shear-fibre

8/8/2019 shear-fibre

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Mater ia ls an d St ructures /Mat6r iaux e t Con st ructions ,Vol.35 , November 002, pp 528 -535

E x p e r im e n t a l a n d t h e o r e t ic a l i n v e s tig a t io n o f t h e s h e a rr e s is ta n c e o f s te e l f ib r e r e i n f o r c e d p r e s t r e s s e d c o n c r e t eX - b e a m s - P a r t I1 : T h e o r e t ic a l a n a l y s i s a n d c o m p a r is o nw i t h e x p e r i m e n t sK. S . E l l io t t 1 , C . H . Peaston 2 and K . A . Pa inea( I ) U n i v e r si ty o f N o t t in g h a m , U K - ( 2 ) A r u p R e s e a r c h , F o r m e r l y a t U n i v e r s it y o f N o t t in g h a m , U K - ( 3 ) U n i v e rs i ty o f D u n d e e , F o r m e r l y a tUniver s i t y o f N ot t ingham, U KPaper r ece ived:Ju ly 30 , 200 1; Paper accepted: October 31 , 20 01

A B S T R A C T R I~ S U M I~T h i s i s t h e s e c o n d p a r t o f t w o p a p e r s o n t h e e x p e r i -m en tal (Pa r t I) and th eore tical (Par t I I) res is tance of s teel

f ib r e r e in fo rced p recas t concre te beam s .H i g h s t re n g t h s te e l w i r e , a n d t h i n a m o r p h o u s m e t a lf i br e s h a v e b e e n i n t r o d u c e d i n t o p r e s t r e s se d c o n c r e t e Xb e a m s i n o r d e r t o s t u d y t h e i r b e h a v i o u r u n d e r s h e a r l oa d s.E x p e r i m e n t a l t es ts h a v e d e t e r m i n e d s h e a r s t r e n g th s a t t h eu l t ima te and c r ack ing loads, and s how n increased duc t i l i tyw i t h u p t o 2 % f i b r e c o n t e n t . F r o m t h e s e te st s t w o d i f fe r -e n t m e t h o d s a re p r o p o s e d f o r p r e d i c ti n g t h e u l t i m a t es h e a r c a p a c i ty - t h e s e a r e t h e f i b r e s u p p l e m e n t a d d i ti v em e t h o d , a n d t h e m o d i f i e d F l k C p r i n c ip a l t e n s il e s tr es sm e t h o d . T h e p r i n c i p a l t e n s il e s t r e n g t h o f t h e f i b r e r e i n -f o r c e d c o n c r e t e i s g i v e n as a f u n c t i o n o f c o m p r e s s i v es t r e n g t h a n d f i b r e v o l u m e . T h e m e a n v a l u e o f t h e r a t io o fthe ca lcu la ted to the tes t s t r eng th i s 0 .89 wi thou t par t ia lsa fe ty f acto rs , and , be in g conserva t ive , i s p ropo sed fo r u sein des ign . A ca lcu la t ion mo dal i s p r esen ted .

C e t te pa t t i e es t l a deux i~me de deu x ar t ic l es s ur l es exp& iences(par t i e I ) e t l a th& r ie (par t i e I I ) s ur la r & is tance des pou t r es enbdto n pr~fabriquO renforcOde ibr es d ' ac#r .D e s a c ie r s a h a u t e a d h & e n c e e t d e m i n c e s f i b r e s m ( t a ll iq u e samo r phes on t ~ t~ in t r odu i t s dans des pou t r es X en b~ton pr &o ntr a in tpo ur ~ tud ier l eur compor tem ent s ou s Fact ion des e f for ts t ranchant s .

t 9 I . I I .D e s e s s a i s e x p e n m e n t a u x o n t d ~ t e r m t n e la r e st st a n c e u l t i m e a uc i s a il lement e t l es char ges de i s s ur a t ion , e t on t mon t r ~ une haus s e dela duc t i l it ( avec un vo lume a l lan t jus qu 'a 2 % de ibr es. Su r la basede ces es s a i s, de ux mO thodes d i f f&en tes s on t pr opos & s pou r la prOdic-t ion de la r &gtance u l t ime au cg a i l lemen t - l a m( thade des f ibr essupplOmentaires addit ionnel les et la m(thode de la contrainte de ten-s ion pr inc ipa le du b & n de ibr es d ' ac ier . La r &is tance pr inc ipa le entens ion du b ( ton r en forc( de f ibr es es t don n& com me fonc t ion de lar &is tance a la compr es s ion e t du vo lum e & f ibr es . La va leur moyennedu r appor t en tr e la r &is tance ca lcu l& e t cd le donne ' e pa r l es es sa i s es tde 0 , 8 9 s ans fac teur d e m ajor a t ion , e t , ( tan t cons er va tr i ce, r i l e es tpr opos & co mm e mO thode de ca lcu l . Le modhle de ca lcu l es t pr &ent~ .

1 . R E V I E W O F E A R L IE R S T U D I E S1 . 1 S h e a r t e s t s o n p r e s t r e s s e d f i b r e r e i n f o r c e dc o n c r e t e ( P F R C ) b e a m s

T h e m a i n p u r p o s e o f t h e s e te s ts h as b e e n t o d e t e r -m i n e t h e e n h a n c e m e n t i n s h e a r r e s i s t a n c e d u e t o t h ep r e s e n c e o f fi b r es , g i v e n b y t h e f i b r e v o l u m e r a t i o V f , t h es h e a r s p a n - t o - e f f e c t i v e d e p t h r a t io a / d , a n d t h e a m o u n to f p re s t re s s . R e s u l t s a r e e x p r e s s e d i n t e r m s o f t h e p e r -cen ta ge increase 11 in shear load a t u l t im ate f a i lu r e V u t totha t at f ir s t crack Vc~.

