FE modeling of inelastic behavior of reinforced high-strength concrete continuous beams

Transcription

1 Structural Enginring and Mchanics, Vol. 49, No. 3 (214) DOI: FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams Tijiong Lou 1, Srgio M.R. Lops 1 and Adlino V. Lops 2 1 CEMUC, Dpartmnt of Civil Enginring, Univrsity of Coimbra, Coimbra , Portugal 2 Dpartmnt of Civil Enginring, Univrsity of Coimbra, Coimbra , Portugal (Rcivd Novmbr 5, 212, Rvisd Dcmbr 18, 213, Accptd Dcmbr 27, 213) Abstract. A finit lmnt modl for prdicting th ntir nonlinar bhavior of rinforcd high-strngth concrt continuous bams is dscribd. Th modl is basd on th momnt-curvatur rlations pr-gnratd through sction analysis, and is formulatd utilizing th Timoshnko bam thory. Th validity of th modl is vrifid with xprimntal rsults of a sris of continuous high-strngth concrt bam spcimns. Som important aspcts of bhavior of th bams having diffrnt tnsil rinforcmnt ratios ar valuatd. In addition, a paramtric study is carrid out on continuous high-strngth concrt bams with practical dimnsions to xamin th ffct of tnsil rinforcmnt on th dgr of momnt rdistribution. Th analysis shows that th tnsil rinforcmnt in continuous high-strngth concrt bams affcts significantly th mmbr bhavior, namly, th flxural cracking stiffnss, flxural ductility, nutral axis dpth and rdistribution of momnts. It is also found that th rlation btwn th tnsil rinforcmnt ratios at critical ngativ and positiv momnt rgions has grat influnc on th momnt rdistribution, whil th importanc of this factor is nglctd in various cods. Kywords: high-strngth concrt; bams; momnt rdistribution; finit lmnt mthod 1. Introduction Th advanc in construction matrial tchnology has mad it possibl to manufactur high-prformanc concrts, faturd by grat improvmnt with rgard to mchanical charactristics, workability and durability. Du to its attractiv advantags, high-strngth concrt has bn broadly usd in spcial structurs, such as cross-sa bridgs, whr th strngth, durability and srvicability ar of particular concrn. Also, th utilization of high-strngth concrt in normal constructions is xpctd to b popular. It is wll known that high-strngth concrt is mor fragil whn compard to normal-strngth concrt. Th fragility of high-strngth concrt gav ris to som doubts of its us in structurs, sinc th ductil capacity of th structurs is vry important from a point of viw of structural safty, particularly in high sismic rgions. Ovr last 15 yars, many xprimntal and thortical studis hav bn conductd to xamin th ductil bhavior and plastic rotation of rinforcd high-strngth concrt bams (Arslan and Cihanli 21, Brnardo and Lops 24, 29, Carmo Corrsponding author, Profssor, srgio@dc.uc.pt Copyright 214 Tchno-Prss, Ltd. ISSN: (Print), (Onlin)

2 374 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops and Lops 25, 28, Cucchiara t al. 212, Kassoul and Bougara 21, Ko t al. 21, Kwan t al. 24, Lops and Brnardo 23, Lops and Carmo 26, Pam t al. 21, Yang t al. 212) and columns (Fostr 21, Fostr and Attard 21, Tan and Nguyn 25, Campion t al. 26, 212). Th studis gnrally showd th high-strngth concrt mmbrs hav sufficint ductility to guarant thir structural safty providd that an appropriat choic of th amount and location of th rinforcmnt is mad, and that th gnral ruls adoptd for normal-strngth concrt structurs can also b applid to high-strngth concrt structurs. So far, most of th prvious works concrning th bhavior of high-strngth concrt structurs focusd on statically dtrminat structurs such as simply supportd bams. In fact, in practical nginring, th statically indtrminat structurs, such as frams and continuous bams, ar mor common. Du to th xistnc of rdundant rstraints, th bhavior of continuous bams may introduc diffrnt aspcts, which implis that th ovrall bhavior of th structurs can b quit diffrnt from that of simply supportd bams. In continuous bams, som rdistribution of momnts might tak plac whn th constitunt matrials bgin to xhibit inlastic bhavior. Th rdistribution of momnts is a vry important charactristic of continuous bams and should b wll takn into considration in th ultimat dsign of such typ of bams. Howvr, th momnt rdistribution bhavior as wll as othr flxural charactristics of continuous high-strngth concrt bams is yt to b wll undrstood, sinc fw studis on this topic ar availabl (Carmo and Lops 25, 28). This articl prsnts a finit lmnt (FE) modl dvlopd to prdict th complt rspons of continuous high-strngth concrt bams from zro loads up to th ultimat. Th FE mthod is basd on th momnt-curvatur rlations and is formulatd using th Timoshnko bam thory. Th proposd analysis is usd to rproduc th xprimntal rsults of a sris of two-span continuous high-strngth concrt bam spcimns. A paramtric study is carrid out on continuous bams with practical dimnsions. Emphasis of th numrical valuation is placd on th ffct of tnsil rinforcmnt and th rdistribution of momnts in continuous rinforcd high-strngth concrt bams. c f cm 1 E c c u c Fig. 1 Strss-strain diagram for unconfind concrt in comprssion

