CRACK CONSTITUTIVE MODEL TO SIMULATE THE BEHAVIOR OF FIBER REINFORCED CONCRETE STRUCTURES FAILING IN PUNCHING

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1 CMNE/CILAMCE 007 Porto, Juho, 007 APMAC, Portugal 007 CRACK CONSIUIVE MODEL O SIMULAE HE BEHAVIOR OF FIBER REINFORCED CONCREE SRUCURES FAILING IN PUNCHING A. Vetura Gouveia 1 *, Joaquim Barros, Álvaro Azevedo 3 ad J. Sea Cruz 1: Civil Egieerig Departmet School o echology Polytechic Istitute o Viseu Campus de Repeses, Viseu, PORUGAL vetura@dcivil.estv.ipv.pt, web: : Civil Egieerig Departmet School o Egieerig Uiversity o Miho Campus de Azurém, Guimarães, PORUGAL barros@civil.umiho.pt, jsea@civil.umiho.pt web: 3: Civil Egieerig Departmet Faculty o Egieerig Uiversity o Porto Rua Dr. Roberto Frias, s/ Porto, PORUGAL alvaro@e.up.pt, web: Keywords: Crack Costitutive Model, Puchig, Steel Fiber Reiorced Sel-Compactig Coete, Material Noliear Aalysis, Fiite Elemet Method, Iverse Aalysis. Summary. For some structural applicatios, maily with redudat supports, addig ibers to ete ca be a eomic strategy, especially whe shear brittle ailure mode has a high risk o occurrece. I the preset work, a ack stitutive model is proposed to simulate localized ailure modes due to puchig. he perormace o the developed stitutive model was appraised by simulatig the behavior observed i puchig tests with lightweight pael prototypes o steel iber reiorced sel-mpactig ete (SFRSCC). he values o the parameters required by racture mode I o the SFRSCC were obtaied by iverse aalysis.

2 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz 1. INRODUCION Recet developmets i the iber reiorced ete techology showed that the additio o a small tet o hooked ed steel ibers to a sel-mpactig ete resulted i a high stregth ad ductile material with a sigiicatly ieased lexural ad puchig resistace [1]. he mechaical perormace o the developed steel iber reiorced sel-mpactig ete (SFRSCC) is maily due to the proposed iovative mix desig strategy. I the determiatio o the proportios o each aggregate i the ial solid skeleto the presece o the steel ibers ad their speciicities (iber tet ad iber aspect-ratio) were take ito acut [1]. he developed SFRSCC was used to mauacture lightweight pre-cast paels or buildig açades, with the geometry schematically represeted i Figure 1. Sice the SFRSCC layer at the lighter parts o the pael is oly 30 mm thick, its resistace to puchig was assessed rom experimetal tests [] S Polystyree 300 S' S-S' 150 Polystyree Figure 1. Lightweight SFRSCC pael (dimesios i mm). Modelig the material oliear behavior o a lamiar structure ailig i puchig is still a challege i the mputatioal mechaics domai. A 3D ack stitutive model ca be implemeted, usig solid iite elemets [3], but the umerical stability is mpromised whe soteig ack stitutive diagrams are adopted or modelig all three racture modes that are activated durig the propagatio o a ack. Furthermore, acrdig to the kowledge o the authors o the preset work, a well accepted strategy or evaluatig the soteig laws that simulate racture modes II ad III is ot yet available. I the preset paper a simple but accurate model is proposed to simulate the behavior o cemet based lamiar structures ailig i bedig ad i shear. Special emphasis is put o the aalysis o structures ailig i puchig. he developed model is based o iite elemet techiques ad was implemeted i the FEMIX mputer de [4]. he Reisser-Midli theory was used i the text o layered shells. By siderig the oliear behavior o each layer, ack propagatio alog the thickess o these structures ca be simulated [5]. Fracture mode I was modeled by a ack stress vs. ack strai tri-liear diagram, whose

