ISET Joural of Earthquake Techology, Paper No., Vol., No., March 7, pp. 9 9 A RESPONSE SPECTRUM-BASED NONLINEAR ASSESSMENT TOOL FOR PRACTICE: INCREMENTAL RESPONSE SPECTRUM ANALYSIS (IRSA) M. Nuray Aydıoğlu Departmet of Earthquake Egeerg Kadll Observatory ad Earthquake Research Isttute (KOERI) Boğazç Uversty, Istabul, Turkey ABSTRACT Respose Spectrum Aalyss (RSA) procedure has become a stadard aalyss tool tradtoal stregth-based desg of buldgs ad brdges uder reduced sesmc loads. RSA has bee recetly exteded to estmate olear sesmc demads. The Icremetal Respose Spectrum Aalyss (IRSA) procedure s based o a straghtforward mplemetato of RSA at each pecewse lear cremetal step betwee the formato of cosecutve plastc hges. The practcal verso of IRSA works drectly wth smoothed elastc respose spectrum ad makes use of the well-kow equal dsplacemet rule to scale modal dsplacemet cremets at each pecewse lear step. IRSA ca be characterzed as a adaptve mult-mode pushover procedure, whch modal pushover aalyses are smultaeously performed for each mode at each cremetal step uder approprately scaled modal dsplacemets followed by a applcato of a modal combato rule. Examples are gve to demostrate the practcal mplemetato of IRSA. KEYWORDS: Icremetal Respose Spectrum Aalyss, Mult-mode Pushover Aalyss, Performace-Based Assessmet ad Desg, Ielastc Spectral Dsplacemet, Equal Dsplacemet Rule INTRODUCTION The Respose Spectrum Aalyss (RSA) procedure has become a stadard desg tool aalyss of buldgs ad brdges uder reduced sesmc loads. I spte of the approxmate ature of modal combato rules volved, mult-mode RSA has prove to be a powerful ad easy-to-use method wth a ratoal represetato of modal dyamc propertes as well as the drect defto of the sesmc put through desg respose spectrum. Today RSA has bee corporated a stadard fasho almost all moder sesmc desg codes as part of the stregth-based sesmc desg process ad t provdes a reasoably accurate estmato of the peak sesmc demad quattes the lear rage. O the other had, durg the course of progress of earthquake egeerg the last few decades researchers ad egeers have become well aware that structural behavor ad evetual damageablty of structures durg strog earthquakes were essetally cotrolled by the elastc deformato capactes of the ductle structural elemets. Accordgly, t has bee cocluded that the sesmc evaluato ad desg of structures should be based o olear deformato demads, ot o lear stresses duced by reduced sesmc forces that crudely correlated wth a assumed overall ductlty capacty of a gve type of a structure. Cosequetly, the last decade has wtessed the advet of the performace-based desg cocept, whch sgfcat progress has bee acheved wth the developmet of practceoreted olear aalyss procedures based o the so-called pushover aalyss. All pushover aalyss procedures ca be cosdered as approxmate extesos of the respose spectrum method to the olear respose aalyss wth varyg degrees of sophstcato. For example, Nolear Statc Procedure NSP (ATC, 99; FEMA, ) may be looked upo as a sgle-mode elastc respose spectrum aalyss procedure where the peak respose s obtaed through a olear aalyss of a modal sgle-degree-of-freedom (SDOF) system. I practcal applcatos, modal peak respose ca be approprately estmated through elastc dsplacemet spectrum (FEMA, ; CEN, 3).
7 A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) Note that sgle-mode pushover aalyss ca be relably appled to oly two-dmesoal respose of low-rse buldg structures regular pla or smple regular brdges, where the sesmc respose s essetally govered by the fudametal mode. There s o doubt that applcato of sgle-mode pushover to hgh-rse buldgs or ay buldg rregular pla as well as to rregular brdges volvg three-dmesoal respose would lead to correct, urelable results. Therefore, a umber of mproved pushover aalyss procedures have bee offered recet years a attempt to take hgher mode effects to accout (Gupta ad Kuath, ; Elasha, ; Atoou et al., ; Chopra ad Goel, ; Kalka ad Kuath, ; Atoou ad Pho, a, b). I ths cotext, Icremetal Respose Spectrum Aalyss (IRSA) procedure has bee troduced as a drect exteso of the tradtoal RSA procedure (Aydıoğlu, 3, ). Despte the fact that pushover aalyss has become extremely popular recet years, there s stll a lack of agreemet o a uversally accepted defto of the procedure. From a hstorcal perspectve, pushover aalyss has always bee uderstood as a olear capacty estmato tool ad geerally called as capacty aalyss. The olear structure s mootocally pushed by a set of forces wth a varat dstrbuto utl a predefed dsplacemet lmt at a gve locato (say, lateral dsplacemet lmt at the roof level of a buldg) s attaed. Such predefed dsplacemet lmt s geerally termed target dsplacemet. The structure may be further pushed up to the collapse codto order to estmate ts ultmate deformato ad load carryg capactes. It s for ths reaso that pushover aalyss has bee also called as collapse aalyss. However, vew of performace-based sesmc assessmet ad desg requremets, the above defto s ot suffcet. Accordg to the ew cocept troduced by Freema et al. (975) ad Fajfar ad Fschger (9), whch was subsequetly adopted ATC (ATC, 99) ad FEMA 73 (BSSC, 997; FEMA, ), pushover aalyss wth ts above-gve hstorcal defto represets oly the frst stage of a two-stage olear statc procedure, where t smply provdes the olear capacty curve of a equvalet sgle-degree-of-freedom (SDOF) system. The peak respose,.e., sesmc demad, s the estmated through olear aalyss of ths equvalet SDOF system uder a gve earthquake or through a elastc dsplacemet spectrum. I ths sese the term pushover aalyss ow cludes as well the estmato of the so-called target dsplacemet. Evetually, cotrollg sesmc demad parameters, such as plastc hge rotatos, are obtaed ad compared wth the specfed lmts (acceptace crtera) to verfy the performace of the structure accordg to a gve performace objectve uder a gve earthquake. Thus accordg to ths broader defto, pushover aalyss s ot oly a capacty estmato tool, but at the same tme t s a demad estmato tool. It s rather surprsg that amog the varous mult-mode methods that appeared the lterature durg the last decade, oly two procedures,.e., Modal Pushover Aalyss (MPA) troduced by Chopra ad Goel () ad Icremetal Respose Spectrum Aalyss (IRSA) developed by Aydıoğlu (3, ) coform to the above-gve cotemporary defto (for refed