2isclosertoone,thecorrelationfactor,predictedpredictedðdmeasuredÞSTD=ðdmeasuredÞAverage;issmallerinthecaseofEq.3(0.94fromEq.3comparedto1.13fromBarton’sequation).

dpredicteddpredicted

ThemagnitudesofðdmeasuredÞmaxandðdmeasuredÞmincalculatedusingEq.3areclosertoone.

dpredicteddpredicted

ThemagnitudesofðdmeasuredÞmaxandðdmeasuredÞmincalculatedusingEq.3areclosertoone.

Forallthreespecimens,Eq.3worksbetterthanBarton’sequation.

d

dd

–

Table6ResultsofdirectsheartestsonroughfracturesRocktypeLimestone1

Specimennumber123

Sandstone

123

Granite

123

Limestone2

123

L(mm)76.976.666.251.0151.0350.4051.151.251.251.150.751.1

rn(MPa)0.51.02.00.51.01.51.01.52.00.51.02.0

sp(MPa)1.081.222.530.821.001.231.562.333.190.921.673.17

dp(mm)0.442.262.782.421.41.361.461.80.920.720.70.73

––

•

•

TheabilityofEqs.3and4inpredictingpeaksheardisplacementsofroughfracturesofthesandstonearealmostthesame.

Barton’sequation(Eq.2)worksbetterthanEq.3inpredictingthepeaksheardisplacementsofroughfracturesofthegraniteandLimestone2.

Ingeneral,itcanbeseeninTables7and8andFig.5thatbothBarton’soriginalandmodifiedmodelsunderes-timatethepeaksheardisplacementofroughfracturesconsideredinthisstudy.However,Barton’smodel

Table7CalculationsofJRCandpeaksheardisplacementofroughfracturesRocktypeandadoptedmagnitudeofJRC

Sampleno.

JRC

Modifiedmodel(Eq.3)dpredicted(mm)

Limestone1(JCS=22.5MPa)

123

Sandstone(JCS=33MPa)

123

Granite(JCS=101MPa)

123

Limestone2(JCS=124MPa)

123

17.110.614.315.08.95.916.217.719.49.19.310.1

0.580.810.960.430.600.860.350.410.450.290.370.47

dpredicteddmeasured

Barton’smodel(Eq.2)dpredicted(mm)0.930.790.800.670.560.660.680.700.730.570.570.59

dpredicteddmeasured

1.320.360.340.180.430.630.240.230.480.400.520.

2.110.350.290.270.400.490.470.390.790.790.810.81

123

608

Fig.5PredictedversusthemeasuredpeaksheardisplacementforroughfracturesLimestone 1 Granite Limestone 2 SandstoneP.Asadollahietal.

(a) Barton’s original model: Peak shear displacement (b) Modified Barton’s model: Peak shear displacement predicted using Equation (2) predicted using Equation (3) Table8ComparisonbetweenBarton’sequation(Eq.2)andEq.3inpredictingpeaksheardisplacementofroughfracturesRocktypeandadoptedmagnitudeofJRCLimestone1(JCS=26.4MPa)Sandstone(JCS=32.8MPa)Granite(JCS=101.3MPa)Limestone2(JCS=138.5MPa)Averageofallrocktypes

PredictedusingEquation3Barton’sequationEquation3Barton’sequationEquation3Barton’sequationEquation3Barton’sequationEquation3Barton’sequation

predicted

ÞAverageÆSTDðdmeasured

d

predicted

ðdmeasuredÞmax

d

predicted

ðdmeasuredÞmin

d

0.67±0.560.92±1.040.41±0.230.39±0.100.32±0.150.55±0.210.53±0.120.80±0.010.48±0.300.66±0.50

1.322.110.630.490.490.790.0.811.322.11

0.350.290.180.280.230.390.400.790.180.28

statisticallyworksalittlebitbetterthanthemodifiedmodel

inpredictingpeaksheardisplacementofroughfracturesinvestigatedinthisresearch,because:••••

predicted

ThevalueofðdmeasuredÞAveragecalculatedusingBarton’sequation(Eq.2)isclosertoone.

ThecorrelationfactorissmallerinthecaseofEq.3(0.63fromEq.3comparedto0.76fromBarton’sequation).

dpredicted

ThemagnitudeofðdmeasuredÞmaxcalculatedusingEq.3isclosertoone.

dpredicted

ThemagnitudeofðdmeasuredÞmincalculatedusingEq.2isclosertoone.

d

directsheartestsonroughfractures.Inaddition,Table9comparesBarton’smodelandthemodifiedmodelaccordingtotheirratioofpredictedtomeasuredratioofs

rforroughfractures.