3 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz deiig parameters were obtaied rom iverse aalysis [6], siderig the orce-delectio curves registered i three-poit otched SFRSCC beam tests carried out acrdig to the RILEM C 16-DF remmedatios [7]. he model is desibed ad its perormace is appraised usig the results obtaied rom puchig tests carried out with represetative parts o the developed SFRSCC pael.. NUMERICAL MODEL.1. Itroductio Previous works showed that the Reisser-Midli theory or shell structures [5] is capable o predictig with high accuracy the behavior o lamiar ete structures up to ailure [8]. However, whe these structures ail i shear, this type o approach seems to be iadequate, sice the model predicts acks that are orthogoal to the middle surace o the shell, which is ot the real orietatio o the acks ormed i the experimetal tests. o explore the possibility o usig the Reisser-Midli shell theory to simulate the material oliear behavior o ete lamiar structures ailig i puchig, a soteig law was itroduced or modelig the two out-o-plae shear stress-strai diagrams. o simulate ete ackig a multi-ixed smeared ack model was implemeted [9]. Fracture mode I is modeled by a ack stress vs. ack strai diagram, whose deiig parameters were determied rom iverse aalysis based o the results obtaied i lexural tests. he specimes used i these tests ad the paels studied with the preset model were o the same age (seve days)... Formulatio Whe the behavior o a material is sidered to be liear elastic the stitutive matrix ( D ) does ot deped o the stress or strai level. However, whe the material behaves oliearly, this matrix ca ot be assumed as stat. Numerically, it may be sidered that D assumes stat values or very small stress ad strai iemets ad, sequetly, the stitutive law may be deied as ollows, σ = D ε (1) where σ represets the stress iemet, ε is the strai iemet ad D is the taget stitutive matrix. I the text o the Reisser-Midli shell theory, the stress vector has ive mpoets, {,,,, } σ = σ σ τ τ τ () where the irst three stitute the i-plae mpoets ( σ m ) ad the last two are the out-o-plae or trasverse shear mpoets ( σ s ). I a similar way, the strai vector also has ive idepedet mpoets, 3

4 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz {,,,, } ε = ε ε γ γ γ (3) Sice the thickess o the tested structural elemets is small, ete is assumed to behave i liear elastic regime i mpressio. I tesio the behavior o the ete is sidered liear util the tesile stregth is reached. he stitutive matrix is mposed o two parts, the irst oe beig associated with the i-plae (membrae ad lexural) deormatio mpoets ( D m, e ), ad the sed oe associated with the out-o-plae (trasverse shear) deormatio ( D s, e ), both icludig ete () properties i elastic regime (e). hereore, the ete elastic stitutive matrix may be desibed as Dm, e 0 D = (4) 0 Ds, e where D m, e ad D s, e may be deied as D 1 ν 0 E = ν ν 1 ν 0 0 c m, e (5) 1 0 Dse, = F G c 0 1 (6) where E c is the Youg's modulus, G c the trasverse elasticity modulus ad ν is the Poisso's ratio, siderig the ete i elastic regime. he actor F is a shear rrective stat, associated with the assumptio o stat out-o-plae shear stresses, whe, i isotropic materials, they assume a parabolic distributio. For rectagular sectios, F = 56. Whe the tesile stregth o ete is reached, the material starts to behave oliearly, ad the stitutive matrix is chaged. I the preset work, to simulate the loss o stregth associated with the strai-soteig behavior o ete i tesio ad i out-o-plae shear, the i-plae deormatio mpoets ( D m, e ) ad the out-o-plae shear deormatio mpoets ( D s, e ) o the stitutive matrix are modiied. he strai-soteig behavior o ete i tesio is simulated by a stress-strai tri-liear diagram, as represeted i Figure, relatig the ack stress ad ack strai orthogoal to the ack plae. he out-o-plae shear strai-soteig behavior o ete is simulated by the diagram represeted i Figure 3, relatig a geeric out-o-plae shear stress with the rrespodig out-o-plae shear strai. 4

