ProceedingsoftheInstitutionofCivilEngineersStructuresandBuildings165March2012IssueSB3Pages111125http://dx.doi.org/10.1680/stbu.2012.165.3.111Paper900101Received30/11/2009
Accepted26/08/2010Keywords:beams&girders/concretestructures/shearfailuresICEPublishing:All
rightsreservedStructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshourShear
capacity of reinforcedconcrete corbels usingmechanism
analysisj1Keun-HyeokYangMSc,PhD,Archi.EngngAssociateProfessor,DepartmentofArchitectural
Engineering,Kyonggi
University,Suwon,Kyonggi-do,SouthKoreaj2AshrafF.AshourMSc,PhD,CEng,FIStructEReaderinStructural
Engineering,EDT1,School
ofEngineering,DesignandTechnology,UniversityofBradford,Bradford,UKj1j2A
mechanism analysis is developed to predict the shear capacity of
reinforced concrete corbels. Based on shear
failureobservedinexperimental tests,
kinematicallyadmissiblefailuremechanismsareidealisedasanassemblageoftworigidblocksseparatedbyafailureplaneofdisplacementdiscontinuity.Shearcapacitypredictionsobtainedfromthedeveloped
mechanismanalysis are in better agreement with corbel test results
of a comprehensive
databasecompiledfromtheavailableliteraturethanother existingmodels
for corbels. Thedevelopedmechanismmodelshows that the shear
capacity of corbels generally decreases with the increase of shear
span-to-depth ratio, increaseswiththeincreaseof mainlongitudinal
reinforcement uptoacertainlimit beyondwhichit remainsconstant,
anddecreases withtheincreaseof horizontal appliedloads. It
alsodemonstrates that thesmaller theshear span-to-overall depth
ratio of corbels, the more effective the horizontal shear
reinforcement.1. IntroductionReinforced concrete (RC) corbels,
generally dened as
shortcantilevershavingshearspan-to-depthratioslessthanorequalto1.0,
arecommonlyusedtotransferloadsfrombeamstocolumnsor walls inprecast
concrete construction. Corbels are primarilydesignedtoresist
vertical loadsandhorizontal actions owingtorestrained shrinkage,
thermal deformation and creep of
thesupportedbeamand/orbreakingofabridgecrane.Owingtotheirgeometricproportions,
thecapacityofRCcorbelsisgovernedbyshear rather thanexure andshear
deformations are not negli-gible, similar to deep beams. RC corbels
are identied
asdiscontinuityregions(Schlaichetal.,1987)wherestraindistribu-tion
is signicantly non-linear and conventional
beamtheorywouldnotbeapplicable.RCcorbels are generally known to
display several modes offailure (ASCEACI Committee 426 (ASCEACI,
1973);
KrizandRaths,1965;Russoetal.,2006),suchasanchoragefailureoryieldingofmainlongitudinalreinforcement,
shearsplittingattheinterface between column and corbel, diagonal
splitting orcrushing of the concrete strut joining the loading
point
andbottompointoftheinterface,andlocalcrushingfailureunderthebearingplate
of the appliedload. Beam-shear failure occurringbetween the
interface and diagonal plane joining the loadingpoint
andbottompoint of theinterfacewasthemost commonlyobserved failure
in experimental tests (Campione et al., 2007;Fattuhi, 1994a, 1994b,
1994c; Foster et al., 1996; Russoet al.,2006; YongandBalaguru,
1994). Prematurefailuremodes, suchas anchorage failure of main
longitudinal reinforcement andbearingfailure(Campioneet al., 2007),
wouldbepreventedbyproperreinforcement detailing,
asspeciedinACI318-08(ACI,2008) or EC 2 (BSI, 2004). On the other
hand, it is notstraightforward to evaluate the capacity of corbels
owing tobeam-shear failureasit
involvesvariousparameterssuchastheamountofmainlongitudinalandhorizontalshearreinforcements,shearspan-to-depthratioandamount
ofhorizontal
load(Fattuhi,1994a,1994c;YongandBalaguru,1994).Inthepresent study,
amechanismanalysis for corbels is
devel-opedusingupper-boundtheoremof concrete
plasticitytocom-plement the strut-and-tie models driven froman
equilibriumapproach and calibrated against limited test results.
