3 t World Conference on Eartquae Engneerng Vancouver, B.C., Canada Augut -6, 4 Paper No. 5 DYNAMIC RESPONSE AND COLLAPSE OF COLUMN MODELS Fumng ZHANG SUMMARY A te collape of renforced concrete column attrbuted to te lo of vertcal bearng capacty accompaned wt cumulatve ncreae of vertcal deformaton, analy tat able to nclude te vertcal repone ndpenable for collape mulaton of buldng tructure. In t paper, te collape of renforced concrete column model tuded, ung a mplet lumped ma model conderng te orontal and vertcal vbraton. Fber model of platc nge for RC column are appled, and factor decdng column collape uc a P-delta effect and tre deteroraton of materal are taen nto conderaton. It ndcated tat column model wll collape at a maller eartquae nput level for a multaneou nput of vertcal moton, te ultmate afety of buldng ould be evaluated conderng te 3-dmenonal dynamc beavor of vertcal member. INTRODUCTION Recently, very large pea acceleraton bot at orontal and vertcal drecton were recorded durng ntene near-fault eartquae and t may gve u two reearc ubject to nvetgate. Frtly, a te large orontal pea value may eceed g n PGA and m/ n PGV, t ncreangly neceary to cec te ultmate afety and capacty of tructure ung eartquae load or record everal tme larger tan condered n emc degn code. At te ame tme, analy ould conder te poble trengt deteroraton and negatve tffne at deformaton far beyond te lmt of non-negatve tffne retorng force model. Secondly, te large vertcal component requre u to nvetgate te 3D dynamc repone of tructure, and te 3D retorng force caractertc and te 3D retng capacte of tructure. Untl now, numerou tude were carred out to nvetgate te effect of vertcal ground moton [], and t wa found tat vertcal moton mgt ncreae te aal force of column and te orontal repone of buldng. However, te frame model not able to nvetgate te tructural repone untl buldng collape, a te retorng force model are uually applcable only to a relatvely mall elato-platc deformaton. Analytcal program by element dcretaton (fber model) enable u drectly ncludng te trengt deteroraton from te tre-tran relaton of materal []. It wa appled for nvetgatng te effect of Reearc Engneer, EdM-NIED (Eartquae Dater Mtgaton Reearc Center, Natonal Reearc Inttute for Eart Scence and Dater Preventon), Kobe, Japan. Emal: ang@edm.boa.go.jp.
vertcal moton on te repone of renforced concrete frame buldng, and found tat te multaneou nput of vertcal moton doe ncreae te damage of concrete element and te damage of buldng [3]. It epected tat t nd of optcated program be appled for a drect dcuon of tructural repone not only wtn a moderate damage level but alo up to collape. Before tudyng te collape of real tructure, mple model and parametrc tude may be ueful for pcng up te man factor decdng tructure collape, and epoe te problem and lmt of current emc degn. Te autor nvetgated te collape and te effect of vertcal moton ung mplfed DOF model, for RC column aumng contant tory deformaton and wtout conderng P-delta effect [4], for RC oft tory buldng [5][6]. Tee tude concluded tat te penomenon of collape mgt be decrbed a te dvergence of vertcal and orontal dplacement. And even te orontal repone may be not affected by te nput of vertcal moton f te damage tll moderate, te vertcal downward dplacement wll be accumulated and te model wll collape at a maller eartquae nput level for a multaneou nput of vertcal moton. Te vertcal retance of column mut be nured for avodng tructure collape. Te gnfcance of vertcal vbraton may be reaoned from te fact tat te collape of column accompaned wt te accumulaton of compreve aal tran; t wa evdenced from numerou tatc tet of renforced concrete column. Te table lmt of RC column for orontal retorng force wa propoed baed on te aal deformaton [7]. T paper nvetgate te dynamc repone and collape of column located at te frt tory of RC buldng. Te analytcal model modfed from a popular loadng etup ntalled n Japan were column ubjected to ant-ymmetrc double curvature fleure deformaton. Reearc focued on te vertcal retance of RC column, and orontal trengt and long-term aal tre are two man