MATERIAL NONLINEAR ANALYSIS OF STEEL FIBRE REINFORCED CONCRETE BEAMS FAILING IN SHEAR

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1 MAERIAL NONLINEAR ANALYSIS OF SEEL FIBRE REINFORCED CONCREE BEAMS FAILING IN SHEAR Joaquim Barros 1, Ravidra Gettu, Brya Barragá 1 Uiversity of Miho, Uiversitat Politeia de Cataluya, Spai Abstrat Experimetal researh has poited out that fibre reiforemet gives valuable tributio for the shear stregth of ete beams. o obtai data to hek the validity of the formulatio proposed by RILEM C 16-DF for the evaluatio of the fibres tributio for the ete shear stregth, sets of ete beams were tested experimetally. o simulate the behaviour of this kid of strutures, a mputatioal de was developed, based o the fiite elemet tehiques. A aurate simulatio of the behaviour of strutures failig i a brittle mode, suh is the ase of the beams failig i shear, is a true hallege i the mputatioal mehais domai. o reprodue with eough auray the fraturig proess of this type of elemets, a multifixed ak model is implemeted. he ability of the post-akig stress-strai diagram proposed by RILEM C 16-DF to simulate the ak propagatio is heked. A strai softeig triliear diagram is also derived from iverse aalysis usig the fore-defletio relatioship obtaied i RILEM three-poit othed beam tests. Its apability to model the frature mode I is also assessed. he umerial strategy developed is desibed i the preset work ad the appropriateess of the model is evaluated simulatig some beams tested experimetally. 1. Itrodutio Oe of the most promisig uses of steel fibres o the ete tehology is i the iease of ete shear stregth. I urret appliatios, sts requiremets advie to avoid total replaemet of steel stirrups by steel fibres. Partial replaemet is urret pratie i parts of the strutures with high peretage of stirrups where, due to ete pourig diffiulties, the desired ete quality might be ot assured. ehial ad eomial beefits, however, were poited out i shallow beams of high stregth ete, where stirrups were totally replaed by steel fibres [1]. o assess the effiay of the fibre reiforemet for the ete shear stregth, series of beams of retagular ad oss-setios reifored with 4 kg/m 3 of hooked eds steel fibres were tested []. Flexural bedig tests with othed steel fibre reifored ete (SFRC) speimes were also arried out ardig to the remmedatios of RILEM C 16-DF [3] for evaluatig the equivalet (f eq ) ad the residual (f R ) flexural tesile stregth parameters of the SFRC applied i the beams []. he values of these parameters ad the ultimate load of the SFRC beams were used to hek the validity of the formulatio proposed by RILEM C 16-DF [4] for the simulatio of the tributio of the fibre additio to the ete shear stregth []. Joaquim Barros, Material oliear aalysis of steel fibre, 1/1 .barros@ivil.umiho.pt

2 o predit the deformability, the load arryig apaity ad the ak patter of elemets reifored with vetioal ad steel fibres, a umerial model based o the fiite elemet tehiques ad iludig stitutive laws for modellig the oliear behaviour of the iterveig materials was developed. Speial are was take i the simulatio of the post-akig behaviour, sie it is the mai property beefited by fibre reiforemet. Due to lak of spae, this work oly leads with the beams of series S1 represeted i Fig. 1 []. he evaluated ete properties are idiated i able 1. F a b Ø Ø3 6Ø5 a = 91 mm; b = 19 mm a = 137 mm; b = 193 mm a = 189 mm; b = 61 mm (a) (b) () Fig. 1 - SFRC-x3_S1 (a); SFRC-x45_S1 (b) ad SFRC-x6_S1 () of beams of series S1 []. able 1 - Coete properties of the beams of series S1. Coete f [MPa] f eq, [MPa] f eq,3 [MPa] f R,1 [MPa] f R,4 [MPa] Plai ete 3.1 (±%) SFRC (±%) 5.85 (±15%) 6.39 (±1%) 5.96 (±1%) 5.99 (±1%). Costitutive model he ete akig is simulated uder the framework of the multifixed smeared ak epts [5]. Ardig to the preset model, the total strai iemet of the aked ete, ε, is the additio of the strai iemet i the frature zoe, ε, with the strai iemet of the ete betwee aks, ε ep : ε = ε + ε (1) ep Joaquim Barros, Material oliear aalysis of steel fibre, /1 .barros@ivil.umiho.pt