3 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams 375 c * f cm Confind concrt.85 f cm f cm Unconfind concrt E c 1 * c u c Fig. 2 Strss-strain diagram for confind concrt in comprssion * u c 2. Matrial constitutiv laws 2.1 Strss-strain law for concrt in comprssion In this study, th strss-strain bhavior for unconfind concrt is simulatd using th modl rcommndd by Eurocod 2 (EC2) (CEN 24) and Mod Cod 199 (MC9) (CEB-FIP 199). Th strss-strain diagram is shown schmatically in Fig. 1, and is xprssd as follows 2 c k (1) f 1 ( k 2) cm whr / (2) c c k 1.5 E / f (3) f cm c c cm f 8 (4) ck.31 ( ).7 c fcm (5) E c.3 22( fcm /1) (6) in which σ c and ε c ar th concrt strss and strain, rspctivly; f cm is th man comprssiv strngth (in MPa); f ck is th charactristic cylindr comprssiv strngth (in MPa); ε c is th concrt strain at pak strss; E c is th modulus of lasticity of concrt (in GPa). Eq. (1) is valid for < ε c < ε u, whr ε u is th ultimat concrt comprssiv strain dtrmind by ( ) 3.5 u for fck 5 MPa (7a)

4 376 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops ( ) [(98 ) /1] 4 u fcm for ck 5 MPa f (7b) To facilitat th numrical modling, it is assumd that, at initial loading, th concrt in comprssion is linar lastic until th lastic strss and strain mt th curv quation rprsntd by Eq. (1), as illustratd in Fig. 1. MC9 also dfind a modl for confind concrt, which is a modification of th unconfind modl by adjusting th magnitud of som ky paramtrs, namly, th concrt strngth, th strain at pak strss and th ultimat strain, as shown in Fig. 2. In th absnc of mor prcis data, th rlations btwn th paramtrs for confind concrt and thos for unconfind concrt may b xprssd as follows * fcm fcm( w) for 2.5 f cm (8a) * fcm fcm( w) for 2.5 f cm (8b) * * 2 ( / ) c c fcm fcm (9).1 (1) * u u w * * * whr f cm, c, and u ar th confind strngth, confind strain at pak strss and confind ultimat strain, rspctivly; ω w is th volumtric mchanical ratio of confining stl; α is th ffctivnss of confinmnt, qual to α n α s, whr α n dpnds on th arrangmnt of stirrups in th cross sction and α s dpnds on th spacing of stirrups; and σ 2 is th ffctiv latral comprssion strss du to confinmnt. 2.2 Strss-strain law for concrt in tnsion Th strss-strain bhavior for concrt in tnsion is modld by a linar-lastic law prior to cracking and by a bilinar tnsion-stiffning law aftr cracking, as shown in Fig. 3, whr f t is th tnsil strngth and ε cr is th cracking strain, qual to f t /E c. Th tnsil strngth is dtrmind in trms of th rcommndation by EC2: for concrt class not highr than C5/6 f t.3 f (11a) 2/3 ck for concrt class highr than C5/6 f t 2.12ln(1 f /1) (11b) cm 2.3 Strss-strain law for rinforcing stl Th strss-strain bhavior for rinforcing stl in both tnsion and comprssion is modld by a bilinar lastic-hardning law, as shown in Fig. 4, whr σ s and ε s ar th stl strss and strain, rspctivly; f y and f su ar th yild strngth and ultimat tnsil strngth of rinforcing stl, rspctivly; E s is th modulus of lasticity of rinforcing stl; and ε su is th ultimat stl strain, which is takn as.75 in this study.

5 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams 377 c f t.2 f t cr 5 15 cr cr c Fig. 3 Strss-strain diagram for concrt in tnsion s f su f y 1 E s su s Fig. 4 Strss-strain diagram for rinforcing stl 3. Momnt-curvatur rlations To stablish th momnt-curvatur rlations of rinforcd concrt sctions, th sctions ar dividd into many concrt layrs and rinforcmnt layrs, as shown in Fig. 5. Th strain in ach concrt layr is assumd to b constant, and qual to th valu at th cntr of th layr. Each rinforcmnt layr rprsnts th stl bars at th lvl of th layr. Th analysis assums that a plan sction rmains plan aftr dformation, and that th rinforcmnt prfctly bonds with th surrounding concrt. Basd on ths assumptions, as shown in Fig. 5, th axial strain at any fibr of th cross sctions is givn by y (12) whr ϕ is th sction curvatur, and y is th distanc from th nutral axis.

