Evaluation of Residual Axial Load Capacity of RC Columns. after Shear Failure

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Procings o th Tnth Paciic onrnc on Earthquak Enginring Builing an Earthquak-Rsilint Paciic 6-8 ovmbr 05, n, Australia Evaluation o Rsiual Axial Loa apacit o R olumns Y.

akano Institut o Inustrial cinc, Th Univrsit o Toko, Toko, Japan ABTRAT: This papr introucs an arch rsistanc mol to valuat th rsiual axial loa carring capacit o a Rinorc oncrt R column, which has sur

Th valuation ormula o th rsiual axial loa carring capacit, which consists o longituinal rinorcmnt an conin concrt contributions, is also riv bas on th propos mol.

ITRODUTIO From th post-arthquak rconnaissanc surv, it is obsrv that rinorc concrt R short columns an R columns with poor transvrs rinorcmnt ar vulnrabl to shar ailur.

1 Procings o th Tnth Paciic onrnc on Earthquak Enginring Builing an Earthquak-Rsilint Paciic 6-8 ovmbr 05, n, Australia Evaluation o Rsiual Axial Loa apacit o R olumns Y. Yang atr har Failur Dpartmnt o Architctur, Th Univrsit o Toko, Toko, Japan K. atsukawa, H. hoi & Y. akano Institut o Inustrial cinc, Th Univrsit o Toko, Toko, Japan ABTRAT: This papr introucs an arch rsistanc mol to valuat th rsiual axial loa carring capacit o a Rinorc oncrt R column, which has sur svr shar ailur u to a larg arthquak. Th propos arch rsistanc mol, bas on th thor o structural mchanics, can giv a bttr unrstaning o th loss o axial loa carring capacit o a R column. Th valuation ormula o th rsiual axial loa carring capacit, which consists o longituinal rinorcmnt an conin concrt contributions, is also riv bas on th propos mol. In aition, th atabas o loaing tsts carri out b othr rsarchrs is also compil to vri th ormula. Th stimation rsults b th propos ormula show a goo agrmnt with tst rsults. ITRODUTIO From th post-arthquak rconnaissanc surv, it is obsrv that rinorc concrt R short columns an R columns with poor transvrs rinorcmnt ar vulnrabl to shar ailur. For svrl shar-amag R columns without ajacnt loa ristribution mmbrs aroun thm, th trioration o th axial loa carring capacit o ths amag columns can la to a global or partial builing collaps Fig.. For xisting columns vulnrabl to shar ailur as mntion abov, it is thror ncssar to valuat th rsiual axial loa carring capacit an to giv avic or sismic rsistanc valuation or sismic rtroit. Until now, svral valuation mols hav bn propos b othr rsarchrs to account or th loss o axial loa carring capacit or such R columns pron to shar ailur. Uchia an Uzono 00 hav prsnt an valuation mol bas on th principl o virtual work to prict th rsiual axial loa carring capacit. Elwoo an ohl 005 hav vlop a shar-riction mol to prict th rit angl at which axial collaps occurs. Takain an Yoshimura 007 hav also propos an valuation mol to stimat th axial loa carring capacit atr shar ailur, bas on th concpt o ruc ailur surac an rgrssion analsis o svral tsts rsults. Howvr, or R columns ail in shar Fig., whn th covr concrt is totall rmov an th conin cor concrt is compltl crush unr arthquak motions, it is not as to obtain an intuitiv unrstaning rgaring th loss o axial loa carring capacit through ths mols. From akano Lab. II, UToko Fig.. Axial collaps 995 Kob Earthquak From akano Lab. II, UToko Fig.. har ailur 995 Kob Earthquak Papr umbr 5