3 he ete betwee aks a be i elasti (e) or i elasto-plasti (ep) behaviour. I the last ase, assoiated elasto-plasti theory is used to obtai plasti deformatio ad to assure the material does ot violate its yield surfae [7]. he ete is govered by the followig stitutive equatio: σ = D ε () where, i the ase of uaked liear-elasti material, desigatio of D, with the followig format: e D 1 υ E = υ 1 1 υ De 1 υ D bemes with the where υ ad E are the Poisso effiiet ad the Youg's modulus of the uaked ete. Whe aked, but with the ete betwee aks i liear-elasti behaviour, D has the followig figuratio [6]: 1 { } e = e e + e e D D D D D D D (4) where is a matrix defiig the orietatio of the aks formed at a samplig poit. If m aks ours at a samplig poit: = 1,, (5) i m where the ak orietatio of a geeri ith ak is defied by the matrix : i s θi si θi siθisθ i i = (6) siθisθi siθisθi s θi si θi with θ i beig the agle betwee x 1 ad the ormal to the ith ak plae (see Fig. ). I (4) D is a matrix iludig the stitutive law of the aks: D where D i D =... Di Dm D is the ith ak stitutive law: i DIi, = DII, i (3) (7) (8) Joaquim Barros, Material oliear aalysis of steel fibre, 3/1 .barros@ivil.umiho.pt

4 with D ad I D beig the ak frature mode I ad mode II stiffess modulus, II respetively. he ak system of a samplig poit is govered by the followig relatioship: σ = D ε (9) where σ is the vetor of the iemetal ak stress mpoets (Fig. ): σ σ,1 τt,1... σ, i τt, i... σ, m τ t, m ε ε ε,1 γ t,1... ε, i γ t, i... ε, m γ t, m = ad is the vetor of the iemetal ak strai mpoets: = (11) he D I of (8) is haraterised by the frature parameters (see Fig. 3), amely, the tesile stregth, σ = 1 f t, the frature eergy, G f, the shape of the softeig law ad the ak badwidth, l b. Fibre reiforemet mehaisms are refleted, maily, o the eergy dissipated i the mode I fraturig proess ad o the shape of the softeig brah. For fibre tets used i urret ete appliatios, the remaiig ete properties are oly margially affeted by fibre additio [5]. he ak mode I stiffess is simulated by the triliear diagram represeted i Fig. 3. (1) x σ τ t γ t t θ + σ ε σ 1 α σ 1 1 α σ 1 D I G f l b ε σ ξ ε 1 u ξ ε u ε u ε x 1 Fig. - Crak stress ad ak strai mpoets Fig. 3 - riliear softeig diagram he frature mode II modulus, D II, is obtaied from the expressio [5, 6]: β DII = G (1) 1 β where G is the ete shear modulus ad, Joaquim Barros, Material oliear aalysis of steel fibre, 4/1 .barros@ivil.umiho.pt

