reinforced concrete corbels-shear strength model and design formulae

ACI Structural Journal/January-February 2006 3ACI Structural Journal, V. 103, No. 1, January-February 2006.MS No. 03-339 received August 6, 2003, and reviewed under Institute publicationpolicies. Copyright 2006, American Concrete Institute. All rights reserved, including themakingofcopiesunlesspermissionisobtainedfromthecopyrightproprietors.Pertinentdiscussionincludingauthorsclosure,ifany,willbepublishedintheNovember-December 2006 ACI Structural Journal if the discussion is received by July 1, 2006.ACI STRUCTURAL JOURNAL TECHNICAL PAPERAnewmodelfordeterminingtheshearstrengthofreinforcedcon-crete (RC) corbels, or brackets, is proposed in this paper. The modelis obtained by superimposing the shear strength contribution of thestrut-and-tiemechanismduetothecrackedconcreteandprincipalreinforcement, and the strength contribution due to stirrups. The firstcontributionisexpressedbymeansofalimitingshearstrengthexpression,whereasthesecondisderivedfromtheequilibriumofthe strut-and-tie mechanism in the presence of stirrups. An explicitformuladependentontwocoefficientsisderivedfortheshearstrength of corbels. These two constants are calibrated on the resultsof 243 test data, which can be found in the relevant literature. Theexpression obtained in this way is compared to the ACI Code and themost recently proposed formulas and computing procedures, and itresults as better fitting the measured shear strengths. On the basis ofresults of this paper, a design formula is proposed.Keywords:bracket;corbel;reinforcedconcrete;shear;strength;stirrup;strut-and-tie.INTRODUCTIONCorbels,orbrackets,arecantileverswithashearspan-depth ratio lower than unity, generally jutting out from wallsor columns. They have the principal function of supportingprefabricated beams or floors at building joints, allowing, atthe same time, the force transmission to the vertical structuralmembers. Corbels are principally designed to resist the ultimateshear force Vu applied to them by the beam, and the ultimatehorizontalactionNuduetobeamshrinkage,creep,ortemperature changes.The principal failure modes1,2 for members without stirrupsare: 1) shear failure; 2) yielding of the principal reinforcement(flexuraltension);3)crushingofconcretestrut(flexuralcompression);and4)diagonalsplitting.Incorbelswithsecondaryreinforcement(stirrups),whichisalwaysrecommended,1-4allthefailuremodesmentionedpreviouslytendtoconvergeintoasingletypologyoffailuremodecalledbeam-shearfailure.Thelastoneischaracterizedbytheopeningofoneormorediagonalcracksfollowedbyshear failure in the compressed zone of the strut.2Duetothevariabilityinthenatureoffailuremodes,theidentificationofmechanicalbehaviorofcorbelsatfailureand the evaluation of their shear strength are very complex,asshownbypreviousstudies.1,2,5-9Thisevaluationis,atpresent,performedbymeansofashear-frictionmethod,3strut-and-tie models,6-8 or an iterative procedure.9Themodelproposedhereinisbasedontheequilibriumconditionsofthestrut-and-tiemechanism,andittakesaccountofasofteningapproximateconstitutivelawforcrackedconcrete,andtheadditionalcontributionofthehorizontal stirrups.RESEARCH SIGNIFICANCETheaimofthepresentstudyistoresolveproblemsinvolved in predicting the shear strength of corbels by meansofasingleexpression,adequatelyaccurate,thatallowstoavoidthecurrenttediousandtime-consumingcomputingprocedures.Theexpressionitselfhighlightstwoprincipal-resistant contributions: one due to concrete strut and principalreinforcement, and the other due to stirrups. A formula basedon the proposed expression and on 243 experimental resultsis also proposed for design.MODEL BASESAtypicalreinforcedconcrete(RC)corbelisshowninFig.1(a).ThecorbelisloadedbytheverticalforceVuapplied at the distance a from the column face and by thehorizontal action Nu. The horizontal principal reinforcementof area As is placed at the distance (h-d) from the support plan,andthesecondaryreinforcementwithoverallareaAhisprovided by horizontal stirrups. Only corbels with stirrups inthehorizontaldirectionareconsideredherein,asthisistheconstructive typology used more often in practice.Inthepresentstudy,itisassumedthatfailurealwaysoccurs from the crushing of the diagonal compressive strut(dottedbandinFig.1(b)),whoseformationisrevealed,atincreasing loads, by the appearance of inclined cracks on thewebofthecorbel.Thefailurebyyieldingoftheprincipalreinforcementisexcludedbecauseyieldingstraindoesnotlead to steel fracture, the last one occurring at a very greatstrain. In fact, it is observed that the currently named flexuralTitle no. 103-S01Reinforced Concrete CorbelsShear Strength Model and Design Formulaby Gaetano Russo, Raffaele Venir, Margherita Pauletta, and Giuliana SommaFig.1(a)GeometryofRCcorbel;and(b)strut-and-tiemodel with forces acting on corbel.ACI Structural Journal/January-February 2006 4tensionfailure(becausefailureisinitiatedbyyieldingoftension steel10) is due to crushing of the concrete strut andnot due to the rupture of the bars. Stirrups contribute to thecorbel shear strength by increasing the compressive strengthoftheconcretestrut,theresistanceduetotheaggregateinterlock,andthedowelactionatthecrackedinterface.Moreover,concreteandstirrupsinteract,andthemutualeffect