versos of MPA, see Heradez-Motes et al. (), ad Kalka ad Kuath ()). Others have actually dealt wth structural capacty estmato oly, although ths mportat lmtato has bee geerally overlooked. It meas that oe of them amed at estmatg the olear deformato demads (such as plastc hge rotatos or story drfts) uder a gve earthquake. Although elastc respose spectrum of a specfed earthquake was utlzed, t was ot for demad estmato, but oly for scalg the relatve cotrbutos of vbrato modes to obta sesmc load vectors (Atoou et al., ; Elasha, ; Gupta ad Kuath, ; Kalka ad Kuath, ; Atoou ad Pho, a) or to obta dsplacemet vectors (Atoou ad Pho, b). Geerally, buldg s pushed to a selected target dsplacemet that s actually obtaed from a olear respose hstory aalyss (Gupta ad Kuath, ; Kalka ad Kuath, ). Alteratvely a pushover aalyss s performed for a target buldg drft ad the earthquake groud moto s scaled to match that drft (Atoou ad Pho, a, b). Therefore the results are always preseted a relatve maer, geerally the form of story dsplacemet or story drft profles where pushover ad olear respose hstory aalyss results are supermposed for a matchg target dsplacemet or buldg drft. Thus, such pushover procedures are able to estmate oly the relatve dstrbuto of deformato demad quattes, ot ther magtudes, ad hece ther role a cotemporary deformato-based sesmc evaluato/desg scheme s questoable. I vew of the above assessmet, the ma objectve of ths paper s to preset the salet features of IRSA procedure (Aydıoğlu, 3, ), whch has bee recetly cluded the Turksh Sesmc Code (Mstry of Publc Works ad Settlemet, ; Aydıoğlu, ) as a practcal tool for performace-
ISET Joural of Earthquake Techology, March 7 7 based sesmc assessmet of exstg buldgs. But a broader framework, the paper ams as well to provde a clear sght to the theoretcal ad practcal aspects of the pushover aalyss methods, geeral. Towards ths ed, t wll start wth explorg the theoretcal roots of the pushover methods, ad wll cotue wth the basc developmet ad mplemetato of adaptve ad varat sgle-mode ad mult-mode pushover procedures, cludg IRSA. EXPLORING THE THEORETICAL ROOTS OF PUSHOVER ANALYSIS As t s stated above, all pushover methods ca be looked upo as olear extesos of the Respose Spectrum Aalyss (RSA). I ths drecto, olear respose hstory aalyss of a MDOF system wll be treated the followg through a pecewse lear process where the olear behavor s modeled by smple plastc hges.. Pecewse Lear Modelg of Nolear Respose Plastc hges are zero-legth elemets through whch the olear behavor s assumed to be cocetrated or lumped at predetermed sectos. A typcal plastc hge s deally located at the cetre of a plastfed zoe called plastc hge legth to be defed at the each ed of a clear legth of a beam or colum. A oe-compoet plastc hge model wth or wthout stra hardeg ca be approprately used to characterze a b-lear momet-curvature relatoshp. The so-called ormalty codto ca be used to accout for the teracto betwee plastc axal ad bedg deformato compoets (McGure et al., ). Plastc hge cocept s deally suted to the pecewse lear represetato of cocetrated olear respose. Lear behavor s assumed betwee predetermed plastc hge sectos as well as temporally betwee the formato of two cosecutve plastc hges. As part of a pecewse learzato process, yeld surfaces of plastc hge sectos may be approprately learzed,.e., they may be represeted by fte umber of les or plaes two- ad three-dmesoal respose models, respectvely. As a example, two-dmesoal yeld surfaces (les) of reforced cocrete ad wde flaged steel sectos are show Fgure. Note that umber of lear segmets may be creased reforced cocrete secto for a ehaced accuracy. N yc N yo N yt M yo M yb, N yb M M y M yb, N yb M (a) (b) Fg. Pecewse learsed yeld surfaces (les) of typcal (a) reforced cocrete secto, (b) wde flaged steel secto. Pecewse Lear Equatos of Moto of Nolear System I a plastc hge model wth mult-lear hysteretc behavor, the dyamc respose would be essetally lear durg a cremetal step betwee a tme t ad a prevous tme stato t at whch the respose s already determed. Thus, pecewse lear cremetal equatos of moto of a olear mult-degree-of-freedom (MDOF) structure subjected to a u-drectoal groud moto ca be wrtte for t > t as
7 A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) g g g G x x x M[ u ( t) u ( t )]+ C [ u ( t) u ( t )]+( K K )[ u( t) u( t )]= MI [ u ( t) u ( t )] where u t represets the relatve dsplacemet vector ad u t refers to the groud accelerato of a g gve earthquake x-drecto. I x s a kematc vector represetg the pseudo-statc trasmsso of the groud accelerato to the structure, whose compoets assocated wth the degrees of freedom x earthquake drecto are uty ad others are zero. I Equato, M deotes the mass matrx, K represets the stataeous (taget) stffess matrx cremetal step ad K G refers to geometrc stffess matrx accoutg for secod-order (P-delta) effects. The stataeous dampg matrx C s assumed to be Raylegh type,.e., t s formed as a lear combato of mass ad stffess matrces. 3. Pecewse Lear Mode-Superposto The stataeous dsplacemet respose durg a pecewse lear cremetal step ca be expaded to the modal coordates as N m = g x u( t) = u ( t); u ( t) = Φ Γ d t () x whch N m refers to the umber of modes to be cosdered the modal expaso, d t s the modal dsplacemet, ad Φ s the stataeous th mode shape vector to be obtaed from a free-vbrato aalyss: ( ) = ω K K Φ ( ) MΦ (3) G where ω represets the stataeous atural frequecy. partcpato factor for a earthquake x-drecto, whch s defed as Γ T g L x Φ MIx x = = T M Φ M Φ Γ x Equato () deotes the stataeous Substtutg Equato () ad tme dervatves to Equato ad pre-multplyg wth Φ followed by applyg modal orthogoalty codtos ad cosderg Equato () result a ucoupled stataeous modal equato of moto the th mode: g g d t + ξ ω d t + ( ω ) d t = [ u x t u x( t )] (5) * ( ) ( ) * ( ) * + d ( t ) + ξ ω d ( t ) + ( ω ) d ( t ) Here, ξ represets modal dampg rato, ad d * ( t ) s expressed as L x d T * Φ ( t ) = Lx Mu( t ) where s as defed Equato (). Equatos (5) ad reveal that each modal equato s depedet upo the past respose hstory of the MDOF structural system terms of dsplacemet vector ad ts tme dervatves developed at the prevous tme stat. Applyg modal expaso to u(t ) as Equato (), d * ( t ) gve Equato ca be expressed as d N m ( T ) ( ) ( ) Φ Φm Γxm dm t * m= ( t ) = Lx ( ) M ( ) from whch t ca be observed that f Φ were close eough to Φ, the above-metoed couplg ( ) would cease to exst. Ideed, f t s assumed that Φ Φ for all modes, whch s expected to hold () (7)
