Shear failure of plain concrete in strain localized area

Transcription

1 Shear failure of plain onree in srain loalized area Y. Kaneko & H. Mihashi Tohoku Universiy, Sendai, Miyagi, Japan S. Ishihara Asanuma Corporaion, Takasuki, Osaka, Japan ABSTRACT: The objeive of his paper is o sudy he shear failure of plain onree in srain loalized area. An experimenal sudy on plain onree subjeed o shear was arried ou. The shape of speimen was a horizonally double-nohed onree blok. In order o sudy he shear sofening haraerisis of plain onree, a mehanial model for he marosopi shear failure is applied o he experimens, fousing on he enire load-displaemen relaion. The mehod makes use of roaing smeared rak onep and russ model, ombined in a simple model. The analysis employs he developmen of muliple diagonal raks and marosopi fiiious shear rak propagaion. The model is found in good agreemen wih he experimens. The analyial resuls poin ou ha he shear sofening haraerisis depend on he size of srain loalized area. Keywords: Dire shear es, Shear sofening, Fiiious shear rak, Srain loalizaion 1 INTRODUCTION The researh on raking behavior of onree has been largely progressed afer he proposal of Fiiious Crak Model (FCM) by Hillerborg e al. (1976). The model desribes he mode I fraure behavior a he raking proess zone by means of ension sofening urve, whih is he funion of ensile srengh and fraure energy. The ension sofening urve is he relaion beween he ohesive fore along he fiiious rak and he rak widh. Afer his proposal, many experimenal and analyial researh works on mode I fraure of onree have been arried ou and i beomes possible o obain experimenally he fraure parameer suh as fraure energy and ension sofening urve. I has been known ha he aual fraure mode observed in onree sruures is omplex behavior assoiaed wih sruural sysem, loading and boundary ondiions and so on. Therefore, i is neessary o omprehend he physial behavior on mixed mode fraure ombined wih mode I and II, o develop he mehanial model o desribe he behavior, and o obain he mehanial parameers o express quaniaively he model based on he sandard es. However, mos of evaluaion mehods on shear fraure behavior of onree sruures are based on experimens, and he heoreial approah suh as a limi analysis does no give he suffiien soluion regarding deformaion behavior. In addiion, here are a few researhes on shear sofening haraerisis and he generalized definiion on shear sofening is no onfirmed. For his ehnial bakground, he objeive of his researh is o expand he onep of so-alled fiiious rak model o mode II shear fraure, o develop a mehanial model and o propose finally a simple es mehod wih whih he neessary mehanial parameers are idenified for modeling. Auhors have onsrued he shear fraure sequene based on he fraure proess observed in he experimens of a dire shear es, developed a mehanial model and idenified he orrelaion beween shear sofening behavior and aggregae ineraion behavior (Kaneko e al. 1). The mehodology of his mehanial model has been applied o he shear-off failure of plain and fiber reinfored onree shear key joins (Kaneko e al. 1993ab, Kaneko 1993) and he shear failure of reinfored onree membrane elemens (Kaneko 1/8

