Numerical modelling of large scale steel fibre reinforcedreinforced concrete beams failing in shear

Transcription

1 Numrical modlling of larg scal stl fibr rinforcdrinforcd concrt bams failing in shar Ali Amin, Stphn J. Fostr Univrsity of Nw South Wals, Sydny, Australia. Abstract Exprimntal and numrical studis on Stl Fibr Rinforcd Concrt (SFRC) ovr th last 5 dcads, or so, hav indicatd that th post cracking strngth of concrt can b improvd by providing suitably arrangd, closly spacd, wir rinforcmnt. Whil th databas of xprimntal and numrical shar tsts of SFRC mmbrs is xtnsiv, th pool of tst data and numrical modls, alik, of SFRC bams containing convntional transvrs shar rinforcmnt (stirrups) ar limitd. Th bhaviour of full scal stl fibr rinforcdrinforcd concrt (SFR-RC) bams ar analysd hrin using a smard crack modl providd by ATENA 2D intgratd with a constitutiv law drivd aftr an invrs analysis from prism bnding tsts. Th numrical modl is validatd against xprimntal rsults obtaind from four larg scal SFR-RC bams and is shown to rasonably modl th xprimntal rsponss. Th modl allows a bttr undrstanding of SFR-RC structurs failing in shar and can b usd as a basis for dvloping nw dsign procdurs for such structurs. Kywords Stl fibrs, concrt, shar, stirrups, ATENA. 89

2 1 Introduction Exprimntal and numrical invstigations hav shown that th inclusion of stl fibrs in concrt, whn adoptd in adquat quantitis, can improv th shar rsistanc of bams by incrasing th post cracking strngth of th concrt. Fibrs mbddd within concrt dlay th propagation and growth of cracks by improving th ffctivnss of th crack-arrsting mchanisms prsnt whn bams ar subjctd to high shar strsss. Many studis hav considrd th possibility of utilizing SFRC by assigning a proportion of th shar rsisting capacity of bams to th fibrs. This has bn ralizd by ACI-318 (2008), and mor rcntly by th fib Modl Cod 2010 (2012) and th Draft Australian Bridg Cod: Concrt (2014). Som inconsistncis in som of ths approachs, howvr, hav bn idntifid (Fostr, 2010; Amin & Fostr, 2014). To accuratly modl th inhrnt nonlinar proprtis of lmnts composd of SFRC, it is ncssary to mak us of advancd softwar which can corrctly modl concrt fractur. This is of particular importanc whn crack propagation at th lmntal lvl can significantly influnc th global, or structural, rspons of th mmbr (Fostr & Voo, 2004, Fostr t al., 2006, Minlli & Vcchio, 2006). To this nd, a numrical invstigation is undrtakn to study, in dtail, th dvlopmnt of shar rsistant mchanisms in stl fibr rinforcd-rinforcd concrt bams with and without convntional shar rinforcmnt using a commrcially availabl Finit Elmnt packag, ATENA 2D, dvlopd by Crvnka Consulting (Crvnka t al., 2002). Th numrical modl is validatd against xprimntal rsults obtaind from four 5 mtr long by 0.3 mtr wid by 0.7 mtr high rctangular simply supportd bams with varying transvrs and fibr rinforcmnt ratios. An valuation of th transvrs stirrup contribution vrsus fibr contribution to th shar capacity is compild and rcommndations ar mad for th xplicit inclusion of th distinct capacitis to xisting dsign modls. 2 Exprimntal Invstigation An xprimntal program was conductd to invstigat th combind ffct of stl fibrs and traditional transvrs rinforcmnt on th rspons of larg scal bams subjctd to four-point loading. Four bams with various fibr and transvrs rinforcmnt ratios wr constructd and tstd to failur. As th invstigation was dsignd for th spcimns to fail in shar, th bams wr dsignd to nsur that flxural tnsil failur did not occur. Th spcimn dimnsions and tsting arrangmnts ar shown in Figur 1. Th spcimns ar dsignatd using th notation BX-Y-Z whr X is th dosag of fibrs (in kg/m 3 ), Y is th diamtr of stirrup (in mm), and Z is th stirrup spacing (in mm) within th critical shar rgions. For xampl, spcimn B rprsnts a bam rinforcd with 25kg/m 3 of stl fibrs and 10mm stirrups spacd at 450 mm c/c. Th longitudinal rinforcmnt was fabricatd from nominally 500 MPa grad, hot rolld, dformd bars. Th bams containd two layrs of thr normal ductility 28 mm diamtr tnsil rinforcmnt (N28), which corrsponds to a flxural rinforcmnt ratio of 1.98%. 90