ISET Joural of Earthquake Techology, March 7 73 relatvely redudat systems, the modal orthogoalty codtos would result the followg smplfcato: d * ( t ) d( t ) () For the sake of smplcty, the followg modfed otato s used all expressos to follow: ( d t d d t d ) ( ) ; ( ) (9) Thus from Equatos (5), () ad (9), typcal th modal equato ca be expressed approxmately a cremetal form as g d + ξ ω d + ( ω ) d = u () x g g g( ) x = x x where u u u s the groud accelerato cremet ad dsplacemet cremet, the latter of whch ca be expressed as ( ) = + d d d d represets the modal Note that the thrd term at the left-had sde of Equato () s called modal pseudo-accelerato cremet, whch s defed as d a = ( ω ) () where ts cumulatve value at the th step ca be wrtte as smlar to the cumulatve modal dsplacemet gve Equato : ( ) = + a a a (3) Thus Equato () ca be rewrtte as g d + ξ ω d + a = u () x Wth respect to the exact cremetal equatos of moto gve Equato, approxmate modal cremetal equatos gve Equato () or () are expected to provde better results relatvely redudat systems due to the assumptos dcated Equato (). Such systems have the potetal of developg large umber of plastc hges ad therefore the formato of a ew hge would oly margally (or eve eglgbly) modfy the mode shapes of the structural system. O the cotrary, structural systems where oly a small umber of hges ca potetally develop, such as brdges wth few solated sgle-colum pers, the use of cremetal equatos of moto (see Equato () or ()) could lead to erroeous results, because sgfcat chages could occur mode shapes successve cremetal steps. Note that these observatos apply as well to those systems whose respose s practcally govered by a sgle mode oly.. Modal Hysteress Loops ad Modal Capacty Dagrams It s show above that cremetal soluto of Equato ca be approxmately reduced to the cremetal soluto of Equato () or (), through whch modal dsplacemet versus modal pseudoaccelerato dagrams ca be costructed. Those hypothetcal dagrams represet the modal hysteress loops, whch are schematcally depcted Fgure (a). The outer hysteress loops should be the fattest the frst mode ad get ther ad steeper as the mode umber creases. Accordg to Equato (), the stataeous slope of a gve dagram s equal to the egevalue (atural frequecy squared) of the correspodg mode at the pecewse lear cremet cocered. The backboe curves of the hypothetcal modal hysteress loops the frst quadrat may be approprately called the modal capacty dagrams, whch are dcated by sold curves Fgure (a). I the specal case where the frst mode aloe s assumed to represet the dyamc respose, the modal capacty dagram s, by defto, detcal to the so-called capacty spectrum defed the Capacty Spectrum Method (ATC, 99). The term modal capacty dagram s troduced by Aydıoğlu (3) by addg the word modal to the termology proposed by Chopra ad Goel (999). Note that learly elastc respose, modal hysteress curves ad modal capacty dagrams degeerate to straght les as show Fgure (b).
7 A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) a =3 a =3 = = = = d d (a) (b) Fg. (a) Schematc represetato of hypothetcal modal hysteress loops ad ther backboe curves (modal capacty dagrams sold curves); (b) Correspodg curves ad dagrams lear respose 5. A Geerc Defto of Pushover Aalyss Wth the framework of the theoretcal bass explaed above, pushover aalyss ca be defed as a mootoc olear aalyss of progressvely yeldg MDOF system wth a smultaeous mootoc costructo of the modal capacty dagram(s) utl the peak respose s obtaed for a gve earthquake groud moto, so that the aalyss procedure ca be used as a essetal tool performace assessmet process. Thus, accordg to the classfcato gve the troductory secto of ths paper, pushover aalyss s ultmately defed as a sesmc demad estmato tool. More specfcally, the aalyss should be able to produce ductle deformato demads, such as plastc hge rotatos or correspodg stras, as well as brttle force demads, e.g., shears reforced cocrete elemets. Wth respect to the above preseted aalytcal formulato, ow the aalyss process s chaged from a dyamc respose hstory aalyss of MDOF system to a mootoc pushover hstory aalyss, whle the cremetal tme step trasforms to a mootoc pushover step, whch s defed as the aalyss step betwee the formato of two cosecutve plastc hges. Sce modal capacty dagrams are defed as the backboe curves of the modal hysteress loops, ther peak values,.e., modal sesmc demad, ca be obtaed from the olear soluto of Equato () uder a gve earthquake groud moto. Alteratvely, elastc respose spectrum ca be utlzed for the same purpose, whch s the preferred opto for route egeerg applcatos. The pushover hstory aalyss ca be performed ether the form of a statc aalyss uder the specfed equvalet sesmc loads wth adaptve or varat dstrbutos, or t may be formulated as a pecewse lear respose spectrum aalyss by cosderg the cotuously chagg propertes of the structure. The latter may be terpreted as performg pushover aalyses varous modes smultaeously. As a geeral backgroud to pushover hstory aalyses, relatoshps betwee the coordates of modal capacty dagrams,.e., modal dsplacemet ad modal pseudo-accelerato of modal SDOF systems versus the correspodg respose quattes of the MDOF system, ca be expressed as the followg: (a) Pecewse lear relatoshp betwee the th modal dsplacemet cremet ad the correspodg dsplacemet cremet of MDOF system at the th pushover step s = u Φ Γx d (5) (b) Pecewse lear relatoshp betwee the th modal pseudo-accelerato cremet ad the correspodg equvalet sesmc load cremet of MDOF system at the th pushover step s