6 378 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops Layr Aci y ci Nutral axis Asj sj Fig. 5 Sction and strain Momnt Yilding Ultimat Loading 1 (EI) la Cracking Unloading (EI) la 1 Curvatur Fig. 6 Momnt-curvatur diagram for rinforcd concrt sction Assuming that th cross sctions ar subjctd to bnding momnt M (zro axial forc), th axial and flxural quilibriums of th sctions can b rspctivly xprssd as (13) ci Aci sj Asj i j (14) M A y A y i ci ci ci sj sj sj j whr σ is th strss and A is th ara; th subscript ci rprsnts ach concrt layr and sj rprsnts ach rinforcmnt layr; th summation is mad for all concrt or rinforcmnt layrs. Th momnt-curvatur rlations of bam sctions ar gnratd by incrmntally varying th prscribd curvatur starting from zro valu. Th ultimat failur of th sctions occurs whn th comprssiv concrt or tnsil rinforcmnt rachs its ultimat strain. Th procdur for dtrmining th complt momnt-curvatur rlation of a cross sction is summarizd as follows: (1) Incras th valu of th curvatur (initial curvatur of zro) with a small stp;

7 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams 379 (2) Dtrmin th strains in stl and concrt using Eq. (12) basd on an assumd initial or th prvious nutral axis dpth (gnrally, th initial position of th nutral axis can b takn as th position of th cntroidal axis); (3) Assss th strss in ach concrt or rinforcmnt layr basd on th strss-strain laws of th matrials; (4) Calculat th total axial forc contributd by th concrt and rinforcmnt using th right sid of Eq. (13); (5) Chck th axial quilibrium of th sction; if th quilibrium is not satisfid, adjust th position of th nutral axis until th unbalancd axial forc is vanishd; (6) Comput th bnding momnt using Eq. (14); (7) Rpat from stp (1) until failur of th sction. Th typical momnt-curvatur diagram for a rinforcd concrt sction is shown schmatically in Fig. 6. During th whol loading procss, th momnt-curvatur rspons would xprinc thr stags. Th first stag is charactrizd by lastic bhavior and is finishd by concrt cracking. This is followd by th scond stag up to stl yilding and thn th third stag until th ultimat failur. In th cas of unloading, it is assumd that th momnt dcrass linarly with th curvatur, and th slop of dcras is qual to th lastic flxural stiffnss (EI) la, as shown in Fig FE mthod Th FE mthod is formulatd using th Timoshnko bam thory. In this thory, it is assumd that a plan sction normal to th cntroidal axis bfor dformation rmains plan but dos not rmain normal to th cntroidal axis aftr dformation bcaus of th ffct of transvrs shar dformations. In a two-nod lmnt, th transvrs displacmnt w and rotation θ can b xprssd as functions of thir rspctiv nodal displacmnts w N1w1 N2w2, N1 1 N2 2 (15) Whr N1 ( x2 x) / l and N2 ( x x1) / l, in which l is th original lngth of th bam lmnt; x 1, x 2 ar th coordinat valus for th lmnt nods, and x 2 x 1 = l. Th curvatur and shar strain can b xprssd rspctivly as follows: d dn1 dn2 1 2 Bu b (16) dx dx dx whr dw dn1 dn2 w1 N1 1 w2 N2 2 Bu s (17) dx dx dx T u { w1 1 w2 2} (18) dn dx dn dx 1 2 B b [ ] (19)

8 38 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops dn1 dn2 B s [ N1 N2] dx dx (2) Th bnding momnt M and shar forc Q can b xprssd rspctivly as follows: M ( EI) ( EI) Bu (21) b Q ( GA / k) ( GA / k) Bu (22) whr EI is th flxural stiffnss, which is obtaind from th pr-gnratd momnt-curvatur rlation; GA is th shar stiffnss, which is assumd to b unchangd during th loading procss; k is th shar corrction factor to allow for cross-sctional warping. In this study, th valu of k is takn as 1.2 for a rctangular sction. Basd on th virtual work principl, th following lmnt forc quilibrium quations can b stablishd whr P is th lmnt quivalnt nodal loads whr T b T s s P B Mdx B Qdx (23) l l T P { Q1 M1 Q2 M 2} (24) Substituting Eqs. (21) and (22) into Eq. (23) yilds lmnt stiffnss quations P K u ( K K ) u (25) b s K T b Bb ( EI ) B bdx (26) l K T s Bs ( GA / k) B sdx (27) l Th forms of K b and K s ar valuatd using th on-point Gauss quadratur rul. Th stiffnss quations for th structur ar assmbld in th global coordinat systm from th contribution of all th bam lmnts. Aftr applying a propr boundary condition, th nonlinar quilibrium quations ar solvd by th incrmntal-itrativ mthod. Th itrativ schm for an incrmntal stp is summarizd as follows: (1) Form or updat th lmnt stiffnss matrics, and assmbl thm into th structur stiffnss matrix; (2) Solv quilibrium quations for displacmnt incrmnts, and add thm to th prvious total to gt th currnt nodal displacmnts; (3) In th local coordinat systm, comput th lmnt curvatur using Eq. (16) and lmnt shar strain using Eq. (17); (4) Us th momnt-curvatur rlation to dtrmin th bnding momnt M and to updat th flxural stiffnss EI, and us Eq. (22) to comput th shar forc Q; (5) Dtrmin th lmnt nd forcs using th right sid of Eq. (23) and thn assmbl thm into th intrnal rsisting loads;

