DYNAMIC TEST AND ANALYSIS OF AN ECCENTRIC REINFORCED CONCRETE FRAME TO COLLAPSE

Transcription

1 th World Conference on Earthquake Engneerng Vancouver B.C. Canada August -6 4 Paper No. 8 DYNAMIC TEST AND ANALYSIS OF AN ECCENTRIC REINFORCED CONCRETE FRAME TO COLLAPSE Yousok KIM Toshm KABEYASAWA SUMMARY A shakng table test was conducted to nvestgate the torsonal response characterstcs of a renforced concrete frame wth asymmetrc plan consstng of a shear wall and ndependent columns and the shear collapse mechansm of the columns desgned n accordance to the 97s Japanese desgn practce resultng vulnerable to shear falures. The specmen was subjected to fve dfferent earthquake base motons scaled approprately so that the response wll be from elastc to nelastc and fnally to collapse. The observed dsplacement responses whch were consderably dfferent between the stff and fleble frame were presented and the torsonal response rato magnfed from lnear to nonlnear range were also descrbed. In the fnal run the process of shear strength deteroraton of the ndependent columns was llustrated wth observed responses such as forces dsplacements and transverse renforcement stran and fnally showed shear falure followed by aal load collapse. A macro model for RC column was proposed to smulate the epermental results whch s based on the plane stran-plane stress state and smeared rotatng crack approach. The salent features of the proposed model are the capablty of consderng strength deteroratng effect resultng from the softenng behavor n concrete consttutve law and the bendng shear and aal force nteracton formulated from the stress resultants. The nonlnear analyss algorthm usng two teratve schemes s llustrated both of whch are contnued tll the force and the stress equlbrum condton s satsfed. Fnally the analyss result of the proposed model was verfed through comparson wth the observed response by whch the strength softenng effect on the shear collapse process was clarfed. The lmtaton of the model and the future research needs are also dscussed. INTRODUCTION Among the characterstcs of structures that have suffered severe damage or collapse durng past earthquakes the tems to be nvestgated n ths epermental and analytcal study are as follows: the Graduate Student Department of Archtecture Graduate School of Engneerng The Unversty of Tokyo Tokyo Japan. Emal:yskm@er.u-tokyo.ac.jp Professor Earthquake Research Insttute The Unversty of Tokyo Tokyo Japan. Emal:kabe@er.utokyo.ac.jp

2 lack of shear strength n RC columns desgned followng 97 s Japanese renforcement detal practce whch lead to shear falure and the loss of aal load carryng capacty () asymmetrc plan system composed of ndependent column frame and wall frame whch nduce consderable stffness and strength eccentrcty and hence concentrate damage on weak frame. The objectves of ths epermental study therefore are to understand the collapse process of columns wth poor shear capacty and to assess the nfluence of stffness and strength eccentrctes on elastc and nelastc earthquake responses. Analytcal study s performed for the frst objectve of ths study. Several nonlnear analytcal models have been proposed for renforced concrete column takng nto account not only comple nteracton effects but also strength degradaton. These models can be categorzed nto two felds n terms of the hysteretc model used for representng the deteroratng characterstcs although hybrd model combnng two hysteretc models was also proposed. One s the member hysteretc model based on the force-deformaton relaton constructed from the member-level epermental studes whch usually adopted n a lumped plastcty model. Ths analytcal approach s smple and economcal but the way of defnng strength deteroraton and couplng effect between bendng and shear s formulated usng emprcal formulaton that entrely depend on the epermental data rather than based on the closed form equaton. Therefore the etensve epermental data are needed for mprovng the accuracy and relablty of the model. The other one s materal model (.e. concrete and renforcement) that s formulated n terms of stressstran relaton. These models are generally mplemented n the fnte element model whch are capable of representng the detaled local behavor and the more accurate descrpton of nelastc behavor compared to the member hysteretc model. In addton bendng shear and aal force nteracton can be ncorporated n an eplct way based on the stress-stran relaton and strength softenng effect also formulated n a relatvely ratonal manner despte ths s also based on the epermental results. It s well known fact however that fnte element model based on the materal consttutve models requres the refned mesh dvson for the accurate estmaton of the nelastc behavor n RC structures and therefore s not sutable for the nonlnear frame analyss n terms of the computaton efforts. Ths paper compromsng between accuracy and economcal requrement n analytcal model proposes the renforced concrete column model composed of only three elements representng two hnge regons and central part between them the nelastc propertes of whch are based on the two-dmensonal materal consttutve models.. TEST SPECIMEN AND EXPERIMENTAL SETUP Specmen A one-thrd scale renforced concrete specmen was tested on the shake table whch comprses a wall and a column frame n the frst story and wall frames only n the second story as shown n Fgure. Asymmetrc plan n the frst floor generate consderable stffness and strength eccentrcty amount up to.4 and.5 respectvely. The stffness eccentrcty n the frst story s gven by: ek R = ek a + b e k k l = k y Where e k s the dstance between center of stffness and mass l y s the dstance of each frame from center of mass a and b are the dmenson of plan n longtudnal and transverse drecton k s stffness