2 1998) in whih he fore-displaemen haraerisis were predied appropriaely. Based on he researh ahievemen, a mehanial model is applied o he dire shear es (Ishihara e al. 3) fousing on he enire deformaion behavior of onree in his paper. EXPERIMENTAL WORK As shown in Figure 1, he dire shear es was ondued wih he parameers of he nohdisane and wih join or wihou join o obain he pos-peak haraerisis and he deformaion behavior a he loalized shear failure area. In he es, he raio of noh-disane (a) o he speimenheigh (D) was se as.1 or.17, and he fraure behavior a he shear failure area was observed by means of a mirosope. In he speimens, he horizonal wedge-ype noh wih he maximum opening displaemen of 5 mm was insalled a upper and lower posiions, and wo ypes of speimens were adoped suh as he noh disane of 6 mm (a/d=.1, UJ6 and JR6 series) and 1mm (a/d=.17, UJ1 series). Here, UJ sands for he unjoined speimens and JR for he joined ones wih surfae roughness of. mm. The ompressive srengh f' and spliing ensile srengh f of onree obained wih he same mixing proporion are summarized as follows: onree C1 (f' =3.4MPa, f =.47MPa) for UJ series; onree C1 and onree C (f' =35.5MPa and f =3.MPa) for JR series. The maximum aggregae size was 15 mm for relaively narrow noh-disane. The basi suppor (loading poin) ondiion was he pin-suppor a he upper and he fixed one a he lower. The speimens of UJ6-8 and 9 were suppored a boh pin-suppors. In UJ6 series, he verial and horizonal displaemens beween A1-bol and A-bol were measured a boh fronage and reverse, as shown in Figure1. Regarding he fraure behavior, he following phenomena were observed. In he speimens of UJ6-1, and 3, he flexural raks iniiaed a boh upper and lower nohes. Eah flexural rak propagaed o he opposie noh ip and finally reahed o i. In he speimens of UJ6-4, 5, 6 and 7, only one flexural rak iniiaed and reahed o he opposie noh ip. On he oher hand, in he speimens of UJ6-8 and 9, he flexural raks firs iniiaed and propagaed similar o he speimens of UJ6-1, and 3. Subsequenly, he shear raks were observed near he ener of shear plane beween boh nohes wih he mirosope. Speifially, in he speimen of UJ6-8, he shear raks dominaed he failure mehanism and he ompression srus beween boh nohes finally rushed as shown in Figure. In he speimen of UJ6-9, he shear raks sopped propagaing and a flexural rak propagaed o he opposie noh ip and finally reahed o i. In he speimens of UJ1-1 and, only one flexural rak firs iniiaed and propagaed o he opposie noh ip. Subsequenly, he shear raks were observed a he shear plane beween boh nohes wih he mirosope. Finally, he shear raks dominaed he failure mehanism and he ompression srus beween boh nohes rushed. In he speimens of UJ1-3, he similar behavior was observed o he speimens of UJ1-1 and. However, he final failure was aused by no shear raks beween boh nohes bu flexural raks ha reahed he noh ip. Figure 1. Speimens and Loading Sysem of UJ 6 (Fronage) Flexural Crak Shear Crak Shear Fraure Figure. Craking Paerns of Speimen UJ6-8 (Fronage) /8

3 3 SHEAR FRACTURE MODEL 3.1 Modeling for Srain Loalized Area The damage area (widh: W da ) assoiaed wih shear failure is modeled by a single fiiious marosopi shear rak as shown Figure 3(a) and (b). The raking behavior is modeled based on he fraure sequene defined in he previous researh (e.g., Kaneko e al. 1). The following simple fraure sequene of diagonal muliple rak is marosopially onsrued and is shemaially shown in Figure 3(). 1. A he damaged area, diagonal muliple raks iniiae along he prinipal sress axis, and finally he shear fraure zone is formulaed as disribued raks. The diagonal muliple raks are assumed evenly disribued along he shearfraured zone wih a erain angle of inlinaion.. Wih furher shear loading, he diagonal muliple raks are assumed o roae following he prinipal sress axis under mode I ondiion. 