5 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz τ OP σ σ,1 σ, σ,3 D 1 D sec D I G g = G / l b l b D 3 -γ OP u -γ OP max -γ OP p τ p OP τ max OP G c -τ max OP γ OP p D III,sec liear behavior γ OP max G III l b γ OP u γ OP ε, ε,3 ε,u ε -τ p OP Figure. ri-liear stress-strai diagram. Figure 3. Geeric out-o-plae (OP) shear stress-strai diagram. Whe the tesile stregth is reached at a itegratio poit (IP) o a iite elemet, the portio o ete icluded i its iluece area chages rom the uacked to the acked state. Usig the multi-ixed smeared ack cept [10], the mpoet Dm, e o the ete stitutive matrix D is replaced by D m, determied with the ollowig equatio [9], ( ) 1 m, e m, e m, e m, e Dm = D D D + D D where represets the trasormatio matrix rom the ack local ordiate system to the elemet local ordiate system, ad D represets the ack stitutive matrix, as desibed i the ollowig equatios s θ si θ siθsθ = siθ sθ siθsθ s θ si θ DI 0 D = (9) 0 DII I Eq. (8), θ is the agle betwee the ack local ordiate system ad the iite elemet ordiate system (see Figure 4). I Eq. (9), D I ad D II represet, respectively, the stitutive mpoets rrespodig to the ack opeig mode I (ormal) ad ack slidig mode II (i-plae shear). he racture mode I modulus, D I, is deied i Figure, where α i ad ξ i are the parameters that deie the shape o the ack ormal stress vs. ack ormal strai diagram. he ultimate ack strai ( ε u, ) is deied as a uctio o α i ad ξ i parameters, o racture eergy ( G ), tesile stregth ( ct ) ad ack bad width ( l b ), as I (7) (8) 5

6 θ A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz ollows [5], ε,u I G = ξ + α ξ α ξ + α l ct b (10) whereα = σ / σ, α = σ,3/ σ,1, ξ1 = ε, / ε, ad ξ = ε,3/ ε,. 1,,1 he racture mode II modulus, D II ult, is obtaied rom, D II ult β = Gc (11) 1 β ε β = 1 (1) ε ult, where β is the shear retetio actor, deied as a uctio o the actual ack ormal strai ( ε ) ad the ultimate ack ormal strai ( ε ult, ). Whe a liear deease o β with the iease o ε is assumed, the p1 = 1. Larger values o the expoet p 1 rrespod to a aster deease o the parameter β [5]. he use o a soteig stitutive law to model the i-plae ack shear stress traser ca improve the accuracy o the simulatio o structures ailig i shear [11]. However, the simultaeous presece o soteig laws to model racture modes I ad II itroduces additioal diiculties o assurig vergece durig the loadig procedure o a material oliear aalysis. p1 x s t τ t σ Crack σ τ t w x 1 Figure 4. Crack stress mpoets, relative displacemets ad local ordiate system o the ack. he out-o-plae shear behavior is assumed to be liear elastic util the tesile stregth is reached. Whe the portios o ete associated with the IP chage rom uacked to 6