3 F F W W F F Loadng Drecton.7 Weak Frame (a) plan (b) elevaton Fgure : Test Specmen (unt : m) load cell : Accelerometer : Dsplacement transducer Fgure : Epermental setup and nstrumentaton of frame n loadng drecton whch was calculated from pushover analyss n elastc range and k s the sum of stffness of all frames n loadng drecton. Each frame s strength q was calculated usng materal propertes from concrete cylnder test and tensle test of sample bars whch can be found n Km [] and then used to calculate the strength eccentrcty followng Equaton () where e q s the eccentrc capacty of the frames to the mass and C s the base B shear coeffcent. eq q l y R = e = C () -D D4@ eq q B a + b q Fgure : Column detal 5 5 Total heght of specmen s 54mm whch s the summaton of base (mm) load cell (4mm) the frst story (8mm) W (mm) the second story (8mm) W (mm) and steel plates (8mm) (Fgure ). Two concrete masses W and W (84.6 KN ) and steel plates (48. KN ) on the specmen produced aal load stress.5 A f ( f = 8MPa) ) n the frst story column whch g c c correspond to that of s-story buldng. The frst story ndependent columns were desgned n accordance to 97 s Japanese renforcement detal practce as shown n Fgure whch are vulnerable to shear falure after fleural yeldng. Base Moton Input Plan and Instrumentaton The specmen was subjected to the seres of base moton wth selected fve levels as shown n Table. The levels of the base motons were determned on the bass of prelmnary analyss results from whch the RC specmen was epected to collapse at the stage 5 (CHI). The duraton tme of the base motons was scaled by / to satsfy the smltude law. The aal stresses and the shear coeffcents

4 corresponded appromately to those of the proto-type s-story buldng by mposng the addtonal mass (steel plates) on the specmen. Before and after the nput of base motons a whte nose nput wth low acceleraton level was run to observe the change of the natural frequency of the damaged specmens. The responses of the specmens such as acceleratons dsplacements strans n steel bars and shear and aal forces n the frst story columns were recorded n Hz samplng rate wth accelerometers ( channels) dsplacement transducers ( channels) electrcal resstance stran gages (6 channels) and load cells (4 channels) respectvely. The epermental setup and locaton of measurng nstruments are shown n Fgure. Table : Base moton nput plan Earthquake data Mamum target velocty Rato to the prototype Mamum acceleraton of prototype Mamum velocty of prototype Mamum acceleraton nput to specmen Mamum velocty nput to specmen (kne) (gal) (kne) (gal) (kne) TOH TOH ELC JMA CHI TOH : Myag-ken Ok earthquake recorded at Tohoku unversty n 978 -ELC : Imperal Valley earthquake recorded at El centro n 94 -JMA : Hyogo-Ken Nambu earthquake recorded at Japan Meteorologcal Agency n 995 -CHI : Chle earthquake n 985 TEST RESULTS Damage Process of Specmen The damage dentfcaton of the specmens was estmated wth three methods whch were observaton of cracks generated n specmen the number of yelded stran gage attached to renforcng bars and the change of natural frequency calculated from system dentfcaton method. The detals of the results are dscussed n Km []. Lateral and Torsonal Responses Fgure 4(a) shows the horzontal dsplacement responses of the wall and the column sde durng ELC7.5 whch were measured from the dsplacement transducer nstrumented between the base and the bottom of W. The horzontal dsplacement response of the column sde was much larger than that of the wall sde whch resulted from the torsonal response of the specmen wth consderable eccentrcty. The smlar responses were observed n the other nput stages although not presented here. The etent of the torsonal response n elastc and nelastc range was evaluated by the nde r ndcatng the relaton between lateral dsplacement of center and rotaton angle (Fgure 4(b)). For nstance the value of r becomes zero n case of pure torsonal mode and nfnte n case of parallel translaton mode. Namely the torsonal responses become domnant as the value of r decreases. As shown n Fgures 4(c) the nde r becomes small wth the specmen damaged by ncreased load level whch suggests that the torsonal response became more domnant n nelastc range rather than n elastc. These results may be eplaned n terms of the fact that strength eccentrcty of ths specmen governng the characterstc of torsonal response n nelastc range s so hgh that the wall sde was not yelded n spte of yeldng of columns.