parameter for nvetgaton. Fber model of platc nge for RC column are appled, and factor decdng column collape uc a P-delta effect and tre deteroraton of materal are taen nto conderaton. Numercal ntegraton performed ung te correct ntantaneou tffne of element by predctng te tme ncrement compatble wt tffne varaton. For mplcty, only one of te two orontal eartquae component condered. SIMPLIFIED DYNAMIC MODEL FOR REINFORCED CONCRETE COLUMNS Analytcal Model Safety of buldng tructure largely ndebted to te capacte of vertcal member at te frt tory. Uually, tatc tet mantan a contant aal force durng te lateral loadng reveral; ere a dynamc analytcal model wt DOF n Fgure aumed for nvetgatng te eartquae repone of RC column, and only one orontal component condered. Altoug te aal tre of column may be trongly related to te overturnng moment [5], t paper dcue te dynamc collape of column located at te central part of buldng frame. In Fgure, RC column deform ant-ymmetrcally n double curvature fleure, and te deformaton wll concentrate at te end of member at large deformaton. Repone and collape of te analytcal model wll be determned not only by te caractertc of te column but alo te rgdte and v of upper tore at orontal and vertcal drecton. Let EI and EA be te bendng and aal elatc tffne of RC column, ten te rgdte of upper tore are aumed a follow n proportonal to te egt H of te lamped ma.
N,Q m, v H l EA EI N,M (nge) D l / (a) Hnge model for RC column and tran dtrbuton e l Aal tran e φ Curvature φ e Ground moton EA,EI Fgure. An analytcal model for tudyng te dynamc collape of RC column & & (nge) & & l e (b) Contrbuton to column deformaton Fgure. Platc nge model and deformaton of one alf of a column e v EI () ( H ) EA () ( H ) Actually, te value of lamped ma rele on te deformaton dtrbuton of te frt tory and upper tore. Aumpton of contant ma could not account te effectve ma aocated wt te deformaton concentraton at te frt tory after eperenced large deformaton. Platc Hnge Model for RC Column Platc nge model appled for RC column wt unform ecton and renforcement. In Fgure, te deformaton dtrbuton of one alf of a column aumed a own n Fgure (a), and deformaton wll be concentrated at te platc nge after yeldng of column. For te regon outde te platc nge, a unform aal tran and a trangle curvature dtrbuton are aumed correpondng to te aal force and bendng moment. Fgure (b) ow te component of orontal and vertcal dplacement {, } T, and tey are epreed n ncremental formaton by Equaton (3) a te um of deformaton reulted by platc nge and deformaton by te regon outde of platc nge. Here, te egt of column, l repreent te lengt of te platc nge, { φ, } T are te curvature and aal tran at platc nge. EI and EA are aumed a contant by ung ter elatc rgdte. ( l ) 3 l φ M EI (3) l l N ( ) EA Increment of retorng force { Q, N} T of column are related wt te tree { M, N} T at column end by Equaton (4), were te econd and te trd term account te P-delta effect. Q N M N M N N (4)
Fber Model at Platc Hnge At te poble platc nge, column ecton dvded nto dcrete element. Applyng te aumpton tat plane cro ecton reman plane after deformaton, tran of te t element wt a dtance of from te center of ecton wll be epreed by curvature φ and aal tran of te ecton a follow. φ (5) Te local tffne matr [ ] for platc nge ecton epreed a follow, were A E, are te tangent modulu and area of dcrete element. [ ] Σ Σ Σ Σ A E A E A E A E N M φ φ φ (6) Compatblty and Equlbrum of Column Subttutng Equaton (6) nto Equaton (3), we ave te compatblty matr [ ] T between column dplacement and deformaton at platc nge a follow. [ ] T φ or [ ] T φ (7) [ ] [ ] EA l EI l l l T ) ( ) ( 3 (8) Subttutng Equaton (7) nto Equaton (4), we ave te tffne matr [ ] c of column a follow. [ ] N Q c, were [ ] [ ][ ] N T c (9) Equaton of Moton and Overall compatblty Let te dplacement and retorng force at te orontal and vertcal drecton to be, and N Q,, and ntal vcou dampng factor, are ndependent and proportonal to ter ntal elatc crcular frequence ω ω,, te ncremental equaton of moton are epreed a follow. Were, repreent ground moton n eac drecton. Aal force N wll be mnu value at compreon. m N Q m m && && & & && && ω ω () Dplacement of te lamped ma are epreed by Equaton (), and ung Equaton (9) and (), te overall tffne [ ] K wll be epreed by Equaton (). / / N Q v () [ ] K N Q, were [ ] [ ] / / K c v () Furter, ung Equaton (7),(9) and (), we ave te compatblty matr [ ] T between overall dplacement and deformaton at platc nge a follow. [ ] T φ, were [ ] [ ][ ] [ ] K T T c (3)