9 2mm 2mm 22mm FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams 381 As2 P 1mm Cntr lin As1 Sction 1 As3 Sction 1 As4 6mm Cntr support 5mm. 145mm 3mm 15mm 12mm Fig. 7 Dtails of xprimntal bams Tabl 1 Amount of longitudinal rinforcmnt in xprimntal bams Bam A s1 (mm 2 ) A s2 (mm 2 ) A s3 (mm 2 ) A s4 (mm 2 ) V (2Ø 1) 157 (2Ø 1) 383 (2Ø 1+2Ø 12) 157 (2Ø 1) V (2Ø 1+2Ø 1) 157 (2Ø 1) 383 (2Ø 1+2Ø 12) 157 (2Ø 1) V (2Ø 12+2Ø 12) 226 (2Ø 12) 559 (2Ø 1+2Ø 16) 157 (2Ø 1) V (2Ø 16+2Ø 12) 42 (2Ø 16) 854 (2Ø 12+2Ø 2) 226 (2Ø 12) V (2Ø 16+2Ø 16) 42 (2Ø 16) 854 (2Ø 12+2Ø 2) 226 (2Ø 12) V (2Ø 16+2Ø 2) 42 (2Ø 16) 854 (2Ø 12+2Ø 2) 226 (2Ø 12) (6) Comput th out-of-balanc loads for th nxt itration. 5. Numrical application 5.1 Exprimntal bams In an xprimntal program conductd in Coimbra (Carmo 24, Carmo and Lops 25, 28), a group of 6 rinforcd high-strngth concrt continuous bams, dsignatd as V1-.7, V1-1.4, V1-2.1, V1-2.9, V1-3.8 and V1-5., wr tstd up to failur. Th main variabl of th tst was th amount of tnsil rinforcmnt at th ngativ momnt rgion, which varid from 157 (V1-.7) to 13 mm 2 (V1-5.). Th spcimns wr of a rctangular sction with width of 12 mm and dpth of 22 mm, and wr 6 mm long with two spans symmtric with rspct to th cntr lin crossing th cntr support, as illustratd in Fig. 7. Th layout and amount of rinforcmnt (A s1, A s2, A s3 and A s4 ) ar prsntd in Fig. 7 and Tabl 1. Th longitudinal rinforcmnt consistd of 1, 12, 16 or 2 mm dformd stl bars having avrag yild strngth of 569 MPa and tnsil strngth of 669 MPa. Th shar rinforcmnt consistd of 6 mm stirrups with spacing of 1 mm in th outr shar spans, and of 8 mm stirrups

10 382 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops with spacing of 1 mm in th innr shar spans. Th avrag concrt strngth at 28 days for th spcimns is 71 MPa. Th product αω w in Eqs. (8) and (1) is takn as.16 in th analysis to considr th confinmnt of concrt. Mor dtails about th spcimns hav bn rportd lswhr (Carmo 24, Carmo and Lops 25, 28). 5.2 Analysis and comparison Th proposd modl is usd to prform th analysis of th xprimntal bams. Du to symmtry, half of th bam is considrd and dividd into 19 bam lmnts (1 lmnt for th part outsid th nd support, 9 lmnts with qual lngth for th part btwn nd support and loading point, and 9 lmnts with qual lngth for th part btwn cntr support and loading point). Th slf-wight of th bams is convrtd into uniform load (.66 kn/m). It is obsrvd that ovr th ntir loading procss, all th analyzd bams but V1-5. xprinc squntially fiv typical phass, namly, th onst of concrt cracking at th cntr support (first cracking), th onst of concrt cracking at th span critical sction locatd at th loading point (scond cracking), th bginning of rinforcmnt yilding at th cntr support (first yilding), th bginning of rinforcmnt yilding at th span critical sction (scond yilding), and crushing of concrt. For Bam V1-5., th first yilding appars at th span critical sction, followd by th scond yilding at th cntr support. Fig. 8 shows th comparison btwn prdictd and xprimntal rsults in rlation to th load-dflction rspons of th bams. Th finit lmnt analysis (FEA) xhibits an ovrstimation of th pr-cracking stiffnss and cracking loads of th bams. This can b xplaind that in th pr-cracking stag, th concrt may not display prfctly lastic bhavior as assumd in th analysis, and that th concrt tnsil strngth calculatd by Eq. (11b) may b ovrstimatd. Howvr, th ovrall rsponss prdictd by th analysis agr wll with th xprimntal ons for all bams xcpt for V For Bam V1-2.1, th prdictd ultimat load is obviously lowr than th xprimntal valu. This may b attributd to that th actual matrial proprty of tnsil rinforcmnt in this bam is diffrnt from th valu adoptd in th analysis. 5.3 Momnt-curvatur rspons Th numrical rsults rgarding th momnt-curvatur rspons at th cntr support sction of th bams ar shown in Fig. 9. Th ntir rspons is charactrizd by thr stags sparatd by two points corrsponding to concrt cracking and rinforcmnt yilding, rspctivly. Bcaus th bhavior in th first stag (lastic stag) is mainly controlld by th concrt, th rsponss in this stag and th cracking momnts for th bams ar almost th sam. Aftr cracking, th contribution of rinforcmnt bcoms incrasingly important, so th rsponss for th bams diffr. Th highr th amount of rinforcmnt, th stiffr th bam, as xpctd. Th flxural curvatur ductility, μ ϕ, of a sction can b dfind by u (28) y in which ϕ u, ϕ y ar th curvaturs at th ultimat limit stat and at th bginning of tnsil rinforcmnt yilding, rspctivly. A list of valus of ϕ y, ϕ u and μ ϕ at th cntr support sction of th bams is givn in Tabl 2. It