5 d Dsplacem ent (m m ) 5-5 colum n wal Tm e (sec.) r (a) Dsplacement response θ d θ r (b) Defnton of r D splacem ent r=.99m r=.89m r=.85m r=.78m TOH5 ELC 7.5 JM A CHI R otaton angle (c) Relaton between lateral dsplacement and rotaton angle Fgure 4 : Torsonal response Shear Force Dstrbuton shear force (KN ) - Wall colum n TOH.5 ELC7.5 CHI rato (%) rato (%) (a) column vs. wall (b) rato of column to base h Fgure 5 : Shear force dstrbuton carred by columns and wall 5 5 TOH.5 ELC7.5 CHI The base shear force was computed summng up the eternal forces calculated by multplyng masses of W and W to the acceleraton record of them. Subsequently shear forces carred by wall was calculated by subtractng shear forces recorded at the load cells nstrumented at the base of the ndependent columns from base shear force. Fgure 5(a) llustrates the shear forces carred by the columns and the wall n the st story and the rato of the column shear force to the base one s shown n Fgure 5(b). Note that all the shear forces presented n Fgure.5 are the values at the tme when the base shear force attaned the peak n both drectons. From these fgures t s seen that the shear force carred by the columns s relatvely smaller than that of wall and degrade gradually wth ncreasng nelastcty. Horzontal Dsplacement vs. Shear Force The hysteretc relatons between the horzontal dsplacement and the shear force of the two ndependent columns are presented n Fgure 6. The sold and dotted lnes are calculated shear strength (.9KN) and shear at calculated fleural strength (5.5KN) of two columns respectvely. In TOH.5 and TOH5 the relaton between two responses s almost lnearly elastc and as the load level ncreases the stffness degrades and the lateral drft becomes larger. The mamum shear forces were attaned durng JMA whch was almost the same as that of the calculated strength. Durng the response to CHI nput the stffness and strength degradatons of the specmen became rapdly sgnfcant under reversed cyclc loadngs and resulted n collapse when the elapsed tme was around sec.

6 Shear force(kn ) T O H.5T O H 5 ELC7.5 J M A 5 C H I Horzontaldsplacem net(m m ) Fgure 6: Shear force and dsplacement relaton Collapse Process of Columns To nvestgate the process of RC column falure the tme-hstory responses and ther relatons observed for seconds are llustrated n Fgure. 7 whch are from sec. to sec. after CHI base moton was run. The stran hstory of the transverse renforcement shown n Fgure.7 was measured at the md-heght of the column. Two reference tmes were selected to dvde the responses nto three parts. At frst reference tme 6.7 sec marked wth black trangles the large peak n shear force was recorded and then both the stffness and strength degraded consderably and lateral renforcement bar started to epand. Base m oton 6 (gal) - Shear force Shear force (K N ) 4 - (KN) -4-6 (KN) (m m ) - 4 Horzontaldsplacem ent -4 A al forc e - V ertcaldsplacem ent Vertcaldsplacem ent (m m ) Horzontaldsplacem ent (m m ) Aal force(kn) (mm) (μ) - -6 S tran gauge(hoop) Vertcaldsplacem ent(m m ) sec sec sec Tme (sec) Horzontaldsplacem ent(m m ) Fgure 7 : Responses of east column n RC specmen durng CHI