3 Two 20 mm diamtr longitudinal rinforcing bars (N20) wr locatd at th top sction of th bam. Th masurd yild strngth of th longitudinal bars was 540 MPa. Th 2-lg stirrups wr fabricatd from hard-drawn wir rinforcmnt and wr spacd at 450 mm cntrs. Th 10 mm diamtr stirrups had a yild strngth of 450 MPa; th 6 mm stirrups had a yild strngth of 550 MPa. Th stl fibrs usd in this study wr th doubl nd-hookd Dramix 5D-65/60-BG fibr. Th fibrs wr 0.9 mm in diamtr and 60 mm long. Th spcimns wr cast using a singl batch of concrt obtaind from a local rady mix supplir. Th concrt had a prscribd charactristic comprssiv strngth of 32 MPa and th aggrgat usd was basalt with a maximum particl siz of 10 mm. All bam spcimns wr tstd in an Instron 5000 kn stiff tsting fram and tstd undr a ram displacmnt control of 0.3mm/min. A linar strain convrsion transducr (LSCT) was placd at th bam mid-span to masur displacmnt (Figur 1). Th radr is rfrrd to Amin and Fostr (2014) for furthr dtails on th xprimntal study. 2V 2N Figur 1: (a) LSCT 2250 Spcimn dimnsions (in mm): (a) Tst stup; (b) Cross sction 3N28 3N (b) 3 Mchanical Proprtis of SFRC Th masurd mchanical proprtis of th SFRC at th tim of tsting ar prsntd in Tabl 1. Th fractur proprtis of SFRC wr dtrmind dirctly by obtaining th tnsil strngth of th concrt matrix, f ct, and th rsidual tnsil strngth, f 1.5 (takn at a crack width of 1.5mm), from six uniaxial dogbon spcimns tstd to DR AS (2014). Th tnsil proprtis wr also dtrmind indirctly from th rsidual flxural strngth, f R,j, through fiv thr-point notchd prisms tstd to EN (2007). Th man comprssiv strngth (f cm ) and Young s modulus (E c ) wr dtrmind by tsting thr 150 mm diamtr x 300 mm cylindrs. 91

4 Tabl 1: Mchanical proprtis of SFRC f cm (MPa) E c (GPa) f ct (MPa) f 1.5 (MPa) f R1 (MPa) f R2 (MPa) f R3 (MPa) f R4 (MPa) To dfin th tnsil strss vrsus crack opning displacmnt fractur constitutiv rlationship for SFRC, th simplifid invrs analysis procdur dvlopd by Amin t al. (2013) was usd in this study. Th formulation of this modl is foundd upon a sctional analysis of a prism in bnding and considrs th influnc of fibrs on th momnt carrid by th spcimn from th point in th tst whr th un-crackd concrt has littl influnc on th prism capacity and considrs rigid body rotations. Th nominal strss carrid by th fibrs, f(w), is takn as: æ f ö = ç + - ³ è 3 ø f ( w) k R2 ( f f ) x( w) x ( w) b R4 R2 0 ( D- hsp) w = - (1b) 3 0.7h 4 sp whr D is th total dpth of th prism and h sp is th dpth lss th notch dpth. For prisms tstd to EN (2007), x(w) (w in mm) simplifis to: ( w) 0.43w 0.25 x = - (2) Th modl of Equation 1 is prsntd gnrically in Figur 2(a) and corrsponding to th xprimntal data of this study in Figur 2(b). Th orintation factors k t and k b ar applid to th dirct tnsion and prism bnding data, rspctivly, to convrt th rsults to that of an quivalnt 3D fibr distribution fr of influnc of th boundaris (Amin t al., 2013) and takn as: (1a) k t = l b f (3a) k b p = l b f (3b) whr l f is th lngth of th fibr and b is th width of th cross sction of th matrial tst. 92