2 Hnc, in this papr, an arch rsistanc mol is prsnt or shar amag R columns with crush cor concrt, which can giv a bttr unrstaning o th loss o axial loa carring capacit. This mol is uc bas on th thor o structural mchanics an obsrvations ma on R columns havil amag in shar. Th rivation o th valuation ormula is also prsnt in this papr. In aition, th atabas o svral loaing tsts carri out b othr rsarchrs is also compil to vri th valuation ormula. DEVELOPET OF THE ARH REITAE ODEL. Dinition o limit stat o axial collaps In this sction, or a R column with crush cor concrt, as shown in Fig., th limit stat o axial collaps is in b analsing th chang in intrnal orc Q, an, at both n sctions o shar ail portion with horizontal isplacmnt. It can b consir that th rotation o th shar ail portion ns is rstrain b th uppr an lowr loor slabs or non-structural walls an th intrnal momnts at th n sctions ar qual to ach othr. Thus, or th shar ail portion, th quilibrium quation o momnt can b writtn as Eq. an th intrnal momnt at th n sctions can b xprss as Eq.. Whn th axial orc is a constant, th chang o intrnal momnt with horizontal isplacmnt can b shown as Fig.. Th bning momnt 0.5 contribut b axial orc linarl bhavs with horizontal isplacmnt. Howvr, th bning momnt 0.5QL provi b shar orc is non-linar bcaus th iling occurs at both ns an it crass along with incras in horizontal isplacmnt. Whn th shar orc gras to zro with incras in horizontal isplacmnt, th total bning momnt acting on th n sctions is qual to th maximum momnt capacit o column sction as shown b A in th igur. I th horizontal isplacmnt incrass urthr, th quilibrium btwn intrnal orcs an momnts will b lost an th axial collaps will tak plac. It can b conclu that th stat o shar orc qual to zro is th limit stat o quilibration at which th horizontal isplacmnt rachs a maximum. In this papr, or a R column unr a constant axial orc with incras in horizontal isplacmnt, th stat o shar orc qual to zro is in as th limit stat o axial collaps. At th in limit stat o axial collaps, th rsiual axial loa carring capacit can b consir qual to th constant axial orc. In this papr, this charactristic will b us to vlop th valuation mol o rsiual axial loa carring capacit. shar orc Q axial orc momnt n sction =0.50.5QL limit stat o axial collaps Q=0 A L0- L= L0 shar ail portion 0.5QL maximum momnt capacit n sction 0.5 Fig.. Fr bo iagram o shar ail portion with crush cor concrt Fig.. Rlationship btwn intrnal momnt an horizontal isplacmnt at th n sctions = QL = QL whr =th axial orc; Q =th shar orc; =th bning momnt o th n sctions; L 0 = th lngth o shar ailur portion in axial irction; an L = th lngth o shar ailur portion in vrtical irction.

3 . Arch rsistanc mol For a shar ail R column at th limit stat o axial collaps, as shown in Fig., whn th covr concrt is totall spall o an th conin cor concrt is compltl crush u to an arthquak, an arch rsistanc mol s Fig. 5 can b vlop unr th ollowing assumptions.. Th bon strngth btwn th rinorcing bars an concrt can b nglct an no transmission o orcs is xpct btwn longituinal rinorcmnt an crush cor concrt.. o rotation is xpct at th ns o th shar ail portion.. Evr longituinal stl bar has th sam intrnal orcs an momnts at n sctions.. Th buckling o longituinal rinorcmnt os not happn prior to th axial collaps. 5. onin cor concrt cannot rsist an bning momnt as th cor concrt is crush compltl. Q=0 Q Q arch ct Q Q Th smbols o intrnal orcs at longituinal bar n sctions ar onl inicat or on o bars. Fig. 5. Arch rsistanc mol Accoring to th inition o th limit stat o axial collaps, th sum o shar orcs o longituinal stl bars an conin cor concrt quals to zro Eq. an th quilibrium quation o momnt is shown in Eq.. It rvals that i th axial orc o conin cor concrt c, th ccntricit o axial orcs carri b conin cor concrt, an th rlationship btwn axial orc an momnt o longituinal stl bar ar givn, th axial orc o longituinal stl bar can b obtain. It shoul b not that in this mol th axial orcs acting with an ccntricit at th n sctions o conin cor concrt part can vlop a orc coupl, which rsists in th momnt inuc b th P- Δ ct o longituinal stl bars Eq.. Th rsistanc o th orc coupl c acting on th crush cor concrt can b consir as th intraction btwn crush cor concrt an longituinal stl bars an it is call arch ct in this rsarch. nq Q = 0 n = n whr =th rsiual axial loa carring capacit; =th axial orc carri b ach longituinal stl bar; Q =th shar orc carri b ach longituinal stl bar; =th bning momnt carri b ach longituinal stl bar; =th axial orc carri b conin cor concrt part; Q =th shar orc carri b conin cor concrt part; =th ccntricit o axial orcs carri b conin cor concrt; =th horizontal isplacmnt; n=th numbr o longituinal stl bars.. Axial orc carri b conin cor concrt part Th ormula o axial orc carri b conin cor concrt is uc bas on th quilibrium o orcs shown in th r bo iagram Fig. 6 an th ollowing assumptions.