3. The ensile srain of raks and he ompressive srain of srus beween eah rak and he nex inrease oninuously, and he marosopi shear sofening sars assoiaed wih he rushing failure of ompression srus. Speifially, diagonal muliple raks are oalesed ino he marosopi fiiious shear rak aused by he highly loalized srain disribuion. In order o evaluae a quaniy of energy, he srain a he damaged area is ranslaed o boh he shear slip displaemen and he shear rak opening displaemen of marosopi fiiious shear rak. In he presen approah, he fraure proess of diagonal muliple raks is modeled by means of a ombinaion of a roaing smeared rak onep and a russ model. The inen of his approah is based on he fa ha i is ofen desirable o find general analyial soluions, whih are muh easier o handle han numerial soluions produed by nonlinear FEM analysis. 3. Roaing Smeared Crak Model The presen mehanial model saisfies hree basi requiremens: equilibrium, ompaibiliy and maerial onsiuive laws. Sress ransformaion ondiions (equilibrium) in a raked elemen a he damaged area are formulaed based on he works of Vehio & Collins (1986) and Hsu e al. (1987). In his modeling, sress and srain are assumed uniformly disribued as averaged ones over he enire damaged area. Afer diagonal raking ours, a series of diagonal ompression srus is formed in he ompression direion (-direion). The elemen akes only ompressive sress σ in he -direion of ompression srus and only ensile sress σ in he ension direion (-direion) ransverse o ompression srus. Shear sress τ along he raked elemen is assumed zero. Thus, σ and σ are always prinipal sresses of his sysem. The angle beween he x-y and - oordinae sysems is designaed as θ as shown in Figure 4. This angle is also he angle of inlinaion of ompression srus wih respe o he x-axis. The averaged sresses and srains of onree elemen in he wo oordinae sysems, x-y and -, are ransformed aording o he following equaions. σx = σos θ + σsin θ (1a) σ y = σsin θ + σ os θ (1b) τ = ( σ σ )sinθ osθ (1) xy h P δ x δ xy σ x τ xy (1) Crak Iniiaion y () Crak Roaion (3) Crushing of Sru Compression Sru Fiiious Shear Crak d W da Fiiious Shear Crak h P (a) Damaged Area Winged Crak Roaing Crak (b) Modeling of Crak W da x () Fraure Sequene Figure 3. Modeling for Shear Fraure a Damaged Area 3/8

4 ε = ε os θ + ε sin θ (1d) x ε = ε sin θ + ε os θ (1e) y ( ) γ = ε ε sinθ osθ (1f) xy σ y σ σ y τ xy θ θ σ x x - oordinae x -y oordinae Figure 4. Sress Transformaion Sysem where E = Young's modulus; f = ensile srengh of onree; ε r = raking srain; ε u1 and ε u = pos-raking haraerisi srains; and G F = fraure energy. h is he inerval beween eah diagonal rak and he nex as shown in Figure 3 (Kaneko e al. 1). The assumed ompressive sress-srain relaion in he direion of ompression srus is onsrued based on he works of Soroushian e al. (1986) and Hognesad (1951) as shown in Figure 5(b). In addiion, sofening of onree srus relaed o ensile srain in he direion perpendiular o srus is onsidered based on he work of Vehio & Collins (1986). Thus, assumed sress-srain relaions are desribed by he following equaions (Kaneko e al. 1). For he onree elemen beween he uniformly disribued diagonal raks, whih roae along he prinipal sress axis, he following onsiuive laws are applied. The assumed ensile sress-srain relaion of plain onree in he direion perpendiular o he ompression srus is formulaed by he following equaions. The bilinear ensile sress-deformaion relaion originally