7 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz acked state the out-o-plae shear stresses are stored ad the relatio betwee each out-o-plae shear stress-strai ( τ 3 γ 3 ad τ 31 γ 31 ) ollow a idepedetly soteig behavior as show i Figure 3. he mpoet D s, e o the ete stitutive matrix D o Eq. (4) is replaced with D s, which is deied by where D s 3 3,max D III,sec = ; γ 3,max 3 D,sec 0 III = 31 0 DIII,sec τ τ (13) 31 31,max D III,sec = (14) γ 31,max acrdig to a secat approach (see Figure 3). For each out-o-plae shear mpoet, the peak shear strai is calculated usig the stored peak shear stress at ack iitiatio ad the ete elastic shear modulus III τ τ 31, p γ = ; γ = (15) 3, p 3, p G c Each out-o-plae ultimate shear strai is deied as a uctio o the out-o-plae peak shear OP strai ( γ p ), the out-o-plae shear stregth ( τ OP p ), the mode III (out-o-plae) racture eergy ( G ) ad the ack bad width ( l b ), as ollows γ 3, u 3, P G III 3, p b 31, p G c III G = γ + ; γ31, u = γ31, P + τ l τ l I the preset approach it is assumed that the ack bad width used or assurig mesh idepedece whe modelig racture mode I ca also be used to deie the dissipated eergy i the out-o-plae racture process. 3. ASSESSING HE FRACURE MODE I CRACK CONSIUIVE LAW FROM INVERSE ANALYSIS o obtai the values o α i, ξ i, G, 31, p b (16) ct that deie the tri-liear stress-strai soteig diagram (see Figure ), a iverse aalysis was perormed usig the orce-delectio relatioships rerded i the three-poit otched SFRSCC beam tests, carried out acrdig to RILEM C 16-DF remmedatios at the age o seve days [6]. he iverse aalysis sists o the evaluatio o the values o these parameters, leadig to the miimizatio o the ratio betwee the area limited by the experimetal ad the umerical curves ad the area udereath the experimetal curve. he umerical curve rrespods to the results obtaied 7

8 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz by meas o a FEM aalysis (see Figure 5a), where the specime, the loadig ad the support ditios were simulated i agreemet with the experimetal lexural test setup. I this text, the specime was disetized usig a mesh o 8 oded seredipity plae stress iite elemets. he Gauss-Legedre itegratio scheme with poits was used i all elemets, with the exceptio o those located at the specime symmetry axis, where 1 poits were used. Liear elastic material behavior was assumed i all the elemets, with the exceptio o those located above the otch ad alog the specime symmetry axis, where elastic-acked behavior i tesio was sidered. he ack bad width, lb, was assumed to be 5 mm, which rrespods to the width o the elemets above the otch. I Figure 5b the results experimetally obtaied i the lexural tests are mpared with those resultig rom the umerical model or the same test setup. Although ot exactly icidet, there exists a good agreemet betwee the experimetal ad the umerical curves. he values assumed or the racture parameters, α i, ξi, ad G, that resulted i the obtaied umerical curve represeted i Figure 5b, are listed i able 1. (a) 8 ode seredipity plae stress elemets 1 itegratio poits elastic-acked behaviour x y (b) 0 Load [kn] 8 ode seredipity plae stress elemets itegratio poits elastic behaviour scatter o experimetal results experimetal average results umerical simulatio Delectio [mm] Figure 5. hree-poit otched beam lexural test at 7 days: (a) FEM mesh used i the umerical simulatio ad (b) obtaied results. 4. MODEL APPRAISAL he perormace o the proposed stitutive model is assessed by simulatig the behavior observed i a puchig test with lightweight pael prototype o SFRSCC. he test layout ad the test setup are represeted i Figure 6. he iite elemet idealizatio, load ad support ditios used i the umerical simulatio o the puchig test are show i Figure 7. Oly oe quarter o the pael was used i the simulatio due to double symmetry. Seredipity 8 oded Midli shell layered elemets with Gauss-Legedre itegratio scheme were used. he pael thickess o 110 mm was demposed i 11 layers o equal thickess. I the lightweight zoe (shaded elemets i Figure 7) the irst 9 layers rrespod to the polystyree material ad oly the last 3 layers rrespod to SFRSCC. he dashed lie 8

9 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz represets the support o the pael, which was simulated with lie sprigs with iiite stiess i mpressio ad ull stregth i tesio, i order to simulate the loss o tact betwee the pael ad the support durig the loadig process. Steel Plate (100x100x10) 150 Q-Q' Actuator Q Q' 300 Polystyree Steel Plate (100x100x10) SFRSCC (x300) (a) (b) Figure 6. (a) est pael prototype or the puchig resistace ad (b) test setup (dimesios i mm). x Poit load 6 50 mm + + Lie sprigs hickess mm hickess mm 6 50 mm Figure 7. Geometry, mesh, load ad support ditios used i the umerical simulatio o the puchig test. he ete properties used i the simulatio o the puchig test are listed i able 1. o evaluate the perormace o the proposed model two umerical simulatios were carried out. he ormer siders a liear behavior or both out-o-plae shear mpoets. he latter siders a soteig behavior i both out-o-plae shear mpoets whe the SFRSCC acks. x 1 9