Based onexperimental testscarriedout
bymanyresearchers(Campioneetal., 2007; Fattuhi, 1994a, 1994b,
1994c; Foster et al., 1996;Mattock, 1976; YongandBalaguru, 1994),
kinematicallyadmis-sible failure modes are idealised and studied.
The effect ofdifferent parameters on the shear capacity of corbels
is alsoinvestigated using developed mechanismanalysis, existing
em-pirical equationsbyFattuhi (1994a), theshear-frictionmodel byACI
318-08 (ACI, 2008), a simplied strut-and-tie model by111Russoet al.
(2006), asoftenedstrut-and-tiemodel byHwangetal. (2000)andtest
resultsofacomprehensivedatabasecompiledfromavailableliterature.2.
Review of existing modelsCurrently available theoretical models to
evaluate the shearcapacity of corbels would be classied into three
categories:empirical equations (Fattuhi, 1994a) calibrated against
test re-sults, formulas(ACI318-08, (ACI, 2008); Mattock,
1976)devel-oped fromshear-frictiontheory(HermansenandCowan,
1974;Mattock,1976),andstrut-and-tiemodels(Hagberg,1983;Hwanget al.,
2000; Russoet al., 2006; Siao, 1994; Solanki andSabnis,1987)
including plastic truss models (Campione et al.,
2007).Existingmodelsfor estimatingtheshear capacityof
corbelsarebrieysummarisedbelow.2.1 Empirical
equationBasedonextensivetest results, Fattuhi
(1994a)combineddiffer-ent parametersinuencingtheshearcapacity, Vn,
ofcorbelsanddevelopedthefollowingformula:Vn k1(bd)k2fct
k3(a=d)k4rst k53(10)k6( N=V)fy=fcu k7(d=h)k81:whereb, dandh(inmm)
arewidth,
effectivedepthandoveralldepthoftheinterfacebetweencolumnandcorbel,
respectively,
aisshearspanmeasuredfromtheloadingpointtotheinterfaceasshowninFigure1,fct(inMPa)isindirecttensilesplittingstrengthofconcrete,fcu(inMPa)iscubecompressivestrengthofconcrete(
1:23 f 9c, where f 9cis cylinder compressive strength),rst(Ast=bd)
is ratio of main longitudinal reinforcement, Astand fyare area and
yield point stress of main longitudinalreinforcement, respectively,
and N=V is the ratio
betweenhorizontalandverticalloadsappliedtocorbels.Thevaluesoftheconstants,
k1tok8,obtainedfromregressionanalysisoftestdataare 611, 0.7298,
0.3569,0.8204, 0.5745,0.1644,0.0261,0.1342, respectively. The above
equationtakes noaccount
oftheinuenceofhorizontalshearreinforcement.Althoughit
isrelativelyeasytouseempirical equationssuchasEquation1,
itsapplicationtocorbelshavingparametersdeviatedfromtherangeusedfor
calibrationisdoubtful. Inaddition, thisequation may cause overtting
owing to the high number ofconstantsusedtodevelopit.2.2 Formula
using shear-friction theoryACI 318-08(ACI, 2008) species theshear
capacityof
corbelsconsideringtheloadtransfercapacityofhorizontalreinforcementbyshear
frictionandexural yieldingof the
mainlongitudinalreinforcementattheinterfaceasfollowsVn minAstfy
Ahfyh N ,Astfyjd Nh d jd a264375< min0:2f 9cbd,5:5bd 2:where is
coefcient of friction which is taken as 1.4 formonolithic
construction, Ahand fyhare total area and yieldstrength of
horizontal shear reinforcement, and jd[d 0:5(Ast fy N)=(0:85f 9cb)]
is moment lever arm. The rst termof theright-handsideof
Equation2indicates theshear frictionresistance of longitudinal
tensile reinforcement and horizontalshear reinforcement across the
interface between corbel andcolumn, while the second termrefers to
exural resistance oflongitudinal tensilereinforcement.
