Available nline at www.sciencedirect.cm Prcedia Engineering 14 (2011) 1447 1454 he welfth East Asia-Pacific Cnference n Structural Engineering and Cnstructin Nnlinear Mdeling f RC Beams Subjected t rsin using the Smeared Crack Mdel Mstfinejad, D. a *, alaeitaba, S. B. b, a Department f Civil Engineering, Isfahan University f echnlgy b Department f Civil Engineering, Isfahan University f echnlgy Abstract he behavir f RC elements under trsin is a challenging field. Numerical methds have been extensively used fr this purpse preferably t experimental studies that are assciated with rather high csts and difficulties. he present study aims t examine the pssibility f the nn-linear analysis f RC beams subjected t trsin using the smeared crack mdel. Fr this purpse, 14 experimental samples previusly studied by ther researchers were selected and mdeled using ANSYS. he specimens included 12 rectangular sectins, ne hllw (bx-like) sectin, and ne -shaped sectin. he specimens were mdeled in FE sftware and were subjected t mntnic lading up t failure. he results indicated that the adpted smeared crack mdel is capable f making relatively acceptable estimatins f trsinal capacity f the specimens such as their cracking and failure mments; hwever, it underestimates the ultimate rtatin. Discussin n inadequacy f the smeared crack mdel fr nnlinear mdeling f RC beams under trsin is the main target f this paper. 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Keywrds: RC Beam, rsin; FE Mdeling; Smeared Crack 1. Intrductin he study f trsin as a prblem f elasticity dates back t mre than a century ag (Ameli 2005). Analysis f cncrete elements under trsin has been paid due attentin fr abut 70 years by researchers (Rahal 1993). rsin prblem is a rather difficult task in analytical wrks as well as experimental attempts cmpare t the ther type f lads (Ameli and Rnagh 2007; Salm et al. 2004; Ghbarah et al. 2002). On the ne hand, perfrming experimental tests t study pure trsin prblem r trsin and shear simultaneusly is very hard wrk. On the ther hand, experimental specimens are limited in number * Crrespnding authr Eail address: dmstfi@cc.iut.ac.ir 1877 7058 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. di:10.1016/j.preng.2011.07.182
1448 D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 because f the cstliness f tests. hus, befre perfrming experiments, finite-element analysis as a mre ecnmic study helps have ptimal specimens befre underging experiments. hat s why many researchers have s far intended t analyze cncrete elements numerically. Althugh many studies have been made regarding nnlinear analysis f elements under bending, shear, and cmbinatin f bending and cmpressin, i.e. clumns (Kachlakev et al. 2001; alaeitaba 2003), fewer have been made abut trsin (Wei 2005). On the ther hand, since the behavir f cncrete under trsin is similar t that f shear in mst brittle elements even thugh bending has much ductility in beams, studying the influential factrs n trsin is f great significance. In the present research, in rder t get familiar t the factrs f effect n the 3D nnlinear analysis f cncrete elements under trsin, 14 experimental specimens studied under trsin by preceding researchers were mdeled and analyzed nnlinearly, making use f ANSYS. Prcessing the utcmes, all results were cmpared with the relevant cnclusins btained by thse researchers. 2. Specimens As mentined befre, 14 specimens were made in this research, which were tested under trsin in the labratry. able 1 cntains the prperties f the crss sectin, length, and the lngitudinal and transverse reinfrcement f specimens, and als thse f cncrete and reinfrcing steel. 3. Mdelling and nnlinear analysis f specimens In rder t mdel and analyze the specimens with nnlinear finite element, ANSYS sftware was used. All specimens were mdeled with Slid elements dimensinally similar t the labratry specimens. SOLID65, ne f the elements mstly used fr the mdeling f cncrete vlumes with r withut reinfrcement, was used fr the cncrete material because f the capability f mdeling cracks in tensin as well as crushing in cmpressin (ANSYS 2007). In this element, there exists 8 ndes with 3 DOFs at each nde, and reinfrcement can be included in three perpendicular directins, identified by the vlume rati f the reinfrcing steel. his prperty is used t mdel the transverse reinfrcement f specimens under pure trsin. Fr the sake f mre precisin, the vlume f each specimen in the mdel was brken int small parts such that the transverse reinfrcing bar was separately mdeled. he crss sectin f the beam cnsists f smaller parts arund the transverse reinfrcing bars such that the majrity f the vlume f each part is ccupied by the bar. he lngitudinal bars were defined using Link8 element (ANSYS 2007) and cntinuusly cnnected t slid elements representing cncrete with cmmn ndes. 3.1 Study f reinfrced cncrete with sftware Many methds have been develped t study the nnlinear behavir, i.e. plasticity and fracture f cncrete (Chen 1982). Sme methds t distinguish cracking and t analyze specimens up t rupture threshld include smeared cracks and the predetermined cracks mdel (Kachlakev 2001). In the present study, the smeared cracks mdel is elucidated. Because f the inherent nnlinearity in the mechanical behavir f cncrete, the analysis f a cncrete element is nnlinear frm the very beginning. Hwever, in the nnlinear analysis prcedure, a reinfrced cncrete element can be analyzed either linearly r nnlinearly.
