CONCRETE LIBRARY OF JSCE NO. 26, DECEMBER 1995

EVALUATION OF SHEAR STRENGTH OF CONCRETE BEAMSREINFORCED WITH FRP

(Translation from Proceedings of JSCE, No.508/V-26, 1995)

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Hikaru NAKAMURA Takeshi HIGAI

rare‘:We

The shear strength of concrete beams reinforced with FRP was studied through experiment andanalysis. The effect of longitudinal tensile reinforcement stiffness on the diagonal tension failurestrength was calculated using the extended modified compression fields theory, and an evaluationmethod was proposed based on the analytical results. Furthermore, shear strength of beams failingdue to FRP stirrup rupture was well evaluated by considering the lower stiffness and the strengthreduction of FRP at bent corners. A

Key Words : FRP, shear strength, bent bar strength, stijjiaess of reinforcement, modifiedcompression fields theory

Hikaru Nakamura is an associate professor in the Department of Civil and EnvironmentalEngineering at Yamanashi University, Kofu, Japan. He received his Doctor of Engineering Degreefrom Nagoya University in 1992. His research interests include failure mechanism of RCstructures and application of finite element analysis. He is a member of ISCE and JCT.

Takeshi Higai is a professor in the Department of Civil and Environmental Engineering atYamanashi University, Kofu, Japan. He received his Doctor of Engineering Degree in 1972 fromUniversity of Tokyo. His research interests relate to the shear failure of reinforced concretestructures, dynamic behavior of concrete structures and maintenance and strengthening existingconcrete structures. He is a member of JSCE and ICI.

—11l—

1. NTRODUCTION

In recentyears, the use of FRPqiber ReinforcedPlastic)rods in place of steel bars or prestressingbars as reinforcementfor concretestructureshas been activelystudied. As a result, many usefulreports were presented[1],and both deformationand strength were evaluated for the nexurebehaviorof the concretebeams reinforcedwith FRP[2].However,the shear behaviorof concretebeamsusing FRP has not everbeen established.

Many reports have made clean that the shear strength of concrete beams without shearreinforcement.reinforcedwithFRP as the longitudinaltensile reinforcementis less than those withsteel bars[3][4].Tsuji et al.[3]proposeda methodto evaluatethe shear strengthfor suchbeams byconsideringa ratio of Young's modulusof FRP to that of steelbar in a term of the reinforcementratio of shear strengthequationof JSCE.Sincethe methodwas, however,obtainedon the basis ofa small number of experimentalresults, its applicabilityis not always clear. One report[5]showthat the shear strengthcan be calculatedclosely or safely in comparisonwith the experimentalresultsusing the methodproposedby Tsuji et al. Onthe contrary,anotherreport[6]showsthat thecorrection considering a ratio of Young's modulus does not necessarily evaluate the actualstrength.

At the same time, it is reportedthat the shear strengthof concretebeams using FRP for shearreinforcement is small, as compared with the calculated value by a modified truss theory(vc+vs)[2]. Althoughseveralevaluationmethodsof the strengthof such beams are proposed, itseems that, at present, no methodgives satisfactorysolutions.The main reason why the shearstrength of such beams has never actuallybeen evaluatedappearsto be thaf the ruptureof FRPstirrupscorrespondingto yieldingof the steelbars occursat a crackintersectionOrbent corner?fa beam due to local stress intensitybeforethe uniaxialstrengthis reached[7]-[12].Therefore,1tlSimportantto considersuchphenomenawhen evaluatingthe shear strength.

The purpose of this paper is to evaluatethe shear strengthof concretebeamsreinforcedwith FRProds as the tensile and shear reinforcement.First, the shear resistancebehavior of the concretebeamsreinforcedwithFRP rod is confirmedby experimentalobservation.Second,the bent cornerstrengthand the diagonaltensionstrengthof FRP rods,whichare typicalmechanicalpropertiesofFRP in concretebeams,are investigatedtheoretically,and the effectof Young's modulus of thereinforcementson shear behavior is analyzed using an extended modified compressionfieldstheory. Finally, the shear strength of the beams is evaluated-using the results of the aboveinvestigations.

2. OUTLINEOF EWERmNTS

GR4afgiai&

FRP used consistsof continuousglass fibersimpregnatedwith resin (GFRP).that the effect of Young's modulus of the reinforcementon the shearinvestigated,since the Young's modulusof GFN is very small compared

GFRP was used sostrength could be

with the steel bars.

=cain.nre!nff2T.c;Tln2tS.(sGdlA)p;Ss6)foE#ai:,t:ssgnied:tl:ith.fSP.Sc;ncgmo2fioercemu:endd,tAedaEeearZ5iuC:OSsfthe bent comer was 5mm or 10mm.The materialpropertiesof GFRP and steel bars are showninTable-1. The tensile strengthand Young's modulusof GFRP were obtainedby the test proposedby Idemitsu[14].The compressivestrengthof the concrete(f'c)is shownin Table-2.

bAn outline of the beam specimenstested in this study is shown in Table-2. The cross secdon,without stirrups, was 30cm wide, 20cm high and with an effectivebeam depth of 15cm.Thebeams were tested on a two-pointloadingsystemwith monotonicallyincreasingloads.

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The cross section with stirrupswas a width of 20cm, height of 30cm, and an effectivebeam depthof 25cm. Parameters of the experimentare (1) material of longitudinalreinforcement(GFRP,steelbar), (2) material of stirrup(GFRP,steel bar), (3) spacing of the stirrups and (4) the radius of thebent corner of GFRP stirrups. The beams were tested on a one-point loading system withmonotonically increasing loads. Fig:1 gives details and its loading condition of specimens ofGGO5-20and GG10-20. This experimentsare characterizedby the radius of the bent corner ofGFRP stirrups and allow investigation of the effect of the bent bar strength on beam shearstrength.