Equation2alsoassumes thatappliedhorizontal forces
canberesistedbylongitudinal tensilereinforcement.
Theaboveformulabasedonshear-frictiontheoryneglects the shear
transfer capacity of concrete, as the
criticalfailuresectionisalwaysassumedat
theinterfacebetweencorbelandcolumn, as showninFigure1. However,
failureof corbelshaving a very small shear span-to-depth ratio or
sufcienthorizontalshearreinforcementseldomoccursalongtheinterface,aspointedoutbyHwangetal.(2000)andHermansenandCowan(1974).
In addition, ACI 318-08 limits concrete strength ofcorbelsto f
9c< 27:5MPawhentheshear capacityof
corbelsisgovernedbythesecondpartofEquation2, whichdoesnotallowfull
utilisation of higher strength concrete. For corbels havingshear
spantodepthratiogreater than1.0, ACI
318-08recom-mendstheuseofastrut-and-tiemodel
describedinACI318-08,AppendixA.aVNdhAstAhPotential failureplane in
strut-and-tiemodelPotentialfailure planein
shear-frictiontheoryFigure 1. Potential failure planes of corbels
considered in
existingmodels112StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshour2.3
Strut-and-tie modelConsidering equilibrium, compatibility and
constitutive laws ofcrackedreinforcedconcrete, Hwanget al. (2000)
predictedtheshear capacity of corbels based on the softened
strut-and-tiemodelshowninFigure2(a)asfollowsVn FD sin Fh tan
3:where FDand Fhare compression force in concrete strut andtension
force in horizontal shear reinforcement, respectively,[ tan1(
jd1=a)] is the angle of concrete strut to the long-itudinal axis of
corbels as showninFigure 2(a). Hwanget
al.usedthelinearbendingtheoryofreinforcedconcretebeamswithonly
tensile reinforcement neglecting shear deformations andnon-linear
distributions of strains and stresses of corbels
toestimatethemomentleverarm, jd1,asjd1 1 k3 d4:where k[(nrf)2
2nrfqnrf] is the ratioof compressionzone depth to effective depth
at the interface, rf[(Ast An)=bd] is equivalent main longitudinal
reinforcementratio, An(N=fy) is area of main longitudinal
reinforcementresistingtheappliedhorizontal force, N,
n(Es=Ec)ismodularratio of elasticity, Esand Ec( 4700f 9cpinMPa) are
elasticmoduli of reinforcement andconcrete, respectively.
InEquation3, Fhis assumedtobeafunctionof thearea, Ah,
andaveragestrainof horizontal shear reinforcement, andFDisdependent
ontheconcretestrutwidth, ws,assumedtobeequaltokd, angle,
,andmaximumallowablecompressive stress, d,max, of concretestrut.
Hwanget al. adoptedthe softenedstressstraincurve
ofcrackedconcreteproposedbyZhangandHsu(1998)toestimated,maxthatisdependentontheaverageprincipalcompressive(d)andtensile(r)strainsinconcreteandasofteningcoefcient,
:The compatibilityconditionconsideredbyHwanget al. relatesthe
average principal strains tothe average horizontal (h)
andvertical(v)strainsasbelowr d h v5:Toavoiditerativeprocedure,
Hwanget al.
proposedreasonablevaluesfortheaveragehorizontal(h)andvertical(v)strainsandhence,thevaluesof
dand rcanbedeterminedusingnumericalanalysis. As a result, the shear
capacity of corbels using thesoftened strut-and-tie model proposed
by Hwang et al. can beobtained, although several assumptions based
on linear elasticbeamtheoryareimposed.Ontheotherhand, Russoetal.