D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 1449 able 1: Specimens Prperties Specimen Name Length (mm) Shape & Sizes rans. Reinf. Lngit. Reinf. (Mp a) CH1* 2500 Rec. (Bx) 350*500+250*400: Hle 48.9 CS1* Rec. 350*500 mm 52.5 N1* N2* 2440 Rec. 150*350 mm 37 N3* RC* RG* 1900 Rec. 150*350 mm 39 36 Ra-c* ---- Ras5.5/75* Rec. 100*200 mm 27.5 Ras5.5/150* Rb-c* Rbs5.5/160* 1000 Rec. 150*300 mm ---- 28.8. c* / ---- 26.5 Ref.* 3960 Rec. 279.4*279.4 mm 34 Mpa * CH1, CS1 (Hi and Almahaidi 2006); N1, N2, N3 (Ghbarah et al. 2002); RC, RG, Ra-c, Ra-s5.5/75, Ra-s5.5/150, Rb-c, Rbs5.5/160, c (Charilis 2006); Ref (Panchacharam and Belarbi 2002). 3.2 Material descriptin define cncrete material in ANSYS, the fllwing parameters are necessary: Shear transfer cefficient in pen crack, shear transfer cefficient in clsed crack, uniaxial tensile cracking stress, and Uniaxial crushing stress.as recmmended by previus researches, the shear transfer cefficient in clsed cracks must be taken clse t unity, e.g. 0.99(Kachlakev et al. 2001; alaeitaba 2003). he stress-strain relatin fr reinfrcing steel can be apprximated as bilinear r multilinear curve. In the present study, the multilinear relatin with istrpic hardening was used. he crrespnding strengths f steel and cncrete, the characteristics f the stress-strain relatins such as yield-pint and rupture strains, elasticity and rupture mduli were extracted frm the relevant references and standards (ACI 2005; alaeitaba 2003).
1450 D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 3.3 Crack prpagatin When the stress at a Gauss pint alng either f the directins reaches the cracking stress alng that directin, the element is cnsidered as cracked alng the directin perpendicular t that directin. Parameter βt is very imprtant parameter that indicates the relatin between shear stresses and strains in any cracked element. his is a very effective parameter in the analysis f a specimen under trsin, which is als highlighted by ther researchers in this field (Salm et al. 2004). Based n the utcmes btained by researchers (Kachlakev et al. 2001; alaeitaba 2003), βt ranges between 0.05 t 0.25 fr pen cracks. hus, shear transfer n the crack surface, i.e. shear stiffness faces significant reductin. Mrever, trsinal stiffness f the crss sectin cncrete, in a real specimen, is severely reduced after cracking. 3.4 Supprt cnditins, meshing, lading, and analysis f specimens he specimens were meshed after gemetric mdeling. Since the size f elements directly affects the duratin and arithmetic wrk dne by the cmputer, this effect was assessed in a number f mdels. he maximum size f the elements was cnsidered 30 mm. In fact, the sizes assigned t elements ranged frm very small t big sizes. he end f each specimen was assumed t be trsinally clamped. hus, all mdels, either finite-element r labratry mdels, were cantilever beams. In the labratry, the clampedend cnditin was adjusted s that the lngitudinal mvement f the beam was cmpletely free. reach this end, all jints at the end f the beam were fixed against mving alng the tw perpendicular directins in the plane f the supprt, and released alng the length f the beam. exert trsin n the specimens, the lad was applied n the ndes f the plane cntaining the free end f the beam in the frm f tw frces equal in magnitude and with ppsite senses, as a cuple. In rder t prevent premature fractures due t stress cncentratin at either end f the beam, the elements which included the laded jints were defined t be devid f cracking and crushing while ther prperties are the same as in ther cncrete elements. perfrm the analysis, the lad was applied incrementally in several substeps by using the Autmatic Sub stepping capability in sftware. 4. Results and discussins After preprcessing the mdels and adjusting the nnlinear analysis parameters, including the number f substeps, the cnvergence criterin, and the tlerance f calculatin, the mdels were analyzed. have assurance abut accuracy f the results, the fllwing hints are wrth being reminded f: he mst influential parameters in the analysis prcedure were recgnized based n the cmprehensive studies wrked in Ref. 6, 8, and 18. Numerus analyses were carried ut fr each specimen, in which the mst imprtant factrs f influence such as β t, mesh size, and the lading manner were changed t a large extent (in the allwable regin). All results were cmpared and verified with thse btained frm experiments. he utcmes were satisfactrily dependable. 4.1 Cracking status: Fig. 1 shws the nset f cracks in a specimen n any f the fur sides f the crss sectin. As shwn in the figure, cracks begin t prgress n the greater side. his phenmenn can als be bserved in the previus wrks. On the ther hand, pening and fracture ccurs in ne f the initial cracks.