DYNAMIC ANALYSIS OF REINFORCED CONCRETE COLUMNS Contructve Rule of Materal Te tre-tran ( σ ) relaton of renforcement and concrete are aumed a own n Fgure 3. Tenon trengt of concrete gnored, and µ c c te ultmate tran of concrete. For renforcement, buclng mplfed a a tragt lne wt negatve tffne. Young modulu of renforcement and concrete are aumed to be E MPa, Ec Fc / MPa repectvely. σ / E Non-Iteratve Integraton Metod Ung Intantaneou Stffne A epreed by Equaton () and (3), bot compatblty matr [ T ] and overall tffne matr [ K ] are related wt te ntantaneou matr [ ] of platc nge. Terefore, repone analy ould ue te correct value of [ ],.e., te correct tangent modulu of dcrete element, to enure te equlbrum and te compatblty condton. y renforcement Wtn te record tme nterval t of an eartquae wave, Newmar β metod ( β / 4) appled temporally ung an approprate contant tme nterval. If a tffne varaton confrmed for any element, ten te tme ncrement t compatble to te frt of tffne varaton calculated for repone regreon [8]. T varable tme ncrement wa made poble by ung eplct formula about overall dplacement, and about element tran from te relaton of Equaton (5). For eac element, a quadratc equaton for loadng and a lnear equaton for unloadng are obtaned for olvng te tme ncrement t compatble to tffne varaton. Te correct t wll be te mnmum value tred for all fber element. Eartquae Record and Inputted Eartquae Moton Vertcal (SCS-UP) and one orontal (SCS5) component of Sylmar-converter Staton, January 7, 994 Nortrdge eartquae are ued, record tme nterval t. 5, and duraton 4 econd. Te pea value of eartquae record are PGA UP 5.6m /, PGA 6.m / repectvely. In order to obtan te mamum nput level leadng to column collape, eartquae record of bot drecton are multpled by a ame multpler Ω, and te analy to fnd te mnmum of Ω leadng to collape for dfferent analytcal parameter combnaton. Multpler Ω ued for convenence of dcuon; t may contradct wt te fact tat te rato of pea acceleraton between vertcal and orontal component a well a te wave form may vary for dfferent eartquae ntenty. Analytcal Parameter For column of a n-tory RC buldng, aumng te egt of lamped ma H ( / 3)n, te tory egt 3.5m 35cm, te wegt of a ngle tory W 4tf 4gf, ten ung te long-term aal force rato n of column defned by Equaton (4), te e of ecton wll be decded by Equaton (5). mg n W n (4) DF D c F c y σ σ y 5 y E E 5 y σ y E y Fgure 3. Contructve rule of materal µ c c F / E c c 5 c / c concrete c E c c 4 σ / F c F c
n W D (5) n F c Te number of tory ( n 3,5,7,9) and te long-term aal force rato ( n. to.4, wt a ptc of.5) are coen a te analytcal parameter. Two nput condton are nvetgated about, () multaneou nput of vertcal ground moton, () vertcal ground moton not nputted but varable aal force by vertcal dplacement condered. Tme tory analy performed for an nput wave tat are Ω tme of te eartquae record. Computaton termnated wen meetng te followng condton aumed for column collape. After tat tme, te column wll accompany wt a dvergence of orontal and/or vertcal dplacement. () Lo of orontal retance wen / > qy. T a condton for column long t orontal retorng force due to P-delta effect, wen aumng te f end avng a perfect elato-platc moment-rotaton relatonp under contant aal force. () Lo of vertcal retance due to te falure of core concrete at platc nge wen < µ c c. T mean te aal tran at te center of ecton eceed two tme of te ultmate tran of concrete materal. For analytcal model, ntal dampng at orontal a well a vertcal drecton are taen a.,., a we ave lttle nformaton untl now about te vertcal dampng. Te lengt of poble platc nge aumed be L D, te ecton dvded nto concrete and renforcement element. For a quare RC ecton, renforcement rato aumed be p g.