3. ENERWNT RESULTS

The experimentalresults are summarizedin Table-3.All test specimensfailed in shear mode andthe longitudinalreinforcementsdid not yield or break at failurein any specimen.

ubFor beams withoutstirrups,the nemral crackpropagatedto the upper partof the crosssectionassoon as the crack initiated. The ultimatefailure was determinedby the diagonaltension crackwhich occurredfrom the flexural crack at a distanceof about 1.5d fromthe loadingpoint. Thefailure modeof the specimenswas diagonaltensionfailure.&Crack formationin the specimenswith stirrupswas, before failure, almost the same as in thespecimenswithout stirrups.The diagonalcrack occurredfTromthe nexural crack at a distanceof1.5d- 2.Od from the loading point. The crack formationof a specimenof GGO5-20at theultimate is shown in Fig:2. The load carrying capacity increasedby the effect of the stirrupsafter the diagonal tension crack occurred.The specimensreinforcedwith steel bars as stirrupsfailed after the stirrupsyielded,while the failureof specimensreinforcedwith GFRPstirrups,wascausedby the breakingof the stirrups.GFRPbreakingoccurredat the upper or lowerbent cornernear the diagonaltensioncrack.One purposeof this experimentsis invesdgationof the effect ofthe bent bar strengthon shear strength.It is understoodfrom the experimentalresults that theshear strength of the specimenswith a bent comer radiusof 5mmis obviouslylowerthan that of10mm;the strengthof the bent corneris an importantfactorfor evaluadngthe shear strengthofthe beam reinforcedwithFN.

Table-2 Outline of Specimens

Table-1 MaterialProperties

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A re a o f T e n sil e Y o u n g 's S t rai n a t

c ro s s s e c tio n s tre ,n g th m o d ul u s b re a k h g

2C m M P a M P a %

G 1 6 2 . O l 7 5 l . 4 2 . 9 4 x l O 42 . 5 5

G 6 O . 3 5 8 2 8 . 0 3 . l 4 x l O 4 2 . 6 4

D 19 2 . 8 7 3 7 l . 6 i . 8 O x lO 5

D 10 0 . 7 l 3 2 4 . 6 i . 9 0 x lO 5

D 6 0 . 3 2 3 7 0 . ii . 8 0 x l O 5

S p e c im e n L o n gi tu d in a l

re i n f o r c e m e n t

sp a c in g

o f s tirr u p

C m

r

m n

P t

%

a / d f'c

X P a

C o l 3 - G I G 1 . 3 4 4 . 0 2 2 . 7

'G O 2 4 - G 1 6 i . 7 9 4 . 0 2 7 . 8

G G O 5 - 1 0 4 - G l 6 G 6 ( 1 0 ) 5 I. 6 l 3 . 0 3 5 . 4

G G l O - 1 0 . 4 - G 1 6 C 6 ( 1 0 ) 1 0 i. 6 1 3 . 0 3 3 . 4

G G O 5 - 2 0 4 - G 1 6 G 6 ( 2 0 ) 5 I , 6 1 3 . 0 3 5 . 2

G G 1 0 - 2 0 4 - G 1 6 C 6 ( 2 0 ) l O 1 . 6 l 3 , 0 3 5 . 2

D G O 5 - 1 5 3 - D 1 9 G 6 ( 1 5 ) 5 1 . 7 2 3 . 0 3 4 . 7

D C l O - l 5 3 - D l 9 G 6 ( 1 5 ) 1 0 i . 7 2 3 . 0 3 4 . 4

D G O 5 - 2 5 3 - D 1 9 G 6 ( 2 5 ) 5 1 . 7 2 3 . 0 3 5 . 6

D G 1 0 - 2 5 3 - D 1 9 C 6 ( 2 5 ) lO i . 7 2 3 . 0 3 5 . 8

C D - 1 5 4 - G I G D 6 ( l 5 ) 1 . 6 l 3 . 0 3 8 . 6

C D - 2 5 4 - G 1 6 D 6 ( 2 5 ) 1 . 6 1 3 . 0 3 7 . i

Table-3ExperimentalResults

vv::clip;BTla;;,eSqtueaetiobna,rs,vm::f:fE.?.t83RPVs,):Truss TheoryVyAs(U1.15)/s), Vs,2:Eq.(1l)

Vco/:Vs.I for steel bar, vs.2for stirrup

lp

.i.

75 3@ 20 l5

20

Cni?_.lsItL)

Figure-i Detail and Loading Condition of Specimen(GGO5-20,GGlO-20)

Figure-2 Crack Fomution at Ulumate SLate(GGO5-20)4@a&As shown in experimentalobservations,bent bar strengthis a major factor for shear strength.Therefore,we must first Clarifythe bentbar strengthin orderto evaluatethe shearstrengthof theconcrete beams reinforced with FRP. The evaluation equation of the bent bar strength istheoretically.derivedas follows.

We assumean bent cornerFRP cast in concreteas shownin Fig:3. We considerthe deformationat the cross section varied from straight part to bend, since it is experimentallyobservedthatbreakingat the bent corneralwaysoccurat that location[8].Whena tensileforceis appliedto thebar as shown in Fig.-3, and there is no bond between concrete and FRP, the FRP stretch 8xuniformlyat the cross secdonin the straightpart subjectedto uniformaxial force. It is assumedthat the cross sectiondeformrotationangleof 4)(fad)maintaininga radiusof the bent cornerof r.Then, if Bemoulliassumptionis used,the straindistributionin the crosssectionis representedbya followinghyperboliccurve at any point y in the cross section sinceFRP shoulddeform6x atthe bent part.