is almost indistinguishable.In this study, it is assumed that the corbel strength is duetothesumoftwoindependentresistingcontributions:theoneprovidedbystrutandtie,andtheotherbystirrups.ItfollowsthatfortheshearstrengthvuofanRCcorbel,thegeneral expressionvu = vc + vh(1)might be proposed, where vc is the shear strength contributionoffered by the strut-and-tie mechanism created by the diagonalcompressedstrutandtheprincipalreinforcement,andvhshows the contribution given by the secondary reinforcement.The expression for vc is analytically derived herein fromthe corbel equilibrium equations. In particular, it is directlyrelatedtothevalueofthecompressionforceCcintheinclinedstrutofacorbelwithoutstirrups(Fig.1(b)).Thevalue of Cc is a function of an unknown biaxial strain statedependentoncorbeldimensions,principalreinforcementandstirrupsamount,concretecompressivestrength,andfailure modes.1 It follows that the vc value is linked to all themultiple variables mentioned previously, which are difficultto quantify.To obtain an expression for the shear strength vc, one canstartbyconsideringatheoreticalupperlimitvalueofvc,vc,lim. Therefore, vc is assumed to be a fraction of vc,limvc = c1 vc,lim(2)where c1 (< 1.0) is a factor to be determined on the basis ofexperimental results.Itfollowsthattheexpressionfortheshearstrengthvuisobtained from Eq. (1) by means of Eq. (2)vu = c1 vc,lim + vh(3)SHEAR STRENGTH CONTRIBUTION LIMIT DUE TO STRUT-AND-TIE MECHANISM vc,limFor determining vc,lim, the authors refer to a corbel withoutstirrups (Fig. 1(b)). The strength contribution can be deducedin a way similar to that proposed for deep beams,11 but takingaccountofthehorizontalforceatfailureNu.AccordingtoHwang et al.,9 it is assumed that Nu is directly applied to thecentroidofthereinforcement(Fig.1(b)).Theconsideredstrut-and-tiemechanismleadstothefollowingequilibriumequations (rotation around Point O)Ts Nu = Ccsin (4)Vc = Cccos (5)(6)whereVcistheultimateshearforcecarriedbyacorbelwithoutstirrups;istheanglebetweenthecompressedconcrete strut and the vertical direction; Cc is the compressionforce in the inclined strut of a corbel without stirrups; l is itswidth; a is the shear span; d is the corbel effective depth; andTs represents the yield force oftheprincipalreinforcement(Fig. 1(b)).The mean shear strength at the corbel-column interface isgiven by(7)where b is the width of the corbel.Using Eq. (5), one obtains(8)Strut-load inclination Substituting Eq. (5) into Eq. (6) yields(9)According to Hwang et al.,9 the width of the compressedstrut might be given by the depth to neutral axis of the crosssection at the column interfacel = kd (10)where k is obtained from the classical bending theory ofreinforced concrete beams with only tensile reinforcement(11)inwhichnistheratiooftheelasticmoduliofsteelandconcrete, n = Es/Ec, and the flexural reinforcement ratio f isassumed 9 to be given by(12)withAn=Nu/fys,wherefysistheyieldingstrengthoftheprincipal reinforcement.It can be observed that, by using ultimate strength insteadof linear analysis for estimating l, a shear strength formula isVca Cc dl sin2------------- \ | |Ccl cos2-------------- cos sin =vcVcbd------ =vcCc cosbd------------------ = tanal cos2-------------- +dl sin2------------- -----------------------\ ] ] ]| |=k nf( )22nf+ nf =fAsAnbd----------------- =ACImemberGaetanoRussoisaprofessorofstructuralanalysis,HeadoftheDepartmentofCivilEngineering,andProvostforBuildingoftheUniversityofUdine,Italy.Hisresearchinterestsincludenonlinearbehaviorofreinforcedcon-crete structures.RaffaeleVenirisacivilengineerwhocollaborateswiththecivilengineeringdepartmentoftheUniversityofUdine.Hisresearchinterestsincludeshearbehaviorinreinforced concrete members and strut-and-tie modeling for discontinuous regions.MargheritaPaulettaisapostdoctoralstudentattheUniversityofUdine.Shereceived her PhD in civil engineering at the Department of Civil Engineering, UniversityofUdine.Herresearchinterestsincludebondbehaviorinreinforcedconcretestructures and strut-and-tie modeling of nonflexural members.Giuliana Somma is a researcher at the University of Udine. She received her PhD instructuralengineeringattheUniversityofFirenze.Herresearchinterestsincludeshearbehavior in reinforced concrete elements, beam-column joints, and earthquake engineering.ACI Structural Journal/January-February 2006 5obtainedthatapproximatestheexperimentalresultsworsethan that obtained from the linear analysis.The value of n is obtained by assuming, from ACI 318-02,3that Es = 200,000 MPa andEc = 47,000MPa (13)it follows that n = 42.6/ .Equation(9),usingEq.(10)andtrigonometricrelations,yields(14)Because Eq. (14) is an explicit expression of the only parameterk,whichisgivenbyEq.(11),andaanddareknown,noeffort is required in the calculation of .Compression force in strut, CcHwangetal.9definetheeffectiveareaofthediagonalstrut, Astr, asAstr = l b (15)where l is provided by Eq. (10). The same expression (Eq. (15))isusedhereinforAstrandaconstantstressdistributionissupposedinthestrut.Hencethemaximumvalueofthecompression force Cc can be computed asCc = d,maxbl (16)whered,maxisthemaximumvalueoftheconcretecompressionstressdintheprincipald-direction(