EFFECT OF DESIGN VARIABLES ON DISPLACEMENT DUCTILITY OF FRP- REHABILITATED RC SQUAT COLUMNS. K. Galal 1 ABSTRACT

Transcription

1 Proceedings o he 8 h U.S. Naional Conerence on Earhquake Engineering April 8-,, San Francisco, Caliornia, USA Paper No. 9 EFFECT OF DESIGN VARIABLES ON DISPLACEMENT DUCTILITY OF FRP- REHABILITATED RC SQUAT COLUMNS K. Galal ABSTRACT The seismic perormance o reinorced concree (RC) squa columns in exising uildings or ridge piers indicaes ha hey are vulnerale o non-ducile shear ailure due o heir low shear span o deph raio. Wrapping RC columns wih caron or glass ire-reinorced polymer (FRP) shees proved o increase heir shear capaciy and consequenly he lexural displacemen duciliy wihou signiican increase in he column s siness. The displacemen duciliy capaciy o a srucural elemen is one o he widely used perormance parameers. A designer o a rehailiaion scheme using FRP wraps or a recangular RC squa column should develop an undersanding or he inluence o dierenacors on he column s displacemen duciliy capaciy. Unlike in laoraory-esed columns, designing he required numer o FRP wraps o reach a argeed displacemen duciliy or a squa column in an exising old srucure involves several uncerainies in he design variales. A oal o eleven variales ha inluences he design o FRP jacke are grouped ino our caegories, namely; maerial properies, geomerical properies, reinorcemen conen and axial load level. The sudy evaluaes he eec o changing hese variales on he displacemen duciliy capaciy o FRPrehailiaed RC recangular squa columns. Inroducion Recangular reinorced concree (RC) columns are widely used in ridge pier design, and hey make up he majoriy o uilding columns. Columns in need o srenghening and reroi are very common. Among hose are squa columns -wih low shear span o deph raio- ha are suscepile o shear ailure during an exreme loading even such as a major earhquake. The radiional wisdom has een o avoid he consrucion o squa columns in seismically acive zones. However, here exis a large numer o squa columns ha are a risk o rile ailure modes. These columns may have een originally designed as long columns and hen parial supporing non-srucural walls were laer consruced hereore creaing a shor (squa) column. Squa columns may also have een he resul o recen design ollowing curren codes. Perormance-ased seismic engineering is he modern approach o earhquake resisan s. Proessor, Dep. o Building, Civil & Environmenal Eng., Concordia Universiy, Monreal, Queec, Canada

2 design. Seismic perormance (perormance level) is descried y designaing he maximum allowale damage sae (damage parameer) or an ideniied seismic hazard (hazard level). Perormance levels descrie he sae o a srucure aer eing sujeced o a cerain hazard level as: Fully operaional, Operaional, Lie sae, Near collapse, or Collapse (FEMA 7/7, Vision ). Overall laeral delecion, duciliy demand, and iner-sorey dri are he mos commonly used damage indices. The duciliy o he column pas iniial seel ar yielding has ecome he argeor good design. This approach is expeced o decrease he proailiy o ailure o he srucure, and increase is energy dissipaing capaciy, when sujeced o he design ground moions. Fire composies are used o increase he shear srengh o exising concree eams and columns y wrapping or parially wrapping he memers. Addiional shear srengh conriuion is inroduced y oriening he ires normal o he axis o he memer or o cross poenial shear cracks. Increasing he shear srengh can aler he ailure mode o e relaively more ducile compared o shear ailure. Shear srenghening using exernal ire reinorced polymer (FRP) may e provided a locaions o expeced plasic hinges or sress reversal or enhancing posyield ehaviour o columns sujeced o seismic loads y compleely wrapping he secion. A designer o a rehailiaion scheme using FRP wraps or a recangular RC squa column should develop an undersanding or he inluence o dierenacors on he column s displacemen duciliy capaciy. Unlike in laoraory-esed columns, designing he required numer o FRP wraps o reach a argeed displacemen duciliy or a squa column in an exising old srucure migh involve several uncerainies in he design variales. The variales are grouped ino our caegories, namely; maerial properies, geomerical properies, reinorcemen conen and axial load level. The maerial properies include he compressive srengh o concree, yield sress o longiudinal and ransverse rears. The geomeric properies include he raio o he disance eween longiudinal reinorcemen-o-he overall deph, and he shear spano-deph raio o he column. The reinorcemens' conens include hose o he ransverse reinorcemen, longiudinal reinorcemen and is arrangemen, he FRP wrap conen, and he coninemen eeciveness coeicien. The level o axial load acing on he column arising rom he possile variaions in verical load during a seismic even is also sudied. The ojecive o his sudy is o evaluae he eec o changing hese variales on he displacemen duciliy capaciy o FRP-rehailiaed RC recangular squa columns. Displacemen duciliy capaciy, µ, o FRP-rehailiaed columns Figure shows ypical idealized lexural and shear laeral orce-dri capaciies o a reinorced concree column. The column s response will ollow he lexure envelope unil i reaches he shear capaciy envelope. Susequenly, degradaion in he shear srengh will occur and he column loses is laeral orce capaciy ollowing a negaive siness, unil he residual shear capaciy is reached. Several models have een developed o represen he degradaion o shear srengh o reinorced concree columns wih increasing deormaions (Waanae and Ichinose 99, cheim and Moehle 99, Priesly e al. 99, Kowalsky e al. 999, Sezen, Elwood and Moehle ). The models do no consider columns rehailiaed wih FRP wraps. Galal e al. () veriied a model or he shear capaciy-displacemen duciliy envelope or axially and laerally loaded FRP-rehailiaed RC recangular squa columns. Figure shows he comined lexural and shear response o he model. The analyical predicions o he model showed reasonale correlaion wih he experimenal resuls availale in he lieraure.