11 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams V V FEA Tst (lft span) Tst (right span) FEA Tst (lft span) Tst (right span) Dflction (mm) Dflction (mm) V V FEA Tst (lft span) Tst (right span) FEA Tst (lft span) Tst (right span) Dflction (mm) Dflction (mm) V V FEA Tst (lft span) Tst (right span) FEA Tst (lft span) Tst (right span) Dflction (mm) Dflction (mm) Fig. 8 Comparison btwn prdictd load-dflction rspons and xprimntal rsults for th spcimns is sn that a highr amount of tnsil rinforcmnt mobilizs a highr valu of ϕ y but a lowr valu of ϕ u, lading to a much lowr curvatur ductility.

12 Momnt (knm) 384 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops V1-.7 V1-1.4 V1-2.1 V1-2.9 V1-3.8 V Curvatur (1-6 rad/mm) Fig. 9 Momnt-curvatur rspons for th spcimns according to numrical prdiction Tabl 2 Valus of curvatur ductility and curvaturs at yilding and ultimat for th spcimns according to numrical prdiction Bam ϕ y (1-6 rad/mm) ϕ u (1-6 rad/mm) μ ϕ V V V V V V V1-.7 V1-1.4 V1-2.1 V1-2.9 V1-3.8 V Nutral axis dpth (mm) Fig. 1 Variation of nutral axis dpth with applid load for th spcimns according to numrical prdiction

13 Momnt (knm) Momnt (knm) Momnt (knm) Momnt (knm) Momnt (knm) Momnt (knm) FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams Span critical sction Cntr support 1-1 Span critical sction Cntr support -2-3 V V V1-2.1 Span critical sction Cntr support V1-2.9 Span critical sction Cntr support Span critical sction Cntr support 3-3 Span critical sction Cntr support -4 V V Fig. 11 Momnt volution with applid load for th spcimns according to numrical prdiction

14 386 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops 5.4 Variation of nutral axis dpth Fig. 1 shows th numrical rsults rgarding th variation of th nutral axis dpth at th cntr support sction with th applid load. Th volution of th nutral axis dpth can b charactrizd by four stags sparatd by thr points corrsponding to concrt cracking, stabilization of crack dvlopmnt, and rinforcmnt yilding, rspctivly. In th first stag, th position of nutral axis is at th plac of cntroidal axis of th transformd sction and rmains unchangd. This is followd by th scond stag, charactrizd by a significant drop of th nutral axis dpth with a small incras in th applid load. Th bhavior in th third stag is opposit to that in th scond stag, that is, th movmnt of th nutral axis is insignificant whil th incras in th applid load is grat. Aftr th tnsil rinforcmnt in th cntr support sction yilds (in th last stag), a quick drop of th nutral axis dpth with th applid load is obsrvd. It is also obsrvd that at th ultimat limit stat, a highr amount of tnsil rinforcmnt rsults in a largr nutral axis dpth, as xpctd. 5.5 Momnt rdistribution Fig. 11 shows th numrical rsults rgarding th dvlopmnt of bnding momnts at th span critical sction and cntr support with th applid load. Th straight lin in th graph rprsnts th lastic valus, which ar calculatd assuming th constitunt matrials ar linar lastic. Du to rdistribution of momnts, th momnt volution in a bam might b influncd by cracking of concrt and yilding of rinforcmnt (formation of plastic hings), whil th xtnt of influnc is dpndnt on th contnt of tnsil rinforcmnt. For bams with low amounts of tnsil rinforcmnt (Bams V1-.7 and V1-1.4), th actual momnt, computd by FEA, bgins to dviat from th lastic valu at th onst of cracking. This is followd by a stabilizing trnd until yilding of th tnsil rinforcmnt at th cntr support, which causs an accntuatd trnd of dviation btwn actual and lastic momnts du to furthr rdistribution of momnts from th cntr support rgion to th span critical rgion. In comparison with Bam V1-1.4, th abov obsrvation is mor obvious for Bam V1-.7 du mainly to th largr stiffnss diffrnc btwn critical positiv and ngativ momnt rgions. On th othr hand, for bams with high rinforcmnt ratios, concrt cracking (for V1-2.1, V1-2.9 and V1-3.8) and/or rinforcmnt yilding (for V1-5.) may not hav a noticabl ffct on th momnt rdistribution, as shown in Fig. 11. Th dgr of momnt rdistribution, β, can b xprssd as M M u 1 (29) whr M u is ultimat actual momnt, computd by FEA in this study; and M is th lastic momnt corrsponding to th ultimat load. A summary of valus of th actual and lastic momnts and th dgr of momnt rdistribution at th span critical and cntr support sctions of th xprimntal bams is givn in Tabl 3. It is obsrvd that at th ultimat limit stat, all spcimns but V1-5. hav positiv (ngativ) rdistribution at th cntr support (span critical sction). Bam V1-5., in which th tnsil rinforcmnt ratio at th ngativ momnt rgion is highr than that at th positiv momnt rgion, on th othr hand, shows a ngativ (positiv) rdistribution at th cntr support