N a r a y a n a n a n d D a r w i s h [1 ] t e st e d p re s t re s s e d b e a m s( P F R C ) u s i n g c r i m p e d s te e l fi b re s o f V f = 0 .3 % t o 3 .0 % ,a/c l = 2 .0 and 3 .0 , w ith par t ial an d fu ll pr~estress . Par t ly pre-

s t r e s se d b e a m s ( w i t h a d d i t i o n a l r e b a rs ) f a i l e d a t h i g h e rl o a d s t h a n t h e i r c o u n t e r p a r t s s u c h t h a t r I r e d u c e d w i t hincreas ing prest res s. Fo r bea m s f a i l ing in w eb shear ten -s i o n a t a id = 2 . 0 , t h e p l a i n c o n c r e t e b e a m s c o l l a p s e di m m e d i a t e l y a f t e r f i r s t c r a c k i n g , w h i l e f i b r e r e i n f o r c e db e a m s c r a c k e d a t s i m i l a r l o a d s b u t w e r e a b l e t o s u s t a i nconsiderable loads beyo nd the f irs t crack g iv ing r I = 40% fori_~fS=1%. Ab dul-W aha b [2] reco rded r l = 18% for V f = 1 %f i b r e s a t a /d = 2 . 2 5 . F l e x u r a l s h e a r f a i l u r e ~ w e r eo b s e r v e d a t a id = 3 .0 by bo th these researcher s. Lo re n tsen[3] r ec o rded r 1 = 5 0% us ing V f = 1 .5% HS f ib res on fu l lyp r e s t r e s s e d I - b e a m s t e s t e d a t a i d = 3 . 7 . B a l a g u r u [ 4 ]r e c o r d e d r l = 6 % u s i n g V ( = 1 . 5 % H S f ib r e s o n P F I L C T -b e a m s w i t h m i n i m u m s h ea r f in ks . T h e r e w a s a n o p t i m u mv a l u e o f / ~ = 0 . 7 5 % , b e y o n d w h i c h t h e r e w a s a n i n s ig n i f-

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E l l io t t , P e a s t o n , P a i n e

icant increase in s t rength . I t i s l ikely that th is low v alue o fr 1 s d ue to th e effect of the shear links .Tes ts by Sh in [5] have show n tha t re in fo rced beamsc o n t a i n i n g s h e a r l i n k s a t t a i n l o w e r v a l u e s o f r 1 w i t hincreas ing V f a s those tha t do no t co n ta in l inks . L inks a rem u c h m o r a l e f f e ct iv e a t p r o v i d i n g s h e a r r e i n f o r c e m e n tthan f ib res , s ince even 50 % o f the requ i re d shear linksr e n d e r s f i b r e r e i n f o r c e m e n t i n e f f e c ti v e . S i m i la r b e h a v -i o u r h as b e e n o b s e r v e d b y F u r l a n J r . [ 6 ] o n P F R C b e a m sw i t h v a r i o us a m o u n t s o f s h e ar r e i n f o r c e m e n t . I n a d d i -t i o n , f i b r e s p e r f o r m b e t t e r t h a n s h e a r l i n k s i n l i m i t i n gcrack wid ths by b ind ing the c racks . L i t t le advan tage i s tob e g a i n e d f r o m t h e c o m b i n e d u s e o f l in k s a n d f i br e s , o rthe u se o f f ib res wh ere shear links cou ld be u sed .Balagu ru [4 ] and Rajagopa l [7 ] t es t ed par t i a l ly -p re -s t ressed beams us ing s t raight ro un d s teel f ibres wi th 1~ =0 .75%, and found s ign i f i can t increases in bo th V ul a n dV cr These t es t s showed q = 71%, a t t r ibu ted to the c rackar res t mech an i sm o f fibres. Tan [8] t es t ed par ti a l ly p re -s t r e s s e d T - b e a m s u s i n g V f - - 1 % H S f i b r e s a n d f o u n d23% increases in web c rack ing s t reng th . The behav iou ro f these beams d i f fe red f rom a l l the o ther t est s in tha t thep l a i n c o n c r e t e b e a m s h a d a c o n s i d e r a b l e l o a d - c a r r y i n gcapac i ty af t e r w eb c rack ing .I n c o n c l u s i o n , f i b r e s i n c r e a s e d t h e u l t i m a t e s h e a rs t re n g t h o f b e am s w i t h o u t s h e ar r e i n f o r c e m e n t u p t o 1 .8t imes the i r c rac k ing s t reng th .

1.2 U l t im ate shear capaci ty pred ictive methodsF r o m t h e v a r i o u s s h e a r t e s t s o n p r e s t r e s s e d F R Cb e a m s , t w o d i f f e r e n t m e t h o d s h a v e b e e n s u g g e s t e d f o rp red ic t ing th e u l t imate shear capac ity:(a) the f ib re supp lem en t add i t ive met hod(b) the BS8 110 m od i f i e d m ethod .