D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 1451 Left side p side Right side Btumn side Figure 1: Cracks at each face f the specimens he cracking shape and the crack develping trend alng the length f the specimen were studied fr all mdels. It was bserved that, in all specimens, frm the cracking initiatin until the fracture instant, i.e. the last sub step befre divergence f the slutin, the principal stresses and strains increased arund the initial cracked zne. his was explained in the wrks f many ther researchers amng the references used in this research. Mrever, the crack-line angle cnfrmed quite well t the results btained by ther researchers since it was bserved t be abut 45º. 4.2 Ultimate lad he cracking and fracture trques in each experimental and finite-element specimen are included in able 2. Furthermre, in the present research, the cracking and fracture trques are calculated in cnfrmity with ACI318 equatins. It can be bserved frm this table that the mean amunts f the tw ratis are 0.78 and 0.79 fr the cracking trque, and 0.85 and 0.88 fr the ultimate trque (trsinal capacity). On the ther hand, bth ratis in table 2 are clse t unity in the case f FEM slutin. herefre, bth cracking and fracture trques are evaluated satisfactrily crrect. 4.3 -trsinal rtatin curve he trque vs. trsinal rtatin curves fr the labratry specimens as well as FEM mdels are shwn in Figure 2. he abve figures reveal the fllwing results: he linear part is cnsnant with experiment in mst cases. In mst cases the cracking trque cnfrms satisfactrily t that btained frm experiment. In all curves, the pst-cracking znes pertaining t reduced trsinal stiffness have been btained with mre errr than thse frm experiment. his is because the reductin in the stiffness matrix f reinfrced cncrete elements.
1452 D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 herefre, the behavir f FEM specimens after the peak f trque-rtatin curves is prblematic. It seems that ne f the reasns behind this prblem is cnsidering large defrmatins in ANSYS because this ptin des nt functin fr the SOLID65 element. It is ntewrthy that nncnstrained trsin, especially trsin f reinfrced cncrete members must be studied based n the large defrmatin thery [19]. able 2: magnitudes f the cracking and ultimate trques accrding t ACI318. Specimen Name Exp. Cracking Exp. Ultimate FEM Cracking FEM Ultimate ACI Cracking ACI Ultimate CH1 CS1 N1 N2 N3 RC RG Ra-c Ra-s5.5/75 Ra-s5.5/150 Rb-c Rb-s5.5/160 c Ref. 15.80 68.40 5.32 5.88 6.59 9.50 8.00 2.39 2.25 2.30 6.95 6.92 8.48 17.00 49.40 62.90 12.88 12.80 12.59 15.00 14.80 2.40 3.15 2.30 -------- -------- ------- 18.00 15.00 36.00 5.22 5.93 6.40 7.65 6.72 1.34 1.57 1.48 5.48 5.69 5.50 13.47 21.07 54.06 12.10 11.81 12.01 11.80 13.09 1.49 2.25 2.29 7.38 7.21 9.10 19.42 41.99 43.51 5.59 5.59 5.59 5.63 5.63 1.17 1.17 1.17 4.02 4.02 3.94 5.63 63.85 65.37 10.86 14.63 9.52 9.51 9.51 1.17 2.73 1.95 4.02 6.36 3.94 25.75 ) N e u rq ) N e u rq Rtatin (Degrees) Rtatin (Degrees) ) N e u rq ) N e u rq wist *10^-6 (rad/mm) wist (Degrees)
D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 1453 ) N e u rq Angle f twist per length (rad/m) Figure 2: -twist curve f the specimens 5. Cncluding remarks he cnvenience f mdeling and nnlinear analysis f reinfrced cncrete beams under pure trsin was studied using ANSYS sftware. he study was carried ut by mdeling an analyzing 14 specimens which had been studied by ther researchers, making use f the SOLID65 element in sftware. Each specimen was laded mntnically up t the fracture threshld. he same specimens were als analyzed using the methd put frward by ACI318 t btain their trsinal capacities. he