Spacing of Cracks in Reinforced Concret

SPACINGOFCRACKSINREINFORCEDCONCRETE ByZdenekP.Bazant,1F.ASCEandByungH.Oh2 ABSTRACT:Thespacingandwidthofcracksinaparallelcracksystemisap-proximatelyanalyzedusingtheenergycriterionoffracturemechanicsaswell asthestrengthcriterion.Theenergycriterionindicatesthatthecrackspacing isafunctionoftheaxialstrainofthebarsandalsodependsonbarspacing, bardiameter,fractureenergyofconcrete,anditselasticmodulus.Boththe energyandstrengthcriteriayieldaminimumstrainnecessarytoproduceany cracks.Theenergycriterionandthebondslipconditionsfurtheryieldalower boundonpossiblespacingofcontinuouscracks.Therulesfortheformation ofshorter,partiallength,cracksarealsosetup.Approximateexpressionsfor thecrackwidthat,andaway,fromthebarsarederived.Numericalcompari-sonsindicatesatisfactoryagreementwithexistingtestdata,andlendtheoretical supporttooneaspectoftheempiricalGergely-Lutzformulaobtainedbysta-tisticalregressionanalysisoftestdata.Finally,formationofskewcracksina biaxiallystressedandbiaxiallyreinforcedplateisanalyzedandacrackspacing formulaisderivedforonetypicalcase. INTRODUCTION Thewi dt handspacingofcracksinparallelcracksyst emsinreinforced concretestructureshaveamajorinfluenceonstructuralperformance, includingshear,tensileandbendi ngstiffnesses,energyabsorptionca-pacity,ductility,andcorrosionresistanceofreinforcement.Muchhas beenlearnedalreadyaboutcrackwi dt handspacing(1,2,5-14,16,18-20,22-25,29,30),butacompletelyrationalandgeneralmet hodofpredictionis stillunavailable.Wewillat t empthereanadvanceinthisdirection,pay-ingparticularattentiontocrackspacingsinceitisalreadyknownrea-sonablywellhowtocalculatet hewi dt honcet hecrackspacingisde-termined,e.g.,asisshowninanearlierst udyoftension-stiffening(5). Theexistingformulasfort hespacingofcracks(2,11-13,18)arebased onthest rengt hcriterion.Theneedforempiricalrelationssuggest s, however,thatt hest rengt hconceptmaybeinsufficient.Theoretically, crackingisfractureandsot heenergyrequiredtoformt hecracksshoul d bealsotakenintoaccount.Aswewillsee,thisgivesreasonableresults wheret hest rengt hconceptdoesnot .Theenergyapproachhasalready beenusedtoderiveaformulafort hespacingoft hermalcracksinrock (8,10)andofdryingcracksinconcrete(7).Wewillempl oyhereasimilar approach,restrictingourattentiontocrackscausedbyappl i edloads. CRACKINGCRITERION Applicabilityoffracturemechanicstoconcretehasbeendeniedbymost specialistsbecauseofnegativeexperi ment alevidence.However,some ^rof.ofCiv.Engrg.andDir.,CenterforConcreteandGeomaterials,Tech-nologicalInst.,NorthwesternUniv.,Evanston,111. 60201. 2Asst.Prof.,Dept.ofCiv.Engrg.,TechnicalUniv.,Seoul,Korea. Note.DiscussionopenuntilFebruary1,1984. Toextendtheclosingdateone month,a writtenrequestmustbefiledwiththeASCE ManagerofTechnicaland ProfessionalPublications.Themanuscriptforthispaperwassubmittedforre-viewandpossiblepublicationonApril12,1982. This paperis partoftheJournal ofStructuralEngineering,Vol.109, No.9,September,1983.ASCE,ISSN0733-9445/83/0009-2066/$01.00.PaperNo.18238 2066 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.recentresearchers,e.g.,Hillerborg,etal.