3D NONLINEAR FINITE ELEMENT ANALYSIS OF CONCRETE UNDER DOUBLE SHEAR TEST

Transcription

1 - Technical Paper - 3D NONLINEAR FINITE ELEMENT ANALYSIS OF CONCRETE UNDER DOUBLE SHEAR TEST Ha Ngoc TUAN *1, Hisanori OTSUKA *2, Eizo TAKESHITA *3 and Shinichiro ABE *4 ABSTRACT This paper presens a sudy on shear srengh of lighweigh and normal aggregae concree, morar and pase specimens under double shear es. A linear relaionship beween compressive and shear srengh has been found for all maerials. 3D nonlinear finie elemen analysis closely prediced experimenal shear srenghs. Some aspec of failure mechanism was discussed by invesigaing deformaion, sress and srain disribuion of he numerical model. Keywords: Shear srengh, double shear, smeared crack model, nonlinear finie elemen analysis 1. INTRODUCTION Concree in srucural members rarely failed due o shear sress. This is because concree maerial is weak in ension and comparaively srong in shear. This leads many researchers hink ha unlike srengh in ension and compression, concree has no characerisic value for shear [1]. In principle, shear can be considered as a biaxial ension-compression sae of sress. Fig.1 illusraes his idea. The closed curve obained from experimenal sudy is a ypical biaxial srengh envelop for concree subjeced o proporional biaxial loading. This concep of muliaxial behavior considered concree as elasic-fracure maerial so before limi sress is reached concree remains inac. However he siuaion is no as simple as ha. I is also well known from experimens ha in he microlevel when sress is high, micro-cracks have occurred and he behavior of concree is nonlinear. In cerain sress regime hese micro-cracks will disribue in a very narrow band, so in he macro-scale afer concree has fracured, shear failure may occur in a weak plane in maerial along he shear band. Several researchers believe in he exisence of shear fracure in concree. The mechanism underlying heir claims is shown in Fig.2. Shown in Fig.2(a) is Bazan [2] definiion of a shear crack. A precursor o he developmen of macro shear is an array of ensile (mode I) mircocracks propagaed in a very narrow region. A failure, he inclined cracks are supposed o join, in order o form a single shear plane as indicaed. A second approach o explain Fig.1 Biaxial srengh of concree Fig.2 Definiion of shear failure shear failure is an idea of Ficiious Crack Model proposed by Hillerborg [3] as shown in Fig.2(b). In his approach a process zone, which is a region in fron of sress free macrocrack where microcracks disribued, is subjeced o in plane shear. The hird siuaion occurs when he shear zone is no sufficienly confined ha means a mixed mode sae of sress developed in concree. A curve crack will propagae possibly like a skech in Fig.2(c). Shear in exising cracks has led o vivid discussions in he pas decade. Some shear esing mehods have been proposed and applied o concree such as single shear es, double shear es, indirec single or double shear ess and punch-hrough shear es ec. However researchers raised doub abou possibiliy o realize pure shear (mode II) damage and concluded ha all of he proposed esing mehods yielded a mixed mode failure of concree [4]. While *1 Graduae school of Engineering, Kyushu Universiy, M.E, JCI Member *2 Prof. Dep. of Civil and Srucural Engineering, Kyushu Universiy, D.E, JCI Member *3 Research Engineer, Chiba Research & Developmen lab., Taiheiyou Maerial Corp., M.E *4 Graduae school of Engineering, Kyushu Universiy (During he research)