1 .2 .1 F i b r e s u p p l e m e n t a d d i t i v e methodA s h e a r s u p p l e m e n t Vb due to the f ib res is added toc o n v e n t i o n a l e q u a t i o n s f o r p l a i n r e i n f o r c e d c o n c r e t eb e a m s , s u c h t h a t t h e u l t i m a t e s h e a r c a p a c it y o f R F R Cbeam s i s g iven as:V. = Vc + Vb (1)

w her e V~ i s the c oncre te con t r ibu t ion to shear capac ity, acombina t ion o f concre te compress ive res i s t ance Vcc , p r e -s t ress ing V .. and dow el res is tance V~ Giv en that afteru"c r a c k i n g , w e b s h e a r t e n s i o n i n P F R C a c ts s i m i l a rl y t oshear t ens ion in RF R C [1] i t m ay be sui tab le to adop tm e a s u r e s o f V b p r e v i o u s l y d e f i n e d f o r u s e i n R F R C .T h e s e m e t h o d s h a v e l a rg e l y c o n c e n t r a t e d o n t h e t e n si les t ress provide d by fibres cro ss ing a shear crack.Vd , wh ich , depends o n the t ens i l e sp l it t ing and bo nds t reng th o f concre te i s l i tt l e a f fec ted by f ibres. Ho we ver ,i t has been sugges ted by Swamy [9 ] tha t f ib res increasethe s ti ffness o f the dow el zone he lp ing to co n ta in do welc r a c k g r o w t h , r e s u l t in g i n m o r e e f f i c ie n t d o w e l c o n t r i -bu t ion as fo l lows:V d = Pwb ,2 (O.156 fL , lt - 0.25) (U nits offfl, ult a re in N / m m 2) (2)

w he re Pw is the perc entage o f tension s teel based on bwd, b ,i s the ne t co ncre te wid th a t the l eve l o f the s t eel cen t ro idand S, u/t is the u l t ima te tensi le f lex ural s t rength of FR C.H o w e v e r , i n g e n e r a l , f i b r e s h a v e l it t le a f fe c t o n V ca n d t h e r e f o r e i t c a n b e a s s u m e d t o b e e q u a l f o r n o m i -na l ly iden t i ca l PF RC and p la in p res tressed beam s . Th ea p p r o a ch t a k e n i n t h e A C I B u i l d i n g C o d e [ 1 0 ] g i v e n b ythe fo l lowing equa t ion , where a sa fe ty fac to r o f 1 .5 hasbee n re mo ved , i s the re fo re app l icab le :

V ~ = V cc + V p = [ 0 . 4 ~ - fs + O . 4 5 % p ~ ] b w d (3 )w h e r e 0 . 4 ~ f ~ i s a d e s ig n a p p r o x i m a t io n t o t h e te n s il es t reng th , and % x is the pres tress at the cri t ical sect ion .T h e t e r m ~ b is u s u a l ly t h e s o le m e a s u r e o f t h eimp roved shear res is tance due to f ib res . In me asu r ing Vb,the add i t iona l pos t -c rack ing shear res is tance o f R F R C i sas sumed to be due to the f ib res t rans fe r r ing t ens il e s tressa c r o s s a c r a c k a n d n o t d u e t o a n y a d d i t i o n a l s h e a r i n ge f f e c t . C a s a n o v a [ 1 1 ] s u g g e s t e d c a l c u l a t i n g t h e p o s t -c rack ing f ib re t ens i l e s t res s fu d i rec t ly f rom un iax ia l t en -s i o n t es ts a t a c r i ti c a l c r a c k w i d t h o f 1 % o f t h e i n n e rlever a rm 0 .9d . Fo r d iagona l c racks inc l ine d a t 45 ~ Vb isg iven as :

Vb = f . b J (4)ftu can b e calculated th eore tically as ~10 V[ z[)V acc ord -ing to Li ra [12] , wh ere V[ i s the f ib re vo lum e f fa&ion , ~sthe f ib re mat r ix in te r fa~ ia l bond , )V i s the f ib re aspec trat io , and 110 is f ibre o rien tat io n fac2or. I t i s recogn isedthat a crac k crossed by the f ibres r un nin g d iagonal ly at 45 ~

f r o m t h e l o n g i tu d i n a l r e i n f o r c e m e n t t o t h e t o p o f th ebeam never ac tua l ly reaches the top o f the beam. Fo r th isreason the te rm 0 .9d is used in preference to d .Th e 'Dra m ix ' gu ide l ines [13 ] sugges t the f ib re sup -p lem en t i s due to a co ns tan t t ens i l e s t ress ac t ing a long ad i a g on a l cr a c k o f h e i g h t 0 . % . Vb i s ca lcu la ted by as sum-i n g t h e b e n d i n g m o m e n t c a r r ie d b y t h e f i b re s is to e q u a lt h e b e n d i n g m o m e n t d u e t o a n e q u i v a l e n t p o s t - c r a c k in gf lexu ra l s t reng th , fa e 30o Us in g em pi r i ca l re l a t ionsh ips

. 7 t , q , . .co rre la t ing thxs f l exu ra l s t reng th to the d i rec t a rea l t en -sile s t r e n g t h f t , o , the shear supp lem en t i s g iven as :

V b = 0 . 5 4 f c t , , , x R t b d (5 )w h e r e R t i s the ra t io o f the t ens i l e s treng th befo re andaf te r c rack ing and i s g iven by N em ege er [14 ] as :

R t : ( 1 8 0 C + W I E I ) ( 6)wh ere Wfi s the f ib re co n ten t ( in kg /m 3 ) and C i s a func-t ion o f the anchorage e f fec t o f the f ib res (g iven as 20 fo rH S fibres , and 35 for s t raight roun d fibres) . A com par i-s o n b e t w e e n E q u a t i o n s ( 4 ) a n d ( 5 ) i s g i v e n i n F i g . 1 .The resu l t ag rees wi th Ba lagu ru [4 ] where there i s l i t t l ei m p r o v e m e n t i n s t r e n g t h b e y o n d a V f o f 0 . 75 % . O t h e rt es ts o n R F R C b e a m s [ 1 5- 17 ] s h o w e d r e d u c e d i m p r o v e -m e n t b e y o n d 1 % . T h e a p pe a ra n ce o f a n o p t im u m V f isp a r t i a l l y d u e t o r e d u c e d c o n c r e t e w o r k a b i l i t y w i t h

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Elliott, Peaston, Paine

Fig. 4 - Equilibriumof tensile and compressive forces in acracked section.

