Open Access Prediction of the Seismic Response of Steel Frames with Concentric Diagonal Bracings

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1 Send Order for Reprnt to 11 Te Open Contructon and Buldng Tecnology Journal, 13, 7, 11-1 Open Acce Predcton of te Sec Repone of Steel Frae wt Concentrc Dagonal Bracng M. Boco *, A. Ger, E.M. Marno and P.P. Ro Departent of Cvl and Envronental Engneerng, Unverty of Catana, Italy Abtract: Concentrcally braced frae and eccentrcally braced frae are effcent ec retant yte but tey are generally prone to develop torey collape ecan. A underlned n oe prevou paper wt regard to eccentrcally braced frae, t tendency depend on te overtrengt factor and daage dtrbuton capacty factor reultng fro te ue of coon degn procedure. In t paper a prevouly propoed procedure wc predct te egtwe dtrbuton of te daage at collape of eccentrcally braced frae appled to concentrcally braced frae. To apply t procedure, proper defnton of te overtrengt factor and daage dtrbuton capacty factor are derved. Te effectvene of te propoed procedure teted on frae wt concentrc dagonal bracng caractered by dfferent torey nuber and degned by coon degn procedure. Te target repone provded by nonlnear dynac analye. Te ec nput conttuted by ten artfcal accelerogra. Te paper prove tat te propoed procedure able to predct accurately te nonlnear dynac repone of yte n wc te daage anly retrcted to a few torey. In te oter cae, oe not neglgble catterng between actual and expected value of daage can be found at oe torey. Keyword: Approxated etod, concentrcally braced frae, daage dtrbuton capacty factor, dual braced yte, overtrengt factor, ec repone. INTRODUCTION Eccentrcally braced frae and concentrcally braced frae endowed wt eter tradtonal or bucklng retraned brace are generally caractered by low lateral pot-elatc tffne. Several reearc tude ave own tat tee yte, even toug degned n coplance wt capacty degn prncple, can ow a tendency to a oft torey collape ecan [1-] and tat t beavour accentuated by a non-oogeneou dtrbuton of te overtrengt of dpatve eber [9-11]. In tradtonal concentrcally braced frae, a concentraton of nelatc deforaton at a few torey furter prooted by te decreae of te brace retance n te pot-bucklng range of beavour and by te reducton of te brace dpatve capacty under cyclc loadng condton [1, 13]. Soe reearcer ave deontrated, owever, tat te preence of contnuou colun can reduce axu and redual drft deand [1, 15] and caue a ore unfor dtrbuton of te drft deand along te egt of te buldng [, 1-1]. To aceve a global dpatve beavour of te tructure, Eurocode (EC) [] tpulate tat te axu value of te overtrengt ould not exceed te nu value by ore tan 5% (oogenety trengt condton of dpatve eber). Pat tude ave deontrated, owever, tat eccentrcally and concentrcally braced frae ay develop a torey collape ecan even f te *Addre correpondence to t autor at te Departent of Cvl and Envronental Engneerng, Unverty of Catana, Italy; Tel: ; Fax: ; Eal: boco@dca.unct.t and ng..boco@gal.co oogenety trengt condton verfed n degn [3]. Furter, t a been deontrated tat te ec repone at collape of tradtonal and ted eccentrcally braced frae [] related to te overtrengt factor and alo to anoter paraeter called daage dtrbuton capacty factor n reference [5]. An analytcal forulaton of te daage of lnk bea a been propoed a a functon of bot te torey overtrengt and daage dtrbuton capacty factor (DDC) to predct te egtwe dtrbuton of te daage at collape of eccentrcally braced tructure and to degn dual eccentrcally braced frae [, 7]. Te DDC factor copleentary to te overtrengt factor becaue t calculated on te ba of te deforaton capacty of te dpatve eber and on te ec beavour of lnk n te nonlnear range of beavour. Te procedure can be appled to yte tat are not caractered by a gnfcant toronal beavour [-31]. In t paper a proper defnton of te brace overtrengt and daage dtrbuton capacty factor provded wt reference to frae wt concentrc dagonal bracng. Furter, te relaton propoed wt reference to eccentrcally braced frae to predct te daage at collape extended to ple and dual concentrcally braced frae. PREDICTION OF DAMAGE INDEX To quantfy te egtwe dtrbuton of te daage of braced frae at falure, a torey daage ndex defned by ean of te followng relaton µ d! 1 DI = (1) µ! 1 f 17-3/13 13 Benta Open