(17), Petersson(26),A.Ingraf-fea(privatecommunications,1981,1982),andthewriters(3),indicate thatasatisfactoryagreementwithessentiallyallfracturetestdatacan beachievedprovidedonetakesintoaccountthefactthat,duetoma-terial heterogeneity,thefractureprocesszoneis long,spanningalength ofatleastseveralaggregatesizes.Asaconsequenceofthisfact,the linear fracturemechanicscannot be used(except forvery largestructures suchasdams)andtheenergycriterionoffracturemechanics,aswellas thestrengthcriterion,mustbeappliedgloballyoverregionsthatmea-sureatleastseveralaggregatesizes.Thisisclearevenwithoutexperi-mentalevidence,sincestructuralanalysisimpliesthehypothesisof smoothingofaheterogeneousmaterialbyanequivalenthomogeneous continuuminwhich,ifoneusesthelanguageofthestatisticaltheory ofrandomlyheterogeneousmaterials,thestressesandstrainsmustbe understoodastheaveragesoftheactualstressesandstrainsinthemi-crostructureovertheso-calledrepresentativevolumewhosesizemust betakento beatleastseveraltimesthesizeoftheinhomogeneities.On asmallerscale,thecontinuumconceptsaremeaningless. Whenthefractureprocesszoneis notnegligiblysmall,onemusttake intoaccountthestress-strainrelationofthematerial,includingthede-cline ofstressto zero at very large strains(strain-softening).Its twoprin-cipalcharacteristics,whichseemsufficientformostpracticalpurposes, arethetensilestrength,/,'(determinedasthepeakstressindirectten-siontests),andthefractureenergy,b/2, Fig.1(g)] orverydensecracks[s b/2,Fig.l(z)j,but notthecaseinFig.1(h)forwhichs/b-1.Forthissituation,wewill thereforehavetodeviseacertainapproximateinterpolationtechnique. Itmustbeadmittedthatthemethodofstresslinesisoftencorrect only intheorderofmagnitude,anda twoor three-folderrorispossible. However,therelativeerrorremainsaboutthesameforsimilargeome-tries,andthemethodgivescorrectstructureoftheexpressionforthe energyrelease,exceptfora numericalfactor.Sincewewillcalibratethe numericalfactorbytestdata,themethodseemsacceptable. It is well knownthatthebondshearstressesmaycausethesteelbars toslip.Tomakeexplicitformulasattainable,wewillassumethateither thereisnobondsliporthebondslipoccursovertheentirelengthof thebarandthebondshearforceperunitlength,Fb,isconstantand equalto itsultimatevalue,FJ,. Inreality,Fb is ofcoursevariable; itequals 2069 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.zeroatintersectionswiththecracksandatthemidlengthbetweenthe cracks,causingthelengthLb ofbondsliptobecomeinrealitylessthan thecrackspacing.Furthermore,thevalueofF'b neednotbethesame asthatfrompullouttests.Becausewedealherewithamuchshorter bondsliplengthandsmallerslipdisplacements,F'b canbehigherthan inthesetests. Doweneedtoconsiderthatatransitionfromano-slipsituationto bondslipcanoccurduringcrackformation?Withinourassumptionswe donot,aslongasweassumethebondsliptooccurovertheentire lengthofthebarbetweenthecracks.Weconsiderthecracktoformat