ItcanbeseeninFig.6andTable9thatbothmodelsworkverywellinpredictingthestress–displacementcurve.Forsheardisplacementssmallerthanabouteighttimesofthepeaksheardisplacement,bothmodelsunderestimatetheshearstressesand,afterthat,bothoverestimatetheshearstresses.Itcanbeconcludedthatthemodifiedmodelisalittlebitbetterthantheoriginalmodelduetothefollowingreasons:•

ss

ThevalueofððrÞpredicted=ðrÞmeasuredÞAverageobtainedusingthemodifiedmodelisclosertoonecomparingtothosecalculatedusingBarton’smodel.

4.3.2PredictionofShearStress-DisplacementCurveFigure6comparesthestress–displacementcurvespre-dictedusingBarton’soriginalmodelandthemodifiedmodelwiththestress–displacementcurvesobtainedfrom

•Thecorrelationfactor,

ssððrÞpredicted=ðrÞmeasuredÞSTDssððrÞpredicted=ðrÞAverageÞAverage;issmaller

inthecaseofthemodifiedmodel.

123

ExperimentalValidationofModifiedBarton’sModelforRockFracturesFig.6ComparisonbetweenBarton’soriginalmodelandthemodifiedmodelinpredictingstress-displacementcurveforroughfractures1.5609

1.5/(a) Limestone 1; Specimen 1 /(b) Limestone 1; Specimen 2 1Experimental results 10.5Modified model Barton's model 0.5Experimental results Modified model Barton's model /000.5 1 1.5 2 2.5P 3 0 0.5 1 1.5 2 2.521.510/P 33 .54 3/(c) Limestone 1; Specimen 3Experimental results Modified model Barton's model /(d) Sandstone; Specimen 1 21Experimental results0.5Modified model Barton's model/0P0/P0 5 10 15 20 25 30 0 1 2 3 4 11.5/(e) Sandstone; Specimen 2 (f) Sandstone; Specimen 3 10.50.5Experimental results Modified model Barton's model Experimental results Modified model Barton's model0/P0 1 2 3 4 5 6 7 8 0/P0 2 4 6 8 10 12 14 21.510.50/(g) Granite; Specimen 1 2/1.51Experimental results Modified model Barton's model (h) Granite; Specimen 2 Experimental resulst 0.5Modified model Barton's model /P0 2 4 6 8 10 12 14 0/P0 1 2 3 4 21.510.5/(i) Granite; Specimen 3 Experimental results Modified model Barton's model 21.510.5/(j) Limestone 2; Specimen 1 Experimental results Modified model Barton's model /0P/0 5 10 15 20 25 0 5 10 15 200P25 21.510.50/(k) Limestone 2; Specimen 2 Experimental results Modified model Barton's model 21.510.5/(l) Limestone 2; Specimen 3 Experimental results Modified model Barton's model /P0 5 10 15 20 25 30 0/P0 5 10 15 20 25 30 123

610P.Asadollahietal.

Table9ComparisonbetweenBarton’soriginalmodelandthemodifiedmodelinpredictingtress-displacementcurveforroughfracturesRockTypeLimestone1SandstoneGraniteLimestone2

Averageofallrocktypes

ConstitutivemodelBarton’smodelModifiedmodelBarton’smodelModifiedmodelBarton’smodelModifiedmodelBarton’smodelModifiedmodelBarton’smodelModifiedmodel

predicted

ÞAverageÆSTDððrsÞrmeasured

ðsÞpredictedððrÞmaxsÞrmeasured

ðsÞpredictedððrÞminsÞrmeasured

ðsÞ1.09±0.381.02±0.321.09±0.421.02±0.411.11±0.821.23±0.871.52±1.011.70±1.351.20±0.711.29±0.88

2.362.072.682.584.194.116.457.046.457.04

0.690.590.740.670.40.460.780.840.590.69

•

ss

ThevalueofððrÞpredicted=ðrÞmeasuredÞmaxobtainedusingthemodifiedmodelissmallerthanthosecalculatedusingBarton’smodel.

Thesenegativedilationsareincludedinthemodifiedmodel.

4.3.3PredictionofNormalDisplacement–Shear

DisplacementCurveFigure7comparesthenormaldisplacement–sheardis-placementcurvespredictedusingBarton’soriginalmodelandthemodifiedmodelwiththenormaldisplacement–sheardisplacementcurvesobtainedfromdirectsheartestsonroughfractures.Inaddition,Table10comparesBarton’smodelandthemodifiedmodelbydisplayingtheratior¼j

ðdvÞpredictedÀðdvÞmeasured

jðdvÞmeasured

5Conclusions

Theexperimentalstudypresentedinthispapervalidated

themodificationsproposedbyAsadollahi(2009)toBar-ton’soriginalmodel.Thefollowingconclusionscanbedrawnbasedontheresultsofourtesting:•

Forsawed(orplanar)fractures,themodifiedmodelworksmuchbetterthantheoriginalBarton’smodel.