5 Tnsil Strss (MPa) FRC 2014 Joint ACI-fib Intrnational Workshop s 3 25kg/m3 5D EH Fibrs Simplifid Modl (this study) f ct 2 f(w) 1 Data rang dirct tnsion tst x kt G F (a) w COD, w (mm) (b) Figur 2: Tnsil constitutiv law of SFRC: (a) Gnric rspons; (b) Application of invrs analysis procdur of Equation 1 to tst data. 4 Finit Elmnt Modl A st of numrical analyss wr prformd on th four bams of this study utilizing th softwar ATENA (Crvnka t al., 2002). Th matrial modls CC3DNonLinCmntitious2 (Usr) ar usd in this study to modl th concrt and ar basd on a fractur-plasticity approach which combins constitutiv modls for tnsil (fractur) and comprssiv (plastic) bhaviour. Th fractur modl is basd on th classical orthotropic smard crack formulation and crack band modl. It utilizs th Rankin failur critrion and is not rstrictd to any particular shap of hardning or softning laws. A fixd crack approach is usd in this study; although thr is som dbat on whthr a rotatd crack approach (Minlli & Vcchio, 2006), or combination of both (Fostr, 2006) is mor appropriat. Th plasticity modl is foundd on th Mntry-Willam failur surfac (Mntry and Willam, 1995) using a rturn mapping algorithm for th intgration of th constitutiv quations. To combin th fractur and plasticity modls, a strain dcomposition approach introducd by d Borst (1986) is usd; that is: p f = + + (4) whr, p, and f ar th lastic, plastic and fracturing strains, rspctivly. Th nw strss stat in th plastic modl is computd using a prdictor-corrctor formulation whr th plastic corrctor is computd dirctly from th yild function of th rturn mapping algorithm. Th crack opning, w, is computd from th summation of th fracturing strain and incrmntal fracturing strain multiplid by th charactristic lngth. Th full drivation of th matrial modl can b found in Crvnka and Papanikolaou (2008). 93

6 For SFRC in comprssion, no siz ffcts wr considrd and th concrt is modlld using th nonlinar strss-stain rlations givn by ATENA; that is: c ( ) s = ( - ) + ( E E - 2) f E E cm c o c c c o whr E c, E o and c ar dfind in Figur 3(a). This choic rflcts th fact that th addition of stl fibrs to concrt in convntional dosags, as is in this study, dos not hav significant influnc on th pr-pak comprssiv proprtis of th concrt; th post-pak rspons, howvr, has bn shown to b influncd by th fibr dosag (Mansur t al., 1999). In this study, th post-pak softning curv bars littl influnc on th numrical study as no phnomna rlating to concrt crushing ar bing studid. For SFRC in tnsion prior to cracking th strss-strain rlation of SFRC can b dscribd as linar lastic with Elastic Modulus, E c. A crack is thn initiatd whn th tnsil strss xcds th cracking strngth of th matrix, f ct (Rankin critrion). To tak account of th rsidual tnsil strsss du to shrinkag, th in-situ tnsil strngth of th concrt, f ct, is takn as 0.3 f = 1.75MPa. cm Th most critical paramtr whn considring th dsign of a structural mmbr manufacturd with SFRC is its post cracking, or rsidual, tnsil strngth. In this papr th constitutiv law dscribing SFRC in tnsion (Equation 1) is implmntd into th ATENA modl. A basic assumption of smard crack modls is that th componnts of th softning strss-strain rlationship ar uncoupld in th local coordinat systm associatd with a crack. Taking this assumption into account, th tnsil softning bhaviour is xprssd in trms of strss vrsus crack opning displacmnt (s-w) rlationship (Figur 3b). With th matrial proprtis givn in Tabl 1 substitutd into Equation 1, th tnsil rlationship for th concrt, f t, is givn as: ì Ec pr cracking ï ft = í w post cracking, w < 0.04mm ï î w post cracking, w³ 0.04mm Th modl allows for an implicit coupling of th componnts by using a cracking shar modulus that is a function of th Mod I cracking strain; thrfor shar transfr across cracks is possibl. (5) (6) 94