4 . At th inclin an horizontal cutting plan, th rlationship o orcs paralll an prpnicular to th cutting plan is subjct to th oulomb s law o riction.. Th plastic strngth o all transvrs rinorcmnt o th shar ail portion can b achiv ull unr th tnsil action inuc b th crush conin cor concrt. Th quilibrium quations o horizontal an vrtical irction o orcs shown in th r bo iagram Fig. 6 can b givn as Eq. 5 an Eq. 6. Th axial orc o conin cor concrt can b givn as Eq. 7 b solving Eq. 5 an Eq. 6. Also, th quilibrium Eq. 8 o momnt at point O shown in Fig. 6, can b vlop to obtain th ccntricit o th axial loa acting on th n sctions o conin cor concrt, as shown in Eq. 9. L0 o Q=μ θ L0 Att s o ' μ' inclin cutting plan θ horizontal cutting plan s L0- Fig. 6. Fr bo iagram o conin cor concrt L0 µ sinθ = µ cosθ At 5 t s = µ sinθ cos θ 6 L0 µ sin θ cos θ = At 7 t s µ sinθ sinθ L0 = µ 8 = µ L 9 0 whr θ= th inclination o cutting plan; s= th spacing o transvrs rinorcmnt; A t = th sction ara o transvrs rinorcmnt; t = th il strngth o transvrs rinorcmnt; = th comprssiv orc prpnicular to th inclin cutting plan; μ= th riction actor.. Axial orc carri b longituinal rinorcmnt bars As mntion abov, xcpt or th axial orc carri b conin cor concrt an its ccntricit obtain in th abov sction, th rlationship btwn axial orc an momnt o longituinal stl bar is also rquir to obtain th axial orc carri b longituinal rinorcing bars. Thus, in orr to riv th rlationship btwn axial orc an momnt o longituinal stl bar, as shown in Fig.7, it is assum that at th limit stat o axial collaps, th n sctions o longituinal bars rach a ull plastic conition. B strss intgration mtho, axial orc an bning momnt can b obtain. For asir vlopmnt o th rlationship btwn axial orc an momnt or longituinal bars, th ial circular cross sction is appli nglcting th ct o stl ribs. Th lasto-plastic mchanical proprt is aopt nglcting strain harning. Th bning momnt an axial orc acting on th n sctions o longituinal bars can b xprss as Eqs. 0 an. From thm, th initial rlationship btwn axial orc an momnt o longituinal bars can b obtain, as shown in Eq.. Th istanc h rom cntroi o ara A to th cntrlin in Eq. an Fig. 7 can b trmin b Eq.. Th axial orc can also b obtain b strss intgration ovr comprssiv part o n sction, as

5 shown in Eq.. B liminating th paramtr x an h, an b combining th thr quations Eqs., an, th inal rlationship btwn axial orc an momnt o longituinal bars can b shown as Eq. 5. B solving th quations Eqs. 6, 9 an 5, th axial orc o longituinal bars can b trmin. Howvr, it is iicult to obtain an xplicit unction or axial orc o longituinal bars in trms o horizontal isplacmnt an it is too complx or practical applications. o, or th actual practic o sismic rsistanc valuation or sismic rtroit, th intraction rlationship btwn axial orc an momnt ns to b simplii to obtain an xplicit unction or axial orc o longituinal bars. Thus, b using th lliptic unction Fig.8, Eq. 6 to approximat th intraction rlationship btwn axial orc an momnt Eq. 5, an xplicit unction or axial orc o longituinal bars can b obtain Eq. 7. As shown in Fig.8, compar with linar approximation introuc b EIwoo an ohl, 005, th mtho propos in this papr has a rlativl high accurac, although th ormula itsl is complx. Q Q n sction tnsil bar n sction nutral axis x h ara A cntroi o ara A cntrlin tnsion strss istribution in ull plastic sction comprssion Fig. 7. Ial plastic sction at th ns o longituinal bars or th shar ail portion π Eq. 6 Eq. 5 linar approximation EIwoo an ohl, Fig. 8. Axial orc an momnt intraction rlationship = A h 0 = π A = π h x h = π 5