7 Subsequently at 9.77sec marked wth whte trangle larger lateral drft was developed comparng to the prevous one and lateral stffness and strength was lost entrely and fnally the loss of aal load-carryng capacty led the specmen to collapse. From these fgures the process and the cause of the column aal falure may be nterpreted as follow: the column response at the frst peak nduced the crtcal crackng assocated wth the yeldng of the hoop whch caused the resdual hoop strans and the shear strength decay. The second peak drft eceeded the prevous mamum. Here the hoop mght be fractured snce the resdual stran fall down and the loss of the nterface shear transfer along the shear crackng mght cause the fatal loss of the aal capacty. It should be noted that the nelastc stran of the hoop was accumulated wth cyclc load reversals n the second tme regon and ths could be the man cause of the shear and aal falure of the column. ELEMENT FORMULATION A column member n frame analyss s generally dealzed by one lne element wth two-end nodes as shown n Fgure 8(a). In the proposed model however the element s dvded nto three lne elements by nsertng two nternal nodes (4) located at α L from two eternal nodes () (Fgure 8(b)) whch represent the boundares of the plastc hnge regons. Furthermore as shown n Fgure 8(c) each lne element s transformed to plate element wth 4 nodes. The procedure of dervng member stffness matr s descrbed below. y z D m θ y y f z d z f d α L 4 () 7 8 () 5 6 m θ y y f z d z f d ( α ) L () () f z z f z 4 z 4 α L 4 f f 4 f 4 4 f (a) (b) (c) f Fgure 8 : Proposed column model Fgure 9 : Lne vs. Plate element Dervaton of the member stffness matr The member stffness matr s derved by assemblng the stffness matr of three lne elements under the drect stffness approach based on the nodal force equlbrum condton (Equaton ()). z z f z z { F { F { F { F 4 [ k ] [ k ] () () [ k ] [ k ] 4 () () [ k ] + [ k ] [ k ] 4 (). [ k ] + [ k ] = symm 44 { D { D { D { D4 ()

8 The superscrpts n parenthess denote element number and the subscrpts are for node number. Epressed usng nternal ( ) and eternal ( e ) node notaton Equaton () becomes { F e { F = [ K ] [ K ] ee e [ K ] [ K ] e { D e { D (4) On the bass of the assumpton that no eternal force s appled to the nternal nodes Equaton (5) s obtaned. { D = [ K ] ( { F [ K ]{ D ) { = e e F (5) On substtutng Equaton (5) nto Equaton (4) fnally we can obtan the member stffness matr n the form of Equaton (6) whch relates only the eternal nodal dsplacements to forces. { = [ K ]{ D F (6) { { [ ] [ ] F = F K K { F 6 e e Where [ K ] = [ K ] [ K ] [ K ] [ K ] 6 6 ( ) e e e (7) The stffness matr dervng procedure descrbed above s based on the drect stffness method and the statc condensaton method whch assembles and reduces the stffness matr respectvely. Plate Element Formulaton Equaton (9) shows the relatonshp between lne element and plate element whch have 6 and 8 DOF respectvely. The lne element can be transformed to plate element based on the two assumptons: one s plane secton hypothess and the other s the stress assumpton that transverse stress s zero. The dsplacement relatonshp between two elements s therefore obtaned as Equaton (8) and rewrtten n the matr form Equaton (9). 4 z z z z 4 z z z z + ( D / ) θ ( D / ) θ + ( D / ) θ ( D / ) θ y y y y (8) { T d = [ T ] { d T (9) { { T = 8 z z z 4 z 4 Where { d = { d d θ d d θ T 6 z y z y () The plate element s consdered as lnear plane element wth two nodes along an edge and based on the soparametrc formulaton whch uses the same shape functons to defne the element shape as are used to defne the dsplacements wthn element. A stran-dsplacement matr [ B ] and a plane stran-stress 8 relatonshp are shown n Equaton () and () respectvely. The plate nodal dsplacements n lateral drecton transformed from the lne element are dentcal whch makes the transverse ncremental stran

9 ε become zero n Equaton (). Therefore the lateral stran cannot be found n an eplct way usng Equaton () but evaluated from Equaton (4) whch s consstent wth the assumpton descrbed above. { = [ B ] { d ε () 8 8 { σ = [ ]{ ε T { ε = { ε ε γ z z where [ D] = [ D] T. + [ D] { σ = { σ σ τ D () D D ε = ε γ + = z z D D z z σ D D D D = con steel D D D () D D D σ (4) Once the ncremental transverse stran s found complete plane stran components are obtaned and then plane stresses can be found. In ths study smeared rotatng crack approach s adopted for evaluatng stresses and materal tangent stffness matr from gven strans whch s based on averaged stress and stran ncludng the effect of crack and coaalty between prncpal stran and prncpal stress (Veccho [] Stevens []). T [ k ] = [ B ] [ D][ B ]dv (5) T { f [ B ] { σ = dv (6) 8 8 The ntegrals to obtan the force and stffness matr of plate element (Equaton (5) and (6)) are numercally evaluated usng the two dmensonal gaussan quadrature. Consttutve Model The plate element s a basc analytcal unt n the proposed model. The nelastc propertes are determned from the materal consttutve laws and therefore the accuracy of the analytcal results s to a great etent dependent on the materal models. The concrete model n prncpal compressve and tensle drecton s shown n Fgure. whch takes nto account the compressve strength softenng effect due to tensle stran and the tenson stffenng effect respectvely. The compressve strength reducton factor c s σ c f c f c c σ = c c f c ε c ε c ε c ε c ( ε c σ ) ma c ma ( c ) c fc σc = ( εc εc) ( c ) ε c σ t f cr ( ε t σ ) ma t ma σ = f ( ε / ε ) t cr cr t.4 c c f c c ε ε c c ε c ε cr ε cr ε t c =. /(.8.4ε t / ε c ) c =. c = 5 (a) Compresson response (b) tenson response Fgure : Consttutve model for concrete