7 -s C f cm s (Eq.5) c s C f ct E O E C c E C f (Eq.6) t w (a) (b) b (0,1) s S f sy (1-5,0.6) (2-4,0.3) (1-3,0.15) (1-2,0) Figur 3: (c) E S (d) (a) Strss-strain rspons of SFRC in comprssion; (b) Rspons of SFRC in tnsion; (c) Shar rtntion factor usd in this study; (d) Strss-strain rlationship of rinforcing stl For fixd cracks, th crack systm may not b paralll to th principal strain systm and shar strsss can volv along th crack dgs. Walravn (1980) showd a dpndnc btwn th shar stiffnss and th crack width which is takn into account by shar rduction factors. Shar rtntion functions, b(), ar somtims assumd to b constant. Th advantag of such a modl is that it consrvs th indpndnc of th shar and th normal strsss in th cracking zon. Howvr, it is physically not ralistic sinc th shar rsistanc along a crack is producd by friction, aggrgat intrlock and fibr bridging; all of which ar a function of th crack opning. Morovr, it would imply that vn thos cracks which hav opnd up compltly (no furthr strss transfr in tnsion) would still transmit shar. A mor ralistic choic is to rduc th matrials shar rsistanc linarly or xponntially with incrasd crack opning. An xponntially dcrasing shar rsistanc factor is givn in Equation 7, whr is th Mod I strain in th coordinat systm associatd with th crack, and u is th ultimat strain at which th crack bcoms strss fr. æ ö b( ) = ç 1- è u ø g (7) Largr valus of g corrspond to a mor pronouncd dcras of b in ordr to simulat highr in-plan shar strss dgradation with incrasd crack opning. In this study w adopt th linarizd vrsion of Equation 7 as shown in Figur 3(c). All rinforcing stls wr modlld as simpl lastic prfctly plastic matrials with prfct bond to th concrt (rfr to Figur 3(d)). 95