6 6 x x x arcsin 0 = θ arcsin = 5 = π 6 = n n n π π 7 whr =th iamtr o longituinal bar; =th il strngth o longituinal bar. A= th ara o comprssiv part o n sction. h= th istanc rom cntroi o ara A to th cntrlin. x=th istanc btwn nutral axis an cntrlin..5 Rsiual axial loa capacit o R columns atr shar ailur Thror, th rsiual axial loa carring capacit can b valuat as Eq. 8, consisting o longituinal rinorcing bars an conin cor concrt contributions. n = 8 APPLIATIO OF THE ARH REITAE ODEL To vri th accurac o th ormula o rsiual axial loa carring capacit propos abov, a atabas o tst rsults rom svral shar ail R column spcimns ar compil Tabl. In th vriication, th ollowing assumptions ar ma.. Th critical shar crack angl th angl btwn th crack surac an longituinal irction o column whn th column ails in shar or shar-critical column can b takn as 60 Kato D., Li Z., akamura Y., an Hona Y., 007. To stimat th concrt contribution to rsiual axial loa carring capacit Eq. 7, th lngth L 0 o shar ail portion is rquir an it can b trmin b th critical shar crack angl an th pth o column sction.. Th riction actor μ is aopt as 0.6 Architctural Institut o Japan, 00 nglcting th intrlocking ct o crush concrt. Although it is rport Elwoo an ohl, 005 that th riction actor as shown in Fig. 6 is a unction o th rit angl whn axial ailur occurs, it is takn as a constant valu o 0.6 to simpli th ormula in this papr. Th comparison btwn th calculat an masur rsults o th rsiual axial loa carring capacit is shown in Fig. 9a. As can b oun in th igur, th calculat rsults hav a goo agrmnt with th masur rsults xcpt or spcimns with transvrs rinorcmnt ratio mor than 0.5% PG.7 an PG.0. For th spcimns PG.7 an PG.0, th propos ormula o rsiual axial loa carring capacit las to ovrstimat rsults. It is possibl bcaus th il capacit is not achiv ull in th transvrs rinorcmnt o shar ailur portion at th limit stat o axial collaps an this is not consistnt with th assumption mntion abov. Th vali rang o transvrs rinorcmnt ratio or this assumption shoul b urthr iscuss in th utur.

7 Tabl. Databas o shar ail columns pcimn D b h n ρ t s t ρ t R mm mm mm mm Pa % mm mm Pa % k mm Ru Y., akamura T., an Yoshimura., Ishigami., Owa ira., akamura Takaa., an Yoshimura., Yoshimura., Takain Y., an akamura T., 00 O O O Yamanaka., an Yoshimura., Kato D., Li Z., akamura Y., an Hona Y. 006 DW akamura T., uto., Ito., an Yoshimura., 0 PG PG ot: D=th column sction pth; b=th column sction with; h=th column intrior nt hight; =th iamtr o longituinal bar; n= th numbr o longituinal bar; =th il strngth o longituinal bar; ρ=th longituinal rinorcmnt ratio; t =th iamtr o transvrs rinorcmnt; s=th spacing o transvrs rinorcmnt; t =th il strngth o transvrs rinorcmnt; ρ t =th transvrs rinorcmnt ratio; R =th rsiual axial loa carring capacit at th in limit stat qual to constant axial orc; =th horizontal isplacmnt convrt rom th rit angl at which th shar orc qual to zro In aition to th application o th arch rsistanc mol propos in this papr, th shar-riction mol introuc b EIwoo an ohl 005 is also xamin using sam atabas shown in Tabl. In th application o th shar-riction mol, th pth o column cor c cntrlin to cntrlin o transvrs rinorcmnt is takn as 0.8 tims o th pth o column; Th critical shar crack angl is aopt as 65 EIwoo an ohl, 005; Th riction actor μ is aopt as a constant valu o 0.6 to simpli th application procss o th mol, although it is rport Elwoo an ohl, 005 that th riction actor is a unction o th rit angl whn axial ailur occurs. Th rsult o application Fig. 9b shows that th shar-riction mol givs unrstimations o th rsiual axial loa capacit or most o th spcimns. This is primaril bcaus that th intraction btwn crush cor concrt an longituinal stl bars is not consir in th shar-riction mol. As scrib in sction., this intraction is takn into account in th valuation mol propos in this rsarch, which can giv a bttr stimation o th rsiual axial loa capacit. OLUIO Bas on th thor o structural mchanics an th obsrvation o R columns svrl amag in shar, th arch rsistanc mol to prict rsiual axial loa carring capacit is propos an its accurac is iscuss b comparing ata provi b th prcious tsts. For most o th spcimns inclu in th compil atabas, th stimat rsiual axial capacit has a goo agrmnt with th 7