aconstantoveralldeformation,andsothecrackformationgenerallyre-lievesthestresses.Afterthecrackspacingishalved,thesituationis geometricallysimilarandstressdistributionisthenalsosimilarforour assumptions.Thus, if there is no bondslip beforecracking,there isnone aftercracking inourmodel.Anexceptionistheformationoffirstcracks duringwhichthebondslipcanoccur.Althoughitwouldbepossibleto analyzethiscasewithourmethod,weomititforthereasonsalready explained. ConditionsforInitiationofMicrocrackBands.Ifthereisnobond slip,thestrengthcriterionrequiresthat es >|(no slip)(2) in which/',=tensilestrengthofconcrete;andEc =its Young'smodulus. (Thisconditionalsoappliestotheformationoffirstcracks.) Ifthereisbondslip,thecracksinitiateonlyifthestressproducedin segment01[Fig.l(g,i)]bythebondforceaccumulatedoverthebond sliplength,Lb,equalsf't.Wenowdistinguishthecaseofverysparse cracks(sb/2)andverydensecracks(sb/2).Forsb/2[Fig. 1(g)],theequilibriumconditionisF'bLb=-n{b2-D2)/',/4,andsinceLb ^s,wemaywriteforsb/2: IT/;4 s^'h^^ For s b/2[Fig.l(i)],theequilibriumconditionis F'bLb= tr[{2ks+Df -D2]/'(/4,andsinceLb

2(s),inwhich2(s)=asmoothfunctionwhichasymptotically approachesh2whens>>b/2,andg2(s)whensb/2[Fig.2(a)].A simplefunctionwhichhasthesetwoasymptoticpropertiesiscj>2(s)= \hl+g2(s)"]v"-(Fromtestdatafitting,thevaluen= 4 wasfoundtobe reasonable.)Thus,foranys/b V ALZ =- Ece2;V= 247c tr(& -D)2(2b+D) \trks2(ks+D) n-i-1/n (no slip)(10) isobtained. Supposenowthatthesteel barsslip. Thestress,o-j, inconcreteinthe crosshatchedregionsofFig.l(g,i)beforecrackingnolongerisEcesand mustbefiguredoutfromtheultimatebondforces,Fj , ,appliedoncon-cretealongthebar[Fig.1(/)].Theirresultantis F'bs, andsotheaverage stressonthecrosssection,01,inFig.1(/)orFig.l(g,i)is,approxi-mately,fors>>b/2: 0(fulllength)(23) Thisconditiondoesnotinvolvees,andsothesolutionisafixedlower limitonthespacingofthecracksoffulllength[smlninFig.2(b,c)].The left-handsideofEq.23hastheformP(s)/s3inwhichP(s)=asixth-order polynomial in s.Thesolutionof Eq.23 can beobtainedbyNewton iteration. AnapproximationtothesolutionofEq.23 can,however,beobtained byreplacingEq.13 withthelimitingvalueforsparsecracks,SpecimenNo. S2S a / (c)SpecimenNo. S3/ it1iIIt (d)SpecimenNo. S4S So 0OO040.00080.00120.00160.00040.0008 SteelSt r ai n,es 0.00120.0016 (e)BeamNo.30 _Theory 0Hognestad(1962) -""o (g)BeamNo.1-22 _Theory 0Mathey, Watstein (I960)jy' -? 3-***' aooo,/es inwhiches=steelstrain[Fig.7(b)]. Thecoefficientofvariation oferrorsinthistypeofregressionisfoundtobew =0.126. The 95% confidencelimitscorrespondingtowareplottedas thedashed curvesinFig.7(a,b). Thesecurvesarehyperbolas,butduetothelarge 2080 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.TABLE1.ParametersforTestData TestSeries (D 1.Clark 2.Chi,Kirstein 3.Kaar,Mattock 4.Hognestad 5.Mathey,Watstein Ec, in thousand poundsper squareinch (2) No.13,212 23,192 33,457 41,912 No.14,277 23,713 33,422 41,500 No.12,891 23,149 33,028 43,120 No.12,514 23,312 No.13,246 23,272 %,in pounds perinch (3) 0.505 0.411 0.412 0.393 0.684 0.661 0.611 0.204 0.287 0.304 0.524 