8 To simulat th bams, a 2D modl was dvlopd. On half of th bam was modlld with symmtry. Th msh snsitivity of rsults of th FEA has bn chckd. For this purpos, th bam was analysd using two diffrnt FE mshs (quadrilatral lmnts limitd to 50mm and 100mm). Th diffrnc btwn rsults of th analyss was ngligibl. Th FE rprsntation of th bams including th msh and boundary conditions is illustratd in Figur 4. For th notional 50 mm msh, th modl was discrtizd using nod isoparamtric mmbran lmnts intgratd on a 2x2 quadratur. Th non-linar modl was solvd th using Nwton-Raphson mthod with convrgnc dtrmind using displacmnts, out of balanc forcs and nrgy and with a solution tolranc of 0.01 for ach. V Y X Figur 4: FE rprsntation of shar critical spcimns 5 Comparisons with Tst Data Th rsults of th FE modlling ar compard with th xprimntal data in Figur 5 for applid shar vrsus mid-span dflction. Th pak strngths obtaind from th FE modl ar compard to th xprimntal valus and prdictions from th altrnativ modl for shar in SFRC in th fib Modl Cod (modl apparing on th lft hand sid of th cod) along with th modl of th Draft Australian Bridg Cod (DR AS5100.5, 2014) in Tabl 2. In gnral, th FE modl capturd th bhaviour of th bams wll. Th modl accuratly prdicts th first cracking load, it modls th post cracking stiffnss wll and th failur loads ar rasonably prdictd. In Tabl 2, two rsults ar prsntd for th cod analyss. In th lft column, V us is takn as na sv f sy,f whr n is th numbr of stirrups that wr actually crossd by th shar crack, as obsrvd in th xprimnt, A sv is th ara of th shar rinforcmnt for on stirrup and f sy,f is th masurd man strngth of th transvrs rinforcmnt. In th right column, th valus ar thos prdictd by th cods (thory) with th stirrups takn as smard along th sction, as is usual practic. Th FEA compars rasonably with prdictions from th cods whn th actual numbr of stirrups crossd is considrd; th lowr rsult for spcimn B is rflctiv of th fact that in th FEA th failur crack crossd just on stirrup, not two as obsrvd in th tsts. In th cod analyss with smard shar stl, th strngth may b ovrstimatd. This is bcaus of th significant snsitivity, in th cas of larg ligatur spacing, in th stl contribution if on, two or thr stirrups ar crossd. For q 34 o for th SFRC bams, as is th cas hr, thory suggsts that 1.8 ligaturs will cross th crack. 96

9 Shar, V (kn) Exprimnt FEA Shar, V (kn) Exprimnt FEA Midspan Dflction (mm) (a) Midspan Dflction (mm) (b) Shar, V (kn) Figur 5: (c) Exprimnt FEA Midspan Dflction (mm) Shar, V (kn) Comparison of FE rsults against xprimntal data for bams: (a) B ; (b) B25-0-0; (c) B ; (d) B (d) Exprimnt FEA Midspan Dflction (mm) Tabl 2: Comparison of FE rsults with xprimnt and cods Spcimn ID n V u,fea (kn) V u,xp (kn) V u,fib (kn) V u,as (kn) With n Thory With n Thory B B B B Figur 6 xamins th dvlopmnt of strain and crack formation in bam B At a mid-span dflction of 7 mm (Figur 6a), which corrsponds to an applid shar of approximatly 210 kn, cracking in th shar span is noticabl. At pak load, (Figur 6b) flxural shar cracks ar dominant. Th critical crack appars in th modl aftr pak load (Figur 6c) and its location and inclination is comparabl to th actual dominant crack obsrvd in th xprimnt (Figur 6d). As a fixd crack modl is usd, th dirction of cracking is dtrmind upon th first crack initiation and its orintation is kpt fixd aftrwards. This is of particular intrst in intrprting th rsults of bam B In th xprimnt, it was obsrvd that th dominant crack had ngagd two stirrups, as opposd to just on in bams B and B In th FEA, th dominant crack passd ovr just on stirrup in bam B (not two). Adopting th approach of th altrnativ modl for shar in SFRC in th fib Modl 97

10 Cod, th prdiction of th shar strngth of bam B for a failur crack ngaging on stirrup is 321 kn and compars wll to th FEA rsult E E E-03 Y Y X X (a) 1.950E-02 (b) Y X Figur 6: (c) Nonlinar rspons and crack pattrns of bam B : (a) Maximum principal strain at 7mm; (b) Maximum principal strain at pak load; (c) Maximum principal strain at 15mm; (d) Actual crack pattrn of bam at failur. Figur plottd for 50mm msh. (d) 6 Conclusions In this papr a validation of th bhaviour of larg scal stl fibr rinforcd-rinforcd concrt bams is mad through a comparison of numrical analyss and shar xprimnts. Th numrical analysis was prformd using a commrcially availabl softwar packag, ATENA, which was adaptd to SFRC using constitutiv laws drivd from an invrs analysis of prism bnding tsts. Th rsults of th numrical analysis indicat that th bhaviour of SFRC bams can b modlld using ATENA whn an appropriat constitutiv law in tnsion is usd. Th modl was shown to compar wll with th xprimntal data in capturing th linar and non-linar rsponss of th bams. 7 Rfrncs ACI , Building Cod Rquirmnts for Structural Concrt and Commntary, ACI Committ 318, 456pp. Amin, A.; Fostr, S.J.; and Muttoni, A. (2013), Evaluation of th Tnsil Strngth of SFRC as Drivd from Invrs Analysis of Notchd Bnding Tsts. Procdings of th 8 th Intrnational Confrnc on Fractur mchanics of concrt and concrt structurs (FraMCoS-8), Toldo, Spain, pp