ISET Joural of Earthquake Techology, March 7 75 f = MΦ Γx a Note that the above expresso s adopted from the mootoc couterpart of the thrd term o the left-had sde of Equato, whch ca be expressed as =( f ) K KG u (7) I fact, substtutg Equato (5) to Equato (7) ad utlzg Equatos (3) ad () results Equato. SINGLE-MODE PUSHOVER ANALYSIS: PIECEWISE LINEAR IMPLEMENTATION WITH ADAPTIVE AND INVARIANT LOAD PATTERNS Sgle-mode pecewse lear pushover procedure s applcable to low-to-medum rse regular buldgs whose respose s effectvely cotrolled by the frst (predomat) mode. Slght torsoal rregulartes may be allowed provded that a 3-D structural model s employed. Regardg the adaptve patter, the frst-mode couterpart of equvalet sesmc load cremet gve Equato ca be wrtte for the th pushover step as = ; = a x f m m MΦ Γ () where m represets the vector of partcpatg modal masses effectve the frst mode. Superscrpt o the partcpatg modal mass ad mode shape vectors as well as o the modal partcpato factor dcates that stataeous frst-mode shape correspodg to the curret cofgurato of the structural system s cosdered followg the formato of the last plastc hge at the ed of the prevous pushover step. I adaptve case, a fully compatble modal expresso ca be wrtte from Equato (5) for the cremet of dsplacemet vector as well: = ; = d x u u u Φ Γ (9) Sce both u ad f are based o the same stataeous modal quattes, there s a oe-tooe correspodece betwee them. Thus, adaptve mplemetato of the sgle-mode pushover aalyss ca be based o ether a mootoc crease of dsplacemets or equvalet sesmc loads. However, ths s ot the case whe the load patter s kept varat durg pushover hstory,.e., a compatble modal dsplacemet expresso caot be provded. I the followg paragraph, pushover aalyss wll be treated o the bass of mootoc crease the equvalet sesmc loads where both adaptve ad varat patters wll be cosdered a commo framework. I the case of varat load patter, Equato () s modfed as = a ; = x f m m MΦ Γ () where the vector of frst-mode partcpatg modal masses, m, s defed at the frst pushover step ( = ) ad retaed varat durg the etre course of pushover hstory. Note that verted tragular or eve heght-wse costat ampltude mode shapes are beg used practce (FEMA, ) place of Φ.. Pushover Hstory Aalyss I pecewse lear pushover hstory aalyss equvalet sesmc load vector of the MDOF system, whch could have ether adaptve or varat patter, s creased mootocally the cremets of f where modal pseudo-accelerato cremet, a, s smultaeously calculated as the sgle ukow quatty at each th pushover step leadg to the formato of a ew hge. I ths respect ay respose quatty of terest, such as the cremet of a teral force, a dsplacemet compoet, a story drft, or a plastc hge rotato of a prevously formed hge, to be developed at the ed of the th pushover step may be wrtte a geerc form as ( ) ( ) = + = + q q q q q a ()
7 A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) Here, q ad q ( ) are the geerc respose quattes to develop at the ed of curret ad prevous pushover steps, q s the respectve cremet, ad q represets a geerc respose quatty to be obtaed for a =,.e., from the applcato of m or m as equvalet sesmc loads, represetg the adaptve or varat patter, respectvely. Now, the above geerc expresso s specalzed for the respose quattes that defe the coordates of the yeld surfaces of all potetal plastc hges, e.g., baxal bedg momets ad axal forces a geeral, three-dmesoal respose of a framed structure. I the frst pushover step ( = ), respose quattes due to gravty loadg are () cosdered as q Equato (). As part of the pecewse learzato process of pushover aalyss as well as to avod teratve operatos the hge detfcato process, yeld surfaces are approprately learzed a pecewse fasho as metoed above (Fgure ),.e., they are represeted by fte umber of les or plaes two- ad three-dmesoal respose models, respectvely. As a example, plaar yeld surfaces (les) of a reforced cocrete or steel secto (j) as show Fgure where a typcal le (s) ca be expressed as α js, M jp+β js, N jp= () Here, M jp ad N jp represet the yeld bedg momet ad correspodg axal force, respectvely, at the secto j whle α js, ad β js, refer to the coeffcets defg the yeld le (s). For the th pushover step, Equato () s specalzed for bedg momet ad axal force as j = ( ) j + j j = ( ) j +,,, ;,, j, M M M a N N N a (3) whch are the substtuted to Equato (), ad a s extracted as ( ) ( ) αjs, M j, βjs, N j, js, α js, M j, +βjs, N j, ( a ) = () The yeld le (s) at the secto (j) that tersected wth a mmum postve ( a ) js, amog all yeld les of all potetal plastc hges detfes the ew hge formed at the ed of the th pushover step. Oce a s determed, ay respose quatty of terest developed at the ed of that step ca be obtaed from the geerc expresso of Equato (). As the formato of the ew hge s detfed, the curret global stffess matrx of the structure s locally modfed such that oly the elemet stffess matrx affected by the ew hge s replaced wth a ew oe for the ext pushover step. Normalty crtero s eforced colums ad walls for the couplg of teral forces as well as plastc deformato compoets of the ewly formed plastc hge. Provded that the load patter s adaptve ad therefore resultg dsplacemet cremets are always compatble wth the equvalet sesmc load cremets, modal dsplacemet cremet, d, s related through Equato () to the correspodg modal pseudo-accelerato cremet, a, obtaed at each pushover step: a ( ) d = ω Here, ω represets the stataeous atural crcular frequecy calculated at the th pushover step. I the case of a varat patter, however, sce modal equvalet loads ad resultg dsplacemet cremets are ot compatble, two procedures ca be suggested to estmate the modal dsplacemet cremets. (a) The frst procedure volves the approxmate calculato of the stataeous egevalue, ( ω ), as a Raylegh quotet (Aydıoğlu, 5): (5)
ISET Joural of Earthquake Techology, March 7 77 u ( ω ) () mk, uk, k mk uk, k whch k, represets the dsplacemet compoet at the kth DOF uder the equvalet sesmc loads m k, wth varat patter that are defed through Equato () for a =. Thus, the modal dsplacemet cremet, d, s obtaed by substtutg Equato () to Equato (5). (b) The secod procedure s the oe already appled practce (ATC, 99; FEMA, ), where modal dsplacemet cremet s calculated through Equato (9),.e., by specalzg t for the roof dsplacemet cremet wth the correspodg frst-mode shape ampltude of the frst pushover step: un, d = (7) Φ N Γ, x It s worth otg that the sgle-mode pushover procedure preseted here, there s o eed to plot the covetoal pushover curve, wth vertcal axs represetg the sum of equvalet sesmc loads,.e., base shear. Accordgly, coverso of the base shear cremets to pseudo-accelerato cremets s ot requred, because those are obtaed drectly by Equato () at each pushover step. I fact, t ca be show that eve f the covetoal approach had bee appled the same results would be obtaed,.e., the base shear x earthquake drecto s obtaed by summg up the equvalet sesmc