0.217 0.109 0.328 0.251 0.259 /',in poundsper squareinch (4) 385 357 367 290 476 447 422 207 259 273 329 240 170 286 299 303 d>in inches (5) 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.0 1.0 Fi,in pounds perinch (6) 3,876 3,499 3,199 3,783 5,950 6,426 6,391 2,771 3,470 3,504 3,868 3,735 2,830 4,173 3,226 3,187 F?, l n pounds perinch (7) 3,895 3,619 3,040 3,743 5,805 6,299 6,322 2,328 3,819 3,676 4,189 3,885 2,888 4,142 3,344 3,287 Note: psi=6,895 N/m2; lb/in.=175.1 N/m;in.=25.4 mm; ksi=1,000psi. All numbers except d, hadto be estimated,in order to get optimum fit,because they werenot reportedby theexperimentalists. size ofourstatisticalsamplestheyarealmoststraight,inwhichcasethe 95%confidencelimitsrelativetomeanY areapproximately1.96co. WeseefromFig.7thatourtheoryachievesasatisfactoryagreement withtestdata. Asfortheerrorsinthecrackspacing,wemaycharacterizethemby thecoefficientofvariation,cos,ofthepopulationofthevaluesofsm/st, s, andst beingthemeasuredandtheoreticalspacings.Fromallthedata pointsinFig.4wegetcos=0.145.Weshouldnote,however,thata statisticalcomparisonoftheerrorsinspacingbasedonFig.4isoflim-itedrelevancesince,as wementionedbefore,thespacingdataavailable involvesomeirregularspacings,donothaveasufficientrangetoshow thehalvingofspacing,anddonotpermitshowingthechangesofreg-ularspacing withsteelstrain.Because of probableirregularspacing,the firstandthe lastdatapoints in Fig. 4 werenot includedin thecalculation ofo)s givenpreviously. Tocalculatethefractureenergy,"fy,therecentlydevelopednewfor-mulafromRef.3hasbeenused. Thevaluesofultimatebondforce,F'v,obtainedastheoptimaforfit-tingthedata(Table1) maybeanalyzedtoseewhethertheyshowsome systematicpatternandcanbeapproximatelypredicted.Assumingthat F[ dependsonthecompressivestrength,f'c,ofconcrete,andcarrying outa least-squareregressionanalysisoftheoptimumvaluesofFj, listed inTable1,wecanobtainthefollowingapproximateformulafortheul-timatebondforceperunitlengthofadeformedsteelbar: 2081 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.00.00240.00480.00720.00960.012024681012 X = wtX = t /eB FIG.7.PlotofMeasured versusTheoreticalValues forCrack WidthBased on Figs. 5-6 Ft= 0.95/^(34) inwhichF% =predictedvalueofF{,f'c mustbeinpsi,andF%isinl b/ in.(1psi=6,895Pa,1lb/in.=175.1N/ m).Notethatthevaluesof FJandF(,aregenerallylarger(roughlyby70%)thantheusualbond force,U'h,determinedbypullouttestsandusedindeterminingthede-velopmentlengthofsteelbars.Thisisbecausewedealherewithalo-calizedbondslipoccurringonlyoverashortbarlength. Forall16 datasetslistedinTable1,theerrorsofEq.34,i.e.,Ft-F{, (withFtandF'hlistedinTable1),havethecoefficientofvariationf= 0.051,whichseemsacceptable. FormationofSkewCracks.Generalloadingcanproduceasystem of parallel cracks whichare not normal to thereinforcingbars.Auniaxial appliedstressinthebardirectionwillobviouslyalwaysproducecracks normaltothebars,andsotheskewcrackscanonlybeproducedunder multiaxialappliedstress,inwhichcasethereinforcementwouldnor-mallybedesignedasanorthogonalmeshratherthanasystemofpar-allelbars. Consideranorthogonalregularreinforcingnet