BoththemodifiedmodelandBarton’soriginalmodeldisplaysubstantialapproximationsinpredictingthedilatantbehaviorofroughfractures.However,duetothefollowingreasons,itisbelievedthatthemodifiedmodelshouldbeusedforpredictingthedilationbehaviorofrockfractures:Barton’sempiricalequationforpeaksheardisplacementofrockfracturescannotconsidertheeffectofnormalstressontheincreaseofthepeaksheardisplacement.––

Barton’smodelcanpredictdilationdisplacementonlyatthepeaksheardisplacement.

ThemodifiedmodelworksbetterthanBarton’smodelinpredictingstress-displacementcurvesfor

hhighvaluesoftheddratio.peak

Thenegativedilation(orcontraction)wasfoundintheexperimentalstudydocumentedhereandintheliteraturereview.ContractionisnotconsideredinBarton’smodelandthismaycauseoverestimationofthefactorofsafetyinstabilityanalysessuchasthestabilityofrockblocksaroundtunnels.

atthesameshear

•

displacements.

Figure7andTable10showthatbothBarton’smodelandthemodifiedmodelareaffectedbymanyerrorsinpredictingthedilationdisplacement.Anidealmodelhastheratioofrequaltozero.However,inbothmodelsrisbetween1and2.5.FromastatisticalpointofviewBarton’smodelworksalittlebitbetterthanthemodifiedmodel.However,duetothefollowingreasons,itcanbeconcludedthatthemodifiedmodelshouldbeusedforpredictingthedilationbehaviorofrockfractures:••

Barton’soriginalmodelcanpredictdilationdisplace-mentonlyatthepeaksheardisplacement.

In13outof15sheartestsonroughfractures,negativedilation(compression)wasmeasuredatsmallsheardisplacements.ThesenegativedilationsarenotconsideredinBarton’smodel,whichcancauseoverestimationofthefactorofsafetyinsomeanalysessuchasstabilityofrockblocksintunnels.

–

123

ExperimentalValidationofModifiedBarton’sModelforRockFracturesFig.7ComparisonbetweenBarton’soriginalmodelandthemodifiedmodelinpredictingdilationdisplacementforroughfractures0.60.40.20611

1v/P(a) Limestone 1; Specimen 1 Experimental curve Modified model Barton's model v/P(b) Limestone 1; Specimen 2 0.5/0P/0 P1 2 3 4 Experimental results 0 0.5 1 1.5 2 2.5 3 -0.29v/P(c) Limeston 1; Specimen 3 Experimental results Modified model Barton's model 4/P-10 5 10 15 20 25 30 1.5v/(e) Sandstone; Specimen 2 P1Experimental results Modified model Barton's model 0.500 1 2 3 4 5 6 7 -0.5/P6v/P(g) Granite; Specimen 1 Experimental results 4Modified model Barton's model 200 2 4 6 8 10 12 14 -2/P8v/P(i) Granite; Specimen 3 6Experimental results 4Modified model 2Barton's model 00 2 4 6 8 10 12 -2/P6v/P(k) Limestone 2; Specimen 2 Experimental results 4Modified model Barton's model 20/P0 5 10 15 20 25 30 -2-0.5Modified model Barton's model 1.5v/P(d) Sandstone; Specimen 1 1Experimental results Modified model Barton's model 0.500 1 2 3 /4 P-0.51.3v/P(f) Sandstone; Specimen 3 0.8Experimentalresults0.3/P-0.20 2 4 6 8 10 12 14 4v/P(h) Granite; Specimen 2 3Experimental results Modified model 2Barton's model 10/P0 1 2 3 4 -16 Limestone 2; Specimen 1 5v/P(j)Experimental results 4Modified model 3Barton's model 210-10 5 10 15 20 /P6v/P(l) Limestone 2; Specimen 3 Experimental curve 4Predicted curveBarton curve 2/P00 5 10 15 20 25 30 -2123

612

Table10ComparisonbetweenBarton’soriginalmodelandthemodifiedmodelinpredictingdilationdisplacementforroughfracturesRocktype

ConstitutiveJRC

rAverageÆSTDrmax

rmin

model

Limestone1Barton’smodel

22.51.96±1.68

5.940.23Modifiedmodel

2.12±1.817.460.15SandstoneBarton’smodel332.50±2.5611.030.45Modifiedmodel2.39±2.268.470.30Granite

Barton’smodel1012.25±2.3210.500.51Modifiedmodel

1.85±1.796.900.11Limestone2Barton’smodel124

1.52±1.797.110.06󰀇Modified󰀇model

1.10±1.73

7.790.01

r¼󰀇󰀇ðdvÞpredictedÀðdvÞmeasuredðd󰀇vÞmeasured

󰀇AcknowledgmentsSimoneAdottoandMarcoInvernizziconductedthisresearchaspartoftheirMSthesesovera6-monthperiodspentattheUniversityofTexas(UT)atAustinwhentheyweresponsoredbythePolytechnicofTurin,Italy.ProfessorDanielePeila,DITAG,PolytechnicofTurinco-advisedSimoneAdottoandMarcoInvernizziandfoundthefundsnecessaryforsupportingtheirstayatUTAustin.

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