3 Laeral resisance V V P V n V lex Vµ= V µ = Vr Displacemen duciliy capaciy µ Shear capaciy envelope Flexural capaciy envelope s Elevaion M V Anchored FRP wrap A FRP v A s A s γ d Cross-secion In his sudy, he laer model is used o develop he laeral resisance (V) displacemen duciliy (µ) poins ha deine he shear capaciy and lexure capaciy o FRP-rehailiaed RC recangular squa columns. Given his, he displacemen duciliy capaciy, µ, is deined as he duciliy when he lexural capaciy envelope inersecs he shear capaciy envelope. This represens he poin o ormaion o local mechanism, which is ollowed y degradaion in he laeral resisance o he column. The equaions ha were used o calculae he poins ideniying he shear capaciy and he lexural capaciy envelopes are given in he ollowing wo susecions, respecively. The equaions are simpliied o e expressed in he leas possile numer o variales. Moreover, i was ound ha he laeral shear resisance or oh shear and lexural capaciies can e expressed in erms o he cross-secional dimensions (.) o he column, hus eliminaing heir (i.e..) eec on he displacemen duciliy capaciy o FRPrehailiaed RC recangular squa columns. This implies ha he size o he column is assumed o have no eec on is displacemen duciliy capaciy. Shear capaciy envelope: The nominal shear capaciy, V n, o a RC column reroied wih FRP composies is equal o he sum o he conriuions o our mechanisms, namely; concree V c, axial load V p, ransverse seel V s, and FRP V. i.e. V n = V c + V p + min [ (V s +V ) and. β c d ] () The model assumes ha he column shear capaciy decreases i-linearly wih he increase o he laeral displacemen duciliy, µ, aer reaching µ= such ha: V µ= = ⅓(V c +V p ) + min [ (V s +V ) and. β c d ] a µ = (a) V µ= = min [ (V s +V ) and. β c d ] a µ = () where c is he unconined concree compressive srengh; is he widh o he column; d is he column secion deph o he ensile seel (as shown in Figure ); β is he conined concree compressive srengh muliplier β=' cc /' c and is deined laer; and ' cc is he conined concree compressive srengh. The conriuions o he our mechanisms reduce o: V =. A =.K β () c cc e e c. Experimenal ackone envelope Displacemen duciliy (µ = / y ) 7 8 Figure. Shear capaciy-displacemen duciliy model or FRP-rehailiaed columns (Galal e al. ). Figure. FRP-rehailiaed recangular RC sujeced o laeral displacemen.