10 adopted from Veccho [] and the descendng branch representng the tenson stffenng effect s from Isumo [4]. And the reloadng and unloadng rules under cyclc loadng are also presented together wth the envelope curve. The consttutve model for both longtudnal and transverse renforcement used n ths study s b-lnear type but ncorporatng the effect of bond to the concrete. Further detals on the renforcement consttutve model as well as the concrete model can be found n Chen [5]. Iteratve Procedure for Numercal Soluton Eternal Nodal Dsplacements loop Internal Nodal Dsplacements m= Nodal Dsplacements of Lne Element Nodal Dsplacements of Plate Element = n Longtudnal and Shear Stran loop () Transverse Stran Consttutve Law Concrete Steel Stresses and Materal Tangent Stffness A procedure for the nonlnear analyss of the proposed model s summarzed n Fgure whch s establshed by ncorporatng the materal and element formulaton descrbed before. In addton two teratve schemes mposng nternal force equlbrum (Equaton (7)) and transverse stress equlbrum (Equaton (8)) whch resulted from the assumptons made n the proposed element formulaton are ntroduced n the algorthm. Both of the teratve procedures are contnued untl the predefned convergence tolerance s satsfed and the updated materal and element stffness s used for evaluatng the resdual dsplacement n the nternal nodes (Equaton (9)) and the transverse resdual stran (Equaton ()) respectvely. { F u () { f + { f () ( ) { f + { f = ( = ) (7) 4 4 Transverse Stress Convergence Check YES Plate Element Forces and Stffness Matr NO u σ σ + ρ σ (= ) (8) = c s s { D = [ K ] { F u (9) u ε = / D () σ Lne Element Forces and Stffness Matr Internal Nodal Force Convergence Check Member Forces and Stffness Matr m : element number n : number of ntegraton ponts YES : ntegraton pont number NO Fgure : Analyss algorthm for proposed model However the dsplacements assgned at eternal nodes should not be changed n the teraton loop as the dsplacement compatble condton at the eternal nodes should be nsured. In the same manner eplctly calculated strans (.e. longtudnal and shear stran) are not updated also n the teraton loop () n whch only the lateral strans at each ntegraton pont are computed usng renewed materal stffness and resdual stress obtaned by mposng the equlbrum between transverse steel stress and concrete stress (Equaton (8)). In partcular ntal values for the nternal dsplacements { D and transverse stran ε can be found accordng to the Equaton (5) and Equaton (4) respectvely whch correspond to the Equaton (9) and () after ntalzaton.

11 ANALYSIS AND RESULTS shear force (KN ) Dsplacement eperm ent analyss Self-weght Fgure : plane frame subtracted from specmen drft rato (% ) Fgure : force vs. drft-rato ε ε z Fgure 4 : stran dstrbuton γ z For the epermental results from TOH.5 to JMA where lttle strength deteroraton was occurred three-dmensonal dynamc analyss has already been performed wth two analytcal models such as classcal fber model and lumped plastcty model (.e. one-component model) n Km [6]. The fber model showed a better correlaton between the calculated and the observed data rather than one-component model. Ths s manly because the aal-bendng and the baal bendng nteracton were ntroduced n the former but not n the latter. Both of the models however faled to smulate the post-peak response durng CHI nput. As noted prevously the proposed model s avalable only n the two-dmensonal problem at current stage. However the test specmen eperenced torsonal response under threedmensonal effect despte undrectonal sesmc load was appled and therefore t s beyond the scope of ths model to smulate the epermental results. As the alternatve to the three-dmensonal dynamc analyss of the specmen twodmensonal statc analyss s performed on a weak frame depcted n Fgure. Obvously ths could be the source of error and the model mght not be sutable for and even ncapable of smulatng the -D epermental results snce so many characterstcs affectng the epermental results are gnored and not ncluded n analytcal procedure. Nonetheless ths analytcal study has the meanng n terms of provng the stablty n teratng nonlnear numercal soluton and the capablty of ncorporatng the strength degradaton effects. Thus the dsplacement controlled statc analyss was performed wth the applcaton of the lateral dsplacement record at the st story whle the constant compresson aal load correspondng to the self-weght was mantaned (Fgure.). Takng nto account the resdual deformaton and the stffness degradng effect observed n the epermental results at the end of each nput the frame was subjected to the dsplacement record connected from TOH.5 to CHI n successon.