10 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz Poisso s ratio ν = 0.15 Iitial Youg s modulus Compressive stregth ri-liear tesio soteig diagram o plai ete ri-liear tesio soteig diagram o SFRSCC Fracture eergy (Mode III) used i the out-o-plae shear stress-strai diagram Parameter deiig the mode I racture eergy available to the ew ack E = N mm c = 5.0 N mm c I ct = 3.5 N mm ; N mm G = ; ξ 1 = 0.07 ; α 1 = 0.15 ; ξ = ; α = N mm G = ; I ct = ; 4.3 N mm ξ 1 = ; α 1 = 0.5 ; ξ = 0.15 ; α = 0.59 G = 3.0 N mm III p = Shear retetio actor Expoetial ( p 1 = ) Crack bad width hreshold agle α th = 30º Square root o the area o the itegratio poit able 1. Coete properties used i the simulatio o the puchig test. I Figure 8 the umerically obtaied relatios betwee the orce ad the delectio i the ceter o the test pael is mpared with the oe rerded i the experimetal test. I this igure it ca be observed that both umerical simulatios have practically the same pre-peak respose. However, the post-peak respose diers sigiicatly. Whe liear behavior is assumed or both out-o-plae shear mpoets, the orce ieases up to 60 kn ad oly or a delectio o 9.6 mm a structural soteig occurs, but with a very smooth load decay. he behavior predicted by this umerical simulatio ater a delectio o about 3 mm diers sigiicatly rom the experimetal respose. Whe a soteig behavior i both out-o-plae shear mpoets is adopted the umerical model predicts with high accuracy the behavior that was experimetally observed. he value o the mode III racture eergy used to deie the out-o-plae shear stress-strai soteig diagram has o experimetal support. his value was estimated i order to assure the abrupt load decay observed experimetally at a delectio o about 3 mm. A value o G = 3.0 N mm was adopted. A iease o G causes the III occurrece o the abrupt load decay at a larger delectio. o estimate the tributio o iber reiorcemet to the puchig resistace, a umerical simulatio was perormed adoptig or the racture mode I the parameters idicated i able 1, which rrespod to plai ete o mpressive stregth matchig the developed SFRSCC. Comparig the curves i Figure 8 it ca be cluded that ibers ot oly ieased sigiicatly the puchig resistace, but also, ad especially, improved the ductility. III 10

11 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz Force (kn) SFRSCC - experimetal SFRSCC - liear out-o-plai shear SFRSCC - soteig out-o-plai shear Plai ete Displacemet (mm) Figure 8. Relatioship betwee the orce ad the delectio at the ceter o the test pael. Figure 9 represets the vertical displacemet ield or a delectio o 10 mm i the ceter o the pael or the case o the umerical simulatio siderig out-o-plae shear soteig. he accetuated gradiet o vertical displacemets matches with high precisio the experimetally observed locatio o the iterceptio o the puchig ailure surace with the top pael ace (see Figure 10). his evideces the capability o the developed approach i the simulatio o this mplex ailure mode. Figure 9. Vertical displacemet ield (i mm) or the umerical simulatio with out-o-plae shear soteig (or a delectio o 10 mm i the ceter o the pael). Figure 10. Puchig itical tour. 11