4 P / ( ζ c) /. ζ c V p = k p = = H ( M / V) M / V () Av yvd + γ Vs = = ρ v yv ( ) s () ( + γ ) V =.9( )( ε ee ) d =.9ρ Fε e yv () cc l l Such ha β = = (Mander e. al 988); ' c c c l = K e ( ρ v + ε e ρ F ) yv is he eecive laeral conining pressure; K e = A e / is he coninemen eeciveness coeicien; ζ = P / ( ' c ) is he axial orce level; M/V is he shear span-o-deph raio o he column; ρ v = A v / s is he shear sirrups conen; E ρ F = is he FRP wrap conen; and yv γ = (d-d') / is he raio eween longiudinal seel o he overall deph o he column. Where A e is he area o eecively conined concree core; k p = or columns in doule curvaure and. or columns in single curvaure; A v, yv, s are he oal cross secional area, yielding srengh and spacing o ransverse reinorcemen; is he oal ransverse design hickness o FRP shees (i.e. or wo opposie sides); ε e is he design srain or FRP: ε e =. or un FRP shees and ε e =. or FRP shees; and E is he Young s modulus o he FRP composie maerial. The aove equaions show ha he shear capaciy-o-cross secional area, V/, o an FRP-rehailiaed RC squa column can e expressed as a uncion o eigh variales, namely; c ', K e, ζ, yv, M/V, ρ v, ρ F, γ. Flexural capaciy envelope In he curren analysis, a ilinear lexural capaciy envelope o he column is assumed. The idealized lexural envelope is deined o envelope he laeral orce-displacemen duciliy response o he irs loading cycles, in case o cyclically loaded columns. Similar ilinear idealizaion or un-rehailiaed RC columns has een used y several researchers (e.g. Elwood and Moehle). Now, deining he lexural capaciy envelope is reduced o deermining he lexure ulimae capaciy o he column, M u, which can e easily ormulaed. Several researchers sudied he deormaional ehaviour and capaciy o RC columns in lexure (Mirza 99, Panagioakos and Fardis, Biskinis e al. ). In his analysis, M u is calculaed y conducing secion analysis considering orce equilirium and srain compaiiliy aailure. Equivalen recangular sress lock is assumed when considering he orce equilirium. The analysis accouns or he axial load on he column. Thereore, he lexural capaciy, M u, o a RC column wih longiudinal reinorcemen conen ρ = /, where is he oal area o longiudinal reinorcemen, can e expressed as a uncion o. Consequenly, he shear orce V lex ha corresponds o he column s lexural capaciy can e expressed as:

5 M u V lex. ( M / V ) M u = = (7) ( M / V) The aove equaion shows ha he column s shear capaciy-o-cross secional area ha corresponds o is lexural capaciy, V lex. /, is uncion o he variales ha deine M u /, and hese are: c ', ρ, y, ζ, γ, in addiion o he arrangemen o longiudinal reinorcemen in he cross-secion and M/V. From he ormulaion o he shear and lexure capaciies' envelopes, i is shown ha he displacemen duciliy capaciy o FRP-rehailiaed recangular RC squa columns depend on eleven variales ha conrol he lexure and shear envelopes' capaciies. These variales are grouped ino our caegories as shown in Tale. Tale. Sudied variales ha aec he displacemen duciliy capaciy o FRP-rehailiaed recangular RC squa columns. Caegory Variale Deiniion Maerial properies Geomerical properies Reinorcemens' conens c ' yv y γ M/V unconined concree compressive srengh yielding srengh o ransverse reinorcemen yielding srengh o longiudinal reinorcemen Raio o he disance eween longiudinal reinorcemen-o-he overall deph Shear span-o-deph raio ρ v Shear sirrups conen [ ρ v = A v / (.s) ] ρ F FRP wrap conen [ ρ F = E / (. yv ) ] ρ Longiudinal reinorcemen conen [ ρ = / (.) ] K e Coninemen eeciveness coeicien [ K e = A e /(.) ] a percenage o he inermediae reinorcemen-o-he oal longiudinal reinorcemen Axial load level ζ Axial orce level [ ζ = P / ( c '..) ] Maximum inelasic dri capaciy m shown y Paulay (), or dieren reinorced concree elemens he yield displacemen, y, is more or less consan, such ha an increase in srengh auomaically resuls in a proporionae increase in siness. This implies ha he maximum inelasic dri capaciy, m = µ y, can e esimaed i he displacemen duciliy capaciy µ is known. Eec o design variales on he displacemen duciliy FRP conen relaionship In order o sudy he eec o design variales on he displacemen duciliy o FRPrehailiaed RC recangular squa columns, he displacemen duciliy capaciy (µ ) FRP conen (ρ F ) relaionship is considered. A ypical RC recangular squa column wih speciic properies as shown in Tale is considered; hence he eec o changing hese properies on µ -ρ F relaionship is sudied. The properies were chosen o represen an exising RC recangular squa column ha is designed according o pre-97 codes (ACI 98).