Fig. 2 - Shear force vs crack width.PFRC beams with a significant ly improved duct i l i ty.Calculation of this compressive resistance is as follows.

Fig. 3 - Definition of the subtraction method.

to shear strength by tensile bridging as derived from Li[24] where the contribution to shear strength is negligi-ble. At w > 4 mm, the crack has extended through thecompressive zone and only kinking of the reinforcementoffers shear resistance.2.1.2 Fibre bridging

Fibres cont r ibute to shear s t rength by the f ibresbr idging across a c rack formed by pr inc ipa l t ens i lestresses. Increased ductility of the FRC beams cannot bee xp l a i ne d by t he f i b re b r i dg i ng me c ha n i sm a l one ,altho ugh the theoretical ultimate streng th is similar to thevalue obser ved in the tests. There fore, altho ugh fibresmay increase the shear strength, there is a further actionresulting in the ability of the beams to carry shear forcesover large deflectio ns and at large diagonal crack width s.The shear crack never ful ly develops through the ful ldepth into the compressive zone as there is a crack arrestmech anism due to fibre pinch ing forces at the crack tip.The compressive resistance of FR C is therefore retainedat greater deformations than in the plain concrete. It isthis ability of the fibres to arrest the shear crack, andmainta in a compress ive res i s tance tha t provides the

2.1.3 Shear force due to compressive resistanceTh e additio nal shear force V.~ carried b y shear fric-c3tion in the co mpression zone is due to fibre crack arrest.This varies from zero at cracking to a ma xim um value ata crack width of about 5 m m w here m any of the fibreshave pulled out and the only other resistance to shear isby d owel action. Ref erring to Fig. 3, this is calculated byfirst subtract ing the total area under the shear load -deflection cu rve (refer to Part I of this paper) of a plainbeam from that ofa FR C beam, for specified deflect ionsof//200,//300,//400 and 1/500, where 1 s the span. Thisarea is dMded by the deflection to give the average addi-tional shear force due to fibres. Th e average V( is c om -puted by subtracting the average fibre bridging contri-but ion Vb, given in Equation (4), from the above. Theconstant tensile bridging stres sf = qo Vf'~f)~f in Equation(4) was used.

At large deflections the beam tends to hinge abou t thecompression zone enabling a couple of forces to be con -sidered in the equilibrium of the vertical shear load versusthe horizontal compression in the con cret ef M and reac-tive tension in th e fibres f M- R efe rri ng to' Fig. 4, thevalue of 0.1h for the heigh~ of the compression block isconsistent with that observed in these X-be am tests, andwith that observed by Casanova [11]. The reactive tensilestress is assumed constant. The total fibre contribution istherefore the combination of a fibre bridging stress andthe reactive tensile stress. Referring to the equilibrium offorces in Fig. 4, f , M and f , M are given as:

Vci , ,f~,M -- 0.0 26h Zbl (11)

where bfis the w idth of the flange, a is the shear span; and:Vcra

s -- 0.468hZb d (12)wher e bf is the average w idth of the be am in the tens ionzone.

The values of f M and f, M are shown in Table 1. Thedef lec t ion a t whi 'ch the ul t imate load occurs var iesbetw een //45 0 an d//35 0. Average values calculated attwice these deflections approximate to the values o f f , M

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M a t e r i a l s a n d S t r u c t u r e s / M a t 6 r i a u x e t C o n s t ru c t io n s , V o l . 3 5 , N o v e m b e r2 0 0 2

T a b l e 1 - A v e r a g e r e s u lt s fo r f c ,M a n d f t , M f ro m a n a l y s is o f c o m p r e s s iv er e s i s t a n c eDeflection: 1/500 1/400 1/300 1/200

Vf (%) a / d0.5 2 .01.0 2 .01.5 2 .0

average0 .5 2 .81.0 2 .81.5 2 .8

a v e r a g e

Tensileand compressive tresses N/mm 2)f ~ , M f t ,M t , : , u

1 7 . 1 3 2 . 6 8 2 0 , 8 11 1 . 9 5 1 . 8 7 1 9 . 1 41 0 . 2 5 1 .6 1 1 9 .5 71 3 . 3 7 2 . 0 9 1 9 . 8 71 7 . 4 2 2 . 7 4 1 8 . 4 31 1 . 7 7 1 . 8 5 20 . 232 . 7 5 0 . 4 3 1 3 . 5 3

1 2 . 2 2 1 . 9 2 1 7 . 1 7

f t , M f c , M f t , M f c , M3 .26 25 .17 3 .94 30 .63 .10 25 .75 4 .03 29 .63 .07 22 .75 3 .56 30 .513 .11 24 .72 3 .87 30 .212 .90 24 .74 3 .89 24 .603 .18 25 .64 4 .03 31 .761.34 18.21 2 .86 25.102 . 7 0 2 3 . 7 9 3 . 7 4 2 7 . 5 6