5 p ε β = 1 (p=1, or 3) (13) εu is the shear retetio fator, with ε beig the ultimate ormal ak strai (Fig. 3). u If ete is uaked ad plasti deformatios ourred due to mpressio, D of () is replaed by the sistet taget operator, D (assoiated flow rule ad isotropi ep hardeig approahes were assumed [6, 9]): f f H H σ σ D Dep = H (14) f f h+ H σ σ where 1 1 f H = D e + λ σ (15) is a matrix that iludes effets of plasti flow, λ is the (fiite) amout of plasti flow withi a loadig step, ad 1 f ( σκ, ) = ( σ Pσ) + q σ σ( κ) = (16) is the yield futio, where a b P b a, = q = dq1 = d[ 1 1 ] (17) are the projetio matrix ad the projetio vetor [6], respetively, with a a = + b, a b b =, = 3b, a d =, a =.355, b = (18) I (13) h is the hardeig modulus, that a be obtaied derivatig the yield futio, f, i relatio to the hardeig parameter, k: f dσ ( κ ) h = = (19) κ dκ I (16) σ ( κ ) is the effetive mpressive stress ad k is the hardeig parameter. he fibre ifluee o the ete uiaxial behaviour is oly sigifiat o the post-peak softeig phase [5]. herefore, the σ ( κ ) was defied from the stress-strai relatioship proposed by CEB-FIP 1993 [7] for the plai ete, where k is trasformed i a effetive plasti strai usig a work hardeig approah [6]. For the aked ete with ete betwee aks i elasto-plasti behaviour, of () is replaed by: 1 { } ep = ep ep + ep ep D D D D D D D () D Joaquim Barros, Material oliear aalysis of steel fibre, 5/1 .barros@ivil.umiho.pt

6 I the simulatios of the preset work, a maximum umber of three aks per samplig poit a be formed, ardig to the followig iteria: the maximum priipal stress attais the tesile stregth ad the agle betwee the ew ak ad the previous aks is greater tha a threshold agle. I the simulatios arried out a threshold agle of 3 degrees was sidered. A sub-iemetatio proedure o the ε is performed to amplish this iteria, ad to aut for the ak status hages resultig from ak iitiatio, losig ad reopeig. I a samplig poit, whe a ew ak is formed, the frature eergy attributed to this ak is futio of the material frature eergy, the eergy dissipated for previous aks ad the relative orietatio amogst the aks [6]. 3. Assessig the frature parameters from iverse aalysis o assess the ete frature parameters, a iverse aalysis was performed, evaluatig the values of the σ ad i ε of the i σ ε diagram (Fig. 3) that fit the experimetal F δ urves with the miimum error of the parameter exp um exp err = AF δ AF δ A, F δ (1) exp where A ad um F δ AF δ are the areas below the experimetal ad the umerial F δ urves, respetively. he test set up used for this purpose was the same remmeded by RILEM C 16 DF for the SFRC [3]. I the umerial simulatio, the ak developmet was restrited to the fiite elemets above the oth. he speime was disetized i eight ode Seredipity plae stress elemets. o avoid a spurious iterferee of the frature mode II i the speime deformatioal respose, two samplig poits per elemet plaed at the speime's symmetry axis mposed the itegratio sheme of these elemets. I the remaiig elemets, a Gauss-Legedre itegratio rule was applied. he adequay of the umerial strategy adopted is show i Fig. 4, revealig that the proposed triliear σ ε diagram is apable of preditig, with eough auray, the post-akig behaviour of plai ete (PC) ad SFRC speimes. able iludes the data defiig the σ ε diagram, where the frature eergy was evaluated assumig the ak-bad width as beig equal to the oth width (5 mm). hese data is the average of three tests Experimetal Fore (kn) PC SFRC_S1 Experimetal Poit load (kn) Defletio at midspa (mm) Fig. 4 Experimetal vs. umerial urves of PC ad SFRC speimes Load poit defletio (mm) Fig. 5 - Experimetal vs. umerial urves of x3 PC beam Joaquim Barros, Material oliear aalysis of steel fibre, 6/1 .barros@ivil.umiho.pt