2 he debae of shear failure mode coninues, here is a pracical ineres in shear srengh of concree. This paper focuses on shear srengh of differen ypes of concree, morar and pase maerials. Nonlinear finie elemen (FE) analysis has been implemened o verify experimenal resuls and simulae failure process of es specimens. 2. SHEAR STRENGTH DEFINITIONS Shear srengh of a maerial is ofen used o covers several conceps such as 1) srengh agains pure shear, 2) shear sress required for failure wihou normal sress, 3) shear diagram on solid inerface depending on normal sress, 4) Mohr s sress envelop [6]. According o Everling [7] shear srengh τ can be defined as he breaking shear force T applied o an imposed surface A supporing no normal force, ha is: A paricular case of k=0.5 gives Mohr s formula of shear srengh which denoed by τ 0 in Fig.3. I is o noe ha definiion using failure envelope doesn give consisen shear srengh. Differen esing mehods yield differen values of shear srengh. From classical school of linear elasic fracure, in pure shear mode of failure, crack should iniiae and propagae under uniform shear sress. Crack growh should also be confined o a plane. A limi shear sress in such circumsance is shear srengh. In his sudy Formula (1) is used o calculae shear srenghs. The shear force was obained using double shear es recommended by Uomoo [5]. 3. TEST PROGRAMS τ=t/a (1) One common way o define concree shear srengh is based on failure curves of concree in σ τ space. Fig.3 schemaically shows such a curve. This curve is an envelope of a series of Mohr s circles obained from shear es of concree specimens under uniaxial and muliaxial sress sae. The value of shear, where he envelope crosses verical axis, is defined as he shear srengh, which can generally be expressed by he following formula[1]: τ = k f f (2) where, τ c is shear srengh (N/mm 2 ) f c and f are compressive and ensile srengh respecively (N/mm 2 ) k is a coefficien obained from experimen c 3.1 Deail of Tesed Specimens The experimenal work consised of specimens c Tension Series name W/C (%) Q. (uni) P P P NS NS NS LS LS LS NSNG NSNG NSNG NSLG NSLG NSLG LSNG LSNG LSNG LSLG LSLG LSLG τ τ 0 Uniaxial ension Failure envelop Uniaxial compression Table 1 Tes specimens Compression σ Fig.3 Failure envelop in σ τ space Descripion Cemen pase Normal aggregae morar Lighweigh aggregae morar Normal weigh concree. LWA concree of normal fine and LW coarse aggregaes LWA concree of LW fine and normal coarse aggregaes LWA concree of LW fine and coarse aggregaes made by four ypes of concrees, wo ypes of morar and cemen pase maerials. Table 1 shows differen ypes of maerial ogeher wih W/C raio and quaniy of specimens for each series of ess. There are 21 series of ess. Each series has 12 specimens and he oal number of specimens is 252. Shear es specimen is a block of dimensions cm. In addiion, for each series of ess, here were hree sandard cylinder specimens for compressive srengh and Young s modulus ess. All specimens were cas and cured in waer under sandard room emperaure for 28 days. Maerials used for morar and concree mixure are presened in Table 2. The selecion of mix proporions for pase, morar and concree specimens is presened in Table Tes Arrangemens and Procedure Phoo 1 shows loading arrangemen of he doubleshear es. A specimen was symmerically posiioned wihin a shear device. The shear device consiss of wo pars, a base and a op par. The ouer surface of shearing edges of he op par is coincided wih he inner surface of shearing edges of he base. The inersecion of his surface and he specimen is he esed secion, ha is he secion used o calculae shear srengh. Along wo esed secions here were srain gauges glued on he surface of he specimen.