withidentical barspac-ingsanddiametersforbarsofbothdirection[Fig.2(i-n)].Supposethat thecracksformatangle45 withthebars[Fig.2(i-n)].Toproducesuch cracks,theaxialstrainsinthebarsofbothdirectionsmustbethesame. Beforethefirstcracking,theconcretewouldberoughlyinastateof biaxial tension,ax= ay. Formationof a crack in y-directionwouldrelieve someofthestress,ux,butwouldhavenoeffectoncy.Applyingagain themethodofinclinedstresslines(withk=1),wemayconsiderthat theformationofverticalcracksinFig.2(;'-Z)wouldrelievestress,ax, fromtheverticallycrosshatchedtriangularregionsandwouldhaveno effectonthestressintheremainingregions. We may nowcomparetwo possiblecrack locations: thecracksrunning throughthe barcrossings[Fig.2(/)],andthecracksrunninginthemid-dlebetweenthesecrossings[Fig.2(l,m)].Comparingtheareasofthe 2082 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.regionsof reliefof axforbothcases(verticallycrosshatchedregions),we seethattheirareaissomewhatsmallerforthesecondofthesetwocases [Fig. 2(l,m)]. Therefore,thestrainenergyrelease,AU,forcracksrunning throughthebarcrossingsappearstobelarger,andsothisisthetype ofcrackswhichshouldform[Fig.2(j,k)],Thisnowrestrictsthepossible spacingstos=nb/V2inwhichn=1,2,3,Wenotehoweverthat AU percrackisthesameforallthesecrackspacings. Therefore,cracksofthedensestspacing[Fig. 2(k)]: S^ 2:P 5 ) shouldformimmediatelywhentheminimumnecessaryaxialstrain,es, inthebarsisreached.Forthisspacing,thestress,0,andsoalargerstrainisrequiredto producethesecracks. Subsequentincreaseofstraincannot,accordingtoouranalysis,pro-ducefurthercontinuousskewcracks.Itcan,however,produceshorter, discontinuouscracksofthetypealreadyanalyzed[Fig.2(n)].Theseare likelynottobeparalleltothecontinuouscracks. Thereare manymoreinterestingquestionswithregardto skewcracks underbiaxialstress,buttheyarebeyondthescopeofthisstudy. SUMMARYANDCONCLUSIONS Thespacingandwidthofcracksinaparallelcracksystemisapprox-imatelyanalyzedusing theenergycriterionof fracturemechanicsaswell asthestrengthcriterion.Followingpreviousworksonrocks(8,10),the energycriterionisappliedhereinanovelwayintegrally,considering theformationoftheentirecrackasoneevent.Forcracksthatarenot longenoughcomparedtotheaggregatesize,thisapproachappearsto bemorerealisticthantheusualfractureanalysiswhichpertainstoen-ergy balanceat infinitesimalcrack lengthincrements(i.e., balanceofen-ergyreleaseandconsumptionrates).Theenergycriterioninvolvesnot onlythereleaseofstrainenergyandtheenergyconsumedtoproduce thecrack,butalsotheenergyconsumedbybondslipduringcracking (ifany). Theenergycriterionindicatesthatthecrackspacingisafunctionof theaxialstrainofthebarsandalsodependsonbarspacing,bardi-ameter,fractureenergyofconcrete,anditselasticmodulus.Boththe energyandstrengthcriteriayieldaminimumstrainnecessarytopro-2083 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.duceanycracks.Theenergycriterionandt hebondslipcondi t i onsalso yieldalowerboundonpossiblespacingofcont i