7 able - Data for defiig the triliear σ ε diagram, obtaied from iverse aalysis σ 1 = ft G f [MPa] ε ε u σ σ 1 ε3 ε u σ 3 σ 1 [N.mm/mm ] PC SFRC Previous experimetal ad umerial researh [5] has show that, for the type ad amout of fibres used i the SFRC beams of the preset work, the frature eergy would be muh lower tha the value obtaied from iverse aalysis. Oe possible justifiatio resides o the fat that i the iverse aalysis the ak propagatio was restrited to the surfae above the oth while, due to the effet of the fibre reiforemet mehaisms, aks aside of this assumed surfae are formed, resultig a frature surfae that should be muh larger tha the assumed oe i the umerial simulatio. 4. Appliability of the iverse aalysis he appliability of the iverse aalysis for defiig a valid σ ε diagram for the umerial simulatio of beams failig i shear will be assessed aalysig a PC ad a SFRC beams of series S1 (SFRC-x3_S1). he beam size effet was take ito aut, followig the remmedatios of RILEM C 16-DF [4]. he data used i the umerial simulatio is iluded i able 3 (simulatio A). Fig. 5 shows that the model a reprodues with eough auray the deformatioal behaviour of beams without ay shear reiforemet (failig i shear). Sie i the PC othed speime RILEM test, the fore-defletio relatioship was registered up to full frature eergy dissipatio, the frature parameters obtaied by iverse aalysis a be diretly used o the defiitio of the mode I ak stitutive law. Usig the frature parameters idiated i able for SFRC, the test of the x3 SFRC beam of series 1 (Fig. 1) was simulated. A flexural failure mode was predited by the umerial model, while this beam was experimetally failed i shear. Moreover, the umerial model has predited a maximum load larger tha the experimetal oe. his shows that the frature parameters obtaied by iverse aalysis with the F-δ relatioships registered i RILEM flexural tests with SFRC speimes aot be diretly used to defie the σ ε diagram for the umerial simulatio of beams failig i shear. 5. Appliability of the RILEM post-akig stress-strai diagram Fig. 6 represets the σ-ε diagram proposed by RILEM C 16-DF [4] to model the uiaxial behaviour of SFRC. he poits defiig this diagram are determied from the followig relatios: σ 1 =.7f tm,fl (16-d) with 16-d < 1.; E =95(f m ) 1/3 ; ε 1 =σ 1 /E ; () σ =.45f R,1 κ h ; ε =ε ; σ 3 =.37f R,4 κ h ; ε 3 =5 Joaquim Barros, Material oliear aalysis of steel fibre, 7/1 .barros@ivil.umiho.pt

8 where f tm,fl ad E are the SFRC average flexural tesile stregth ad Youg's modulus, i MPa, respetively, d is the effetive beam depth, i mm, ad κ h is the size effet parameter. he ability of this diagram to model the post-akig behaviour of the SFRC will be assessed vertig this relatioship ito a σ ε diagram ad, as a title of example, simulatig the SFRC-x3_S1 beam (Fig. 1). o obtai the frature eergy from the RILEM post-akig stress-strai relatioship it was assumed for the frature proess zoe a value equal to three times the fibre legth (18 mm). he data used i this simulatio is iluded i able 3 (simulatio B). As Fig. 7 shows, usig the RILEM approah a flexural failure mode was predited umerially, while the beam has experimetally failed i shear. he maximum load was larger tha the value registered experimetally. σ σ 1 σ σ 3 ε1 ε ε3 ε [o/oo] Poit Load (kn) Experimetal Fig. 6 - σ-ε diagram for SFRC, ardig to RILEM C 16 DF Load-Poit defletio (mm) Fig. 7 - Experimetal vs. umerial urves 6. Simulatio of the SFRC beams of series S1 able 3 iludes the data (simulatio C) used o the umerial simulatio of the osssetioal retagular beams mposig series S1 (see Fig. 1). able 3 - Data used i the umerial simulatio Coete Commo data: 8 ode FE, 3x3 Gauss poits; l b =sqrt (area of fiite elemet); υ =. Simulatios A B C Commo data: E =31854 MPa; f m =3.1 MPa; p=3 [expressio (13)] E =9336 MPa f m =3.1 MPa frature parameters: i able p=1 [expressio (13)] f t =3.36 MPa ξ 1 =.385; α 1 =.6 ξ =.96; α =.51 G f =8.65 N/mm f t =.5 MPa ξ 1 =.3; α 1 =.4 ξ =.1; α =[.5-.] G f =[1.-1.3] N/mm Steel: ode embedded FE, Gauss poits; E s = MPa; f sy =f su =4 MPa Joaquim Barros, Material oliear aalysis of steel fibre, 8/1 .barros@ivil.umiho.pt