N. B. Each value i s averageof o ur results.

f i b r e b r i d g i n g , t h e n t h e A V f t e r m c a n b ere la ted to the average b r id g ing s t res s ft u = r i0V f ' t f ) v f [12] to g ive:A = 1"10.7., .zy (14)Since q0 and ~,f are constan ts for any fibre,

f t , M a relat ionship bet4~veen c, and zf can be pro -4.79 d u c e d b y c u r v e f i t t i n g a s s h o w n i n F i g . 5 .4 . 6 4 T w o a d d i t i o n a l da t a p o in t s a r e b y M a n s u r4 .7 8 [ 2 5 ] a n d F a t t u h i [ 2 6 ] . T h i s r e l a t i o n s h i p4 . 73 b e t w e e n A V f and the f ib re b r idg ing s t res s i s3 . 8 7 subs tan t i a tedby the sp l i t t ing t es t s per fo rmed4.99 o n t h e A M f i b r e s i n w h i c h t h e f i b r e s r u p -3 . 9 4 t u r e d a t u l t i m a t e a s o p p o s e d t o p u l l i n g o u t .Th e va lue o f z f a t u l t imate fo r AM f ib res i s4.33 t h e r e f o r e r e l f i t e d t o t h e u l t i m a t e t e n s i l es t reng th o f the f ib re ra ther than the s t reng tho f th e m a t r i x . T h e v a l u e o f A i s t h e r e f o r e

g i v e n a s :

( 2 r . f y , " ~A = (15)wh ere , r is the ra t io o f f ib re c ro ss -sec t iona l a rea to f ib rep e r i m e t e r , / f is f i b r e l e n g t h , a n d 2 r. ~, / / f i s th e c r i ti c a lf ibre matri :~-in terfacial bo nd s t re ngth eqfial to the valueo f z f a t r u p t u r e o f t h e f i b r e s . T h i s e q u a ls 2 . 5 N / m m 2 ,as suming tha t J} , = 1900 N / m m 2 as g iven by the f ib rema nufac tu re rs [27 ] . Thus A = 321 t ak ing rl0 = 0.41. T hev a l u e o f A o b t a i n e d e x p e r i m e n t a l l y f r o m t h e A M f i b respl i t t ing tes ts equals 335, and shows that A is a measureo f t h e f i b r e b r id g i n g c o m p o n e n t .Fo r the H S f ib res , the c r i t ica l f ib re -m at r ix in te r fac ia lbon d s t reng th can be ca lcu la ted in a s imi lar m ann er ande q ua ls 9 . 2 N / m m 2 w h e n ) ) , = 1 1 0 0 N / r a m 2 . T h i s v a l u ei s shown in F ig . 5 , and sugges t s tha t f ib res in concre tew i t h a c o m p r e s s i v e c u b e s t r e n g t h g r e a t e r t h a n7 0 N / m m 2 w i l l fa il b y r u p t u r e r a t h e r t h a n b y p u l l i n g - o u to f t h e m a t r ix . U s i n g E q u a t io n s ( 1 3 ) a n d ( 1 4 ) ( r o u n d i n gof ffr om 0.48 to 0 .5)fct,5 is g iven as :

+ n o Z , , v , , (16)w h e r e f r o m F ig . 5 :

, c i = f ( f c u ) = l . 7 e ~ 17 6 17 6 ( 1 7 )T h e m a x i m u m f i b r e b r i d g i n g s t r e s s i n t e n s i o n m a ybe app rox imated as 0 3 7 f a e 3 00 [13 ] . Equ a t ion (16 ) ma y9 " : / l , q ,the re fo re be re -w r i t t en as :

f c , , , p = 0. 5 f~ , + 0.37ffl ,eq,300 ( 1 8 )E q u a t i o n ( 18 ) is c o m p a r e d w i t h t h e e x p e r i m e n t a l

da ta ob ta ined fo r f t , , fae 3oo and f~u in F ig 6 Th e go od9 p j t . " "co rre la t ion sugges ts t ha t~ e 300 ~s a good m easu re o f the9 . q , . .f i b r e b r i d g i n g s tr es s. T ~ s p h t t m g t e n s il e s t r e n g t h o fp l a i n c o n c r e t e c a n a l s o b e e s t i m a t e d f r o m t h e f l e x u r a lc r a c k i n g s tr e n g t h o f pl a in c o n c r e t e p r i sm s a s 0 . 6 f / [ 1 3 ]to g ive the u l t imate sp l i t t ing t ens i l e s t reng th pu re ly inte rms o f f l exu ra l p roper t i es as :f c t , s p = 0 .6f~ + 0.37fyt,eq,30o (19)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . // f l / " ~f = 9 . 2 N / m m 28 C r i t i c a l f i b r e - m a t r i x i n t e r f a e i a l /b o n d s t r e n g t h

7 / x f = 1 .6 9e 0" 024fcu5 M a n s u r [ 2 51

~, * T e s t r e s u l t sF a t t u h i [ 2 6 ] 3 0 x 0 . 5 m m H S f i b r e sI

i . i ~ ; i i P fI0 20 30 40 50 60 70 80 90 100

fc u (N/mm2)F i g . 5 - F i b r e m a t r i x b o n d s t r e s s x f v s c o n c r e t e s t r e n g t h f c u .