8 masur rsults. Howvr, it ovrstimat th rsults or spcimns with high latral rinorcmnt ratios PG.7 an PG.0 an th strss istribution assumption or transvrs rinorcmnt is n to improv th mol or R columns with a high transvrs rinorcmnt ratio. alculat axial loa k ρ 0.5% t ρ>0.5% t PG.0 PG asur aixal axial loa k asur axial loa k a Arch rsistanc mol alculat axial loa k b har-riction mol Fig. 9. omparison o calculat-to-masur rsiual axial loa carring capacit 5 AKOWLEGET Th stu was support b JAPA OIETY FOR THE PROOTIO OF IEE unr Grant o. 609 Principal invstigator: Y. akano an LIXIL J Founation unr Grant o.-5 Principal invstigator: K. atsukawa. This support is gratl acknowlg. All opinions xprss in this papr ar soll thos o th authors an o not ncssaril rprsnt th viws o th sponsors. REFEREE: Architctural Institut o Japan. 00. AIJ Guilins or th Dsign o tructural Prcast oncrt Emulating ast-in-plac Rinorc oncrt. Toko: Architctural Institut o Japan. in Japans EIwoo K.J., ohl J.P Axial capacit mol or shar-amag columns. AI tructural Journal Ishigami., Owa ira., akamura Takaa., an Yoshimura. 00. Axial loa carring capacit o sharailing R short columns: Part Outlin o tsts, axial ormation-latral ormation rlations an collaps bhaviro. ummaris o Tchnical Paprs o Annual ting Architctural Institut o Japan in Japans Kato D., Li Z., akamura Y., an Hona Y Tsts on axial loa capacit o shar ailur R/ columns consiring rinorcing tails: rlationship btwn axial loaing tst an latral loaing tst. Journal o tructural an onstruction Enginring in Japans Kato D., Li Z., akamura Y., an Hona Y Exprimntal stu on rsiual axial loa capacit o R/ columns. Journal o tructural an onstruction Enginring in Japans akamura T., uto., Ito., an Yoshimura. 0. Ect o longituinal rinorcmnt ratio on sismic prormanc o R columns with shar mo: collaps tst o R short columns with larg hoop ratio. ummaris o Tchnical Paprs o Annual ting Architctural Institut o Japan in Japans Ru Y., akamura T., an Yoshimura. 00. Axial loa carring capacit o R columns subjct to sismic actions. Procings o th Japan oncrt Institut in Japans Takain Y., an Yoshimura Damag valuation o R/ shar columns bas on concpt o ailur surac contraction. Journal o tructural an onstruction Enginring in Japans Uchia Y., an Uzono Y. 00. Juging collaps o R an R columns ail b shar. Journal o tructural an onstruction Enginring in Japans Yamanaka., an Yoshimura ollaps o Flxur-shar an shar ailing R columns subjct to low axial loa. Procings o th Japan oncrt Institut in Japans Yoshimura., Takain Y., an akamura T. 00. Axial ollaps o Rinorc oncrt olumns. Th th Worl onrnc Earthquak Enginring, Papr o

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