3 These gauges were used o monior and adjus eccenric loading. Monoonic, force-conrolled loading was applied wih speed of incremen of shear sress of 0.1N/mm 2 per second. 3.3 Selecion of Tes Daa Daa of hose only specimens wih damaged surface closed o plane, ha means saisfied condiion ha crack should confined o a plane, were seleced for furher analysis. A number of specimens wih curved damaged surface or inclined surface, which is no coincided wih es secion, were excluded. Deail discussion of damage paerns of esed specimens and procedure for selecion of he es resuls can be found in our previous published paper [4]. 4. TEST RESULTS AND DISCUSION 4.1 Shear srengh vs. compressive srengh Fig.4 shows plos of shear srengh versus compressive srengh for all seleced specimens. I can be seen from he disribuion of daa poins ha shear srengh increases wih incremen of compressive srengh. Linear regression analysis showed ha concree and morar specimens have Shear srengh N/mm 2 ) Phoo 1 Tes arrangemen Pase NS LS NSNG NSLG LSNG LSLG Series9 Morar & Concree Pase Compressive srengh N/mm 2 ) Fig.4 Compressive-Shear srengh relaionship Table 2 Maerials for concree mixure Name Symbol Type Noes Cemen C Normal Porland cemen Densiy 3.16(g/cm 3 ) Fine Naural sand Dry densiy 2.60(g/cm 3 ) S aggregae Lighweigh (Asanolie) Dry densiy 1.90(g/cm 3 ), absorpion 20.2 Coarse Naural crushed rock Dry densiy 2.63(g/cm 3 ) G aggregae Lighweigh (Asanolie) Dry densiy 1.58(g/cm 3 ), absorpion 28.9 Admixure LS P lasicizer Densiy 2.70(g/cm 3 ) High performance waer SP SP8N reducing agen Admixure AF Anifoam agen Micro-air 404 MC Vicious agen 90EMP Series nam e W /C (% ) s/a (% ) Table 3 Mix propoion W eigh per uni volume (kg/m 3 ) W C LS S N S L G N G L SP P P P NS NS NS LS LS LS N SNG N SNG N SNG N SLG N SLG N SLG LSNG LSNG LSNG LSLG LSLG LSLG

4 nearly idenical relaionship beween shear and compressive srengh which can be presened by he solid line, whereas pase specimens show slighly difference as shown by doed line. 4.2 Relaionship beween compressive, ensile and shear srenghs Relaionship beween compressive, ensile and shear srengh of concree is shown in Fig.5. In his figure he coefficien k in Formula (2) was ploed agains compressive srengh of esed specimens. I can be seen ha disribuion of coefficien k is roughly in a range from 0.4 o 0.9.Average value of k for all specimens in he experimen is Shown in he same figure solid lines presening wo cases of coefficien k proposed by Morsh and Mohr. Mos of he daa poins are bounded by hese wo values of k. Similar resul was repored for differen shear es mehods in a previous experimenal sudy [8] K Compressive srengh N/mm 2 ) Fig.5 Coefficien K P Morsh formula Mohr formula Pase NS LS NSNG NSLG LSNG LSLG 5. FINITE ELEMENT SIMULATION Failure process of concree under increasing shear load would be well undersood if we could measure sress disribuion of he esed specimen. However such a measuremen is exremely difficul. FE analysis is anoher approach o simulae his process. In his sudy hree-dimensional nonlinear FE analysis has been implemened for esed specimens. Concree behaviour and crack developmen are described using he roaing crack model [9]. In his model he orienaion of crack is aligned o he principal sress direcion, only normal sress occurred on he crack surfaces, and he crack experienced pure mode I. Therefore, no addiional hypoheses are necessary o characerize he shear behaviour, which is generally difficul o model. However, like any oher smeared crack model where cracks are idealized as being disribued or smeared over he whole elemen, he roaing crack canno describe exac size and locaion of cracks. There is also no such definiion like crack widh in his model. Only a region or elemens where cracks occurred can be deeced. 