9 he umerial ad the experimetal urves of the Load-poit defletio-poit load relatio are represeted i Fig. 8. he ak patter at failure of the SFRC_x6_S1 beam is show i Fig. 9. he umerial simulatios were iterrupted whe vergee was ot possible to assure. Not oly the maximum load was aurately predited, as well as, the shear failure mode of the experimetally tested beams. he model was ot, however, apable of simulatig the deease of stiffess ourred experimetally ear the ultimate load. he high level of defletio at failure observed experimetally is a sequee of the stress trasfer apability betwee ak surfaes that is assured by fibres ossig these aks. More umerial researh is beig doe to ehae this model defiiey. 5 Experimetal 3 5 Experimetal Poit load (kn) 15 1 Poit load (kn) Experimetal Load-poit defletio (mm) (a) Load-poit defletio (mm) (b) Poit load (kn) Load-poit defletio (mm) () Fig. 8 - Experimetal vs. umerial urves of beams: SFRC-x3_S1 (a); SFRC- x45_s1 (b) ad SFRC-x6_S1 () Fig. 9 Crak patter of SFRC-x6_S1 (oly aks i opeig proess are depited) Joaquim Barros, Material oliear aalysis of steel fibre, 9/1 .barros@ivil.umiho.pt

10 7. Colusios o verify if post-akig stress-strai relatioship, σ ε, a be defied by iverse aalysis with the model developed, the ete frature parameters were obtaied arryig out a iverse aalysis with the fore-defletio relatioships, F-δ, obtaied i three poit othed beam speimes. Usig these frature parameters, the behaviour of plai (PC) ad steel fibre reifored ete (SFRC) beams failig i shear was simulated. With the frature parameters obtaied ardig to this strategy, the behaviour of PC beams was aurately simulated. Sie the F-δ relatioship was obtaied up to full eergy dissipatio of the PC speimes, the frature parameters are represetative of the PC post-akig behaviour. It is oly eessary to sider a size effet fator, to take ito aut the size of the struture i aalysis. I ase of SFRC speimes, however, the test is iterrupted at about 5 mm, whe the eergy dissipated is oly a fratio of the total eergy sumed i the fraturig proess. herefore, the frature parameters aot be diretly used o the defiitio of the σ ε. o simulate the behaviour of experimetally tested SFRC beams, a triliear σ ε diagram was defied. he developed model was able of estimatig the ultimate load, the most relevat aspets of ak patter ad the deformability up to ear the maximum load. Akowledgmets he first author wishes to akowledge the grat SFRH/BSAB/91/-POCI, provided by FC ad FSE. Referees 1. Casaova, P., 'Bétos reforés de fibres métalliques du matériau à la struture', PhD hesis, LCPC, Paris, Frae, (996) 3 pgs.. Barragá, B.E., 'Failure ad tougess of steel fiber reifored ete uder tesio ad shear', PhD hesis, UPC, Bareloa, Marh (). 3. RILEM C 16-DF, 'est ad desig methods for steel fibre reifored ete - Fial Remmedatio, Materials ad Strutures', Vol.35, November, pp RILEM C 16-DF, est ad desig methods for steel fibre reifored ete - σ - ε desig method - Fial Remmedatio, Materials ad Strutures, Vol.36, Otober (3), pp Barros, J.A.O., 'Behaviour of fibre reifored ete - experimetal ad umerial aalysis', PhD hesis, Civil Eg. Dept., FEUP, Portugal, (1995) (i Portuguese). 6. Sea-Cruz, J.M.; Barros, J.A.O., Azevedo, A.F.M., "Elasto-plasti smeared ak model for ete", Report 4-DEC/E-5, Civil Eg. Dep. Uiv. Miho, (4). 7. CEB-FIP Model Code, Comite Euro-Iteratioal du Beto, Bulleti d Iformatio º 13/14 (1993). Joaquim Barros, Material oliear aalysis of steel fibre, 1/1 .barros@ivil.umiho.pt