12 Fgure. shows the calculated force-drft rato relatonshp together wth the observed one. Although the analytcal model dd not provde a satsfactory accuracy n predctng the mamum shear strength the model adequately represented the strength deteroraton feature. The dfference of strength deteroratng rate between the model and the epermental data mght be attrbuted to the varable aal load acceleratng the strength degradaton n epermental case whle t was not consdered n analytcal one. In addton baal effect mght also be assocated wth rapdly degradng strength observed n the epermental results. Usng the stran values such as transverse longtudnal and shear stran calculated at ntegraton ponts each stran dstrbuton s llustrated n Fgure.4. It should be noted that the level of the strans developed n the md-heght regon were sgnfcantly low compared to those of hnge regons and remaned wthn elastc range throughout the whole response and the shear stran dstrbuton s constant along the secton as a consequence of the stress assumpton made n the element formulaton. Although these features resultng from the proposed element formulaton are nevtable and could be apparent lmtatons n further realstc evaluaton of nelastc behavour of RC columns these effects on the analytcal results would not be so sgnfcant n most cases that they may be dsregarded. CONCLUSIONS Based on the results of the epermental and the analytcal nvestgaton presented heren the followng conclusons can be drawn. Throughout all the loadng stages from elastc to nelastc range the lateral dsplacement responses of the column sde (weak or fleble frame) were much lager than those of the wall sde (strong or stff frame) whch was epected and attrbuted to the consderable stffness and strength eccentrcty. In partcular the torsonal response was a lttle lager n nelastc responses than n elastc whch may be due to the large strength eccentrcty. The collapse process of renforced concrete columns durng CHI nput shear strength deteroraton resultng n aal load falure along wth nelastc load reversals was nterpreted wth the detaled local responses such as transverse steel stran lateral and vertcal dsplacements shear and aal forces and ther relatonshps. An analytcal member model of column n frame analyss s proposed based on stress-stran relaton formulated from the materal consttutve model. Statc analyss of the test specmen was carred out usng the new model where the stress and force equlbrum condton obtaned from the dynamc test was appled wth numercal teratve procedure. A far correlaton was obtaned between the test and analyss. The model may be used to ncorporate the bendng shear and aal force nteracton and the strength deteroraton under the cyclc loadng. The proposed model s lmted to two-dmensonal analyss and needs to be etended to three-dmensonal model smulatng ncludng as confnement effect and mult-aal nteracton. The materal consttutve law should also be verfed further to mprove the accuracy of the analytcal model. REFERENCES. Km Y Kabeyasawa T. Shakng Table Test of Renforced Concrete Frames wth/wthout Strengthenng for Eccentrc Soft Frst Story. Proceedngs of the Fourth US-Japan Workshop on

13 Performance-Based Earthquake Engneerng Methodology for Renforced Concrete Buldng Structures Toba Japan pp.9-4. Veccho F Collns MP. Modfed Compresson Feld Theory for Renforced Concrete Elements Subjected to Shear. ACI Journal Proceedngs 986; 8() : pp.9-. Stevens NJ Uzumer SM Collns MP. Renforced Concrete Subjected to Reversed Cyclc Shear Eperments and Consttutve Model. ACI Structural Journal 99; 88() : pp Izumo J Shma H Okamura H. Analytcal Model for RC Panel Elements Subjected to In-plane forces. Concrete Lbrary Internatonal JSCE 989 No. : pp Chen S Kabeyasawa T. "Modelng of renforced concrete shear wall for nonlnear analyss." Paper No. 596 Proceedngs of the th World Conference on Earthquake Engneerng CD-ROM New Zealand January 8pp. 6. Km Y Kabeyasawa T. Nonlnear earthquake response analyss on eccentrc plot renforced concrete structure. Summares of Techncal Papers of Annual Meetng Archtectural Insttute of Japan pp (n Japanese)