proposed by Hillerborg (1985) as a ension sofening onep is adoped for he desending branh based on he rak band heory (Bazan and Oh 1983) as shown in Figure 5(a). σ = Eε : ε εr (a) εr ε εu1 3 3 σ = f : εr < ε εu1 ( εu1 εr ) (b) f( εu ε) σ = : εu 1 < ε εu 3 ε ε () ( ) f u u1 ε r = (d) E f ε ε σ = : ε ε λ ε ε (3a) f σ = 1 Z ( ε εo ) : εo < ε εu1 λ (3b). f σ = : εu1 < ε λ (3) ε λ = ε 1. (3d) f ε = E (3e).8 εu1 = + ε Z (3f).5 Z = (3g) ε f ε 145 f 1 where f' = ompressive srengh; ε = assoiaed srain; and λ = oeffiien o ake are of he sofening phenomena. ε ε u1 u 4GF = εr + (e) 5 fh 18GF = εr + (f) 5 fh a) σ 3 ε r ε u1 ε u ε b) σ f' λ ε o ε u1. f' / λ ε W h = da (g) 5 Figure 5. Sress-srain Relaionship (a)tension; (b)compression Young's modulus of onree was esimaed by he following relaion (Chen 198, Kaneko e al. 4/8

5 1). The fraure energy was esimaed as.1 N/mm, a value ofen used by many researhers (e.g., Ros & Blaauwendraad 1989, Balakrishnan & Murray 1988) f E = (4).8 In he presen modeling, a prinipal srain raio ν a (=ε /ε, or apparen Poisson's raio) is onsrued o evaluae simply a omplex fraure sequene governed by a ensile srain. I is assumed ha he relaion beween he ensile srain and ompressive srain of a ompression sru are relaed by he raio ν a defined as: νa = ε / ε =. : ε (5a) νa =.5 : ε < ε (model 1) (5b) ν =. : ε < ε (model ) (5) a The raio ν a is a salien feaure in he presen mehanial model and is defined as ε /ε for he ase in whih he ensile srain onrols he deformaion of he sruure. This is beause he oalesene of diagonal raks ould be ahieved by high srain loalizaion beween eah diagonal rak and he nex. The aim of his model is o eliminae a numerial ieraion in he alulaion of he sress and srain in boh ension and ompression (see he deail in Kaneko e al. 1993ab, Kaneko e al. 1). Speifially, he ompressive srain (ε ) is alulaed by Equaion (5) for monoonially inreasing ensile srain (ε ). The ensile and ompressive sresses an be hen alulaed by subsiuing he known values of ensile and ompressive srain ino eah onsiuive model wihou numerial ieraion. The raio ν a was formulaed based on sruural experimens (Vehio & Collins 1986, Mansure & Ong 1991). The model 1 gives a mean value of saered experimenal daa and he model is defined as a onsan value of ν a wihou seep inrease of ompressive srain. In he analysis for deep beams (Kaneko & Mihashi ), he model gave sable onverged soluions assoiaed wih he seep drop afer he maximum load. Therefore, in his paper, he model is adoped o sudy he pospeak haraerisis. The load P, he shear sliding displaemen (δ xy ) and he rak opening displaemen (δ x ) orhogonal o shear plane a he marosopi fiiious shear rak are alulaed for he speimens wih he widh (b) by he following equaions. P = τ bd (6a) xy δ = γ W (6b) xy xy da δ = ε W (6) x x da Using he prinipal srain raio (ν a =ε /ε ) and he speified onfined sress (σ x =.), he preeding 11unknowns (σ x, σ y, τ xy, ε x, ε y, γ xy, σ, σ, ε, ε, and θ) are redued o 9. By seleing one of hem (ε ) as a known value, he remaining 8 unknowns an be obained from a se of 8 equilibrium, ompaibiliy and onsiuive equaions. Hene, one an develop he relaion beween he average shear sress τ xy and he average shear srain γ xy by he following seps: a) Sele a value of ε ; b) Assume ν a =. (model ); ) Calulae ε from ν a =ε /ε ; d) Calulae σ, σ and λ from Equaions () and (3); e) Calulae θ from Equaion (1a) wih speified σ x (=.); f) Calulae τ xy, γ xy and ε x from Equaion (1); g) Calulae he parameers assoiaed wih load and displaemen from Equaion (6). 