15 6mm 5mm 55mm FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams 387 Tabl 3 Valus of actual, lastic momnts and th dgr of momnt rdistribution for th spcimns according to numrical prdiction Bam M u (kn m) M (kn m) β (%) Span critical Cntr support Span critical Cntr support Span critical Cntr support V V V V V V As2 P 5mm As1 P As3 As4 1667mm 1667mm 375mm 375mm 375mm 375mm L=75mm L=75mm 3mm Fig. 12 Dtails of th rfrnc bam usd for paramtric valuation (span critical sction). This obsrvation can b attributd to th fact that th momnts ar pron to rdistributd from th wakr sctions to th strongr sctions. In addition, it is gnrally obsrvd that th dgr of momnt rdistribution dcrass with th incras of th tnsil rinforcmnt ratio. Howvr, a highr tnsil rinforcmnt ratio (V1-2.9) may xhibits highr rdistribution than do lowr rinforcmnt ratios (V1-2.1 and V1-1.4). This can b xplaind that th stiffnss diffrnc btwn critical positiv and ngativ momnt rgions in V1-2.9 is largr than thos in V1-1.4 and V1-2.1 Th largr th stiffnss diffrnc, th highr th rdistribution of momnts. 6. Effct of tnsil rinforcmnt on th valu of β In this sction, a paramtric study is carrid out to xamin th ffct of tnsil rinforcmnt on th valu of β. A two-span continuous bam with practical dimnsions as shown in Fig. 12 is

16 388 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops / s3 =.67-2 / s3 =1. / s3 = (%) Fig. 13 Effct of ρ s1 and ρ s1 /ρ s3 on th dgr of momnt rdistribution according to numrical prdiction usd to illustrat th rsults obtaind from th analysis. Two important factors rlatd to th tnsil rinforcmnt ar valuatd, namly, th tnsil rinforcmnt ratio at th cntr support ρ s1 or at midspan ρ s3, and th ratio ρ s1 /ρ s3. Thr lvls of ρ s1 /ρ s3 ar considrd, namly,.67, 1. and 1.5; th factor ρ s1 varis from.73% to 4% for ρ s1 /ρ s3 of 1. and 1.5, and to 3.18% for ρ s1 /ρ s3 of.67. Th comprssiv rinforcmnt ratios at th cntr support ρ s4 and at midspan ρ s2 ar takn as.36%. Th concrt is assumd to b unconfind. Th matrial proprtis ar assumd to b th sam as thos of th xprimntal bams dscribd abov, that is, th concrt strngth is takn as 71 MPa, and th stl yild strngth is takn as 569 MPa. Fig. 13 illustrats th influnc of ρ s1 and ρ s1 /ρ s3 on th dgr of momnt rdistribution at th cntr support sction. It is sn that th factor ρ s1 affcts th dgr of momnt rdistribution diffrntly, dpnding on th lvl of th ratio ρ s1 /ρ s3. For ρ s1 /ρ s3 of.67 (that is, th tnsil rinforcmnt ratio at th cntr support is obviously lowr than that at midspan), th dgr of momnt rdistribution dcras slightly as ρ s1 incrass up to 1.55%; with continuing incras of ρ s1, howvr, th dgr of momnt rdistribution dcrass quickly. For ρ s1 /ρ s3 of 1. (that is, th tnsil rinforcmnt ratio at th cntr support is comparabl to that at midspan), th dgr of momnt rdistribution dcrass slowly as ρ s1 incrass. For ρ s1 /ρ s3 of 1.5 (that is, th tnsil rinforcmnt ratio at th cntr support is obviously highr than that at midspan), on th othr hand, th variation of th dgr of momnt rdistribution with varying ρ s1 is not so obvious. Also, it can b obsrvd from Fig. 13 that th ratio ρ s1 /ρ s3 has grat influnc on th rdistribution of momnts. At a givn lvl of ρ s1, a highr valu of ρ s1 /ρ s3 las to a much lowr valu of β at th cntr support sction. Th diffrnc btwn th valus of β for diffrnt ρ s1 /ρ s3 lvls is particularly significant at low lvls of ρ s1. Thr can b positiv or ngativ rdistribution of momnts at th cntr support, dpnding on th lvl of ρ s1 /ρ s3. Whn ρ s1 /ρ s3 is not gratr than 1., th momnt rdistribution at th cntr support is positiv. On th othr hand, whn ρ s1 /ρ s3 is gratr than 1., thr may appar ngativ rdistribution of momnts at th cntr support.