a n d f M a t u l ti m a t e. H e r e f , M = 4 . 6 6 a n d 3 .9 7 N / m m 2 a ta / d = ' 2 . 0 a n d 2 . 8 , r e s p e c t i v e l y , w h i c h i s 0 . 5 5 " ~ f , a n d0.47~1f~ , i . e . approx imate ly equa l to the c rack ing t ens i l es t res s observed in the sp l i t t ing t es t s . Assuming a shearc rack ang le o f 45 ~ these va lues can be d i rec t ly descr ibedas the ad di t ional tensi le s t ress carrie d across the c rack as ac o n s e q u e n c e o f t h e f i b r e s p r e v e n t in g p r o p a g a t i o n o f t h ec r a c k i n t o t h e c o m p r e s s i v e z o n e . T h e t e n s i l e s tr e ssacross the c rack i s the re fo re a com bina t ion o f th i s s tressand the f ib re b r idg ing s tress .

The fo l lowing sec t ion d i scusses the f ib re con t r ibu -t ion to the sp l i tt ing tens i l e s t reng th an d a bas i s on wh icht h e f i b r e r e i n f o r c i n g m e c h a n i s m i n s p l i t t i n g m a y b eut i l ised in the design o f pres tressed F1LC beam s in shear.

2 . 2 T h e o r e t i c a l e q u a t i o n sT h e s p l i tt in g t e n si le s t r e n g t h f s o f F R C c o m p r i s i n gc t,HS f ib res i s g iven in Par t I o f th i s paper as :f~ ,.~ p = O . 4 8~ u ~ u + A V f (13)wh ere A = 237 fo r MS f ib res and 335 fo r AM f ib res .I f t h e m a t r i x c o n t r i b u t i o n t o p o s t - c r a c k in g s t r e n g t hi s cons tan t , a nd the increase in sp l i t ting s t reng th i s due to

5 3 2


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Elliott, Pea ston,Paine8

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I I J I3 4 5 6 7

0 . 5 f c u 0 5 + 0 . 3 7 f c t . e q . 3 0 0

F i g . 6 - U l t i m a t e c y l i n d e r s p l i t t i n g s t r e n g t h f c t , s p: t e s t v a l u e s v sE q u a t i o n ( 1 8 ) .

1 9 7 m m ]

~ ' ] - - - ~ _ _ - 7 " 2 5 . 4 m mF t Tt - - 7 I 2 5 . 4 m me-h

2 5 . 4 2 5 . 4 t u r nA s = 2 2 2 4 9 m m 2Z t = Z b = 1 .2 2 x l O 6 r a m 3I - - 67 8 . 5 x 1 0 6 m m 4

F i g . 7 - C r o s s s e c t i on o f x - b e a m u s e d i n c a l c u l a t io n m o d e l .

2 . 3 S e m i - e m p i r i c a l e q u a t i o n sI n B S 8 1 1 0 a n d m o s t E u r o p e a n c o d e s o f p r a c ti c e t h e

m e t h o d o f c a lc u l a t in g t h e s h e a r r e s is t a n c e o f a p r e st r e s s e dc o n c r e t e b e a m i s t o r es t ri c t th e m a x i m u m p r i n ci p a l t e n -s il e s tr e ss a t t h e g e o m e t r i c a x is o f t h e s e c t i o n t o t h e t e n -s il e s tr e n g t h o f t h e c o n c r e t e . F o r P F P , . C b e a m s , a si m i la ra p p r o a c h c a n b e u s e d b u t f o r t h e s e b e a m s t h e m a x i m u mt e ns i l e s t r e s s t ha t c a n be c a r r i e d a c r o s s t he s he a r c r a c k a tt h e g e o m e t r i c a x is i s e q u a l t o t h e s u m o f t h e f i b r e b r i d g -i n g s t r e s s a n d t h e r e a c t i v e t e n s i l e s t r e s s d u e t o c o m p r e s -s iv e f o rc e s a t t h e t o p o f t h e b e a m . T h e m a x i m u m p r i n -c i p a l t e n s i l e s t r e s s m a y t h e r e f o r e b e e q u a t e d t o f < s p tog i ve t he u l t i m a t e s he a r s t r e s s % , as :

V c o u = " ] f i~ , sp + 0 . 6 7 f c ,, , p O c p , (20)w h e r e f t s p = 0 . 5" 4 fu + 0 3 7 f z e 3 0o W h e r e f l e x u r a l p r o p -, j ~ , . q , e r t m s a r e no t kn ow nJ ~ , e 3oo i s t a ke n a s t l0 ) ~ ,. ~ V~ Th e71 q , " T 'J " ) "v a l u e o f o a t u l ti m a t e w a s f o u n d t o b e a p p ro x i m a t e l yc px0 6 7 0 x o f t h a t at c r a c k i n g , i n d e p e n d e n t o f V~9 cpE q u a t i o n ( 2 0 ) c o u l d a l s o b e r e w r i t t e n i n t h e f o r m r e c o f i a-

m e n d e d b y G i r h a m m a r [ 28 ] a n d u s e d b y P C I [ 21 ] as :Vco u = 0 . 5 f ~ c u + 0 . 3 7 f f t , e q , 3 0 0 + 0 . 3 ( 0 . 67Oc px) ( 21 )

w i t h t h e s h e a r s t r e n g t h o f p l a i n p r e s t r e s s e d c o n c r e t eg i ve n a s :