4 VERIFICATION STUDY 4.1 Load-displaemen relaion The appliabiliy of he mehanial model is mainly examined wih hree unjoined speimens of UJ6-8, UJ1-1 and UJ1-, whih showed obvious shear failure beween wo nohes. Figure 6(a) shows he omparison of load-displaemen relaion beween experimen and analysis. In he analysis, he damaged area widh (W da ) of 15 mm observed in he experimen of UJ6-8 was adoped. I is realized ha he prediion is in relaively good agreemen wih he experimenal resuls for he enire range of loading onsising of he pos-peak region. Speifially, in he speimen of UJ6-8, he siffness in he experimen redues around he loading level of kn. This is aused by he flexural raks near he noh ip, whih is no onsidered in he mehanial model. In he speimen of UJ1-, good agreemen beween he experimen and he analysis is observed, exep ha he experimenal resul keeps he loading level awhile afer he peak load. This is 5/8

6 aused by he roaion of speimen due o he onesided flexural rak. Figures 6(b) and () show he omparison of load-displaemen relaion beween he experimen and he analysis employing several widhs (W da ) of damaged area. I is lear ha he larger widh (W da ) gives he larger pos-peak duiliy wih a sligh reduion of he peak load. Figure 6(d) shows he omparison of loaddisplaemen relaion beween he experimen for joined speimens and he analysis employing W da =15mm and he lower value of f' in he joined speimens (onree C1). The prediions are in good agreemen wih he experimenal resuls as well as he analyses for unjoined speimens, exep he speimen of JR6-1, whih gave exremely low peak load. I was observed ha he raking sequene of JR6-1 deviaed from he join-plane, whih was ompleely differen from he oher joined speimens. Thus, i is larified ha here exiss he srain loalized area even in he joined speimens wih suffiienly roughened surfae as well as unjoined speimens of UJ6-8, UJ1-1 and UJ1-. Figure 7(a) shows he omparison of shear sofening haraerisis and dissipaed energy a he damaged area employing several widhs (W da ) of damaged area. The dissipaed energy is defined here as he area under he shear sress-shear displaemen urve up o he onsidered shear displaemen. I is lear ha he larger widh (W da ) gives he larger pos-peak duiliy and larger dissipaed energy wih a sligh reduion of he peak sress. Figure 7(b) shows he sress-ensile srain urves (a) 6 Load P(kN) Tes-UJ6-8 Tes-UJ1-1 Tes-UJ1- Prediion for UJ6-8 (Wda=15mm) Prediion for UJ1 (Wda=15mm) Shear Dsiplaemen δxy(mm) (b) 4 Load P(kN) Tes-UJ6-8 Prediion (Wda=15mm) 5 Prediion (Wda=5mm) Prediion (Wda=1mm) Prediion (Wda=mm) Shear Dsiplaemen δxy(mm) () 6 Load P(kN) Tes-UJ1-1 Tes-UJ1-1 Prediion (Wda=15mm) Prediion (Wda=5mm) Prediion (Wda=1mm) Prediion (Wda=mm) Shear Dsiplaemen δxy(mm) (d) 4 Figure 6. Comparison of Load-Displaemen Curves beween Experimen and Analysis Load P(kN) Tes-JR6-1 Tes-JR6- Tes-JR6-4 Tes-JR6-5 Tes-JR6-6 Prediion (Wda=15mm) Shear Dsiplaemen δxy(mm) (a) 6 Shear Sress τxy(mpa) Sress (Wda=15mm) Sress (Wda=5mm) Sress (Wda=1mm) Sress (Wda=mm) Energy (Wda=15mm) Energy (Wda=5mm) Energy (Wda=1mm) Energy (Wda=mm) Shear Dsiplaemen δxy(mm) Dissipaed Energy (N/mm).5 (b) 1 Sress (MPa) Tensile Srain ε σ τ xy σ Prediion (Wda=15mm) Figure 7. Shear Sofening Charaerisis and Consiuive Laws (a) 1.5 Shear Srengh in Tes / Predied Shear Srengh 1.5 Speimens wihou Join +1% -1% UJ6-8, UJ1-1, UJ1- UJ6-1 o UJ6-7 UJ6-9, UJ1-3 +1%, -1% Compressive Srengh (MPa) (b) 1.5 Shear Srengh in Tes / Predied Shear Srengh 1.5 Speimens wih Join JR6 Series +1% -1% +% -% +1% -1% +% -% Compressive Srengh (MPa) Figure 8. Comparison of Shear Srengh 6/8