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S p e c im e n D ia g o n a l c ra c k in g lo a d S h e a r str e n g th

V c. eq ' V c. ) V c, 2 v c, ca[i ,c. c,p V exp V c2+ V s) V c2+ V s2 v co[/v eg(a (a ) (b ? (fO (a ) ( a )

C o l 6 . 7 5 9 . 9 1 6 . 1 3 0. 9 1

G O 2 7 . 4 l l 1 . 6 7 7 . 2 2 0 . 9 7

G G O 5 - l O 1 0 . 0 0 l 3 . 2 1 8 . 1 8 0 . 8 2 l7 . 0 0 3 3 . 3 8 1 6 . 0 8 0 . 9 5

G G lO -1 0 9 . 5 0 l3 . 0 4 8 . 0 7 0 . 8 5 2 0 . 4 0 3 3 . 2 7 1 7 . 4 8 0 , 9 1

G G O 5 -2 0 7 . 5 0 1 3 . 2 0 8 . l 7 1 . 0 9 l l. 4 6 2 0 . 7 7 1 2 . 0 1 1. 0 5

G G 10 -2 0 8 . 5 0 1 3 . 2 0 8 . 1 7 0 . 9 6 1 3 . 5 0 2 0 . 7 7 12 . 8 8 0 . 9 5

D G O 5 - 1 5 1 6 . 5 0 l 3 . 6 1 0 . 8 2 l 9 . 6 6 3 0 . 4 1 l8 . 8 7 0 . 9 6

D G lO - l 5 14 . 9 0 1 3 . 5 7 0 . 9 l 2 l . 6 8 3 0 . 3 7 1 9. 8 4 0 . 9 2

D G O 5 - 2 5 1 3 . 9 0 1 3 . 7 3 0 . 9 9 1 6 . 2 7 2 3 . 8 1 1 6 . 7 2 1 . 0 3

D G l O _2 5 1 4 . 0 0 1 3 . 7 6 0 . 9 8 1 6 . 2 7 2 3 . 8 4 l 7 . 5 2 i . 0 8

G D - l5 1 0 . 0 0 1 4 . 0 4 8 . 6 9 0 . 8 7 1 5 . 7 3 1 4 . 7 l 0 . 94

G D -2 5 8 . 5 0 l 3 . 8 6 8 . 5 8 1 . 0 1 l l . 16 1 2 . l 9 1 . 0 9

/ Rupture of G FRP

～Ji## 5" _J l l I I I ( i L T T ～

A A

0

1Before

Deformadon

AfterDeformation

- -+TTI/I/

//

/

FiglEe-3Model of Bent Bar in Concrete

i0.9

i.' 0.8

>o.70.6

0.5

0_4

0.3

0.2

0.1

0

+ 4

t i pE;x&ns%ent

+

+

i 2 3 4 3rn1

Figure-4 Evaluadon of Bent Bar Strent,dh

a(y) -8x/((r+y)0) (1)

Stress distributionis obtainedby multiplyingYoung's modulusi to strain,whenFRP behaviorisstrictly elastic.Then, the stress distributionat breakingis representedby the following equation.Here it is assumed that the breaking of FRP occur when a stress in cross section reach thebreakingstress of q.

o-oy.r/(r+y) (2)

Therefore,the evaluationequation of bent bar strength can be derivedby integratingthe stressdistributionofEq.(2) in the crosssection.

T -JAG,.r/(r.y)dA (3)

The equationis rewrittento Eq.(4) as the functionof the radius of the bent.corner,r, and theheightof the cross section,h, whenthe cross sectionis rectangularwith the heightofh.

i7= Tu.,/hlm(1+h/r) (4)

in whichTu is the uniaxialstrength,r is the radiusof the bent corner,and h is the heightof thecrosssection.

Fig:4 shows the comparisonbetweenthe evaluatedvaluesusing the proposedequadonand theexperimentalresultsobtainedby Miyataet al.[9].h the figure,the lateralaxis showsa ratioof rto h and the verticalaxis showsthe reductionratio of the strengthof the bent cornerfrom theuniaxialstrength.A solid line showsthe Eq.(4) and a + indicatesan experimentallyobtainedvalue. It is understoodthat the proposedequationcan satisfactorilyevaluatethe bentbar strength.

QiBi494aiW

The effect of the diagonaltensileforce due to a diagop_alcrack,as shownin Fig.-5,is considereda factor of shearstrength.Next, we investigatethe evaluationmethodfor diagonaltensile strengthof FRP as anisotropicmaterial.

Considerthat FRP with a lengthL is appliedto diagonaltensile force T with angle 0. Then,thestressresultantacted in the crosssectionis the axialforceis T.cosOand the bendingmomentis T.L.sine. A strainin the cross sectionreach breakingstrainfrom a smallertensile forcethan withuniaxialstatebecauseof the effectof the straingradient.Therefore,the diagonaltensilestrengthisless than the uniaxialtensilestrengthsincethe cross sectionof FRPwithoutplasticregion,breaksas soon as one part of the cross sectionis broken, Then, the diagonaltensile strengthof FRP isrepresentedby the followingequationsfor a rectangularcrosssectionwithheighth.