5.1 Analyical Model Fig.6 shows hree-dimensional model of he double shear es. Concree specimen is modeled using 3D solid 8 nodes isoparameric elemen having dimension of mm. Upper par of loading device and suppors are modeled using rigid elemen. Acual boundary condiions of he model are quie complicaed. Fricion beween upper and lower Fig.6 Analyical model shearing edges (see Phoo 1) depends on he level of verical loading. However he conac area is comparaively small herefore his fricion effec is ignored in he analysis. Shearing edges have common nodes wih esed block. Laeral expansion of specimen is allowed by modeling suppor edges as rollers in horizonal direcion or x direcion in Fig.6. A forced displacemen is applied in he middle poin of he upper par and he push-over analysis is implemened unil he force P a he same poin decreased 95% compared o is maximum value. The analysis made use of FINAL program developed by he Obayashi Corporaion. 5.2 Consiuive model and failure crieria Concree in he experimen has undergone compression wih almos no horizonal confinemen. For such a low level of confinemen, a modified Ahmad riaxial sress-srain relaionship proposed by Naganuma [10] is promising because i showed a good agreemen wih experimenal resuls. This sress-srain model is seleced for consiuive model of concree. The Oosen four parameers concree failure crierion is used in his sudy. According o Naganuma [10] his crierion also well corresponds wih experimen daa of low confined concree under riaxial sress sae. I is already known ha he cracked concree can sill carry some ensile sress in he direcion normal o he crack. The phenomenon is characerized by concree inrinsic propery called

5 fracure energy, which is an area under ensile sresssrain curve. In his sudy, a model proposed by Izumo [11] was used for concree sress-srain relaionship in ension, which can be wrien as follow: σ f ε = ε cr c Load (kn) (3) Where, f : Tensile srengh under biaxial loading; ε :Average ensile srain in he direcion perpendicular o crack; ε ck :Average ensile srain when crack occurred; c: parameer represening condiion of bond beween rebars and concree (c=1.0 for he case of plain concree was used in his sudy) Firs crack Displacemen (mm) Fig.7 Analyical load-displacemen relaionship of NSNG Failure process To invesigae he failure process of esed specimens a represenaive case of analysis is shown hereafer. NSNG-3 is seleced for his purpose. This is a normal concree series of specimens wih he following properies obained from experimen: Compressive srengh: N/mm 2 Spliing ensile srengh:4.06 N/mm 2 Young s modulus : kn/mm 2 Poisson raio: 0.18 Fig.7 shows analyical relaionship of load P a he middle node of he upper par of he loading device (see Fig.6) and displacemen of he same node. Load-displacemen relaionship is linear up o 60% of is maximum load value hen he firs crack occurred in he middle of boom face of he specimen. The firs crack was iniiaed by ensile sress concenraed in he middle of specimen. Fig.8a shows disribuion of principal sress vecors when he firs crack has occurred. Compressive load from upper shearing edge ransferred o lower shearing edge hrough a compression core. A large compressive sress exiss near shearing edges. However roughly beween suppors here was a flow of pure ensile sress causing verical bending crack as can be seen in Fig.8b (compressive and ensile sress vecors are no in he same scale). The double shear es is obviously influenced by bending sress. The lengh of he bending crack however did no increase bu confined wihin 2cm. The nex crack occurred along he sheared secion in he side faces(y=0mm and Y=100mm) of specimen and closed o he suppors. This crack iniiaed also by he ensile sress concenraed along he sheared secion and grew from he boom o he op of he specimen. The crack however developed under mix mode of failure. Fig.9b shows principal srain disribuion a he peak load. Those elemens wih large horizonal srain (ensile) are places where crack occurred due o mosly ensile sress.however close o shearing edges, especially upper edge, are Y=50 mm a) Compressive b) Tensile Fig.8 Principal sresses a he firs crack a) ZX-Shear sress b) Principal srain Fig.9 Mixed mode failure a peak load Phoo 2. Damaged specimen places where ensile srain is no large. In fac in hese region crack occurred under shear sress. Fig.9a shows disribuion of shear sresses in he same plane of side face of specimen. Shear sress did no uniformly disribue along he shear secion