(2006)derivedasimpleequationusingthestrut-and-tiemodelshowninFigure2(b).Inthismodel,load
transfer mechanisms of cracked concrete and
horizontalshearreinforcementwereconsideredandsimpliedasbelowVn 0:8
kf 9c cos 1 0:65rhfyh cot 1 bd6:whereisanon-dimensional
interpolatingfunctiontoprovideasingle expression for softening
coefcient, , given in thesoftenedstrut-and-tie model above,
1represents angle of
con-cretestruttothecolumnaxisasshowninFigure2(b),whichcanbe
obtained fromthe elastic beamtheory and trigonometricrelations,and
rh(Ah=bd)isratioofhorizontalshearreinforce-ment. Theprincipal
tensilestrain, r, wasassumedtobe
fct=Ectoobtaindirectlythesofteningcoefcient, : Russoet al.
alsoproposed that could be approximately expressed as a
poly-jd12jd12aVNT1ws105 cos ws 105 sin ws 1(a)aVN
TFhFDrvdh(b)Figure 2. Typical strut-and-tie models of corbels: (a)
Hwang et al.(2000); (b) Russo et al.
(2006)113StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshournomialofthirddegreeinf
9c:DifferentconstantsusedinEquation6weredeterminedbycalibratingthestrut-and-tiemodel
against243testresultsofcorbels.Other strut-and-tiemodels (Hagberg,
1983; Siao, 1994;
SolankiandSabnis,1987)weredevelopedforcorbelsbuttheyaresimilarin
principle to those presented above. Strut-and-tie models
areconsidered to be a good and rational tool for the design
ofdiscontinuous regions such as RC corbels as they provide
asystematic load transfer and better understanding of
internalforces. However, several
assumptionsareimposedforsimplica-tion.The
widthandinclinationofconcretestrutisevaluatedfromthe neutral axis
depth of the interface using the conventionalelasticbeamtheorythat
isnot applicabletodeepcorbelswhereboth strains and stresses are
non-linear. In addition, biaxial
stressesincompressedconcretestrutsarechosenarbitrarilyorcalculatedfromtensilestrengthof
concrete. Inthefollowingamechanismanalysis of corbels based on the
upper-bound theorem is developedto complement the strut-and-tie
models presented above.3. Mechanism analysis3.1 Failure mechanism
of corbelsFailureplanesof
corbelscommonlyoccurredalongthediagonalplanejoiningtheinneredgeofloadingplateandbottompointofthe
interface (ASCE-ACI Committee 426, (ASCEACI, 1973);Kriz and Raths,
1965). However, fewtest specimens exhibitedsplitting
cracksstemmedfrom apoint closeto theloading plateatfailure. Thus,
the failure mechanismcan be idealised as anassemblage of two rigid
blocks separated by a yield line
represent-ingthefailurezonealongwhichin-planedisplacement
disconti-nuityoccurs (Nielsen, 1984). It is alsoassumedthat
thefailuresurface joins the base point of corbel interface and the
innercorner of loading plate. It should be noted that other failure
modessuch as anchorage failure of main longitudinal
reinforcement,bearingfailureandfailurealongtheinterfacebetweencorbelandcolumncanbemostlypreventedbyfollowingproper
reinforce-mentdetailing(ACI318-08(ACI,2008),EC2(BSI,2004),Yongand
Balaguru, 1994). Rigid block I undergoes a rotation around
aninstantaneous centre (I.C.), while rigid block II is considered
to bexedwiththesupportingcolumnasshowninFigure3.
TheI.C.canbelocatedanywhereinthevertical planeof corbel
anditspositionidentiestheshapeofthefailuresurface.
Jensen(1982)provedthattheoptimalshapeoftheyieldlineisahyperbola
withorthogonal asymptotes at the I.C., as shown in Figure 3(a), and
theyield line reduces to a straight line to achieve a stationary
value ofthetotalenergydissipationwhentheI.C. approachesinnity.