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Jt

1

Figure-5DiagonalTension due to DiagonalCrack

=0.9

rF 0.8

0.7

0.6

0.5

0.4

0.3

0.2

01

0

t1' 1Stt

t -tSr _

+ : CarbonA : AramidI : Glass

1¥^ Eq.(6)

17b tt•`

Eq.(7)1

10 20 30 40degrea

Figure-6ComparisonwithExperiment

T-Tu/(cosO+6L.sine/h) (5)

in which Tu is uniaxial tensile strength, 0 is the direction of the tensile force, and h is the crosssection height.

h the above equation, L is the unknown value. Now, we assume that L is the length ofintersection between FRP and the diagonal crack with angle of 0. Then, L-h.tanO and thediagonaltensile strengthis derivedas,

T= Tu/(cosO+6tan0.sine) (6)

For circular.sections,

T- Tu/(cosO+8tan0.sine) (7)

Figure-6 comparesthe experimentalresultsfrom Maruyama et al.[10] and the proposed equations.The experimentalresults were obtainedfor carbon(CFRP),aramid(AFRP),and.glass(GFRP)withthe directionof diagonaltensileforce variedfrom 0 to 30line correspondrespectively to Eq.(6) and Eq.(7), while

degree.In this figure, a solid or brokenthe symbols + , A and I showthe

experimentalresults for carbon, aramid,and glass fibers. It is understoodthat strength reductiondue to diagonal tensile force is reasonably evaluated by the proposed equations.However, theequationscan not considerthe effect of fiber type, eventhough the strengthreduction is differentfor differentflbers in experiment.

5. EVjuUATION OF SmAR STRENGTHOF CONCRETEBEAMSRBWORCED WITHFRP

uMIt is known from many experimentalobservationsthat the shear strength of a beam reinforcedwith FRP, which has a lower Young7smodulus than steel bar, is smallerthan those reinforcedwith the steel bar[3][4],This fact is confirmedby this experiment(Got and GO2specimensinTable-3) and the strength is about 30 - 40% smaller than the value predicted by the equationproposedby Niwa et a1.[15].It is reportedthat the reason is mainly the differencein Young'smodulus of reinforcement.Therefore, we will analytically investigate the effect of Young'smodulusof the mainreinforcementon the shear strength.

Analysis is a method based on the extended modified compressionfields theory which canaccuratelyevaluatethe shear strengthof the concretebeams[13].The model used in the analysishas a cross sectionof 20 x 20cm, a beam depth of 16cm,and a longitudinalreinforcementratioof 2.68%, as shown in Fig.-7. The materialproperties are that the compressivestrengthof theconcreteis 28MPa and the tensile strengthis 2.8MPa.Analysisis performedin which Young's

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16

4cm

20

Figtlre-7 An(1l_yLicalModel

I•`S63

LLt•`3

i)

¥5;4

2

V :2.00 X

D:i.00 X

A:0.50 X

0..o.25 X

0 I

o5o5o5

u 1 2 M3.men((Gem)Figure-8 Effect of Young's Modulus of Main Reinforcement

^1.2

iullL)

O

Il o.8X

S. o,6llil

S-o.4i^Luv0.2>

(n6) I/4

i

0 :a/d:i

0:a/d=2

A:a/d=3

V:dd=4

0 0.5 1.5 2 2.5

Ei X 105(MPa)

FiglEe-9yR.eahbieT?Mh.pdbulesTe.efnb?%arRSepnfan.r?eaEoenntand

capacity for two-pointloading and simplyTsuTpportedbeams can be evaluatedfor both shear andnemral failurefrom this figureusing the relationof a/d-M/(V.d).However,this analysiscan notbe applied to shear compressionfailure, since the analysis does not consider the effect oftransverselocalcompressivestress.

modulus of the reinforcementis varied fromo.25 x 105fo 2.0 x 105MPa. Therelationshipsbetween the shear force and themoment in failure state obtained from theanalysis are illustrated in Fig.-8. The results

i?6 xyolu.ng',S.?50dxullu.S5Vaaln3es..20f2.o1.X51Xiare respectivelymarkedwith V , D , A and0. The range in which the shear forcegradually decreasesas the moment increases,correspondsto diagonaltension failure in theanalysis. On the other hand, the range inwhich the moment is constant and the shearforce decrease rapidly corresponds to thenexural failure region. The load carrying

As shownin Fig:8, the shearstrengthdecreasealongwithYoung's modulus.It is understoodthatthe Young's modulusof the main reinforcementis an importantfactor influencingshearstrengthand it can be seen that the Young's moduluscurves are almost parallel.This implies that theeffect of Young's moduluson shearstrengthis independentto a ratio of the shear force to themoment and is a functionof Young's modulusonly.Figure-9showsthe reductionratio of shear,,.+.^n^L C•`^-4_ ._J.__A.,J1_ I_ .L1__ ____ _i. 1T_ _ _)_ 1 1 I1^ ^ . _ 1^ 5 1 Jrn nl1 . .. .1 rT

ArTagtnhdIt was confirmed and similarly concluded the effects of longitudinal reinforcementA;alio,compressivestrength of the concrete,and effectivedepth of the cross secdon in the analysis.Therefore, the effect.of Young's modulus on the shear strength can be introducedby simplymultiplying the prev10uSlyintroduced shear strength equations by the effect of the Young,smodulusof the longitudinalreinforcement.

