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A General Shear Design Method

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ACI STRUCTURAL JOURNAL Title no. 93-S5 TECHNICAL PAPER A General Shear Design Method by Michael P. Collins, Denis Mitchell, Perry Adebar, and Frank J. Vecchio A simple. unified method is presented for the shear design of both prestressed COncrete members and nonprestressed concrete members. The method can treat members subjected to axial tension or axial compression and treats members with and without web reinforcement. The derivation of the method is summarized and the predictions of the met
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  ACISTRUCTURALJOURNALTECHNICALPAPER Titleno.93-S5 AGeneralShearDesignMethod byMichaelP.Collins,DenisMitchell,PerryAdebar,andFrankJ.Vecchio Asimple.unifiedmethod is presentedforthesheardesignofbothpre-stressedCOncretemembersandnonprestressedconcretemembers.Themethodcantreatmemberssubjectedtoaxialtensionoraxialcompressionandtreatsmemberswithandwithoutwebreinforcement.Thederivationofthemethod is summarizedandthepredictionsofthemethodarecomparedwiththoseofthecurrentACICode. Keywords:aggregateinterlock;axialloads;buildingcodes;crackwidthandspacing;reinforcedconcrete;shearstrength;structuraldesign. Thesheardesignprovisionsofthe1995ACICode 1 con-sistofabout43empiricalequationsfordifferenttypesofmembersanddifferenttypesofloading,someofwhichareillustratedinFig. I. In1973,theACI-ASCEShearCommittee/expressedthehopethatthese designregula-tionsforshearstrengthcanbeintegrated,simplified,andgivenaphysicalsignificance. AsshownbythegrowthinthenumberofACIsheardesignequations(seeFig.2),thecodehasnotmetthisdesirablegoal. It isinterestingtonotethat,priorto1963,theACIsheardesignprocedurewassosimplethatonlyfourequationswererequired.MostofthesheardesignequationsgiveninFig.1werein-troducedineitherthe1963or1971editionoftheACICode.v Thesedesignequationsweredevelopedintheperiodfollow-ingthe1955air-forcewarehouseshearfailures''andrelyonthetraditionalconceptofaddingaconcretecontribution Vc totheshearreinforcementcontribution Vs calculatedonthebasisofthe45degtrussequation.Since1971therehasbeenanintensiveresearcheffortaimedatimprovingdesignmethodsforshear(seeFig.3).There-searchhasshownthat,ingeneral,theangleofinclinationoftheconcretecompressionisnot45deg,andthatequationsbasedonavariableangletrussprovideamorerealisticbasisforsheardesign.Inaddition,testsofreinforcedconcretepan-elssubjectedtopureshear improvedtheunderstandingofthestress-straincharacteristicsofdiagonallycrackedconcrete.Thesestress-strainrelationshipsmadeitpossibletodevelopananalyticalmodel,calledthemodifiedcompressionfieldtheory,thatprovedcapableofaccuratelypredictingthere-sponseofreinforcedconcretesubjectedtoshear. 36 Theobjectiveofthispaperistopresentbrieflyasimple,generalsheardesignmethodbasedonthemodifiedcompres-sionfieldtheory.Thisdesignmethod,recentlyintroducedbyCollinsandMitchell,7hasbeenadoptedbytheOntarioHighwayBridgeDesignCode,StheCanadianStandardsAs-sociationConcreteDesignCode,9andtheAASHTOLRFDspecifications. 10 ThemethodissummarizedinFig.1. SHEARRESPONSEOFCRACKEDCONCRETE Testsofreinforcedconcretepanelssubjectedtopureshear(seeFig.4)demonstratedthatevenaftercracking,tensilestressesexistintheconcreteandthatthesestressescansig-nificantlyincreasetheabilityofreinforcedconcretetoresistshearstresses.Crackedreinforcedconcretetransmitsloadinarelativelycomplexmannerinvolvingopeningorclosingofpre-exist-ingcracks,formationofnewcracks,interfacesheartransferatroughcracksurfaces,andsignificantvariationofthestressesinreinforcingbarsduetobond,withthehigheststeelstressesoccurringatcracklocations.Themodifiedcompressionfieldmodelattemptstocapturetheessentialfeaturesofthisbehaviorwithoutconsideringallofthede-tails.Thecrackpatternisidealizedasaseriesofparallelcracksalloccurringatangle8tothelongitudinaldirection.Inlieuoffollowingthecomplexstressvariationsinthecrackedconcrete,onlytheaveragestressstateandthestressstateatacrackareconsidered[seeFig.4(b)and4(c)].Asthesetwostatesofstressarestaticallyequivalent,thelossoftensilestressesintheconcreteatthecrackmustbereplacedbyincreasedsteelstressesor,afteryieldingofsomeofthereinforcementatthecrack,byshearstressesonthecrackin-terface.Theshearstressthatcanbetransmittedacrossthecrackwillbeafunctionofthecrackwidth.Notethatshearstressonthecrackimpliesthatthedirectionofprincipalstressesintheconcretechangesatthecracklocation. ACIStructuralJournal, V. 