%, te yeldng trengt of renforcement be σ y 35MPa, te compreve trengt of concrete be F c 3MPa. For te tre-tran ( σ ) relaton of materal, te compreve tran of renforcement wen fracture occurred µ 5, and te ultmate tran of concrete µ c c aumed a µ c,. Te mamum bendng moment of a RC ecton may vary wt te varaton of aal force. A a roug etmaton, orontal trengt coeffcent q y of te model evaluated appromately by Equaton (6) ung a contant aal force. q y n D n n ( ).4 p g σ F y c (6) Terefore, f te renforcement rato p g and trengt of materal σ y, Fc are gven, q y decded only by te long-term aal force rato n and te number of tory n. In Table, elatc vbraton perod are lted for ome parameter. Tme ncrement for numerc ntegraton lmted a t t / 5., tat meet te requrement for nurng computaton accuracy wen ung varable tme ncrement [8],.e.,. maller tan / of mnmum vertcal vbraton perod ( T.58 for n 3 and n. ). Table. Analytcal parameter for 3-tory and 9-tory column model n H(m) mg(tf) D(m) n qy T () T () n H(m) mg(tf) D(m) n qy T () T () 3 7..63..5.3.58 9. 36.95..866.3. 3 7..56.5.39.48.7 9. 36.894.5.535.48.3 3 7..447...643.8 9. 36.775..384.643.4 3 7..4.5.7.83.9 9. 36.693.5.96.83.59 3 7..365.3.38.964. 9. 36.63.3.39.964.74 3 7..338.35.4.4.9 9. 36.586.35.98.4.88 3 7..36.4.96.85.6 9. 36.548.4.67.85.
DYNAMIC RESPONSE AND COLLAPSE OF RC COLUMNS Dynamc repone of column dcued for µ c, and µ c c te ultmate tran of concrete a llutrated n Fgure 3. Mamum Repone In order to fnd te eartquae nput level leadng to column collape, Fgure.4 and 5 ow te mamum repone v. Ω for 3-tory and 9-tory model repectvely. Ω a multpler for Sylmar-converter taton record, and tarted from.5 to te value leadng to column collape. (a) Mamum orontal dplacement Mamum orontal dplacement X ma (n cm) ncreae wt Ω, and at a certan nput level, X ma become untable and ncreae dramatcally wt a very mall ncreae of Ω. In order to avod te dvergence of orontal dplacement, X ma mut be retrcted wtn ome value. Actually, t te orontal dplacement wen column lot t orontal retance. T dplacement ncreae wt a lower long-term aal force rato n, or a larger orontal trengt coeffcent q y (ee relaton between n and q y by Equaton (6) and Table ). In bot Fgure of 4(a) and 5(a), for n.,.5,.,. 3, te lmt of X ma are maller tan X 5cm,cm,5cm, cm, tey are muc maller tan te deformaton capacty obtaned from conderng te P-delta effect of column ( X qy 75cm,8cm,77cm, 48cm, for 3-tory column model). (b) Mnmum vertcal dplacement Te downward mnmum vertcal dplacement Z mn (n cm) ncreae wt Ω and n, and at a certan nput level correpondng to te table lmt of orontal dplacement, Z mn alo become untable and ncreae dramatcally wt a very mall ncreae of Ω. Te lmt of Z mn about -.5cm for 3-tory column model, and about -cm for 9-tory column model. Te dfference may be aren from te elatc deformaton outde te platc nge. (c) Mamum vertcal dplacement Te upward mamum vertcal dplacement Z ma alo ncreae wt Ω. However, for ger long-term aal force rato n. 3, tere are no tenle deformaton occurred untl column collape. (d) Mamum orontal retorng force Te mamum orontal retorng force Q ma / mg (normaled by wegt mg) ncreae wt Ω f t mall and column wtn elatc tate. For lower long-term aal force rato n.,. 5, Q ma / mg ncreae wt Ω even after yeldng of column. However, for ger long-term aal force rato n.,.3, Q ma / mg reman contant rrelevant wt te ncreae of Ω. (e) Mnmum vertcal retorng force Te compreve mnmum vertcal retorng force N mn / mg (normaled by wegt mg) ncreae wt Ω. For lower long-term aal force rato n.,. 5, N mn / mg may be a large a 6 or 4. However, f t normaled by te compreve trengt of concrete D Fc, ten te mamum of Nmn / DFc near collape wll be N / mn D Fc n N mn / mg.5. 6 for dfferent n. Terefore, te aal compreve force lmted wtn te compreve force capacty of column wen ubjected to large bendng deformaton. (f) Mamum tran rate of fber element Te mamum tran rate of fber element at platc nge etremely large, and may be a large a./-.3/. T paper a not ncluded te effect of tran rate on te contructve rule of materal.