6 Variale Tale. Properies o he sudied recangular RC squa column. c ' yv y γ M/V ρ v ρ K e ζ MPa MPa MPa d/ A v /s / P/ c ' Value.7..% %.. The covered range o FRP conen, inended o e used in he column rehailiaion, is up o ρ F = %, which is equivalen o layers o Caron FRP (wih laminae hickness=. mm, =mm, E = GPa, yv = MPa), or layers o Glass FRP (wih laminae hickness=. mm, =mm, E =7 GPa, yv = MPa). Eec o maerial properies Figure shows he eec o he concree compressive srengh on µ -ρ F relaionship. From he igure, i can e seen haor a arge displacemen capaciy, e.g. µ =, increasing he concree srengh c ' rom MPa o MPa increases he required FRP conen ρ F rom.9% o.% and rom.% o.% or and un FRP jackes, respecively. In oher words, an underesimaion o he acual concree compressive srengh o an exising RC column will lead o an unsae design o he required hickness o he FRP jacke used in he column s rehailiaion o mee a speciied arge displacemen. Figure shows he eec o he yielding srengh o ransverse reinorcemen on µ -ρ F relaionship. From he igure, i can e seen ha a RC column wih yv = MPa (or yv = MPa) requires aou 7% more (or % less) ρ F compared o a column wih yv = MPa. This is ariued o he ac ha he reducion (or increase) in he conriuion o seel ransverse mechanism o he shear srengh should e compensaed y he conriuion rom he FRP mechanism in order o achieve he same arge displacemen duciliy. I should e noed ha according o he ormulaion o he conriuion o he ransverse seel and FRP mechanisms given y equaions and, respecively, he linear variaion o yv or a given ransverse reinorcemen ρ v does no yield a similar linear variaion in he required FRP conen ρ F. From he igure, i can e seen ha anchoring he FRP jacke o he column reduces he required conen o FRP or a arge displacemen duciliy, while he range o he relaive increase (or decrease) in FRP conen ρ F wih respec o he decrease (or increase) in yv, remains he same or and un FRP jackes. Displacemen duciliy capaciy ( µ ) yv = MPa y = MPa ρ = % M/V =. P/ c ' =. K e =. γ c ' = MPa c ' = MPa c ' = MPa c ' = MPa un Displacemen duciliy capaciy (µ ) c' = MPa y = MPa ρ = % M/V =. P/ c ' =. K e =. γ yv = MPa yv = MPa yv = MPa un.. FRP conen (ρ F %) Figure. Eec o he concree compressive srengh on he µ -ρ F relaionship... FRP conen (ρ F %) Figure. Eec o he yielding srengh o ransverse reinorcemen on he µ -ρ F relaionship.

7 Figure shows he eec o he yielding srengh o longiudinal reinorcemen on µ -ρ F relaionship. From he igure, i can e seen ha changing he yielding srengh o longiudinal reinorcemen has an inverse eec on he required FRP conen ρ F a a arge displacemen duciliy µ, compared o ha o changing he yielding srengh o ransverse reinorcemen. This is ariued o he ac ha increasing he yielding srengh o he longiudinal reinorcemen increases is lexural capaciy wihou aecing is shear srengh capaciy, which resuls in a reduced displacemen duciliy and higher required FRP conen. A RC column wih y = MPa (or y = MPa) requires aou % less (or % more) ρ F compared o a column wih y = MPa. (µ ) Displacemen duciliy capaciy c' = MPa yv = MPa ρ = % M/V =. P/ c ' =. K e =... FRP conen (ρ F %) Figure. Eec o he yielding srengh o longiudinal reinorcemen on he µ -ρ F relaionship. γ y = MPa y = MPa y = MPa un Displacemen duciliy capaciy (µ ) Eec o geomerical properies Figure shows he eec o he raio o he disance eween longiudinal reinorcemeno-he overall deph γ on µ -ρ F relaionship. From he igure, i can e seen ha changing γ rom. o.8 does no have a signiican eec on he µ -ρ F relaionship or and un FRP-rehailiaed RC squa columns. I should e noed ha his oservaion does no mean ha γ has no eec on he lexural or shear capaciies, u raher, his implies ha he change in oh lexure and shear capaciies is such ha he displacemen duciliy and he required FRP conen o he rehailiaed column are almos he same. c ' = MPa yv = MPa y = MPa ρ = % M/V =. P/ c ' =. K e =. γ =. γ =.8 un.. FRP conen (ρ F %) Figure. Eec o he raio o he disance eween longiudinal reinorcemen-o-he overall deph on he µ -ρ F relaionship. γ (µ ) Displacemen duciliy capaciy Figure 7 shows he eec o he shear span-o-deph raio on µ -ρ F relaionship. From he igure, i can e seen ha reducing he shear span-o-deph raio o RC squa columns reduces is c ' = MPa yv = MPa y = MPa ρ = % P/ c ' =. K e =. M/V =. M/V =. M/V =. M/V =. un.. FRP conen (ρ F %) Figure 7. Eec o shear span-o-deph raio on he µ -ρ F relaionship. γ