7 obained in he analysis. I is realized ha he ompressive sress σ in a ompression sru beomes highly lose o he peak sress and he sress poin in he ensile onsiuive law is on he firs sofening region when he shear sress τ xy reahes he peak sress. This numerial phenomenon is onsidered in he formulaion of shear srengh in he nex seion. 4. Formula for Shear Srengh In order o verify he proposed mehanial model alernaively, he formula for he shear srengh is developed based on he assumpion: he ompressive sress σ in a ompression sru beomes equal o he peak and he sress poin in he ensile onsiuive law is on he firs sofening region when he shear sress τ xy reahes he peak sress (see Figure 7(b)). Here, he values of σ, f', ε, ε o are onsidered negaive sine he ensile sress and srain are defined posiive and he ompressive sress and srain are negaive. Subsiuing Equaions. (5a) and (3d) wih x = - ε /ε o (>) ino Equaion (3a), one an obain he following equaion. σ + σ x f x+ f x = (7) The ondiions for x o give he maximum (loal minimum) of σ are as follows: x r r ε = = (1) ε Subsiuing Equaion (1) ino Equaion (1), one an see ha he ondiion wih Equaion (8b) is saisfied as follows: d σ.8 f = >. dx (13) Subsiuing Equaion (1) ino Equaion (7), one an obain he maximum (loal minimum) of σ as follows: σ max =.46 f (14) Subsiuing Equaions (3e) and (1) ino Equaion (b), one an obain he ensile sress σ (= σ r ) assoiaed wih σ max as follows: 5hf σ = + + (15) r f 6EG F f f ( 6 ) Subsiuing he ondiion of σ x =. ino Equaion (1a), one an obain θ (=θ r ) assoiaed wih σ max as follows: dσ d σ =., >. (8a,8b) dx dx r θ = os 1 σ σ r r max σ (16) Differeniae Equaion (7) wih respe o x, one an obain he following equaions. dσ dσ σ +.34x.4 f +.8 f x=. dx dx (9) d.8 σ d d +.34 σ +.34x σ +.8 f. = dx dx dx (1) Subsiuing Equaions (7) and (8a) ino Equaion (9), one an obain he following equaion. + = (11).136 fx.64 fx.3 f. Then, x (=x r ) assoiaed wih he maximum (loal minimum) of σ is alulaed as follows: Subsiuing Equaions (14)-(16) ino Equaion (1), one an formulae he shear srengh by he following equaion. Here, f' is onsidered negaive. r ( σ f ).46 max r τxy = sin θ (17) A omparison of experimenal daa wih he prediions by Equaion (17) for every unjoined and joined speimens are shown in Figures 8(a) and (b), along wih a ±1 % and ±% error ranges. In he ase of unjoined speimens, he agreemen beween he measured and alulaed shear srengh is indeed good wihin ±1 % error range. In he ase of joined speimens, he prediions give slighly larger deviaion from experimenal resuls han unjoined speimens and are almos wihin± % error range. Here, i should be noed ha he presen formula for shear srengh is appliable o he geomeri and 7/8

8 loading onfiguraions of he speimens presened in his paper. In order o apply he formula o oher onfiguraions, furher numerial sudy may be neessary. 5 CONCLUSION In his paper, an experimenal sudy on plain onree subjeed o shear was arried ou. In order o sudy he shear sofening haraerisis of plain onree, a mehanial model for he marosopi shear failure is applied o he experimen, fousing on he enire loaddisplaemen relaion. From his sudy, he following onlusions an be drawn. 