3 Xma (cm) n..5..3 Zmn (cm) - n..5..3 Zma (cm) - n..5..3..5..5. eartquae nput amplfer Ω..5..5. eartquae nput amplfer Ω..5..5. eartquae nput amplfer Ω (a) Ma. orontal dplacement (b) Mn. vertcal dplacement (c) Ma. vertcal dplacement.8 -.3 Qma/mg.6.4. n..5..3..5..5. eartquae nput amplfer Ω Nmn/mg -3-5 -7 n..5..3..5..5. eartquae nput amplfer Ω tran rate (/ec)....5..5. eartquae nput amplfer Ω n..5..3 (d) Ma. orontal retorng force (e) Mn. vertcal retorng force (f) Ma. tran rate of fber element Fgure 4. Mamum repone of 3-tory column model under varou nput amplfer Xma (cm) 3 n..5..3 Zmn (cm) - n..5..3 Zma (cm) n..5..3..5..5. eartquae nput amplfer Ω..5..5. eartquae nput amplfer Ω -..5..5. eartquae nput amplfer Ω (a) Ma. orontal dplacement (b) Mn. vertcal dplacement (c) Ma. vertcal dplacement Qma/mg.5.5..5..3..5..5. eartquae nput amplfer Ω n Nmn/mg - -3-5 -7 n..5..3..5..5. eartquae nput amplfer Ω tran rate (/ec).3.. n..5..3..5..5. eartquae nput amplfer Ω (d) Ma. orontal retorng force (e) Mn. vertcal retorng force (f) Ma. tran rate of fber element Fgure 5. Mamum repone of 9-tory column model under varou nput amplfer
Retng Capacty of Column Model Te ultmate retng capacty of tructure for preventng collape may be nterpreted a te capacty to ret gravty load, a well a te eartquae acton bot at te orontal and at te vertcal drecton. In t paper, collape of column model defned a te dvergence of orontal or vertcal dplacement, and a a matter of factor, tey occur multaneouly and uddenly wt a very mall ncreae of eartquae nput level. In Fgure 6(a), te capacty to ret an eartquae nput level Ω plotted agant te orontal trengt coeffcent q y. Ω leadng to collape ncreae wt q y, and alo affected by te number of tory. In Fgure 6(b), te capacty agant collape for retng gravty load and eartquae nput plotted togeter, and tere a trong correlaton between te long-term aal capacty rato n a wt eartquae nput Ω, T mean tat te ultmate capacty of column ould be evaluated conderng te vertcal retance. Te long-term aal capacty rato n a defned by Equaton (7), normaled by aal capacty of RC column. mg n na (7) D( Fc pgσ y ) pg ( σ y / Fc ) In Fgure 6(b), a curve of quadratc functon between Ω and / na plotted to decrbe te tendency of ter relaton. Etremely, te functon mean RC column could not ret eartquae wen gravty load equal to te aal capacty of column ( Ω, n a ), and RC column wll never collape f wtout gravty load ( Ω, n ). a Yeldng coeffcent qy.8.6.4. 3F 5F 7F 9F..5..5..5 Eartquae nput amplfer Ω for collape Invered aal force rato /na 5 5 8Ω n a 3F 5F 7F 9F..5..5..5 Eartquae nput amplfer Ω for collape (a) Capacty of column wt a orontal trengt (b) Capacty agant gravty load and eartquae acton Fgure 6. Retng capacty of column model v. yeldng coeffcent or long-term aal force rato Retorng Force at Horontal and Vertcal Drecton Hyterc at bot drecton are own n Fgure 7,8 for 3-tory model wt n. 5. Wen Ω. 88 n Fg.7, altoug dplacement at te orontal drecton relatve large and degradaton of orontal retance accompaned, no dvergence of dplacement were occurred. However, a lgtly ncreae of nput Ω., collape occurred at bot drecton n Fg.8 were Ω. 89. In Fgure 8, te orontal retorng force of (a) mlar to tat of bendng moment at platc nge of (c), te effect of P-delta effect on te orontal retorng force mall, and te degradaton of orontal retance mgt be condered a te reult of materal deteroraton (ee Fgure 3). Fgure 8(b) and Fgure 8(d) ow large upward dplacement and large upward deformaton, and te upward dplacement te reult of tenle deformaton at platc nge. Tere no clear yteretc rule at te vertcal drecton.