Vc o = 0 - 5 f ~ c u + 0 . 3 ~ c p x ( 22 )T h e s h e a r s t r e s s s u p p l e m e n t v b d u e t o a d d i n g f i b r e s i s

g i ve n a s :P b = Vcou - - P co

= 0 . 3 7 f ~ , e q , 3 O o - 0.1%p~ ~ q0 "Ky "~i "V l - 0" l~ (23)T h e s h e a r s tr e n g t h o f p r e s t r e s s e d F P , .C b e a m s c a n

t h e r e fo r e b e g iv e n b y tw o e q u a t i o n s . O n e is c o n s i s te n tw i t h t h e d e s i g n o f p r e s tr e s se d b e a m s a n d h o l l o w c o r es la b s i n t h e E u r o p e a n c o d e s , a n d a se c o n d e q u a t i o n c o n -s i s te n t w i t h t h e a p p r o a c h e s c u r r e n t l y a d v o c a t e d f o rd e s ig n o f r e in f o r c e d F R C b e a m s i n s h e ar .

3 . C A L C U L A T I O N M O D E LT o d e t e r m i n e t h e s h ea r c a p a c it y o f F R C X - b e a m s

h a v i n g a c r o s s s e c t i o n a n d p r e s t r e s s i n g a r r a n g e m e n t u s e di n t h is p a p er , t h e f o l l o w i n g m e t h o d s m a y b e u s e d :1 . t h e a d d i t i o n m e t h o d g i v e n i n E q u a t i o n s ( 1) t o ( 8) ;2 . B S 8 1 1 0 m o d i f i e d e q u a t io n s .

C o n s i d e r t h e X - b e a m s h o w n i n F i g . 7 r e i n f o r c e du s i n g 1 % H S f i b re s 3 0 m m l o n g x 0 . 5 m m d i a m e t er , a n d

f , = 79 . 5 N / m m 2 , w i t h a / d = 2 .8 9 T he ana lys i s i s to bec o m p a r e d w i t h b e a m P B 7 A ( se e P a r t I) w h e r e % p x = 3 . 5N / r a m 2 . 0 024 x 79 5Te ch nic a l da ta : r io = 0 .41 , )~f = 60 , z f . = 1 .7e 9= 1 1 .4 > 9 . 2 N / m m 2 . I ~ = 7 8 . 5 " k g / m 3 . C = 2 0 .Method 1

F o r s h e a r d u e t o c o m p r e s s i o n r e s i s t a n c e :Eq ua t i o n ( 3 ) V~ = 0 . 4 " ] 79 .5 x 181 x 37 x 10 3 = 23 . 9 k N

Vp = 0 . 45 x 3 . 5 x 181 x 37 x 10 3 ~ 10 .5 k NF o r f i b r e b r i d g i n g , f o u r e q u a t i o n s a r e c o m p a r e d :

( a) Eq ua t i on ( 4 ) V b = 2 . 26 x 37 x 181 x 10 - 3 = 15. 1 k Nw h e r e f t = 0 .4 1 x 0 .0 1 x 9 . 2 x 6 0 = 2 . 2 6 N / r a m 2( b ) E q u a t i o n ( 5) V b = 0 . 54 x 0 . 4 ~ ] 79 . 5 x 0 . 623 x 37 x181 x 1 0 - 3 = 8 .0 k Nw h e r e f r o m E q u a t i o n ( 6 ) R t = 0 . 6 2 3( c ) Equa t i on ( 8 ) V b = 0.6 x 15 .1 = 9 .1 kN( d ) E q u a t i o n ( 2 3 ) V b = [ ( 0 . 41 x 60 x 9 . 2 x 0 . 01 ) - ( 0 . 1 x3 . 5 ) ] x 37 x 181 x 10 - 3 = 12 . 5 k N

T h e n E q u a t i o n ( 1 ) g i v e s :M a x i m u m V u = 2 3 . 9 + 1 0 . 5 + 1 5.1 = 4 9 . 5 k NM i n i m u m V u = 2 3 . 9 + 1 0 . 5 + 8 . 0 = 4 2 . 4 k N .

Method2Eq ua t ion (16)fc t w = 0 .5"479.5 + (0 .41x 0 .01 x 9 .2 x 60)= 6 . 7 2 N / r a m 2 c o m p a r e d w i t h 7 . 4 5 N / r a m 2 i n t h e te s t.E q u a t i o n ( 2 0 )Vco = ~ 2 2 + 0 . 6 7 x 6 . 7 2 x 3 . 5 = 7 .8 N / m m 2Vc ou = 7 .8 x 181 x 37 x 10 -3 = 5 2 . 2 k N

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M a t e r i a l s a n d S t r u c t u r e s / M a t 6 r i a u x e t C o n s t r u c ti o n s , Vol. 3 5 , N ov e m ber 2 0 0 2

T able 2[a] - C om par i son o f theo re t ica l and exper imen ta lu l t imate shear s t r eng th s o f pla in beam sa / d