loads gve by Equato () that drecto: g T g T x = Ix f = I x m a V () O the other had, total partcpatg modal mass of the MDOF system the x-drecto s obtaed by summg up the elemets of the vector of partcpatg masses gve Equato (),.e., M gt Lx x = x = M I m (9) Thus modal pseudo-accelerato cremet at the th pushover step s obtaed from Equatos () ad (9) as Vx = M x a (3) whch s othg but the coverso relatoshp used the tradtoal pushover procedure (ATC, 99; FEMA, ). Wth d ad a determed as above, addg to those obtaed at the ed of the prevous pushover step, modal dsplacemet ad modal pseudo-accelerato are calculated from Equatos ad (3) at the ed of the th step as ( ) ( ) = + = + d d d ; a a a (3) It s oted that essetally d ad a are the elemets of a cremetal modal equato of moto of the frst-mode equvalet SDOF system: g d + ξ ω d + a = u x (3) Thus modal capacty dagram of the fudametal mode s obtaed drectly as show schematcally Fgure 3, whch s othg but the so-called capacty spectrum (ATC, 99) obtaed from the tradtoal pushover curve through a modal coordate trasformato. Accordg to Equato (5), stataeous slope of the lear segmet of the modal capacty dagram at the pushover step betwee the plastc hge pots ( ) ad s equal to the fudametal egevalue of the structural system at that step as show Fgure 3. Note that stataeous slope of the capacty dagram could tur out to be egatve due to the P-delta effects, as dcated Fgure 3, whe accumulated plastc deformatos result a egatve-defte
7 A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) secod-order stffess matrx. I the case of varat load patter, at such a crtcal pushover step, the mootoc load crease process s termated. From such a step owards, aalyss s geerally cotued wth a mootoc dsplacemet cremet process, whle retag a costat dsplacemet patter obtaed at the crtcal step. The accuracy of ths approach s doubtful. d a (ω ) a Fg. 3 Modal capacty dagram of the fudametal mode I the case of adaptve soluto, the aalyss process s ot flueced by a egatve stataeous slope of the capacty dagram. A egatve slope meas a egatve egevalue ad thus a magary atural frequecy, whch leads to a modal respose that resembles the o-vbratory respose of a overdamped system (Aydıoğlu ad Fahja, 3). The correspodg mode shape has a remarkable physcal sgfcace, represetg the post-bucklg deformato state of the structure. Although structural egeers are ot famlar wth the egatve (or zero) egevalues due to egatve-defte (or sgular) stffess matrces, such egevalues ad correspodg egevectors do exst, whch ca be routely calculated by matrx trasformato methods of egevalue aalyss, such as the well-kow Jacob Method (Bathe, 99). As metoed above, a remarkable aspect of the above-preseted adaptve procedure s that t does ot ecesstate the plottg of covetoal pushover curve terms of base shear versus roof dsplacemet. Istead, modal capacty dagram, whch tself s the essetal tool for the estmato of modal dsplacemet demad, s obtaed drectly o cludg drect cosderato of the P-delta effects.. Estmato of Modal Dsplacemet Demad: Ielastc Spectral Dsplacemet The above-descrbed process of pushover hstory aalyss s cotued utl cumulatve modal dsplacemet calculated by Equato (3) exceeds the frst-mode elastc spectral dsplacemet. It meas that the last pushover step has bee reached, ad, therefore, the modal dsplacemet to develop at p the ed of ths step, d ( ) (superscrpt p stads for peak ), s made equal to the elastc spectral dsplacemet, S d, : ( p) = d S d, (33) Ths s followed by the calculato of the modal dsplacemet cremet the last step (p): ( p) ( p ) d = Sd d, (3) The elastc frst-mode spectral dsplacemet, S d,, ca be calculated for a gve groud moto record through olear aalyss of the modal SDOF system accordg to Equato (3) by cosderg hysteress loops defed by the b-learzed modal capacty dagram as the backboe curve (see Fgure (b)). However for practcal purposes, elastc frst-mode spectral dsplacemet, S d,, ca be approprately defed through a smple procedure based o the equal dsplacemet rule (FEMA, ): Sd, = CR, Sde, (35) whch S de, represets the elastc spectral dsplacemet of the correspodg lear SDOF system wth the same perod (stffess) as the tal perod of the blear elastc system. Note that practce d
ISET Joural of Earthquake Techology, March 7 79 cracked secto stffesses are used reforced cocrete systems throughout the pushover aalyss ad therefore the fudametal perod of the system calculated at the frst lear pushover step ( = ) s take as the tal perod of the blear elastc system. Ths s cotrary to the tradtoal approach where the fudametal perod s further legtheed excessvely due to the b-learzato of modal capacty dagram. I Fgure, modal capacty dagram ad the elastc respose spectrum are combed a dsplacemet pseudo-accelerato format. I the case where T > TS (wth T S beg the characterstc spectrum perod at the tersecto of costat velocty ad costat accelerato regos), b-learzato of the modal capacty dagram s eve uecessary as dcated Fgure (a), because spectral dsplacemet amplfcato factor C R, s always equal to uty: R C, = ( T > T ) (3) I the case where T TS, tal perod s stll defed as above; however, a terato s ecessary to calculate the spectral dsplacemet amplfcato factor by usg the followg famlar relatoshp (FEMA, ; MPWS, ): + ( Ry, ) TS / T CR, = ( T TS ) (37) Ry, whch R y, refers to the yeld reducto factor (Fgure (b)): R y, S = a Note that alteratve relatoshps are avalable for the dsplacemet amplfcato factor that ca be used practcal applcatos leu of those gve by Equatos (3) ad (37). For those referece may be gve to Aydıoğlu ad Kaçmaz (), ad to FEMA (FEMA, 5). p Oce modal dsplacemet cremet the last step, d ( ) p, s estmated, the correspodg a ( ) s determed, ad tur, ay respose quatty of terest developed at the ed of that step ca be obtaed from the geerc expresso of Equato (). ae, y, S (3) ω =( π/ T ) a & S a, S S a & S a, S ae, S ae, ( T > T S ) ( T T S ) a y, (ω ) (ω ) S d, = S de, d & S d, S de, S d, d & S d, (a) (b) Fg. Estmatg modal dsplacemet demad MULTI-MODE ADAPTIVE PUSHOVER ANALYSIS: INCREMENTAL RESPONSE SPECTRUM ANALYSIS (IRSA) PROCEDURE Mult-mode pushover procedure s teded for applcato o hgh-rse ad/or rregular buldgs ad brdges where the sesmc respose caot be effectvely represeted by the frst mode oly. These clude torsoally sestve buldgs wth 3-D respose characterstcs.