Tme Htory of Dplacement For 3-tory model wt n. 5 and Ω. 89, Fgure 9 ow te tme tory of (a) orontal dplacement, (b) vertcal dplacement, (c) aal tran of platc nge, (d) vertcal retorng force. A udden ncreae of dplacement n bot drecton occurred at te tme around t ec. Apparently, te dvergence of vertcal dplacement a lgtly fater tan tat of orontal dplacement. For Ω. 88, Fgure ow te tme tory of (a) orontal dplacement, (b) vertcal dplacement, (c) nputted orontal acceleraton, (d) nputted vertcal acceleraton. Te nputted acceleraton were multpled te Sylmar-converter taton record by Ω. 88. Altoug large redual dplacement reulted, tere no dvergence of dplacement at bot drecton. Te large upward vertcal dplacement concde wt te large orontal dplacement ( t 4 ec and t 5ec ), t may be nterpreted a te reult of te movement of neutral a at platc nge. or. force Q/mg.4. -3 - -. -.4 or. dplacement (cm) (a) Horontal retorng force ( or. force Q/mg.4. -3 - -. -.4 or. dplacement (cm) (a) Horontal retorng force moment M/D 3 /Fc. -. -.5 -. curvature D (c) Retorng moment at nge vert. force N/mg - - -3-5 vert. dplacement (cm) (b) Vertcal retorng force Fgure 7. Retorng force.88 vert. force N/mg - - -3-5 vert. dplacement (cm) (b) Vertcal retorng force vert. force N/mg -.. (d) Aal retorng force at nge Fgure 8. Retorng force wt.89 - -3-5 aal tran at nge (a) dp. (cm) - -3-4 tme (ec) 5 5 t.88ec (b) dp. (cm) - tme (ec) 5 5 t.88ec. -. -. (c) tran tme (ec) 5 5 t.88ec (d) vert. force N/mg -4-6 tme (ec) 5 5 Fgure 9. Tme tory of 3-tory model wt n. 5 and.89
(a) dp. (cm) tme (ec) - 5 5-3 -4 (b) dp. (cm) tme (ec) 5 5 - (c) or. Acc. nput 6 (cm/ ) -6 tme (ec) 5 5 (d) vert. Acc. nput 6 (cm/ ) -6 tme (ec) 5 5 Fgure. Tme tory of 3-tory model wt n. 5 and nput acceleraton (.88) INFLUNCE OF VERTICAL MOTION AND VERTICAL VIBRATION ON COLUMN COLLAPSE Dynamc repone of column and te effect of vertcal moton are dcued for µ c, were µ c c te ultmate tran of concrete a llutrated n Fgure 3. Smlar reult and concluon may be obtaned for µ c. Here, HV mean te multaneou nput of orontal and vertcal moton; H mean only te orontal moton wa nputted. Influence of Vertcal Moton on Mamum Repone In Fgure, mamum repone of 3-tory model are plotted agant eartquae nput multpler Ω for (a) vertcal dplacement, (b) vertcal compreve force. Te multaneou nput of vertcal moton reult a larger vertcal downward dplacement and a large vertcal compreve force. Te mnmum vertcal force N mn / mg n compreon proportonal to Ω. For te analytcal parameter dcued, f te vertcal dplacement ece.4cm rrelevant to te nput of vertcal moton, ten a dvergence of dplacement wll be ntated and te column model wll be collaped. Even te vertcal moton not nputted; a larger vertcal compreve force reulted. However, N mn / mg not only affected by te nput level Ω, but alo affected by te long-term aal force rato n. For n.,. 3, N mn / mg not le tan.4,.e., a 4% ncreae of aal compreve force. However, for n.,. 