11 Amin, A.; and Fostr, S.J. (2014), Th Bhaviour of Stl Fibr Rinforcd-Rinforcd Concrt Bams in Shar. Procdings of th 10 th fib Intrnational PhD Symposium in Civil Enginring, Qubc, Canada. Crvnka, V.; Crvnka, J.; and Pukl, R. (2002), ATENA A tool for nginring analysis of fractur in concrt. Sadhana Acadmy Procdings in Enginring Scincs, Vol 27, No 4, pp Crvnka, J.; and Papanikolaou, V.K. (2009), Thr dimnsional combind fractur-plastic matrial modl for concrt. Intrnational journal of plasticity, Vol 24, pp d Borst, R. (1986), Non-linar analysis of frictional matrials. PhD thsis, Dlft Univrsity of Tchnology, Nthrlands. DR AS (2014), Bridg Dsign Part 5: Concrt. Draft for Public Commnt Australian Standard, Standards Association of Australia. EN 14651:2007. Tst Mthod for Mtallic Fibr Concrt- Masuring th Flxural Tnsil Strngth (Limit of Proportionality, Rsidual). Europan Committ for Standardization, 17pp. fib Bulltin 66 (2012), Modl cod for concrt structurs. fédération intrnational du béton (fib), Lausann, Switzrland. Fostr, S.J. (2006), Limitd rotation CMM modl for FE analysis of RC mmbrans. Progrss in Mchanics of Structurs and Matrials, 19th Australasian Confrnc on th Mchanics of Structurs and Matrials, 2006, Christchurch, Nw Zaland, pp Fostr, S.J.; and Voo, J.Y.L. (2004), Finit Elmnt Modlling of Fibr Rinforcd Concrt Structurs, Dvlopmnts in Mchanics of Structurs and Matrials, 18 th Australasian Confrnc on th Mchanics of Structurs and Matrials, Prth, Wstrn Australia, Eds A. Dks and H. Hoa, Taylor and Francis, London, pp Fostr, S.J.; Voo, Y.L.; and Chong, K.T. (2006), Analysis of Stl Fibr Rinforcd Concrt Bams Failing in Shar: Variabl Engagmnt Modl, Chaptr 5 in Finit Elmnt Analysis of Rinforcd Concrt Structurs. Eds. Lows, L., and Filippou, F., ACI SP-237, Fostr, S.J. (2010), Dsign of FRC bams for Shar using th VEM and th Draft Modl Cod Approach. fib Bulltin No. 57, pp Mansur, M.A.; Chin, M.S.; and W, T.H. (1999), Strss-strain rlationship of high-strngth fibr concrt in comprssion. Journal of Matrials in Civil Enginring, ASCE, Vol 11, No 1, pp Mntry, P.; and Willam, K.J. (1995), Triaxial failur critrion for concrt and its gnralization. ACI Structural Journal, Vol 92, No 3, pp Minlli, F.; and Vcchio, F.J. (2006), Comprssion fild modlling of fibr rinforcd concrt mmbrs undr shar loading. ACI Structural Journal, Vol 103, No 2, pp Walravn, J.C. (1980), Aggrgat Intrlock: A thortical and xprimntal analysis. PhD thsis, Dlft Univrsity of Tchnology, Nthrlands. 99