12 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz 5. CONCLUSIONS I the preset work a simple but accurate model whose purpose is the simulatio o the behavior o cemet based lamiar structures ailig i bedig ad i shear is proposed. he developed model is based o the iite elemet method ad was implemeted i the FEMIX mputer de. he Reisser-Midli theory was used i the text o layered shells. he ack propagatio alog the thickess o these structures ca be simulated by siderig the oliear behavior o each layer. he parameters o the racture mode I were determied rom iverse aalysis usig the orce-delectio relatioship obtaied i three-poit otched beam tests, carried out acrdig to RILEM C 16-DF remmedatios. his is a very importat poit sice this type o test is much simpler ad aster to execute tha the direct tesile test. o simulate the out-o-plae strai gradiet that occurs i puchig tests, a soteig diagram was proposed to model, ater ack iitiatio, both out-o-plae shear stress-strai stitutive laws. he perormace o the model is appraised by usig the results obtaied i the puchig test with lightweight pael prototypes o steel iber reiorced sel-mpactig ete (SFRSCC). A very good agreemet betwee the experimetal results ad the proposed model is observed. It ca be cluded that the preset model is capable o simulatig the behavior o cemet based lamiar structures ailig i bedig ad i shear. ACKNOWLEDGMENS he authors wish to ackowledge the support provided by the Portuguese Sciece ad echology Foudatio (FC) by meas o the project POCI/ECM/57518/004 FICOFIRE - High perormace iber reiorced ete o ehaced ire resistace. he irst author ackowledges the iacial support o FC, PhD Grat umber SFRH/BD/336/005. REFERENCES [1] Barros, J.A.O.; Pereira, E.B.; Satos, S.P.F., Lightweight paels o steel iber reiorced sel-mpactig ete, Joural o Materials i Civil Egieerig, 19(4), 007. [] Barros, J.A.O.; Pereira, E.B.; Cuha, V.M.C.F.; Ribeiro, A.F.; Satos, S.P.F.; Queirós, P.A.A.A.V. PABERFIA- Lightweight sadwhich paels i steel iber reiorced sel mpactig ete." echical Report 05-DEC/E-9, Dep. Civil Eg., School o Eg. Uiversity o Miho, 63 pp., 005 [3] Barzegar, Fariborz ad Maddipudi, Sriivas, hree-dimesioal modelig o ete structures. I: Plai Coete, Joural o Structural Egieerig, 13(10), pp , October

13 A. Vetura Gouveia, Joaquim Barros, Álvaro Azevedo ad J. Sea Cruz [4] Azevedo A.F.M.; Barros J.A.O.; Sea-Cruz J.M.; Vetura-Gouveia A., Sotware o esio e o projecto de estruturas (Educatioal sotware or the desig o structures). I: Proceedigs o the III Egieerig Luso-Mozambica Cogress, Maputo, Mozambique, pp. 81-9, 003. (i Portuguese). < [5] Barros, J.A.O., Behavior o iber reiorced ete - experimetal ad umerical aalysis, PhD hesis, Civil Eg. Dept., FEUP, Portugal, (i Portuguese). [6] Pereira, E.B.; Barros, J.A.O.; Cuha, V.M.C.F.; Satos S.P.F., Compressio ad bedig behavior o steel iber reiorced sel-mpactig ete, hird Iteratioal Coerece Costructio Materials: Perormace, Iovatios ad Structural Implicatios, Vauver, CD, -4 August 005. [7] RILEM C 16-DF. est ad desig methods or steel ibre reiorced ete - Fial Remmedatio. Materials ad Structures 35(53), pp , 00. [8] Barros, J.A.O. ad Figueiras, J.A., Noliear aalysis o steel ibre reiorced ete slabs o grade, Computers & Structures Joural, 79(1), pp , Jauary 001. [9] Sea-Cruz, J.M., Stregtheig o ete structures with ear-surace mouted CFRP lamiate strips., PhD hesis, Departmet o Civil Egieerig, Uiversity o Miho, 004, < [10] Rots, J.G., Computatioal modelig o ete racture, Dissertatio, Delt Uiversity o echology, [11] Rots, J.G. ad de Borst, R., Aalysis o mixed-mode racture i ete, Joural o Egieerig Mechaics, ASCE, 113(11), pp ,