5, N mn / mg may be a large a. to 3., a reult of te large movement of neutral a at platc nge. Vertcal vbraton occur even wtout vertcal nput. Influence of Vertcal Moton on Retng Capacty of Column Wt multaneou nput of orontal and vertcal moton, column model wll collape at a maller nput level; averagely te omon of vertcal moton wll over-etmate te eartquae retant capacty by %. It mut pont out tat te over-etmaton may be more gnfcantly f compared wt te reult ung retorng force obtaned from te aumpton of contant aal force. Agan, n Fgure (c), te capacty agant collape for retng gravty load and eartquae nput plotted togeter. Curve of quadratc functon between Ω and / na are plotted to decrbe te tendency of ter relaton. Here, n a te long-term aal capacty rato. Influence of Vertcal Moton on Retorng Force Fgure ow te retorng force of a 3-tory model wt or wtout vertcal moton. Te multaneou nput of vertcal moton reult large deteroraton of orontal retorng force, and a large varaton of aal force.
Zmn (cm) - HV n..5..3..5..5. eartquae nput amplfer Ω Nmn/mg -4-6 -8 HV n..5..3..5..5. eartquae nput amplfer Ω aal force rato /n a 5 HV 7.5Ω n a 3F 5F 7F 9F..5..5..5 3. nput amplfer Ω for collape Zmn (cm) - n H..5..3..5..5..5 eartquae nput amplfer Ω Nmn/mg - -3-4 H n..5..3..5..5..5 eartquae nput amplfer Ω aal force rato /n a H 3F 5F 5 7F 9F 6.5Ω n a..5..5..5 3. nput amplfer Ω for collape (a) Ma. vertcal dplacement (b) Mn. vertcal force (c) Retng capacty Fgure. Effect of vertcal moton on te mamum repone and retng capacty or. force Q/mg vert. force N/mg.5 4 -.5 or. dplacement (cm) (a) Horontal retorng force - HV (c) Vertcal retorng force HV -4 vert. dplacement (cm) or. force Q/mg vert. force N/mg H.5-4 -.5 or. dplacement (cm) (b) Horontal retorng force - H -4 vert. dplacement (cm) (d) Vertcal retorng force Fgure. Effect of vertcal moton on retorng force ( n. and.85) Influence of Vertcal Moton on te Tme Htory Fgure 3 ow te tme tory of orontal dplacement and vertcal dplacement wt or wtout te nput of vertcal moton. Wtout vertcal moton, occaonally, te redual orontal dplacement mall at te end of computaton. However, te multaneou nput of vertcal moton reult a cm drft of orontal dplacement after te tme of 4ec, and te fnal dvergence of dplacement due to te lo of orontal retance. If vertcal moton alo nputted, te vertcal dplacement drft to te compreve drecton, becaue te accumulaton of compreve tran at te platc nge. Te low-frequency vbraton
dplacement (cm) dplacemet (cm) 4 4 HV tme (ec) tme (ec) 5 5-5 5 HV (a) Dplacement wt multaneou nput of vertcal moton H tme (ec) 5 5 dplacemet (cm) dplacemet (cm) 5 5 - (b) Dplacement wtout nput of vertcal moton but wt conderaton of vertcal vbraton Fgure 3. Effect of vertcal moton on dplacement repone ( n. and.85) H tme (ec) of vertcal dplacement due to te movement of neutral a of platc nge, wle te g-frequency vbraton due to te vertcal nput moton and t reult a large vertcal force. CONCLUSIONS Dynamc repone of renforced concrete column model untl collape were tuded ung a mplet lumped ma model tat wa developed from fber model of platc nge, and ear deformaton and ear falure of column wa not condered. Factor tat may affect column repone uc a P delta effect and deteroraton of materal were taen nto conderaton. Dynamc collape of RC column attrbuted to te dvergence of dplacement and te lo of retance at bot orontal and vertcal drecton. Parametrc tude