Onthe other hand, the yield line turns into two straight
segmentswhentheI.C.
ofrelativerotationliesinsideoronacirclewhosediameter is the
straight yield line joining the inner edge of
loadingplateandbottompoint of
theinterfaceasshowninFigure3(b).Therefore,
thefailureplaneshavingeitherahyperbolicyieldlineortwostraight
yieldlinesidentifycrushingofconcretestrut andbeam-shear failure
including the yielding of main longitudinalreinforcement,
respectively. For eachlocationof I.C., anupperbound load to the
corbel capacity can be established. However, theoptimum position of
I.C. corresponds to the minimumloadcapacity as explained below.3.2
Material modellingConcrete is assumed to be a rigid perfectly
plastic materialobeying the modied Coulomb failure criteria
(Nielsen, 1984)withzerotensioncut-off.
Theeffectivecompressivestrengthtobeusedincalculation, fc ,isfc ef
9c7:whereeiseffectivenessfactor that isintroducedtoaccount
forthelimitedductilityofconcreteandtoabsorbothershortcomingsofapplyingtheplasticitytheorytoconcrete.Althoughthereisnouniedapproachfor
evaluatingthe effectiveness factor of con-crete, manyinvestigations
(Ashour andMorley, 1996; Brstrup,(a)Hyperbolicyield lineAstxea
VnNrI.C.Rigidblock IRigid block IIY(Origin ofglobal axes)OX
XI.C.YI.C.(b)AstxeaVnNr I.C.Rigidblock IRigid block IIY(Origin
ofglobal axes)OXXI.C.YI.C.dhFigure 3. Idealised failure mechanism
of corbels: (a) hyperbolicyield line; (b) yield line with two
straight
segments114StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshour1990)
clearly showed that the effectiveness factor depends onconcrete
strengthandgeometrical properties of reinforcedcon-cretemembers.
Themeasureofsuccessofthemechanismanaly-sis presented below would
depend on the extent that theeffectivenessfactor
isreasonablyuniformandpredictableacrossthe range of corbels
considered. Inthe present study, Nielsensmodel consideringtheeffect
ofcompressivestrengthofconcretemodiedbythe
equationproposedbyBrstrup(1990) for theinuenceof shear
span-to-overall depthratiois adoptedfor
theevaluationoftheeffectivenessfactorasgivenbelowe 0:8 f 9c200 1
0:2 ah 8:All reinforcement is considered tocarryonlyaxial tensile
andcompressive stresses and its dowel action is ignored.
Steelreinforcement
inbothtensionandcompressionisassumedtobearigidperfectlyplasticmaterial
withyieldstrength, fy: Yieldingof steel reinforcements is generally
achieved by utilising
somesortofamechanicalanchorageatcorbelend.Strainhardeningofsteelreinforcementisignoredasitrequiresexcessive
widecracksin concrete beyond the failure mechanismconsidered in
thepresentanalysis.3.3 Work equationThe upper-boundtheoremis
basedonthe energyprinciple, byequatingthetotal internal energy, WI,
totheexternalworkdone,WE: Thetotal internal
energymainlydependsonthepositionoftheI.C.andtheamountofinternalstressesinbothconcretealongthe
yield line and reinforcement crossing the yield line.
Theenergy(Wc)ldissipatedinconcreteper unit lengthof
theyieldlineiswritteninthefollowinggeneralform(Nielsen,1984)Wc l ef
9c2b1 sin 9:whereis relative displacement of rigidblockI andis
theangle betweenthe relative displacement at the midpoint of
thechordand yieldline chord as showninFigure 3. The
relativedisplacement, , canbeexpressedas r, where r is
distancebetweenthemidpoint of yieldlinechordandtheI.C.
andisrotation of rigid block I. For a yield line with two
straightsegmentsasshowninFigure3(b),itshouldbenotedthatthetwosegmentsintersect
at theI.C. of relativerotation,
indicatingthatonesegmentoftheyieldlineisundercompressionandtheotheris
in tension owing to the relative rotation between the twoblocks.