1) The analysis employing he proposed mehanial model agrees well wih he experimenal resuls on load-displaemen urves onsising of he pos-peak region. Furhermore, a formula for shear srengh is developed and he prediion wih he formula is found in good agreemen wih he experimenal daa. ) The shear sofening haraerisis depend on he size of srain loalized area. Speifially, he larger widh of srain loalized area gives he larger pos-peak duiliy and larger dissipaed energy wih a sligh reduion of he peak sress. Fuure work mus be direed a furher verifiaion sudies wih experimenal observaions and alernaive analyial sudies onsising of several onsiuive models in order o generalize he analyial onlusion idenified in his paper. 6 PREFERENCES Balakrishnan, S., & Murray, D Conree Consiuive Model for NLFE Analysis of Sruures, Journal of Sruural Engineering, ASCE, Vol.114(7). Bazan, Z.P. & Oh, B.H Crak Band Theory for Fraure of Conree, Maerials and Sruures, RILEM, Vol. 16, pp Chen, W.F Plasiiy in Reinfored Conree, MGraw- Hill Book Company. Hillerborg, A., Modeer, M. & Peersson, P.E Analysis of Crak Formaion and Crak Growh in Conree by Means of Fraure Mehanis and Finie Elemens, Cemen and Conree Researh, Vol.6, No.6, pp Hillerborg, A Numerial Mehods o Simulae Sofening and Fraure of Conree in Fraure Mehanis of Conree: Sruural Appliaion and Numerial Calulaion (ed. Sih, G. C. and DiTommaso, A.), Marinus Nijhoff Publishers, pp Hognesad, E A Sud of Combined Bending and Axial Load in Reinfored Conree Members, Universiy of Illinois Engineering Experimenal Saion, Bullein Series No.399, 18 p., November. Hsu, T. T. C., Mau, S. T. & Chen, B Theory of Shear Transfer Srengh of Reinfored Conree, ACI Sru. J., Vol. 84, No., Mar.-Apr., pp Ishihara, S, Mihashi, H., Kaneko, Y., Mori, K. & Uhii, E. 3. Experimenal Sudy of Nohed Conree Blok Subjeed o Shear - Sudy using Miromehanis Approah -, Journal of Sruural and Consruion Engineering, Arhieural Insiue of JAPAN (Transaions of AIJ), No. 57, pp , Aug. (in Japanese). Kaneko, Y., e al. 1993a. Fraure Mehanis Approah for he Failure of Conree Shear Key: Theory, J. of Engineering Mehanis, Vol. 119, No. 4, ASCE, pp.681-7, April. Kaneko, Y., e al. 1993b. Fraure Mehanis Approah for he Failure of Conree Shear Key: Verifiaion, J. of Engineering Mehanis, Vol. 119, No. 4, ASCE, pp , April. Kaneko, Y Fraure Mehanis based Modelling for Failure of Conree Shear Key and Appliaion o Design of Segmenal Sruure, Conree Researh and Tehnology, Vol. 4, No., pp.31-41, July (in Japanese). Kaneko, Y Prediion for Marosopi Shear Failure of Boh Reinfored Conree Membrane Elemens and Reinfored Conree Deep Beams in erms of Load- Displaemen Charaerisis, Conree Researh and Tehnology, Vol. 9, No., pp.43-51, July (in Japanese). Kaneko, Y., Mihashi, H. & Ishihara, S. 1. ENTIRE LOAD- DISPLACEMENT CHARACTERISTICS FOR DIRECT SHEAR FAILURE OF CONCRETE, Modeling of Inelasi Behavior of RC Sruures under Seismi Loads, Commiee Repor, Amerian Soiey of Civil Engineers, pp Kaneko, Y. & Mihashi, H.. Shear Sofening Charaerisis of Reinfored Conree Deep Beams, Journal of Sruural and Consruion Engineering, Arhieural Insiue of JAPAN (Transaions of AIJ), No. 56, pp.115-1, Aug. (in Japanese). Mansur, M.A. & Ong, K.C.G Behavior of Reinfored Fiber Conree Deep Beams in Shear, ACI Sru. J., Vol. 88, No. 1, Jan.-Feb., pp Ros, J. G. & Blaauwendraad, J Crak Models for Conree: Disree or Smeared? Fixed, Muli-direional or Roaing?, HERON, Vol. 34, No. 1. Soroushian, P., Choi, K. & Alhamad, A Dynami Consiuive Behavior of Conree, ACI Journal, Vol. 83, No., pp Vehio, F.J. & Collins, M.P The Modified Compression-field Theory for Reinfored Conree Elemens Subjeed o Shear, ACI Journal, Vol. 83, No., pp /8