93,No.1,January-Febuary1996.ReceivedJune17,1994,andreviewedunderInstitutepublicationpolicies.Copy-right©1995,AmericanConcreteInstitute.Allrightsreserved,includingthemakingofcopiesunlesspermissionisobtainedfromthecopyrightproprietors.Pertinentdis-cussionwillbepublishedintheNovember-December1996 AC!StructuralJournal ifreceivedbyJulyI,1996. ACIStructuralJournal/January-Febuary1996  Michael P. Collins,FACI,isBahen-TanenbaumProfessorofCivilEngineeringattheUniversityofToronto,Toronto,Canada,HeisamemberofACICommittee 358, Con-creteGuidways,theAC!TechnicalActivitiesCommitteesubcommitteeonHigh-Per-formanceConcrete,andjointACI-ASCECommittee 445, ShearandTorsion,He is amemberoftheCanadianStandardsAssociationCommitteefortheDesignofCon-creteStructures.DenisMitchell,FAC!,isaprofessorintheDepartmentofCivilEngineeringandAppliedMechanicsatMcGillUniversity,HeisamemberofACICommittee408,BondandDevelopmentofReinforcement,andACI-ASCECommittee 445, ShearandTorsion.HeisChairmanoftheCanadianStandardsAssociationCommitteefortheDesignofConcreteStructures.ACImemberPerryAdebarisanassociateprofessorintheDepartmentofCivilEngi-neeringattheUniversityofBritishColumbia,Vancouver,Canada,HeisSecretaryofJAC!Committee 341, EarthquakeResistantConcreteBridges,andjointAC1-ASCEcomputer-aideddesignofreinforcedconcrete,AC1memberFrank J. VecchioisaprofessorintheDepartmentofCivilEngineeringattheUniversityofToronto.HeisamemberofACICommittees 441. ReinforcedCon-creteColumns.and 447. FiniteElementAnalysis.andoftheCEBCommitteeonCon-stitutiveModeling. Theaverageprincipaltensilestrain £1 inthecrackedcon-creteisusedasa damageindicator thatcontrolstheaver-agetensilestress f1 inthecrackedconcrete,theabilityofthediagonallycrackedconcretetocarrycompressivestressesjj,andtheshearstress v ci thatcanbetransmittedacrossacrack.Theprincipalcompressivestressintheconcretejjisrelat-edtoboththeprincipalcompressivestrain £2 andtheprinci-paltensilestrain £1 inthefollowingmanner[seeFig.5(a)] (I) where(2) ACIMethodGeneralMethod V = Vc+ V, vn = Vc + v, + vp Non-Prestressed = (1.9 fi: + V. d)v d :5 {3fi:bvd. Beams Vc 2500pw ~II bwd but M. 1.0 V.~ '::S.@f:~~::{:::~::'t~::::::::::::~:' Vc :5 3.5fi:bwd or V = 2ffb..d V. = Av.t;·d cot6 r .I'ã V. = A.lyd V. :5 8 fl: bwd where {3 and €I arefunctions v V s ofthestrain, ex' shearstress, v, PrestreosedBeams =( 0.6ff + 700 v d)bd but 2fi:b..d .:5 5fi:b w d andcrackspacing sr VcVc .:5 where 11l w Vn-~ If II(r .= or Vc = Vci = 0.6[J;b..,d + Vd +-- but Vci ~ 1.7fl!'b~,dbvdv l~ Mmar andand Vc <; Vcw = (3.5fl!' + O.3~c)b..d+ Vp € = M idv+0.5(N.+V.cotB)-ApJpu UZ@!t; _Avlyda[J;b.,d r I?,A. + I?pAp v V. <; --s- AxialCompression = (:.9/17 + v d)bd ndShear Vc2500pw 'f Mu - N(4h-d)' u S 3.5/17b.,dj1 N Vc :5 +__ u_ SOOA g V = Avfyd sfl: bwd . s s AxialTension V< 2(1 + 50:uA )17: bwd ndShear = ! 1 e -,tmt:~~::::::t?:::'::::::'~:~:~:~:::::f _A.lyd sfl: b,.d V. --s- :5 V v DetailingRulesDetailingRules ã ReinforcementshallextendbeyondthepointatwhichitisnolongerrequiredtoresistflexureLongitudinalsteelmust be foradistanceequaltotheeffectiveclepthofthememberor 12d., whichisgreater,..detailedsothat ã Flexuralreinforcementshallnot be terminatedina!ensionzoneunless MN A,ly + Ap,jp< ~ __II + 0.5 ___!!_ ã shearatcutoffs2/3shearpermitted,or <Pd. tjJ ã stirruparea, A inexcessofthatrequiredforshearandtorsion,isprovided,.. +(~ -o.Sl{-l-;;)COI9 ã..Av~ 60b ,s/j;...s :5 d/8{3b' or ã for#11barsorsmaller:shearatthecutoff <; 3/4shearpermittedandcontinuingreinforcementprovidesdoublethearearequiredforflexureatthecutoff. ã Atsimplesupportsandpointsofinflection,thediameteroftheposltlvemomenttensionreinforcementshall be limitedsothat 'W n ld :5 v+a Fig.L=ComparisonofACIandproposedsheardesignapproaches ACIStructuralJournal/January-Febuary1996 37  50.