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E l l io t t , P e a s t o n , P a i n e

crete ' , in ACI SP105 - Fiber Reinforced Concrete Properties andApplications, (1987) 419-434.[5] Shin, S. W ., O h, J . G. and Ghosh, S. K. , 'Sh ear behaviour of lab-oratory-sized high strength concrete beams reinforced with barsa n d s t e e l f i b e r s ' , A C I S P 1 4 2 - F i b e r R e i n f o r c e d C o n c r e t e -Developments and Innovations, (1994) 187-200.[6] Furlan, Jr , S . and de H anai, J . B. , 'Prestressed fibre rein force dc o n c r e t e b e a m s w i t h r e d u c e d r a t i o s o f s h e a r re i n f o r c e m e n t ' ,Cem ent and Con crete Com pos i tes 21 (1999) 213-221.[7] Rajagopal, R. S. and Siddappa, S., 'Experimental investigation onf i b e r r e i n f o r c e d p r e s t r e s s e d c o n c r e t e b e a m s u n d e r s h e a r ' , i n'F ib re Re in fo rced C em ent and C oncre te ' , P roceed ings o f the 4 thIn te rna t iona l Sympos ium, Shef f ie ld , Ju ly 1992 (RILEM 1992)554-570.[8] Tan , K. , Paramasivam, P. and Murugappan, K. , 'S teel f ibers asshear reinforcement in partially prestressed beam s' , A C I S t r u c tu r a l

_Journal 92 (6) (1995) 643-652.[9 ] Swamy, R . N. and Bah ia, H. M. , ' In f luence o f f iber re in fo rce -ment on the dowel resistance to shear ' , A C I J o u r n a 1 7 6 (2) (1979)327-355.[ 1 0 ] A C I C o m m i t t e e 3 1 8 , ' B u i l d i n g C o d e r e q u i r e m e n t s f o rR e i n f o r c e d C o n c r e t e ' A C I 3 1 8 - 8 9 , A m e r i ca n C o n c r e t eInsti tute , Michigan, USA (1989).[11] Casanova, P. , Rossi, P . and Schaller , I. , 'C an steel fibers replacet ransverse re in fo rcements in re in fo rced concre te beams ' , A C IMater ia l sJo u r n a l 94 (5) (1997) 341-354.[12] Lira, T. Y. , Paramasivam, P. and Lee, S. L. , 'Analytical modelfor tensile behaviour of steel-f iber con crete ' , A C I M a t eri a lsJo u r n a l84 (4) (1987) 286-298.[13] Dramix Guide lines, 'Des ign o f Conc re te S t ruc tures . S tee l WireF ib re Re in fo rced C oncre te S t ructu res wi th o r wi thou t O rd ina ryRein fo rc eme nt ' , F ina l Dra ft , (1995) .[ 1 4 ] N e m e g e e r , D . a n d T a t n a l l , P . C . , ' M e a s u r i n g T o u g h n e s sCharacte ri s tic s o f SFR C - A Cr i tica l View of AS TM C1018 ,AC I SP- 155 ', Tes t ing o f F ib re Re in fo rced C oncre te , (1995).

[15] Swamy, R. N . and B ahia, H. M ., 'T he effective of steel fibres asshear re in fo rcement ' , Concrete n ternat ional 7 (3) (1985) 35-40.[16] Narayanan, R . and D arwish, I . Y. S. , 'Us e o f steel f ibres as shearre in fo rcement ' , A C I S t r u c tu r a l o u r n a l 8 4 (3) (1987) 216-227.[17] Ashour, S . A. , Hasanain, G. S. and Wafa, F. F. , 'Shear behaviourof high-strength fiber reinforced c o n c r e t e beams ' , A C I S t ru c t u ra lJo u r n a l 89 (2) (1992) 176-184.[18] BS 8110, 'Structural U se o f Con crete, Part 1 , Code o f Practicef o r D e s i g n a n d C o n s t r u c t i o n ' , B r i t is h S t a n da r ds I n s t i t u t i o n ,Londo n (1985).[19] Eurocode 2 : 'Des ign o f Concre te S t ruc tu res -P ar t 1 : Genera lRule s and Rule s for Buildings, ' E N 1992-1-1 (2001).[ 20 ] F d d & a t i o n I n t e r n a t i o n a l e d e l a P r & o n t r a i n t e , ' P r e c a s tP r e s t r e s s e d H o l l o w C o r e F l o o r s ' , T h o m a s T e l f o r d , L o n d o n(1988).[21J PC I 'Manua l fo r the Des ign o f Hol low Cored S labs. ' P rest ressedCon crete Insti tute , I l l inois , USA , (1991).[22] Stang, H. and A arre , T . , 'Eva lua tion o f c rack wid th in FR C wi thconven t iona l re in fo rcement ' , C em en t a n d C o n cr e t e C o m p o s i t e s 14(1992) 143-154.[23] Paine, K. , 'S teel f ibre reinforced concrete for prestressed hollowcore s labs ', PhD Thes is , Un iversi ty o f Not t ingham , U K, (1998).[24] Li, V. C. , Stang, H. and Krenchel, H. , 'Micromechanics of crackbr idg ing in f ib re - re in fo rced concre te ' , Mater . S t ruct . 26 (1993)486-494.[25] Mansur , M. A. , Ong , K. C . G. and Paramas ivam, P . , 'Shears t reng th o f f ib rous concre te b eams w i thou t s t ir rups ' , Jo u r n a l o fS t r uc t u ra l E n g i n eer i ng , A S C E 112 (9) (1986) 2066-2079.[26] Fa t tuh i , N. , 'Re in fo rced co rbe ls made wi th p la in and f ib rousconcre te ' , A C I S t r u c tu r a lJo u r n a l 91 (5) (1994) 530-536.[27] Bouchare t , J . -M. , Pe rsona l Com mu nica t ion (1997) .[28] Girham mar , U . -A. , 'Des ign p r inc iples fo r s imply-suppor ted p re -stressed hollow core slabs ' , S t r u c t u r a l E n g i n eer i n g R ev i ew 4 (4)(1992) 301-316.

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