?hhoeeuaTag:lag,leynEfglu!ru;:tuh;Si?cf,ce;eesb;adltl?inee::;n.eracfid;t.r-:9:tEpFr;ePn:tOas:Sa?oecnvu?=fa.o:fhdeelsi;;::T8.,?Ffinaa;1ayb,othOef

v - vc(Ei /Es)1/4 (8)

in which Vc is the shear strength of the concretebeam reinforcedwith steelbar, Ei is the Young,smodulus ofFRP, and Es is the Young's modulus of the steel bars.

from strength in the case ofyoung's moddlus oTf2.0 x 10 5MPa. The symbols 0 , aV correspond to a/d-1, a/d-2, a/d-3, and a/d-4. The difference of a/d does not appear.

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1.6

1H:

tu i.43

>1.2

0.8

0.6

oOoO0

000

i&

:rKA

! :.TsouFsed0 :Niwa

_aA

0OA 0.8 1.2

pwoi(res)(%)Figure-10ComparisonbetweenEstimatedand

ExperimentalValues

6.•`

Y)ut<

Et4.1O

C13

2

rA :2.00 x

D:1.00 x

A :o.50 x0 :o.25 x

0 0.004 0.008 0.012ShearStrain

FiglEe-i i (a)f.eiaceoannsdhighbeearhvseEZiSheaI

Table-4 Comparison betweenEstirnatedand ExperimentalValues

6

g5i)Ut•`

a4i-

34}

i?;3

2 .i :2.00 x

D:I.00 x

A :o.50 x

0..o.25 x

o5oso5o5

0.0002 0.0004CurvaLure(1/cm)

Figure-11O')FR.efgEoAsdhi8E%ereenshear

Tsuji et al. proposeda methodof evaluatingthe shearstrengthof concretebeams reinforcedwithFRP using the transformedarea of reinforcementAsPinEs), consideringa ratio of Young'smodulus of FRP to that of the steel bar[3]. The results in our analysis are identicalwith themethod proposedby Tsuji et al., since the effect of the longitudinalreinforcementratio on theshear strength is also proportionalto the 1/4 power in the analysis[13].However,the effect ofyoung's modulusas proposedby Tsuji et a1.differsfromthis analysis,The effect on the shearstrengthis representedby a 1/3power,sincethey use the JSCE equation.

Figure-10 showsthe ratio of esdmatedand experimentalvalues[3][4][6][7]of shear strength.Theestimatedvalues are obtainedfrom the equationproposedby Niwa et al.[15Lwhich is proposed

yc'l,the method of Tsuji et

). It is assumedthat Vc isevaluatedby Niwa's equatiodadd that the Young's modulus of the steel bars in Eq.(8) is 2.0 x

for the conpcretebeams reinforced with steel bars(indicatedbya1.(indicatedby A ), and the proposedmethodof Eq.(8)(indicatedb

10 5wa. the number-ofspecimenswere hybrids ofaAnd 18 used Garb-on(C)OEi-6v.73x 10 4- 1.4tx 16 4 MPa). table-4 shows the average and acoefficientof variationfor the ratio of estimatedand the experimentalvalues for each method.The equationfor concretebeams reinforcedwith steelbar overestimatedthe experimentalresultsand the differenceappearsremarkablyfor lower Young's modulus.Tsuji's methodunderestimatedthe test results.On the other hand, the proposed methodcan satisfactorilyestimatetest results,regardlessof the differenceof Young's modulus.Thecoefficientof variationis also satisfactory.

Figure-11 show the relationshipsbetweenshear force and shear strain and betweenshear forceand curvaturefor M/(V.d)-3 when the Young's modulusof the main reinforcementis variedin

ea?aesrsimazndtaia,rbeSun1(tLlc8)?aosrn7ef,r&iin-f20.r,Cexd1%t4h-gl3?S3S.(Gx),1Tbi&1a1)

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V c a lN e x D C o e ffic ie n t o f v a r ia t io n %

N i w a T s u ii P ro p o s e d N iw a T s u 1'i P r o p o s e d

G , G + C (7 ) i . 5 0 0 .8 O 0 . 9 4 7 . 4 7 . 2 6 . 9

C (u ) I . 0 9 0 .8 6 0 . 9 l lO . I 7 . 3 8 . 0

A ll s p e c im e n s i . 2 5 0 .8 4 0 . 9 2 l8 . 0 8 . 0 7 . 8

10

I'''T

f'8

?

Rtthtt

Ei=2.Ox100kgcmi

/-<1A/ Ei=2.94xlO5kgucm2

+ : Eq.(8)

0 0.5 1 1.5 2 ?_.5 3 3.5

Vs(tf)

Figure-12EMfFu?;tROefH.EcnegisenTodulusof

7__5

1y.

lJ

>t2.1

(J

>

1.5

0.5

+

0+ +

+ 0 o0

A

+ : r-5mm(GFRP)0 : FlOmm(GFRP)A : steel barstirrup

5 10 15 20 25 30

Spachgof Shp(cm)Figue-14 Estimated

(ignoring !een%orfsE&L?negthqentDan

2,51>:

u

>3?-'J

>

1.5

0.5

25

Cm'

20

Figure-13 AnalyticalModel

8

+ : r-5mm(GFRP)0 : r-lOrrm(GFRP)

0 5 10 15 20 ?_5 30

Spacingof Sdrrup(cm)Figure-l5 EsdmatedResultof ExperimentData

(consideringbent bar Strength)

t2?Saxnall.ys5is,.TT?.! ,1.]5,,A.,.5anxd1? 5STnbdOls.?z;exsp10.nd5tXha?anrealsy,seicsbfvoerlyYou;gh'.SuThOduBuq?(gfsatisfactorilyestimatesthe effect of Young's moduluson shear strength,the mechanicalmeaningwas not clear. However, as shown in Fig.-ll, the shear strain which is identical with sheardeformationincreasesrapidlyin proportionas Young's modulusbecomessmall and the effect ofyoung's moduluson shearstrain is great comparedwiththat on the curvature.Therefore,it seemsthat the inctease of the shear strain due to the lower Young's modulusprobablyinfluencesshearstrength.Qi*p&The shear failuremodesof the concretebeams are distinguishedinto failure causedby yieldingorbreaking of stirrups and compressivefailure of the concrete near the loading point, even ifdiagonalcrackspredominate.Both failuremodesare basicallydifferentand the shear strengthforboth modes can not be evaluatedwith the same estimatingmethod. In this paper, only shearfailure causedby the breakingof FRPstirrupsor the yieldingof steel bar is considered.