A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) I le wth the theoretcal backgroud provded above, the mult-mode case mootoc pushover hstory aalyses have to be performed smultaeously all the modes cosdered. As wth the sglemode pushover, the adaptve case both MDOF system dsplacemet cremets ad equvalet sesmc load cremets are based o the same stataeous modal quattes. Thus the mplemetato of multmode pushover aalyss ca be based o ether a mootoc crease of dsplacemets or equvalet sesmc loads. These may be called dsplacemet-cotrolled ad force-cotrolled pushovers, respectvely. To start wth, usg Equato (5), modal dsplacemet cremets of MDOF system ca be expressed as = ; = u u d u Φ Γ x (39) ad correspodg expressos for the compatble sesmc load cremets ca be wrtte as mult-mode couterparts of those gve Equato () as = a ; = f m m MΦ Γ (). Modal Scalg x I order to defe modal MDOF respose, modal dsplacemet cremets or modal pseudoaccelerato cremets a have to be determed all modes at each pushover step, depedg o whether dsplacemet- or force-cotrolled pushover s appled. Sce just a sgle plastc hge forms ad therefore oly oe yeld codto s applcable at the ed of each pecewse lear step, a reasoable assumpto eeds to be made for the relatve values of modal dsplacemet or modal pseudo-accelerato cremets, so that the umber of ukows are reduced to oe. Ths s called modal scalg, whch s the most crtcal assumpto to be made all mult-mode pushover procedures, cludg IRSA. I ths respect the oly excepto s the Modal Pushover Aalyss MPA (Chopra ad Goel, ) where modal couplg s completely dsregarded the formato of plastc hges ad therefore modal scalg s omtted.. Modal Scalg Based o Istataeous Elastc Spectral Quattes Modal scalg s probably the most crtcal ad, at the same tme, oe of the most cotroversal ssues of mult-mode pushover aalyss. I a umber of studes, such as Gupta ad Kuath (), Elasha (), Atoou et al. (), Atoou ad Pho (a), force-cotrolled pushover s mplemeted where modal scalg s performed o stataeous modal pseudo-acceleratos. Usg the termology ad otato of the preset paper such a modal scalg ca be expressed as S ae = ae d a F S () where represets the stataeous th mode elastc spectral pseudo-accelerato at the th pushover step, ad F refers to a cremetal scale factor, whch s depedet of the mode umber. Thus Equato () meas that modal pseudo-accelerato cremets are scaled proporto to the respectve elastc spectral acceleratos. Note that the above-defed modal scalg s essetally detcal to the scalg of modal dsplacemet cremets proporto to respectve elastc spectral dsplacemets, whch may be expressed as S de = de d F S () where represets the stataeous th mode elastc spectral dsplacemet correspodg to the above-gve S ae,.e., S ae = ( ω ) S de. Such a scalg has bee used recetly a dsplacemetcotrolled pushover procedure (Atoou ad Pho, b). Naturally ths type of modal scalg s exact for a sgle-step lear aalyss wth F = ; however t s doubtful whether t should be mplemeted a olear case. I fact, stataeous elastc spectral parameters have o relato at all wth the stataeous olear modal respose cremets. Whe the structure softes due to accumulated plastc deformatos, the stataeous elastc spectral dsplacemet of the frst mode creases dsproportoately wth respect to those of the hgher modes,
ISET Joural of Earthquake Techology, March 7 leadg to a exaggerato of the effect of the frst mode the hge formato process pror to reachg the peak respose.. Modal Scalg Based o Istataeous Ielastc Spectral Dsplacemets Dsplacemet-cotrolled pushover s the preferred approach the Icremetal Respose Spectrum Aalyss IRSA (Aydıoğlu, 3, ), ad modal pushovers are mplemeted smultaeously by mposg stataeous dsplacemet cremets of the MDOF system at each pushover step accordg to Equato (39). I prcple, modal dsplacemets are scaled IRSA wth respect to the elastc spectral dsplacemets, d S, assocated wth the stataeous cofgurato of the structure (Aydıoğlu, 3). Ths s the ma dfferece betwee IRSA ad other studes referred to above where modal scalg s based o stataeous elastc spectral pseudo-acceleratos or dsplacemets. IRSA s adopto of elastc spectral dsplacemets for modal scalg may be cosdered as a ratoal choce, because those spectral dsplacemets are othg but the peak values of the modal dsplacemets to be reached, as wll be show the followg. I practce, modal scalg based o elastc spectral dsplacemets ca be easly acheved by takg advatage of the equal dsplacemet rule. Assumg that sesmc put s defed va smoothed elastc respose spectrum, accordg to ths smple ad well-kow rule (whch s already utlzed above for the estmato of modal dsplacemet demad sgle-mode pushover), peak dsplacemet of a elastc SDOF system ad that of the correspodg elastc system are assumed practcally equal to each other, provded that the effectve tal perod s loger tha the characterstc perod of the elastc respose spectrum. The characterstc perod s approxmately defed as the trasto perod from the costat accelerato segmet to the costat velocty segmet of the spectrum. For perods shorter tha the characterstc perod, elastc spectral dsplacemet s amplfed usg a dsplacemet modfcato factor,.e., C coeffcet gve FEMA 35 (FEMA, ). However, such a stuato s seldom ecoutered md- to hgh-rse buldgs ad log brdges volvg mult-mode respose. I such structures, effectve tal perods of the frst few modes are lkely to be loger tha the characterstc perod ad therefore those modes automatcally qualfy for the equal dsplacemet rule. O the other had, effectve post-yeld slopes of the modal capacty dagrams get steeper ad steeper hgher modes wth gradually dmshg elastc behavor (Fgure 5). Thus, t ca be comfortably assumed that elastc spectral dsplacemet respose hgher modes would ot be dfferet from the correspodg spectral elastc respose. Hece, smoothed elastc respose spectrum may be used ts etrety for scalg modal dsplacemets wthout ay modfcato. As the sgle-mode aalyss, reforced cocrete buldgs elastc perods calculated at the frst pushover step may be cosdered leu of the tal perods obtaed from the b-learzato of modal