nuouscracks.Therul es fortheformationofshorter,partiall engt hcracksarealsosetup.Ap-proximateexpressionsfort hecrackwi dt hat,andaway,fromt hebars arederived. Numericalcompari sonsindicatesatisfactoryagreementwi t hexisting testdata,andalsol endtheoreticalsuppor ttooneaspectoft heempirical Gergely-Lutzformulapreviouslyobtainedbystatisticalregressi onanal-ysisoftestdata.Finally,formationofskewcracksinabiaxiallystressed andbiaxiallyreinforcedplateisanalyzedandacrackspaci ngformulais derivedforonetypicalcase. ACKNOWLEDGMENT FinancialsupportunderU. S.NationalScienceFoundat i onGrantNo. CEE8009050toNort hwest ernUniversityisgratefullyacknowledged.Mary Hillistobet hankedforherexcellentsecretarialassistance. APPENDIX.REFERENCES 1.Albandar,F.A-A.,andMills,G.M.,"ThePredictionofCrackWidthsin ReinforcedConcreteBeams,"Magazine of ConcreteResearch,Vol.26,No.88, Sept.,1974,pp.153-160. 2.Base,G.D.,Read,J.B.,Beeby,A.W.,andTaylor,H.P.J.,"AnInvesti-gationoftheCrackControlCharacteristicsofVariousTypesofBarinRein-forcedConcreteBeams," Research Report No.18, Parts1, 2,CementandCon-creteAssociation,London,England,Dec,1966. 3.Bazant,Z. P.,andOh,B. H.,"Concrete Fracturevia Stress-StrainRelations," Report No.81-10/665,CenterforConcreteandGeomaterials,Northwestern University,Evanston,111., Oct.,1981. 4.Bazant,Z.P.,andOh,B.H.,"CrackSpacinginReinforcedConcrete:Ap-proximateSolution,"Journalof Structural Engineering,ASCE,Vol.109, No.9, Sept.,1983, pp.2207-2212. 5.Bazant,Z.P.,andOh,B. H.,"DeformationofCrackedNet-ReinforcedCon-creteWalls,"Journal ofStructuralEngineering, ASCE,Vol.109,No.1,Jan., 1983,pp.93-108. 6.Bazant,Z.P.,andOhtsubo,H.,"StabilityConditionsforPropagationofa SystemofCracksinaBrittleSolid,"MechanicsResearchCommunications, Vol. 4,No.5,Sept.,1977,pp.353-366. 7.Bazant,Z.P.,andRaftshol,W.J.,"EffectofCrackinginDryingandShrink-ageSpecimens,"Cement and ConcreteResearch,1982(inpress). 8.Bazant,Z.P.,andWahab,A.B.,"InstabilityandSpacingofCoolingon ShrinkageCracks,"Journalofthe EngineeringMechanicsDivision,ASCE,Vol. 105,No.EM5,Proc.Paper14933,Oct.,1979,pp.873-889. 9.Bazant,Z.P.,andWahab,A.B.,"StabilityofParallelCracksinSolidsRein-forcedbyBars,"InternationalJournal ofSolids andStructures,Vol.16,1980, pp.97-105. 10.Bazant,Z.P.,Ohtsubo,H.,andAoh,K.,"StabilityandPost-CriticalGrowth ofaSystemofCoolingorShrinkageCracks,"International Journalof Fracture, R15'No'hc t"1 9 7 9 ,pp-443~456' 11.moras,B. B., "CrackWidthandCrackSpacingin ReinforcedConcreteMem bers," Journalof the AmericanConcreteInstitute,Proceedings,Vol.62,No.10, Oct.,1965, pp.1237-1256. 12.Brorris, B. B., andLutz,L.A.,"EffectsofArrangementofReinforcementon CrackWidthandSpacingofReinforcedConcreteMembers,"Journal of the American ConcreteInstitute,Proceedings,Vol.62,No.11, Nov.,1965. 2084 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.13.CM, M.,andKirstein,A.F.,"FlexuralCracksinReinforcedConcreteBeams," Journalof the AmericanConcreteInstitute,Proceedings,Vol.54,No.10,Apr., 1958,pp.865-878. 