17 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams (a) EC2 limit ACI and CSA limit 1-1 FEA ACI CSA EC (b) (%) EC2 limit 2 ACI and CSA limit 1 FEA ACI -1 CSA EC (c) (%) EC2 limit 2 ACI and CSA limit 1-1 FEA ACI CSA EC (%) Fig. 14 Variation of β with ρ s1 in trms of FEA and various cods. (a) ρ s1 /ρ s3 =.67; (b) ρ s1 /ρ s3 =1.; (c) ρ s1 /ρ s3 =1.5. To tak advantag of th ductility of continuous concrt mmbrs, th cods allow dsignrs to us a linar lastic analysis with allowabl rdistribution of momnts through th us of th cofficint β. In th ACI cod (ACI Committ ), th modification of factord momnts calculatd by lastic thory shall not b mor than 1 ε t prcnt, to a maximum of 2%, whr ε t is th nt

18 39 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops Tabl 4 Prdictions of momnt rdistribution by FEA and various cods ρ s1 /ρ s ρ s1 M u M ε t c/d β (%) (%) (kn m) (kn m) (%) (%) FEA ACI CSA EC strain in xtrm tnsion stl. Th momnt rdistribution can b mad only whn ε t is not lss than.75 at th sction whr th momnt is rducd. In th CSA cod (Canadian Standards Association 24), th ngativ momnt calculatd by an lastic analysis can b incrasd or dcrasd by not gratr than (3 5c/d) prcnt, with a maximum of 2%, whr c/d is th ratio of th nutral axis dpth to th ffctiv dpth of a cross sction. In EC2 (CEN 24), th momnts at th ultimat limit stat computd by a linar lastic analysis can b rdistributd by not highr than [ (.6+.14/ε u )c/d] for concrt strngth qual to or blow 5 MPa, and by [ (.6+.14/ε u )c/d] for concrt strngth gratr than 5 MPa. Th maximum rdistribution is 3% for high- and normal-ductility stl and of 2% for low-ductility stl. A comparison of th dgrs of momnt rdistribution at th cntr support calculatd by various cods with th FEA valus is summarizd in Tabl 4 and Fig. 14. It is sn in th tabl and Fig. 14(a) that for ρ s1 /ρ s3 qual to.67, all th cods can wll rflct th actual trnd of th variation of β with varying ρ s1, although th CSA cod shows a smallr slop whil EC2 xhibits an obviously sharpr slop. All cods ar consrvativ. Howvr, th ACI cod sms to b ovr-consrvativ ovr th ntir rang of ρ s1, whil EC2 may b ovr-consrvativ at high lvls of ρ s1. On th othr hand, for ρ s1 /ρ s3 qual to 1., th variation trnds of β prdictd by various cods, particularly by EC2, ar mor significant than that by FEA, as shown in Fig. 14(b). In addition, th CSA cod is non-consrvativ ovr th ntir rang of ρ s1, whil th ACI cod and EC2 may b non-consrvativ at low lvls of ρ s1. From Fig. 14(c), it can b obsrvd that for ρ s1 /ρ s3 qual to 1.5, all th cods fail to rflct th actual trnd of th variation of β. Also, all cods ar non-consrvativ, xcpt EC2 at a vry high lvl of ρ s1. Fig. 15 illustrats th ffct of ρ s1 /ρ s3 on th valu of β according to prdictions by various cods. Bcaus ithr th paramtr ε t or c/d, which is usd to calculat th dgr of momnt