capacty dagrams (see Fgure (b)). I le wth the equal dsplacemet rule, scalg procedure applcable to the th mode cremet of modal dsplacemet at the th pushover step s smply expressed as d = F S (3) where F s a cremetal scale factor, whch s applcable to all modes at the th pushover step. represets the tal elastc spectral dsplacemet defed at the frst step (Fgure 5), whch s S de take equal to the elastc spectral dsplacemet assocated wth the stataeous cofgurato of the structure at ay pushover step. Cumulatve modal dsplacemet at the ed of the same pushover step ca the be wrtte as d = F S () whch F represets the cumulatve scale factor wth a maxmum value of uty: F = F ( ) + F (5) Note that the modal scalg expressos gve above correspod to a mootoc crease of the elastc respose spectrum progressvely at each step wth a cumulatve scale factor creasg from zero utl uty. Physcally speakg, the structure s beg pushed such that at every pushover step modal dsplacemets of all modes are creased by creasg elastc spectral dsplacemets, defed at the frst step ( = ) the same proporto (accordg to the equal dsplacemet rule ), utl they smultaeously reach the target spectral dsplacemets o the respose spectrum. Show Fgure 5 are the scaled de de
A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) spectra correspodg to the frst yeld, to a termedate pushover step ( F < ), ad to the fal step ( F = ), whch are plotted the ADRS (Accelerato-Dsplacemet Respose Spectrum) format ad supermposed oto the modal capacty dagrams. Fg. 5 Scalg of modal dsplacemets through mootoc scalg of respose spectrum It s worth warg that the equal dsplacemet rule may ot be vald at ear-fault stuatos wth forward drectvty effect. Aga, t eeds to be stressed that IRSA s a dsplacemet-cotrolled procedure ad, therefore, the above-metoed mootoc spectrum scalg apples to spectral dsplacemets oly, ot to the elastc spectral pseudo-acceleratos. For the sake of completeess, however, a compatble modal pseudoaccelerato cremet, a, correspodg to the cremet of scaled modal dsplacemet may be defed from Equatos () ad (3) as where S a ( ω ) a; a ae ( ) a = F S S = S () ω represets compatble elastc spectral pseudo-accelerato, ad S ae elastc spectral pseudo-accelerato correspodg to the elastc spectral dsplacemet, the frst pushover step. refers to tal S de, defed at. Mult-mode Pushover Hstory Aalyss: Smultaeous Pushovers All Modes ad Modal Combato Substtutg Equato (3) to Equato (5) leads to the followg expresso for the dsplacemet vector cremet the th mode at the th pushover step: u ; = u F u =Φ Γ xs de (7) Utlzg Equatos ad (), equvalet sesmc load vector cremet correspodg to the dsplacemet vector cremet gve above Equato (7) may be wrtte for a alteratve loadcotrolled process: f = f F ; f = MΦ Γ S () x a whch S a s the compatble elastc spectral pseudo-accelerato defed by Equato (). Now, pecewse lear mult-mode pushover hstory aalyss ca be performed at a gve pushover step, by mootocally mposg dsplacemet cremets u of the MDOF system, as defed Equato (7), or alteratvely, by applyg equvalet sesmc load cremets f gve by
ISET Joural of Earthquake Techology, March 7 3 Equato () smultaeously all modes cosdered. I ths process, the cremet of a geerc respose quatty of terest, such as the cremet of a teral force, a dsplacemet compoet, a story drft or the plastc rotato of a prevously developed plastc hge, may be calculated each mode as r = r F (9) where r represets the geerc respose quatty to be obtaed each mode for F =,.e., by mposg the dsplacemet vector u gve Equato (7), or alteratvely, by applyg the load vector f gve Equato (). Icremetal scale factor F s the sgle ukow at each pushover step leadg to the formato of a ew plastc hge. I the ext stage, modal geerc respose quatty cremets are combed by a approprate modal combato rule, such as the Complete Quadratc Combato (CQC) rule as ρ m Nm Nm = m ρ m m= = r ( r r ) (5) where s the cross-correlato coeffcet of the CQC rule. Thus, geerc respose quatty at the ed of the th pushover step ca be estmated as ( r r ) ( ) = + r = r + r F (5) whch r ad r ( ) are the geerc respose quattes to develop at the ed of curret ad prevous pushover steps, respectvely. I the frst pushover step ( = ), respose quattes due to gravty loadg () are cosdered as r. The ext stage of mult-mode pushover hstory aalyss s smlar to the sgle-mode aalyss where the above-gve geerc expresso s specalzed for the respose quattes that defe the coordates of the yeld surfaces of all potetal plastc hges, e.g., baxal bedg momets ad axal forces a geeral, three-dmesoal respose of a framed structure. As part of the pecewse learzato process of pushover aalyss as well as to avod teratve operatos the hge detfcato process, yeld surfaces are approprately learzed a pecewse fasho as metoed above (Fgure ),.e., they are represeted by fte umber of les or plaes two- ad three-dmesoal respose models, respectvely. As a example, plaar yeld surfaces (les) of a reforced cocrete or steel secto (j) are show Fgure where a typcal le (s) ca be expressed as α js, M jp+β js, N jp= (5) whch M jp ad N jp represet the yeld bedg momet ad correspodg axal force, respectvely, at secto j whle α js, ad β js, refer to the coeffcets defg the yeld le (s). For the th pushover step, Equato (5) s specalzed for bedg momet ad axal force as M j M = ( ) j + M j F N j = N ( ) j + N j F,,, ;,,, (53) whch are the substtuted to Equato (5) ad F s extracted as ( ) ( ) αjs, M j, βjs, N j, ( F ) js, = (5) α M +β N js, j, js, j, The yeld le (s) at secto (j) that tersected wth a mmum postve ( F ) js, amog all yeld les of all potetal plastc hges detfes the ew hge formed at the ed of the th pushover step. Oce F s determed, ay respose quatty of terest developed at the ed of that step ca be obtaed from the geerc expresso of Equato (5). Modal dsplacemet cremet d ay mode ca be obtaed from Equato (3), ad tur, modal pseudo-accelerato cremet from Equato (), leadg to the smultaeous estmato of respectve cumulatve quattes,.e., the ew coordates of all modes, whch ca be obtaed through Equatos ad (3).