Figure-12shows the analyticalresult of M/(V.d)-3 obtainedfrom analysisbased on the extenqedmodified compressionfields theory.The modelused in the analysishad rectangularcross section

beam depth of 25ch, longitudinalreinforcementr-atioof 1.61%,stirrup area of 2 xstirrup spacing of 10cm, as illustratedin Fig:13. Analysiswas_pe5rformedin which1^ 4

the Young,s modulus of the longitudinal reinforce&ehtu&as eifhe; 2.0 x Ilo5 or 2.94 x 10v'rvyAAA > YYA-Y UY-J.-Y,I IT"I---O y- - I T--I7 "- MIIlr----TTTI -T

Jl

forcefigure

30cm,2,and

of20 x0.35cm

2TdaihT%1tee,talhatax7fsstiTwSEaess2h;5a,xrels?s4tlnTfa&cTehe.fVShneicsatllLn,ss.shTOhTssythmeb:FpiedinS?Z:r

U

represents the shear strength without stirrupTascalculatedby EqM.(8).The shear resisting force of

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the stirrups is calculatedby the followingequation.

vs - Bwh/n2G,i /tanOE. (9)

in which oyi is the stirrupstress obtainedfrom the analysis,Oiis the directionof principalstress,n is the numberof subdividedlayers in crosssection, and Bwand h are the width and the heightof the crosssection.

As shown in Fig.-12, the shear resistingforce of the stirrupyields before the diagonaltensioncrack occurs and increasesgraduallyaRer that. Also, it increasesin linear proportion after theapplied force reachesthe shear strengthof Eq.(8). This tendencyis the same, independentof theYoung's modulusof the longitudinalreinforcementand/orthe stirrup.This result impliesthat theshear resisdng forceof the concretebeam with stirrupis approximatelyevaluatedby the modifiedtruss theory. h this paper, we tried to evaluatethe shear strengthof such beams by using themodifiedtruss theoryin whichyieldingor breakingof the.stirrupat the ultimatestate is assumed.

V =Vc+Vs (10)

in which Vc is the shear strengthconsideringthe effectof Young's modulusof the longitudinalreinforcement,Eq.(8),and Vsis the shear forcecarriedby the stirrupscalculatedby a trusstheory.

Figure-14showthe estimatedresultsof the experimentdataby Eq.(10).Here, Vs is calculatedbyassumingthat the compressivestress declinesto 45 degree and the stirrup stress is the uniaxialyieldingstressor the breakingstress.The A symbolsin the figure show the resultswith steel barstirrups.The shear strength can now be accuratelyevaluatedby the truss theory. Therefore,theshear strength of the concrete beam with FN as longitudinalreinforcementand steel bar asstirrup can be calculateconsideringonly the effect of the Young's modulusof the longitudinalreinforcementon Vc. On the contrary,the shear strength of that with FRP as the stirrup isoverestimatedby Eq.(10).Therefore,in this case,Vs can not be evaluatedusinguniaxial strengthofFRP in the samewayas with the steel bar stirrup.Onereasonis that the strengthof FRP in theconcrete deterioratesfromthe uniaxial strengthdue to the local stress intensityor the combinedeffect of the tension and the shear at the bent and a crack intersectionpart. Therefore,weattemptedto modifyVs consideringthe innuencingfactors.i&In Fig.-14, the shear strengthof the specimenswith differingradius of bent corner is different.Itis understood that the strength of the bent corner must be consideredcorrectly to evaluateVs.Therefore, we re-evaluateVs consideringthe strengthof the bent cornerusing Eq.(4). Figure-15compares estimated values using Eq.(4) with the experimental results. The differing shearstrengthsfor specimenshaving differentradius of bent comerthen disappears.By using Eq.(4) inVs, the effect of the bent corner on the shear strengthis satisfactorilyevaluated.However,sincethe estimatedvalues are larger than experimentvalues,we showfurther modiflCationof Vs.

hM

The factorrelatedto the shear strength,besidesthe strengthof the bent corner,is the effectof thediagonaltensile force. h this case, Eq.(6) and Eq.(7) proposedfor diagonaltensilestrengthmaybe applicable.However,the equationsrequirethe crack angleat crack intersectionand the angleis not always uniquely determined.Therefore,the evaluationof the shear strength using Eq.(6)and Eq.(7) is not consideredin this study.

The other effecton the shear strengthis Young's modulusof the stirrup. Satoh et a1.[17]reportfrom the analyticalresult of FEM that the shear strengthbecomessmall when Young's modulusof the stirrup is low. We therefore investigatethe effect of Young's modulus of the sdrrupanalyticallyand evaluatethe shearstrengthconsideringthe effectin this section.