19 FE modling of inlastic bhavior of rinforcd high-strngth concrt continuous bams CSA ACI -1-2 / s3 =.67 / s3 =1. / s3 =1.5 EC (%) Fig. 15 Effct of ρ s1 /ρ s3 on th valu of β according to cod prdictions rdistribution in dsign cods, is almost indpndnt of ρ s1 /ρ s3, th grat importanc of th ratio ρ s1 /ρ s3 as mntiond prviously is nglctd in all th cods. Probably, a diffrnt paramtr that would b somhow rlatd with ρ s1 /ρ s3 could b mor ffctiv for practical dsign purposs. Thrfor, th authors rcommnd that an xtnsiv study should b carrid out in ordr to find th bst paramtr to b includd in th nw simplifid ruls in cods. 7. Conclusions A FE modl basd on th momnt-curvatur rlations is dvlopd to prdict th nonlinar bhavior of rinforcd high-strngth concrt continuous bams throughout th loading procss. Th momnt-curvatur rlations of rinforcd concrt sctions ar pr-gnratd through sction analysis by satisfying th forc quilibrium and strain compatibility. Th FE mthod is formulatd using th Timoshnko bam thory so as to tak into account th ffct of shar dformations. A group of six continuous high-strngth concrt bam spcimns, whr th main variabl was th amount of tnsil rinforcmnt, is slctd to calibrat th proposd modl and to illustrat th rsults of a numrical valuation. Th rsults producd by th analysis indicat that th tnsil rinforcmnt has significant influnc on th bhavior of continuous rinforcd high-strngth concrt bams, namly, th flxural stiffnss, ductility, nutral axis dpth and rdistribution of momnts. It is found that a highr tnsil rinforcmnt ratio rsults in a gratr flxural stiffnss aftr cracking and a highr nutral axis dpth at ultimat, but lads to lowr ductil bhavior and lss rdistribution of momnts. A paramtric study is conductd on two-span high-strngth concrt continuous bams with practical dimnsions to valuat th influnc of tnsil rinforcmnt on th dgr of momnt rdistribution. Two important factors rlatd to tnsil rinforcmnt, ρ s1 and ρ s1 /ρ s3, ar xamind. Th study shows that th ffct of ρ s1 on th dgr of momnt rdistribution is quit diffrnt for diffrnt lvls of ρ s1 /ρ s3. Th ratio ρ s1 /ρ s3 is found to b a vry important paramtr influncing th

20 392 Tijiong Lou, Srgio M.R. Lops and Adlino V. Lops rdistribution of momnts. Howvr, th grat importanc of this factor is not rflctd in various cods. Acknowldgmnts This rsarch is sponsord by FEDER funds through th program COMPETE - Programa Opracional Factors d Comptitividad - and by national funds through FCT - Fundação para a Ciência a Tcnologia -, undr th projct PEst-C/EME/UI285/213. Th work prsntd in this papr has also bn supportd by FCT undr Grant No. SFRH/BPD/66453/29. Rfrncs ACI Committ 318 (211), Building cod rquirmnts for structural concrt (ACI ) and commntary, Amrican Concrt Institut, Farmington Hills, MI. Arslan, G. and Cihanli, E. (21), Curvatur ductility prdiction of rinforcd high-strngth concrt bam sctions, J. Civil Eng. Manag., 16(4), Brnardo, L.F.A. and Lops, S.M.R. (24), Nutral axis dpth vrsus flxural ductility in high-strngth concrt bams, ASCE J. Struct. Eng., 13(3), Brnardo, L.F.A. and Lops, S.M.R. (29), Plastic analysis of HSC bams in flxur, Matr. Struct., 42, Campion, G., Fosstti, M., Minafò, G. and Papia, M. (212), Influnc of stl rinforcmnts on th bhavior of comprssd high strngth R.C. circular columns, Eng. Struct., 34, Campion, G., Fosstti, M. and Papia, M. (26), Simplifid analytical modl for comprssd high-strngth columns confind by transvrs stl and longitudinal bars, Procdings of 2nd Intrnational FIB Congrss, Napls, Italy. Canadian Standards Association (24), Dsign of concrt structurs (A23.3-4), Mississauga, Ontario, Canada. Carmo, R.N.F. (24), Plastic rotation and momnt rdistribution in high strngth concrt bams, PhD Thsis, Univrsity of Coimbra, Coimbra, Portugal. (in Portugus) Carmo, R.N.F. and Lops, S.M.R. (25), Ductility and linar analysis with momnt rdistribution in rinforcd high-strngth concrt bams, Can. J. Civil Eng., 32, Carmo, R.N.F. and Lops, S.M.R. (28), Availabl plastic rotation in continuous high-strngth concrt bams, Can. J. Civil Eng., 35, CEB-FIP (199), Modl cod for concrt structurs, Euro-Intrnational Committ for Concrt - Intrnational Fdration for Prstrssing, Thomas Tlford Srvics Ltd., Lausann, Switzrland. CEN (24), Eurocod 2: Dsign of concrt structurs - Part 1-1: Gnral ruls and ruls for buildings, EN , Europan Committ for Standardization, Brussls, Blgium. Cucchiara, C., Fosstti, M. and Papia, M. (212), Stl fibr and transvrs rinforcmnt ffcts on th bhaviour of high strngth concrt bams, Struct. Eng. Mch., 42(4), Fostr, S.J. (21), On bhavior of high-strngth concrt columns: covr spalling, stl fibrs, and ductility, ACI Struct. J., 98(4), Fostr, S.J. and Attard, M.M. (21), Strngth and ductility of fibr-rinforcd high-strngth concrt columns, ASCE J. Struct. Eng., 127(1), Kassoul, A. and Bougara, A. (21), Maximum ratio of longitudinal tnsil rinforcmnt in high strngth doubly rinforcd concrt bams dsignd according to Eurocod 8, Eng. Struct., 32, Ko, M.Y., Kim, S.W. and Kim, J.K. (21), Exprimntal study on th plastic rotation capacity of rinforcd high strngth concrt bams, Matr. Struct., 34,

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