14.Clark,A.P.,"CrackinginReinforcedConcreteFlexural Member,"Journalof the AmericanConcreteInstitute,Proceedings,Vol.52,No.8,Apr.,1956,pp. 851-862. 15.Fung,Y.C,FoundationsofSolid Merchanics, PrenticeHall,Inc.,Englewood Cliffs,N.J.,1965. 16.Gergely,P.,andLutz,L.A.,"MaximumCrackWidthinReinforcedCon-creteFlexuralMembers,"Causes,Mechanism,andControl of Crackingin Con-crete, SP-20, AmericanConcreteInstitute,Detroit,Mich.,1968,pp.87-117. 17.Hillerborg,A.,Mod6er,M.,andPetersson,P.E.,"AnalysisofCrackFor-mationandCrackGrowthinConcretebyMeansofFractureMechanicsand FiniteElements,"Cementand ConcreteResearch,Vol.6,1976,pp.773-782. 18.Hognestad,E.,"HighStrengthBars asConcreteReinforcement,Part 2.Con-trolofFlexuralCracking,"Journalofthe Portland Cement AssociationResearch and DevelopmentLaboratories,Vol.4,No.1,Jan.,1962,pp.46-63. 19.Kaar,P. H.,andHognestad,E.,"HighStrengthBarsasConcreteReinforce-ment,Part7.ControlofCrackinginT-BeamFlanges," Journalof thePortland Cement AssociationResearchandDevelopmentLaboratories,Vol.7,No.1,Jan., 1965,pp.42-53. 20.Kaar,P.H.,andMattock,A.H.,"HighStrengthBarsasConcreteRein-forcement,Part4,ControlofCracking,"Journalofthe PortlandCement Asso-ciation Research and Development Laboratories,Vol.5, No.1, Jan.,1963, pp.15-38. 21.Knott,J.F.,Fundamentalsof FractureMechanics, Butterworths,London,En-gland,1973. 22.Mathey,R.G.,andWatstein,D.,"EffectofTensilePropertiesofReinforce-mentontheFlexuralCharacteristicsofBeams," Journalof the American Con-creteInstitute,Proceedings,Vol.56,No.12, June,1960,pp.1253-1273. 23.Meier,S.W.,andGergely,P.,"FlexuralCrackWidthinPrestressedCon-creteBeams," Journalof the StructuralDivision, ASCE,Vol.107, No.ST2, Proc. Paper16010,Feb.,1981, pp.429-433. 24.Nawy,E.G.,"CrackControlinReinforcedConcreteStructures," Journalof the AmericanConcreteInstitute,Proceedings,Vol.65, No.10,Oct.,1968,pp. 825-836. 25.Park,R.,andPaulay,T.,ReinforcedConcreteStructures, JohnWileyandSons, NewYork,N.Y.,1975(Section10.4). 26.Petersson,P.E.,"FractureEnergyofConcrete:MethodofDetermination," Cementand ConcreteResearch,Vol.10,1980, pp.78-89,and"FractureEnergy ofConcrete:PracticalPerformanceandExperimentalResults,"Cementand ConcreteResearch,Vol.10,1980,pp.91-101. 27.Sneddon,I.N.,andLowengrub,M.,Crack Problemsinthe ClassicalTheoryof Elasticity,JohnWileyandSons,NewYork,N.Y.,1969(p.134). 28.Tada,H.,Paris,P.C,andIrwin,G.R.,"TheStressAnalysisofCracks Handbook,"DelResearchCorp.,Hellertown,Pa.,1973. 29.Watstein,D.,andMathey,R.G.,"WidthofCracksinConcreteattheSur-faceofReinforcingSteelEvaluatedbyMeansofTensileBondSpecimens," Journalofthe American ConcreteInstitute,Proceedings,Vol.56,No.1,July, 1959,pp.47-56. 30.Watstein,D.,andParsons,D.E.,"WidthandSpacingofTensileCracksin AxiallyReinforcedConcreteCylinders," Journalof Researchof the National Bu-reauof Standards,Vol.31, July,1943, No.RP545,pp.1-24. 2085 J. Struct. Eng. 1983.109:2066-2085.Downloaded from ascelibrary.org by Maharaj Vijayaram Gajapathi Raj College Of Engineering (MVGR) on 07/11/15. Copyright ASCE. For personal use only; all rights reserved.