6 bu concenraed close o shearing edges causing crack a he final momen of failure. Maximum shear sress was 7.91N/mm 2 which is 1.8 ime of he spli ensile srengh of he specimen. Acual damaged of esed specimen can be seen in Phoo Average shear srengh Analyical average shear srenghs obained by dividing maximum load by wice he area of shear secion are ploed ogeher wih linear regression line of experimen daa in Fig.10. The solid line, he same line for morar and concree in Fig.4, is ploed here again for convenience. From he figure i is clear ha disribuion of shear srenghs obained from FE analysis disribued quie closely o experimenal line. However analyical resuls showed lower srenghs compared o experimen one. This is probably due o difference beween analyical and acual specimen s boundary condiions. If he fricion effec a he suppors can be aken ino accoun hen consrain of laeral expansion could be higher which could give raise o he srengh of specimen. Also acual experimen could be less influenced by bending sress because of he higher acual laeral boundary confinemen. 6. CONCLUSIONS In his paper hree dimensional nonlinear finie elemen analysis is carried ou for concree and morar failing in double shear experimen. Following can be concluded: (1) Concree and morar has he same linear relaionship beween shear and compressive srenghs which is slighly differen from ha of pase maerial. For low compressive srengh pase has higher shear srengh compared o concree bu for higher compressive srengh opposie endency has been found. (2) FE simulaion showed ha maerial in double shear es failed due o ensile and shear sresses. The es was influenced by bending sress causing crack in he middle of specimen. (3) Shear srenghs obained from analysis are closely approximae acual experimenal srenghs. Analyical srenghs are slighly lower han ha of experimen giving conservaive evaluaion of shear srengh. ACKNOWLEDGEMENT The auhors acknowledge he suppors of Mr. Yamasaki for consan help and suppor during he experimen. REFERENCES [1] H. Tanaka, I. Yoshiake, Y. Yamaguchi and S. Shear srengh N/mm 2 ) Analyical Experimen Compressive srengh N/mm 2 ) Fig.10 Comparision bewwen analyical and experimenal shear srenghs Hamada: Experimenal Sudy on he Shear Srengh of Concree Elemens Subjeced o Pure Shear Sress, Journal of Maerial, Concree Srucure and Pavemen, Vol.746, pp , [2] Z.P. Bazan and P.A Pleiffer: Shear Fracure Tess of Concree, Maerial and Srucure (RILEM) Vol.19, pp , [3] A. Hillerborg, M. Modeer, P.E. Peersson: Analysis of Crack Formaion and Crack Growh in Concree by Means of Fracure Mechanics and Finie Elemen Analysis, Cemen and Concree Research, Vol.6, pp , [4] H.N. Tuan, H. Osuka, Y. Ishikawa and E. Takeshia: A Sudy on Shear Srengh of Concree Under Direc Shear Tes, JCI Annual Convenion Proceeding, Vol.28, No.1, 2006 [5] K. Uomoo and T. Minemasu: Fundamenal Sudy on Tesing Mehod for Shear Srengh of Concree, Concree Engineering, Vol.23, No.3, pp.17-18, March [6] M.P. Luong: Fracure Tesing of Concree and Rock Maerials, Nuclear Engineering and Design, Vol.133, pp.83-95, May [7] G. Everling: Commen up on he Definiion of Shear Srengh, Inernaional Journal of Mechanics, Mining science, Vol.1, pp , [8] Y. Azuma and K. Iso: Proposal of Shear Tesing Mehod of Concree Under Mixed Mode Loading, Concree Journal, Vol.23, No.3, pp March [9] R.J. Cope, P.V. Pao, L.A. Clark and P. Norris: Modeling of Reinforced Concree Behaviour for Finie Elemen Analysis of Bridge Slabs, Numerical Mehods for Nonlinear Problem I, pp , [10] K. Naganuma: Sress-Srain Relaionship for Concree under Triaxial Compression, Journal of Consrucion Engineering, AIJ, No.474, pp , Augus [11] Izumo, J.: Analyical Model for RC Plae Elemen Subjeced o In plane loading. Journal of Concree Research and Technology, JCI, Vol.25, No.9. pp