Therefore, the yieldline segment under tensile
stresseswithpureseparation( =2) cannot
contributetotheinternalenergy distribution owing to the assumption
of zero tensileconcretestrength. If theoriginof global
coordinatesisset tobeatthebottompointoftheinterfaceasshowninFigure
3,thetotalinternal energy, Wc,
dissipatedinconcretealongthehyperbolicortwostraightyieldlineis(Jensen,1982)Wc
ef 9c2bF(O9)10:where F(O9) is a function of the position of the
I.C. and,consequently, the shape of the yield line; Equations 11(a)
and11(b)belowgivesthevalueof
F(O9)forthehyperbolicandtwostraightsegmentsyieldlinesrespectively(Jensen,1982)FO9
r1 sin h= sin for r .h=2 sin 11a:FO9 X2I:C: Y2I:C: for r
0:5,especially for st 0:05: When a=h , 0:3, the strut-and-tiemodels
underestimate the normalised shear capacity, especiallywhen st
0:1,whereastheempiricalequationbyFattuhihighlyoverestimatesthenormalisedshearcapacity.ThepredictionsfromACI318-08remainconstant
uptoa=h 0:6, beyondwhichthepredictions decreasewiththeincreaseof
a=h: As aresult, ACI318-08 generally underestimates the normalised
shear capacitywhena=h , 0:5, but overestimatestest resultswhena=h .
0:7:Ontheotherhand,
theproposedmechanismanalysisisclosertothetestresults,regardlessofthe
variationofa=h:5.2 Main longitudinal reinforcementTheinuenceof
themainlongitudinal reinforcement index, st,on the normalised shear
capacity, Vn=bhf 9c, of corbels
withouthorizontalshearreinforcementandhorizontalloadfortwodiffer-ent
shear span-to-overall depth ratios is presented in Figure 7.The
predictionobtainedfromthe mechanismanalysis
increaseswiththeincreaseof stuptoacertainlimit
beyondwhichthenormalised shear capacity remains constant, agreeing
with thetest results. When the main longitudinal reinforcement
reachesthis limit, it becomes strongenoughnot
toyieldandhencetheI.C. lies at thelevel of mainlongitudinal
reinforcement.
Inthiscase,mainlongitudinalreinforcementwouldhavenocontributiontotheshearcapacityofcorbelsasdepictedinFigure7.Theshearcapacity
predicted fromACI 318-08 is governed by the upperlimit
speciedbyEquation2withthe increase of st: For
themechanismanalysisandACI318-08, thelimittothe stdependsona=h,
indicatingthatthevalueofsttoachievethepeakpointdecreases with the
decrease of a=h: On the other hand, theprediction obtained
fromempirical equation, and strut-and-tiemodels increases with the
increase of stwithout any limits,exhibiting large overestimation of
test results for large st,especiallyfora=h 0:9:5.3 Horizontal
loadFigure8showstheeffectoftheratio, ,oftheappliedhorizontalloadto
yieldforceofmainlongitudinalreinforcementonnormal-isedshearcapacity,
Vn=bhf 9c, ofcorbelswithout horizontal
shearreinforcementfortwodifferentshearspantooveralldepthratios,a=h:
Thenormalisedshearcapacityofcorbelssteadilydecreaseswiththeincreaseofaspredictedbyall
theoretical modelsandexperimental results. However,
thepredictionsobtainedfromthemechanismanalysis are closer to the
experimental results thanother models. In particular, strut-and-tie
models largely over-estimate the effect of on the normalised shear
capacity fora=h 0.56.5.4 Horizontal shear reinforcementThe effect
of horizontal shear reinforcement index,h( (Ah fyh=bhf 9c)),
onthenormalisedshearcapacity, Vn=bhf 9c,of corbels without
horizontal load is shown in Figure 9. Thenormalisedshear capacityof
corbelsincreaseswiththeincreaseof has
predictedbythemechanismanalysis andstrut-and-tiemodels and also
supported by the limited experimental resultsavailable. The effect
of horizontal shear reinforcement on theshear capacityof corbels is
more prominent for lowvalues
ofa=h,namely,ahigherincreasingrateisdevelopedincorbelswitha=h of
0.3 than corbels with a=h of 0.9. However,
predictionobtainedfromFattuhis equations does not account for h,
andthe ACI 318-08 prediction is also independent of hfora=h 0.9 and
slightly increases with the increase of hfora=h 0.3.6.