ung tme tory analy were carred out to fnd te multpler Ω for Sylmar-converter taton record leadng to column collape. Te man fndng about te dynamc repone of column model may be ummared a follow. () Horontal trengt deteroraton of RC column manly due to te tre deteroraton of materal at platc nge; t aggravated by vertcal ground moton. () Te collape of RC column may occur uddenly from a relatvely table tate f te eartquae nput lgtly ncreaed. Retance at bot te orontal and te vertcal drecton are lot after a monotonc ncreae of downward vertcal dplacement. (3) Te capacty of RC column for avodng collape may be better undertood f te gravty load wa plotted agant eartquae nput. If te long-term aal tre rato n or te capacty rato n a doubled, ten te capacty agant eartquae nput wll be reduced to /4. Te ultmate afety of tructure ould be dcued conderng te vertcal retance of column. (4) Te multaneou nput of vertcal moton reult large aal compreve force, and enable te column collape at a maller nput level. Te omon of vertcal moton wll over-etmate te eartquae retant capacty by %. (5) Even te vertcal moton not nputted, vertcal vbraton occur and t may reult a large vertcal compreve force. Te retorng force model obtaned from tatc tet under contant aal force may alo over-etmate te eartquae retant capacty. Te nvetgaton wa made ung a lamped ma model; t could not account te effect of dtrbuted ma
and te vertcal vbraton of beam and floor. Aumpton about te vertcal dampng and te lengt of platc nge were made, and mplfed tre-tran relaton of materal were aumed. It epected tat te 3D retorng force caractertc of RC column wll be tuded by future 3D ang table tet. ACKNOWLEDGEMENTS T reearc wa upported n part by Japan Socety for te Promoton of Scence under Award KAKENHI (No.53634). REFERENCES. Anderon J, Bertero V. Effect of gravty load and vertcal ground acceleraton on te emc repone of multtory frame. EERC Report, No.EERC 733, 973: 75-84.. L K. CANNY-A general purpoe computer program for calculatng nonlnear tatc and dynamc repone of 3-demenonal frame tructure conderng te nteracton among varyng aal load and baal bendng moment. Reearc report, No.CE3, Department of Cvl Engneerng, Natonal unverty of Sngapore, 99.. 3. Kuu K, Naano Y, Oada T, Kumaawa F. Te Effect of vertcal ectaton on repone caractertc of tructure, part, Dynamc nelatc analy under b-drectonal ectaton. Summare of AIJ annual meetng, 994: 679-68 (n Japanee). 4. Zang F. Repone and collape of mplfed buldng ubjected to orontal and vertcal moton., Journal of Structural Engneerng, Vol.4B, Arctectural Inttute of Japan, 996: 4-48 (n Japanee). 5. Zang F. Repone and collape of oft frt tory buldng ubjected to orontal and vertcal moton. Annual Report of Japan Concrete Inttute, Vol.9, 997: 495-5 (n Japanee). 6. Zang F, Yamata T, Kubo T. Energy balance of buldng untl collape. Te Elevent Japan Eartquae Engneerng Sympoum, Toyo, Paper No.84,. 7. Ina E, Hra H. Deformaton capacty beyond fleural yeldng of renforced concrete column, part : Stable lmt and qua-table lmt. Journal of Structure and contructon engneerng, No.44, Arctectural Inttute of Japan, : 67-76 (n Japanee). 8. Zang F, Yamata T, Saa H, Kubo T. Vertcal retance of RC per ubjected to multaneou orontal and vertcal ground moton. Aa Conference on Eartquae Engneerng, Manla, Plppne, Vol., 4:-3.