8 displacemen duciliy capaciy and increases he required FRP conen o he FRP jacke. For a argeed displacemen duciliy capaciy µ = or he sudied column, decreasing he shear spano-deph raio rom. o. increases he required FRP conenrom ρ F % o ρ F =.8% and.% or and un FRP jackes, respecively. This emphasizes he imporance o ideniying he expeced shear span o a RC column due o is impac on he required hickness o he FRP jacke, especially in he case o capive columns (or example, hose creaed due o window openings in parially masonry-inilled rames and a he op and oom ends o he columns in he case o masonry-inilled RC rames). Displacemen duciliy capaciy (µ ) Eec o reinorcemen conen Figure 8 shows he eec o ransverse reinorcemen conen on µ -ρ F relaionship. expeced, increasing he ransverse reinorcemen conen increases he duciliy and reduces he required FRP conen. This is ariued o he increase in he conriuion o he ransverse reinorcemen mechanism and hus he shear capaciy. For he sudied column, anchoring he FRP jacke o he column reduces he required conen o FRP y aou % or a ransverse reinorcemen conen range o.% o.8%. Figure 9 shows he eec o he longiudinal reinorcemen conen on µ -ρ F relaionship. Increasing he longiudinal reinorcemen conen decreases he displacemen duciliy and increases he required FRP conen. This is ariued o he ac ha increasing he longiudinal reinorcemen conen increases he lexural capaciy, which in urn reduces he displacemen duciliy capaciy. For example, or a arge displacemen duciliy capaciy µ = or he sudied column, an underesimaion o he longiudinal reinorcemen conen o he exising column as.% raher han %, resuls in an underesimaion o he required FRP conen y %. Ye, anchoring he FRP jacke diminishes his underesimaion. c ' = MPa yv = MPa y = MPa ρ = % M/V =. P/ c ' =. K e =. ρ v =.% ρ v =.% ρ v =.8% un.. FRP conen (ρ F %) Figure 8. Eec o ransverse reinorcemen conen on he µ -ρ F relaionship. γ (µ ) Displacemen duciliy capaciy Figure shows he eec o he coninemen coeicien K e on µ -ρ F relaionship. For he sudied column, increasing he coninemen coeicien increases he displacemen duciliy capaciy and reduces he required FRP conen. This eec diminishes as he argeed displacemen duciliy capaciy reaches µ =. On he oher hand, a column wih FRP jacke will have higher displacemen duciliy capaciy and less required FRP conen compared c ' = MPa yv = MPa y = MPa M/V =. P/ c ' =. K e =. FRP ρ =.% ρ =.% ρ =.% un.. FRP conen (ρ F %) Figure 9. Eec o longiudinal reinorcemen conen on he µ -ρ F relaionship. A v γ

9 (µ ) Displacemen duciliy capaciy o un one. The eec o changing he ransverse reinorcemen conen ρ v o.% and.% or K e =. is shown on he same igure, where i can e seen ha decreasing ρ v decreases he displacemen duciliy capaciy and increases he required conen o FRP. Figure shows he eec o he arrangemen o he longiudinal reinorcemen on µ -ρ F relaionship. From he igure, i can e seen ha increasing he percenage o he inermediae reinorcemen-o-he oal longiudinal reinorcemen in he cross secion, a, increases he displacemen duciliy and decreases he required FRP conen. This is ariued o he ac ha increasing his percenage reduces he lexure capaciy o he secion hus increasing he displacemen duciliy. ρ v =.8% =.% v ρ c ' = MPa yv = MPa y = MPa ρ = % M/V =. P/ c ' =. K e =. K e =. K e =. K e =. un.. FRP conen (ρ F %) Figure. Eec o he coninemen coeicien K e on he µ -ρ F relaionship. γ A Eec o axial load level s Figure shows he eec o he axial load level on µ -ρ F relaionship. From he igure i can e seen ha, or a arge displacemen γ duciliy capaciy µ = or he sudied column, c ' = MPa increasing he axial load level rom.p yv = MPa o (where P o = c ') o.p o increases he required y = MPa P/ c ' =. ρ = % P/ c ' =. FRP conenrom.% o.%. Furher P/ c ' =. increase in he axial load rom.p o o.p o M/V =. P/ c ' =. does no change he required FRP conen. For K he sudied column, anchoring he FRP jacke o e =. un he column reduces he required conen o FRP ρ F y aou %... FRP conen (ρ F %) Figure. Eec o axial load on he µ -ρ F relaionship. Conclusions The eec o eleven variales ha conrol he lexure and shear capaciies and hus he displacemen duciliy and dri capaciies o FRP-rehailiaed recangular RC squa columns, is (µ ) Displacemen duciliy capaciy (µ ) Displacemen duciliy conen c ' = MPa yv = MPa y = MPa ρ = % M/V =. P/ c ' =. a =. a =. a =. un.. FRP conen (ρ F %) (-a )A s Figure. Eec o he longiudinal ar arrangemen on he µ -ρ F relaionship. A v (-a ) a. γ A v FRP