ConclusionsAmechanismanalysisbasedonupper-bound
theoremisproposedtopredicttheshearcapacity of
RCcorbels.Theeffectofdifferent119StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshourparameters
ontheshear capacityof corbels is alsoinvestigatedusing the
developed mechanismanalysis, empirical equationsproposedbyFattuhi,
ACI
318-08basedonshear-frictiontheory,strut-and-tiemodelsdevelopedbyHwanget
al. andRussoet al.andtest results
inacomprehensivedatabasecollectedfromtheavailable literature.
Various analytical models developed in theliterature and present
study consider only beam-shear failuremodeincludingyieldingof
mainlongitudinal reinforcement andcrushing of concrete strut as
other modes of failure such
asbearingfailureoranchoragefailureoflongitudinalreinforcementcanbepreventedbyfollowingreinforcement
detailsspeciedincodes.Thefollowingconclusionsmaybedrawn.(a)
Comparedwiththeexistingmodels,predictionsobtainedfromthedevelopedmechanismanalysisareinbetteragreementwithtestresultsregardlessofshearspan-to-overalldepthratio,horizontalshearreinforcementandhorizontalload.(b)
ThelargeststandarddeviationandcoefcientofvariationoftheratiobetweenmeasuredandpredictedshearcapacitiesofcorbelsareshownbyACI318-08.Inaddition,ACI318-08isunconservativeforcorbelshavingashearspan-to-overalldepthratioofmorethan0.5.(c)
ThenormalisedshearcapacitiesVn=bhf
9cofcorbelsobtainedfromexperimentalresults,empiricalequations,strut-and-tie0250200150100050Vbhfnc/()01
02 03 04 05 06 07 08 09 10Shear span-to-overall depth ratio /(a)a
hFoster ., 1996 (004 006) et al stKriz and Raths, 1965 (003 007)
stFattuhiACI 318-08STM (Hwang .) et alSTM (Russo .) et alMechanism
analysis0250200150100050Vbhfnc/()01 02 03 04 05 06 07 08 09 10Shear
span-to-overall depth ratio /(a)a hFoster ., 1996 (004 006) et al
stFattuhi and Hughes, 1989a ( 007) stFattuhi, 1994c (003 004)
stKriz and Raths, 1965 (003 007) stFattuhiACI 318-08STM (Hwang .)
et alSTM (Russo .) et alMechanism
analysis0250200150100050Vbhfnc/()01 02 03 04 05 06 07 08 09 10Shear
span-to-overall depth ratio /(b)a hFattuhi, 1994a (008 011) stKriz
and Raths, 1965 (008 012) stFattuhiACI 318-08STM (Hwang .) et alSTM
(Russo .) et alMechanism analysisFigure 6. Inuence of a/h on shear
capacity of corbels:(a) jst 0.05; (b) jst
0.1120StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshourmodelsandmechanismanalysisgenerallydecrease
withtheincreaseofshearspan-to-overalldepthratio.(d)Theshearcapacityobtainedfromthemechanismanalysisincreaseswiththeincreaseofthemainlongitudinalreinforcementindexuptoacertain
valuebeyondwhichitremainsconstant,agreeingwithtestresults.(e)
Thenormalisedshearcapacity, Vn=bhf
9c,ofcorbelsdecreaseswiththeincreaseofthehorizontalload;thedecreasingrateobtainedfromthemechanismanalysisissimilartothatoftestresults.(
f )
Theeffectofhorizontalshearreinforcementonshearcapacityismoreprominentincorbelshavingsmallshearspan-to-overalldepthratios.AcknowledgementsThis
workwas supportedbythe National ResearchInstitute ofCultural
Heritage and the Regional Research Centers
Program(Bio-housingResearchInstitute),
grantedbytheKoreanMinistryofEducation&HumanResourcesDevelopment.0250200150100050Vbhfnc/()005
010 015 020 025 030Main longitudinal reinforcement index(a)stFoster
., 1996 ( / 03) et al a h Fattuhi and Hughes, 1989a ( ) a h / 035
Kriz and Raths, 1965 (026 a h / 033)FattuhiACI 318-08STM (Hwang .)