mst imprtant utcmes are as fllws: By using FEM analysis, the cracking and fracture trques culd be evaluated with reasnable accuracy. he trsinal capacity with this methd was even mre precise than that btained using the methd explained in ACI318. he trsinal cracking shape and the crack develping trend were predicted accurately up t the fracture threshld. In the rising parts, the trque-rtatin curves were cnsnant with thse btained frm experiment fr all specimens. Hwever, in the pst-cracking zne, in which trsinal stiffness f the beam decreases significantly, the analysis encuntered cnvergence truble. his happens because f the strict sensitivity f the smeared cracks mdel in nnlinear analysis under trsin, twards the shear transfer cefficient in pen cracks, β t. Small values f βt induces severe reductin in shear stiffness in the stiffness matrix and shear instability n the crack surface. Anther reasn behind this truble seems t be making use f the large defrmatin ptin in the analyses, which is inapplicable fr the SOLID65 in ANSYS. References [1] ACI 318, Building cde requirements fr structural cncrete and cmmentary, ACI Manual f Practice;2005 [2] Ameli, M. rsinal strengthening f RC beam with FRP cmpsites. Ph. D. hesis; 2005, Department f Civil Engineering, University f Queensland,. [3] Ameli M, and Rnagh H R. Analytical methd fr evaluating ultimate trque f FRP strengthened reinfrced cncrete beams. Jurnal f Cmpsites fr Cnstructin.;2007, 11(4), pp. 1-7. [4] Ameli M, Rnagh H R and Dux P F. Behavir f Strengthened Reinfrced Cncrete Beams under rsin. Jurnal f Cmpsites fr Cnstructin; 2007, 11(4), pp. 192-200. [5] ANSYS User's Manual ;2007. Release 11.0, SAS IP, Inc..
1454 D. Mstfinejad and S.B. alaeitaba. / Prcedia Engineering 14 (2011) 1447 1454 [6] Charilis C. rsinal strengthening f rectangular and flanged beams using carbn fiber-reinfrced- plymers-experimental study. Cnstructin and Building Materials;2006. Internet file: www.elsevier.cm/lcate/cnbuildmat. [7] Chen W F, Plasticity in Reinfrced Cncrete. McGraw-Hill;1982. Secnd Editin. [8] Ghbarah A, Ghrbel M N, and Chidiak S E. Upgrading trsinal resistance f reinfrced cncrete beams using fiberreinfrced plymer. Jurnal f Cmpsites fr Cnstructin (ASCE); 2002, 6(4), pp. 257-263. [9] Hii A K Y, and Al-Mahaidi R. An experimental and numerical investigatin n trsinal strengthening f slid and bxsectin RC beams using CFRP laminates. Cmpsite Structures; 2006. Internet file: www.elsevier.cm/lcate/cmpstruct. [10] Hii A K Y, and Al-Mahaidi R. Experimental investigatin n trsinal behavir f slid and bx-sectin RC beams strengthened with CFRP using phtgrammetry. Jurnal f Cmpsites fr Cnstructin; 2006, 10(4), pp. 321-329. [11] Kachlakev D, Miller, Yim S, and Chansawat Kasidit. Finite element mdeling f reinfrced cncrete structures strengthened with FRP laminates. Final Reprt fr Oregn Department f ransprtatin Research Grup;2001. Internet File. [12] Panchachram S, and Belarbi A. rsinal strengthening f reinfrced cncrete beams strengthened with FRP cmpsites. First FIB Cngress, Osaka, Japan; 2002, Octber 13-19, pp. 1-11. [13] Rahal K N, he behavir f reinfrced cncrete beams subjected t cmbined shear and trsin. Ph. D. thesis;1993, University f rnt. [14] Salm P, Gergely J, and Yung D. rsinal retrfit f spandrel beams with cmpsite laminates. Jurnal f Cmpsites fr Cnstructin (ASCE); 2004. 8(2), pp. 157-162. [15] alaeitaba S B. Ductility enhancement f RC jints with FRP laminates, M. Sc. hesis; 2003, Isfahan University f echnlgy. [16] Wei R. Numerical mdeling f fiber reinfrced cncrete beams under trsin. Final Reprt fr Advanced Cmputatinal Mdeling Center; 2005, the University f Queensland.