-----------------------------------------------------------------~ _v v-bwjd NUMBEROFEQUATIONSFORSHEARDESIGNINACICODE A, (v-v,lb.. T=----- 190019101920193019401950 Fig.2-NumberofACIsheardesignequations 19601970198019902000 (c)Localstressesatcrack £2 = -0.002(1- Jl- f/f2maJ 38 NUMBEROFPAPERSONSHEARDESIGNPUBLISHEDINACIJOURNALINEACH5YEARPERIOD Slater,LordandZipprodtU.S.BureauofStandards 172 tests l RichartGratandBachillinoisSlullgartBullelln166207 tests 139 tests l Talbot'sIllinoisBUlletin 29 Ritter's 188 testsSwissPaper ll AirForceWarehouseFailure l MacGregorACI·ASCE 326 Report 57 references924testsACI.ASCE426 l Report202references l Fig.3-Researchintosheardesignmethods (a)Panelloaded in Shear(b)Calculatedaveragestresses where £c' hasbeentakenas-0.002.Aftercracking,theprincipaltensilestressintheconcrete f, isrelatedtotheprincipaltensilestrain £, asfollows[seeFig.5(b)] t, (4) 1+J500£] wherethecrackingstress fer canbetakenas4 JJ:' psi (0.33JJ:' MPa).Forlargevaluesof £ thecrackswillbecomewideandthemagnitude off] willbecontrolledbytheyieldingofthereinforcementatthecrackandbytheabilitytotransmitshearstresses Vel acrossthecrackedinterface[seeFig.5(b)].Theshearstressthatcanbetransmittedacrossthecrackisafunctionofthecrackwidth w andtheaggregatesize a [seeFig.4(c)],asgivenby 2.16JJ:' psiandin. (5) Fig.4-Reinforcedconcretepanelssubjectedtoshear From Eq. (1), theprincipalcompressivestrainfortheloadingportionofthestress-strainrelationshipis0.3 + 24w a + 0.63 (3) ForMPaandmmunits,replacethe2.16by0.18andthe0.63by16. ACIStructuralJournal/January-Febuary1996  (a)Softeningofcompressivestress-straincurveduetotransversetensilestrain £ 1atcrackslip (b)Averagetensilestressesincrackedconcrete asa functionof £1 Fig.5-Stress-strainrelationshipsforcrackedconcrete If thestirrupshavereachedtheiryieldstressandthecrackbeginstoslip,theaveragetensilestressinthecon-crete fl islimitedto (6) Thepreviousstress-strainrelationships,togetherwithequilib-riumandcompatibility,canbeusedtopredicttheload-deforma-tionresponseofreinforcedconcretebeamssubjectedtoshear. 11 Inaddition,theserelationshipscan be usedasthebasisfornon-linearfiniteelementformulations. 12,13 DESIGNOFSTIRRUPSFORSHEAR In applyingthemodifiedcompressionfieldtheorytothede-signofbeams,itisappropriatetomakeanumberofsimplifyingassumptions.AsillustratedinFig.6,theshearstressesareassumedtobeuniformovertheeffectivesheararea b.d.; Thehighestlongitudinalstrain Ex occurringwithinthewebisusedtocalculatetheprincipaltensilestrain EI' Fordesign, Ex canbeapproximatedasthestrainintheflexuraltensionre-inforcement.Thedeterminationof Ex foranonprestressedbeamisillustratedinFig.7.Foraprestressedconcretemem-ber,theconcretesurroundingthereinforcementwillremainincompressionuntiltheappliedtensionexceedsthepre-stressforce Apsfpo, where.~]Oisthestressinthetendonwhenthesurroundingconcreteisatzerostress.Inlieuofmoreac-curatecalculations, fpo canbetakenas1.10times L; Hence,fordesign ACIStructuralJournal/January-Febuary1996 (a) Cross-section (b) Shearstress (c) Longitudinalstrains (d) Biaxialstrainsinweb (e) Tensioninwebreinforcement Fig.6-Beamsubjectedtoshear,moment,andaxialload MC-C t As -T Moment =Mid v VU1~-O.5NV~-O.5Nv Shear = O.5V ucote ~-O.5NU +;;; 1:bbbdd- O.5Nu AxialLoad Fig.7-DeterminationofstrainExfornonprestressedbeam (M/d) + 0.5N u +0.5V u cote-A p/po > (7) EA+EA_0 ssp ps atibility,theprincipaltensilestrain EI canberelatedtothelongitudinalstrain Ex, thedirectionoftheprincipalcompressivestress e, andthemagnitudeoftheprincipalcompressivestrain E2 inthefollowingmanner (8) Hence,asthelongitudinalstrain Ex becomeslargerandtheinclination e oftheprincipalcompressivestressesbecomessmaller,the damageindicator El becomeslarger.Thenominalshearstrength Vn ofamembercanbeexpressedas 39
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