et alSTM (Russo .) et alMechanism
analysis00300350160120080040020Vbhfnc/()005 010 015 020 025 030Main
longitudinal reinforcement index(b)stMattock, 1976 ( / 09) a h
Fattuhi, 1990 ( ) a h / 09 Campione et al., 2007 (a h / 09)
FattuhiACI 318-08STM (Hwang .) et alSTM (Russo .) et alMechanism
analysis0018020035 040006010014Figure 7. Inuence of jston shear
capacity of corbels:(a) a/h 0.3; (b) a/h
0.9121StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshourAPPENDIX:
Numerical example of mechanismanalysisThe following numerical
example presents the shear
capacitycalculationofspecimenPA2testedbyFosteret al.
(1996)usingthemechanismapproach.ThematerialandgeometricalpropertiesofthespecimenPA2areasfollows:
b 150 mm, h 600 mm,f 9c 53 MPa, d 500 mm, a 300 mm, xe 250 mm, Ast
1884 mm2(Y20), fy 450 MPa, Ah 157 mm2(R10) fyh 360MPaandsh 85 mm.
Therewas nohorizontal loadappliedtothespecimen.
Themeasuredshearcapacity, Vn, was800 kN.
Thefollowingstepssummarisetheprocedure:(a)
Calculatetheeffectivenessfactorforconcrete, e,usingEquation8; ve
0.4815.(b) Calculatethemainlongitudinalreinforcementindex,st (Ast
fy=bhf 9c),andthe webreinforcementindex,s,i (Ah,i fyh,i=bhf
9c),foreachindividualbarcrossingtheyieldline, st 0:1777, s,i
0:0118:0250200150100050Vbhfnc/()02 04 06 08 10Ratio of horizontal
load to yield force of the main reinforcement,(a)FattuhiACI
318-08STM (Hwang .) et alSTM (Russo .) et alMechanism
analysis00300160120080040020Vbhfnc/()02 04 06 08Kriz and Raths,
1965 ( / 056, 007 019) a h stFattuhiACI 318-08STM (Hwang .) et
alSTM (Russo .) et alMechanism analysis001810006010014Kriz and
Raths, 1965(021 / 027) a h(007 017) stRatio of horizontal load to
yield force of the main reinforcement,(b)Figure 8. Inuence of on
shear capacity of corbels (jst 0.12,jh 0): (a) a/h 0.25; (b) a/h
0.56122StructuresandBuildingsVolume165IssueSB3ShearcapacityofreinforcedconcretecorbelsusingmechanismanalysisYangandAshour(c)
Determinetheangle
(tan1(h=xe))ofthediagonallinejoiningtheinneredgeoftheloadingplateandbottompointoftheinterfacetothelongitudinalaxisofthecorbel;
67.388.(d)DifferentpositionsoftheI.C.(XI:C:, YI:C:)havetobeexamined
usingtheMatlabsoftwareoptimiser(Chapman,2004).Foreachposition,theshapeoftheyieldlinehastobeidentiedusingEquation11.TheinternalenergydissipatedinconcretehastobeestimatedusingEquations10and11.Forexample,forXI:C:
150 mmandYI:C: 150 mm,r 152 mmandh=(2 sin ) 325mm ,(r