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8i_1

\7t-

'=6h

O

cn-.5

4

3

2

1

J4I

withoutstirrup

?o.i2h&Xli.Z:kEgcccm:222

0.1 0.2 0.3 0.4 0.5

Curvature(10-3/cm)

Figtue-I6 Raneiasdzensarhip.?cegweencurvahre(young's modulus of mah reirdoecement of 29400MPa)

10

¥ql

U

88t<O

(I•`LJ

=76

Cn

4

?without sdrrup2::a-&xiso6;kEgg:gg2Z

0 0.05 0.1 0.15 0.2

Curyature(1 0p3/cm)Figure-18Reladonshipbetween Curvature

and ShearForce(Young's rnodulus of main reirJoecement of 200000MPa)

8

S7OU

,?•` 6

LdQ -

a) 3

m4

3

2

1

/

Ag3Z- - '- -

;v5Bjo.su6!E%&h:cg2.00x 100kgucrn2

0.002 0.004 0.006 0.008Shear Strain

Figtue-I7 RaneiagE:ihip.brceehveenShearSwain(Young's modulus of mah rehdoecement of 29400MPa)

10

G=

\8L<

Or.•`56a)

A•`)

4

?

// /I/

w

_ithoutsdmp

gis?eq:SigSSgcgi0 0.001 0.002 0.003 0.004

ShearStrain

Figure-i9 Raneiagz=arhiF.brce!weenShearSBain(Young's modulus of main reinfoecement of 200000MPa)

The analyticalresults based on the extendedmodifiedcompressionfields theory are shown inFig.-16 - Fig.-19, for the RC cross section shown in Fig.-13. Figure-16 shows the relationshipbetween the curvatureand the shearforce, while Fig:17 shows the relationshipbetweenthe shear

;arhSaegn:a:dedtth:ae.tuSonhfge,a!refmAr.ie:1fioesriA;r'c:e';'e==3tali;h2,?e;i4ntfh.xe,c1:0:Yenn%5:Hfofd.g?-:TS1.Oa:5dtkgsa7-qhPsehlSew;gteed;effsoult3Young's modulusof the sdrrup does not appearfor the curvatureand shearforce relationshipsasshown in Fig:16 and Fig:18. This means that the Young,s modulus of the sdrrup is not apredominantfactor for nexural deformation.On the other hand, the shear strain and the shearforce relationshipsas shownin Fig.-17and Fig.-19are innuencedby the Young's modulusof thestirrup. The shear strain increasesremarkablyfor lower Young's modulusof the stirrup. If theshear strain is identicalwith the shear deformation,the deformationof the beam is dominatedbythe shear deformationand directly dependson the Young's modulusof the stirrups.Therefore,assumingthat the shear strengthis closelyrelatedto the shear deformation,Vs may be evaluatedby consideringthe effectof Young's modulusof the stirrupon the sheardeformation.The methodconsideringthe effect of the shear deformationin Vs is alreadyproposedby Kobayashiet al.[6].Consideringthe shear deformationindirectlyrepresentsthe strengthreductionof FRP due to thecombinedeffect of the tensionand the shear causedby the local shearstrain in beam.

Since the shear strainand the shear forcerelationshipsare almost linearfor every case,the effectof Young's modulus of the stirrup is investigatedby the shear stiffness.Figure-20 show thereduction ratios of the shear stiffnessfrom the results of Young's modulusof 2.0 x 10 5MPa.

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ill i

O

O

'd:ao.8¥Ba 0.6S2 L.

;gao.40.?_

icEiG)0.4

Young's Modulus of-Main Reinforcement

A :2.Ox106kgucn2

0 0.5 1 1.5. 2

young,s Modulusof Sdrrup( x lO5MPa)

Figue-20 Effect of Young's Modulus of Stirrupon Shear Stiffhess

The symbols with + and A show the resultswhen the Young's modulus of the mainreinforcement is 2.94 x 104wa and 2.0 x 10 5MPa, respectively. The reduction of the shearstiffnessis independentof Young's modulusof themain reinforcementand depends on that of thestirrup.The effect is proportionalto the 0.4 powerof the Young's modulusof the stirrup. Assumingthat the effect of the stirrupis related to the sheardeformationdirectly,Vsis modifiedas

2.5

1•`U

>1_ 2nU

>

1.5

0.5 + : r-5mm(GFRP)0 : r-10mm(GFRP)

0 5 10 15 20_ 25 30

Spacingof Stimp(cm)Figue-21 Esdmated Result of E:(9eeiment Data

Table-5ComparisonwithExperiment[11]

vs* - vs(Ei /Es)0.4 (ll)

in which Ei is the Young's modulus of the stirrupand Es is the Young's modulus of the steelbar(2.0 x 10 5wa). Figure-21showthe comparisonbetweenthe experimentaland the estimatedvaluesusing Eq.(11). The applicabilityof the proposedmethodis foundsatisfactory.

Since the above investigationsare obtained from concrete beams reinforcedwith GFRP, theapplicabilityof this methodwas then checkedby comparingwith test resultscarried out by Wakuiet al.(a/d-3, s-25cm) using other FRP[11]. The mechanicalpropertiesof these FRP stirrupsaregreatly different from those of our experiment,as shown in Table-5. The estimatedresults aresummarizedin Table-5.The accuracyof estimationby the proposedmethodis satisfactory.

6. CONCLUSIONS

(1) Experimentalinvestigationshowsthat the radiusof the bent comeris an importantfactor forthe shearstrengthof a concretebeamwith FRP as the stirrup.

(2) Equations to evaluatethe bent bar strength and the diagonal tensile strengthof FRP areproposedfrom the theoreticalinvestigation.They are useful for evaluatingthe shearstrengthofconcretebeamsreinforcedwithFRP.