A Respose Spectrum-Based Nolear Assessmet Tool for Practce: Icremetal Respose Spectrum Aalyss (IRSA) As metoed the case of sgle-mode pushover, whe the formato of the ew hge s detfed, the curret global stffess matrx of the structure s locally modfed such that oly the elemet stffess matrx affected by the ew hge s replaced wth a ew oe for the ext pushover step. The ormalty crtero s eforced colums ad walls for the couplg of teral forces as well as plastc deformato compoets of the ewly formed plastc hge. Thus t s see that mult-mode pushover hstory aalyss wth IRSA s the exteso of sgle-mode pushover hstory aalyss descrbed earler. Ideed, stead of rug a statc aalyss uder equvalet sesmc loads at each step, a respose spectrum aalyss s performed IRSA at each step where sesmc S de put data s specfed ether the form of tal spectral dsplacemet each mode, (whch s calculated the frst pushover step ad remas uchaged at all pushover steps), or sesmc put s gve terms of compatble elastc spectral pseudo-accelerato S defed by Equato (). 3. Estmato of Peak Quattes: Ielastc Sesmc Demad The above-descrbed pushover-hstory process s repeated for all pushover steps utl cumulatve spectrum scale factor defed by Equato (5) exceeds uty at the ed of a gve pushover step. Whe such a step s detected, whch s dcated by superscrpt (p), cremetal scale factor correspodg to ths fal pushover step s re-calculated from Equato (5) as p p F ( ) ( = F ) (55) I the last pushover step, modal dsplacemet cremet s redefed as p p d = CRS de F ( ) (5) where C represets spectral dsplacemet amplfcato factor the th mode. If C >, the R sesmc put for the th mode s modfed from p r ( ) j S de to C R de a S, ad the geerc respose quatty s recalculated at the last step by repeatg the elastc respose spectrum aalyss. Fally peak value of ay respose quatty of terest s obtaed from the geerc expresso of Equato (5) for = p: ( p) ( p p p p p r r ) ( ) ( ) ( ) ( ) = + r = r + r F (57) Spectral dsplacemet amplfcato factor C R s calculated as show below. If T > TB,.e., ( ω ) <ωb, the C R =. If T < TB,.e., ( ω ) >ωb, the C R s determed approxmately as (MPWS, ) where C C R R y B ( p) Ry + ( R ) T / T = ( λ. ) y B ( p) ( p) λ Ry + ( R ) T / T = ( λ >. ) R y s the th mode yeld reducto factor as defed below,.e., the th mode couterpart of the frst mode yeld reducto factor defed Equato (3) (Fgure (b)). Post-yeld slope defed below: R y S = a ae y ( p) ( p) ( ω ) ; λ = ω ( ) R (5) p λ ( ) s also Note that the secod spectral dsplacemet amplfcato factor gve Equato (5) s teded for hgher modes wth shorter atural perods where elastc spectral dsplacemets would be reduced due to steeper post-yeld slopes of hgher-mode capacty dagrams (Öem, ).. Treatmet of P-Delta Effects IRSA P-delta effects are rgorously cosdered IRSA through straghtforward cosderato of geometrc stffess matrx each cremet of the respose spectrum aalyss performed. Alog the pushover- (59)
ISET Joural of Earthquake Techology, March 7 5 hstory process, accumulated plastc deformatos result egatve-defte secod-order stffess matrces, whch tur yeld egatve egevalues, ad hece, egatve post-yeld slopes the modal capacty dagrams of the lower modes. The correspodg mode shapes are represetatve of the postbucklg deformato state of the structure, whch may sgfcatly affect the dstrbuto of teral forces ad elastc deformatos of the structure. Aalyss of elastc SDOF systems based o blear backboe curves wth egatve post-yeld slopes dcate that such systems are susceptble to dyamc stablty rather tha havg amplfed dsplacemets due to the P-delta effects. Therefore, the use of P-delta amplfcato coeffcet (C 3 ) defed FEMA 35 (FEMA, ) s o loger recommeded (FEMA, 5). The dyamc stablty s kow to deped o the yeld stregth, tal stffess, egatve post-yeld stffess, ad the hysteretc model of SDOF oscllator as well as o the characterstcs of the earthquake groud moto. Accordgly, practcal gudeles have bee proposed for the mmum stregth lmts terms of other parameters to avod stablty (Mrada ad Akkar, 3; FEMA, 5). Further research s eeded for the realstc cases of backboe curves resultg from modal capacty dagrams, whch exhbt multple post-yeld slopes wth both ascedg ad descedg braches. For the tme beg, equal dsplacemet rule s used IRSA, eve whe P-delta effects are preset, as log as a mmet dager of dyamc stablty s ot expected accordg to the above-metoed practcal gudeles. 5. Summary of IRSA The aalyss stages to be appled at each pecewse lear pushover step of IRSA are summarzed below: Ru a lear respose spectrum aalyss (RSA) wth a suffcet umber of modes by cosderg stataeous secod-order stffess matrx correspodg to the curret plastc hge cofgurato. RSA at each step actually correspods to performg smultaeous pushover aalyses all modes for a ut value of cremetal scale factor F. I rug RSA, the sesmc put s specfed terms de of tal spectral dsplacemets S, whch would be the same at all pushover steps accordg to the equal dsplacemet rule. They are calculated oly oce at the frst pushover step as elastc spectral dsplacemets. Alteratvely, compatble spectral pseudo-acceleratos S a defed at each step by Equato () may be specfed as sesmc put. All respose quattes of terest, r, are obtaed by applyg a approprate modal combato rule (e.g., CQC rule Equato (5)). () Specalze the geerc expresso of Equato (5) for the respose quattes that defe the coordates of the yeld surfaces of all potetal plastc hges,.e., baxal bedg momets ad axal forces a geeral, three-dmesoal respose of a framed structure. Respose quattes due to the gravty loadg are cosdered as r () at the frst pushover step. Calculate the cremetal scale factor F accordg to the yeld codtos of all potetal plastc hges ad detfy the ew yelded hge. (3) Calculate cumulatve scale factor F from Equato (5) ad check f t exceeded uty. If p exceeded, calculate the cremetal scale factor F ( ) from Equato (55) for the fal pushover step ad carry o accordg to Equatos (5) (59). If ot, cotue wth the ext stage. () Calculate all respose quattes of terest developed at the ed of the pushover step from the geerc expresso of Equato (5). If the fal pushover step has bee reached, termate the aalyss. If ot, cotue wth the ext stage. (5) Modfy the curret secod-order stffess matrx by cosderg the last yelded hge detfed at Stage () ad retur to Stage for the ext pushover step.. Specal Cases Sgle-mode adaptve pushover aalyss preseted earler ths paper s a specal case of IRSA wth =. Sce o modal scalg s volved the sgle-mode aalyss, the cremetal scale factor F becomes drectly proportoal to the modal dsplacemet cremet d as follows (see Equato (3)):