10 evaluaed. The ormulaion o he lexure and shear capaciies is ased on an exising model. The variales are grouped ino our caegories, namely; maerial properies, geomerical properies, reinorcemen conen and axial load level. In order o sudy he eec o hese design variales on he displacemen duciliy o FRP-rehailiaed RC recangular squa columns, he displacemen duciliy capaciy (µ ) FRP conen (ρ F ) relaionship is considered. Based on he analyical sudy, he ollowing conclusions regarding he rehailiaion o exising RC recangular columns using FRP wraps are reached: - For a arge displacemen duciliy capaciy, he required FRP conen o he FRP jacke increases wih he increase o concree compressive srengh, yielding srengh o longiudinal reinorcemen, longiudinal reinorcemen conen, and axial load level. - For a arge displacemen duciliy capaciy, he required FRP conen o he FRP jacke increases wih he decrease o yielding srengh o ransverse reinorcemen, shear span-odeph raio, ransverse reinorcemen conen, coninemen eeciveness coeicien, and percenage o inermediae longiudinal reinorcemen. Reerences ACI, Manual o Concree Pracice, Commiee 8-R-8, American Concree Insiue, Deroi, 98. chheim, M., and Moehle, J.P., Shear srengh and deormailiy o RC ridge columns sujeced o inelasic displacemens, UCB/EERC 9/, Universiy o Caliornia, Berkeley, 99. Biskinis, D.E., Roupakias, G.K., and Fardis, M.N., Degradaion o shear srengh o reinorced concree memers wih inelasic cyclic displacemens, ACI Srucural Journal, (),, Elwood, K.J., and Moehle, J.P., Dri capaciy o reinorced concree columns wih ligh ransverse reinorcemen, Earhquake Specra, (),, FEMA 7/7, 997, NEHRP guidelines or he seismic rehailiaion or uildings, Federal Emergency Managemen Agency, Washingon D.C. p/9p. Galal, K., Araa, A., and Ghoarah, A., Reroi o RC square shor columns, Eng. Sr. J., 7(),, 8-8. Kowalsky, M.J., Priesley, M.J.N., and Seile, F., Shear and lexural ehavior o lighweigh concree ridge columns in seismic regions, ACI Srucural Journal, 9(), 999, -8. Mander J.B., Priesley, M.J.N., and Park, R., Theoreical Sress-Srain Model or Conined Concree, Journal o Srucural Engineering, (8), 988, 8-8. Mirza, S., Flexural siness o recangular RC columns, ACI Sr. J., 87(), 99, -. Panagioakos, T.B. and Fardis, M.N., Deormaion o reinorced concree memers a yielding and ulimae, ACI Srucural Journal, 98(),, -8. Paulay, T., A redeiniion o siness o reinorced concree elemens and is implicaions in seismic design, Srucural Engineering Inernaional, (),, -. Priesley, M.J.N., Verma, R., and Xiao, Y., Seismic shear srengh o reinorced concree columns, Journal o Srucural Engineering, (8), 99, -9. Sezen, H., Seismic response and modeling o lighly reinorced concree uilding columns, Ph.D. disseraion, Dep. o Civil and Environmenal Engineering, Universiy o Caliornia, Berkeley,. Waanae, F., and Ichinose, T., Srengh and duciliy o RC memers sujeced o comined ending and shear, Concree Shear in Earhquake, Elsevier Applied Science, New York, 99, pp. 9-8.