(3) For the shear strengthof concretebeamsreinforcedwith FRP as longitudinalreinforcement,the Young's modulusof the longitudinalreinforcementis a factorinnuencedto the shearstrengthand the effect can be evaluatedaccuratelyby the term of 1/4 power of the ratio of Young'smodulus.The effectof Young'smodulusratioon the shearstrengthis similarlyrepresentedto theeffectof the longitudinalreinforcementratio.

(4) The shearstrengthof a concretebeam reinforcedwith FRP as longitudinalreinforcementandwith steelbar stirrupcanbe accuratelyevaluatedby the modifiedtruss theory.

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S p e c im e nf y E L. V eg V oa/ v cd /v a cM P a M P a t f t f

C - A 2 - 0 l5 4 0 . 07 . 0 0 x l O 4 8 . 6 5 9 . 5 9 i . l l

C - C 2 -0 14 4 0 . 0I . l 2 x l O 5

8 . 5 0 9 . 8 3 i . l6

C - C S 2 -0 2 0 8 0 . 0i . 4 7 x l O 5

l O . 0 9 . 6 9 0 . 9 7

(5) The strengthof FRPin concretedeterioratesfrom the uniaxial strengthdue to the local stressintensity or the combinedeffect of the tensionand the shear at a bend and a crack intersection.The shear strengthof the concretebeamreinforcedwithFRP stirrups,whichfails by breakingofFN, is satisfactorilycalculatedby the proposedmethodwhen the bent corner strength and theeffect of shear deformationis consideredin Vs.

References

[1]JSCE Subcommitteefor Studieson ContinuousFiber : Present state of technologyconcemingapplicationto the field of civil engineeringstructuresof concrete-basecompositematerialusingcontinuousfiber, 1992(inJapanese).[2]Kakuta Y., et a1. : Establishmentof design method of concrete structuresusing continuousfiber, Report of Ministry of Educationin the form of Grant-in-Aidfor co-operativeresearch(A)No.04302040, 1994(in Japanese).[3]Tsuji Y., Saito H., Sekijima K. and Ogawa H. : Flexural and shear behaviors of concrete beamsreinforced with FRP, Proc. of JCI, Vol.10, No.3, pp.547-552, 1988(in Japanese).[4]Kanekura S., Maruyama K., Shimizu K. and Nakamura Y. : Shear behavior of concrete beamsreinforced by CFRP rods, Proc. ofJCI, Vo1.15, No.2, pp.887-892, 1993(in Japanese).[5]Yokoi K., Shima H. and Mizuguchi H. : Applicabilityof shear strength equation of RC beamfor reinforced concrete beams with FRP rods, Proc. of JCI, Vo1.14, No.2, pp.713-716, 1992(inJapanese).[6]Kobayashi T., Maruyama K., Shimizu K. and Kanekura S. : Shear behavior of concrete beamreinforced by CFRP rods, Proc. ofJCI, Vol.14, No.2, pp.70l-706, 1992(in Japanese).[7]Wakui H., Tottori S., Terada T. and Hara C. : Shear resisting behavior of PC beams using FRPas tendons and spiral hoops, Proc. of JCI, Vol.1 1, No.1, pp.835-838, 1989(in Japanese).[8]MaruyamaT., Honma M. and Okamura H. : Experimentalstudy on the tensile strength at thebending part of fiber reinforced plastic rods, Proc. of JCI, Vol.12, No.1, pp.1025-1030, 1990(inJapanese).[9]Miyata S., Tottori S., Terada T. and Sekijima K : Experimental study on tensile strength ofFRP bent bar, Proc. of JCI, Vol.1 1, No.1, pp.789-794, 1989(in Japanese).[10]Maruyama T., Horma M. and Okamura H. .. Some experiments on diagonal tensioncharacteristics of various flber reinforced plastic rods, Proc. of JCI, Vo1.ll, No.1, pp.771-776,1989(in Japanese).[11]Wakui H. and Tottori S. : Reinforcing effect of spiral FRP rods as shear reinforcement, Proc.of JCI, Voi.12, No.1, pp.1 141-1146, 1990(in Japanese).[12]Terada T., Tottori S., Wakui H. and Miyata S. : Study on behavior of shear failure of concretebeams reinforced with FRP bars, Proc. of JCI, Vo1.10, No.3, pp.541-546, 1988(in Japanese).[13]Nakamura H. and Higai T. : Evaluation of shear strength of RC beam section based onextended modified compression field theory, Concrete Library of JSCE, No.25, pp.93-105, 1995.[14]Idemitsu T., Yamasaki T., Harada T. and Yoshioka T. : An experimental study on the methodoftension test for FN tendons, Proc. ofJCI, Vo1.13,No.1, pp.795-800, 1991(in Japanese).[15]Niwa J., Yamada K., Yokozawa K. and Okamura H. : Revaluation of the equation for shearstrength of concrete beams without web reinforcement, Proc. of JSCE, No.372N-5, pp.167-176,1986(in Japanese).[16]Yokoi K., Shima H., Mizuguchi H. and mrohata H. : Stirrup stress of concrete beams usingFN rods as main reinforcement, Proc. of JCI, Vol.13, No.2, pp.771-776, 1991(in Japanese).[17]Sato Y., Ueda T. and Kakuta Y. : Qualitative evaluation of shear resisting behavior ofconcrete beams reinforced with FRP rods by finite element analysis, Concrete Library of JSCE,No.24, pp.193-209, 1994.

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