2 Predcton of te Repone of Concentrcally Braced Frae Te Open Contructon and Buldng Tecnology Journal, 13, olue were µ d te torey ductlty deand and µ f te ductlty deand correpondng to falure of te torey under exanaton,.e. te ultate torey ductlty. T ndex range fro to 1. In partcular, t equal to one at te torey were te ductlty deand equal to te ultate ductlty. Te ean value of te daage ndex along te egt of te buldng ndcated later a DI. It provde a eaure of te propenty of te frae to develop a torey col- lape ecan. alue of DI cloe to unty are obtaned n frae were te ultate ductlty aceved alot ultaneouly at all torey; uc value are repreentatve of a ductle collape ecan. On te contrary, value of DI cloe to zero are obtaned n frae were te daage localed to a few torey. Equaton (1) could be ued to evaluate te egtwe dtrbuton of daage once te ductlty deand n a frae at falure ave been calculated by ean of nonlnear dynac analy. However, t etod of analy ental experte n correctly odellng te ec nput and te nonlnear cyclc beavour of tructural eber and requre a uge coputatonal effort [3]. Owng to t, t not recoended for every day degn ue. Baed on t conderaton, Boco and Ro [5] recently forulated a ple procedure for te predcton of te daage of eccentrcally braced frae and propoed te followng relaton for te evaluaton of te torey daage ndex DI e DI exp ( 3DDC n ) = D! () In t relaton, DDC n te nu value of te daage dtrbuton capacty factor wtn te tructure and D te torey daage varable; pecfcally, t latter paraeter wa calculated a a functon of bot te overtrengt factor Ω and te daage dtrbuton capacty factor DDC by ean of te followng relaton 5 5 n[! ( 1+ DDC) ] D = (3) 5! ( 1+ DDC ) 5, More detal regardng te evaluaton of te overtrengt factor Ω and daage dtrbuton capacty factor DDC of eccentrcally braced frae can be found n [5]. In te preent paper, to enure Equaton () and (3) are utable for te predcton of te egtwe dtrbuton of te daage ndex of teel frae wt concentrc dagonal bracng, proper relaton and crtera are deterned for te evaluaton of Ω and DDC factor of t tructural type. STOREY OERSTRENGTH OF BRACED FRAMES Te torey overtrengt Ω dentfe te torey were te preature yeldng of te tenon brace occur. In t paper, Ω defned a te rato of te actual lateral torey trengt ( Rd ) to tat requred n degn ( Ed ). In order to evaluate te torey overtrengt factor of frae wt concentrc dagonal bracng, te lateral torey trengt Rd of t type of frae ut be properly defned. To t end, two ngle-torey dagonal braced frae wt dfferent noraled lenderne! are ubjected to an ncreang orzontal top dplaceent. Te noraled lenderne! gven by te followng forula () were λ te brace lenderne rato, E and f y are te odulu of elatcty and te yeld tre of teel, repectvely. Specfcally, value of te noraled lenderne equal to 1.3 and. are elected becaue tey repreent te nu and axu noraled lenderne allowed by EC for dagonal brace [, 33]. Independently of te lenderne of te brace, te bae ear - top dplaceent relaton lnear untl te axal force of te brace n copreon lower tan t bucklng retance N b,rd (Fg. 1). Te torey ear correpondng to te bucklng of te brace n copreon naed b,rd and obtaned by te expreon ( N " ) co! b, Rd = b,rd N (5) q were N q te axal force due to te gravty load and θ te angle of nclnaton of te brace wt repect to te orzontal ax. Wen te top dplaceent ncreaed, te contrbuton to te orzontal reacton gven by te brace n tenon ncreae wle te contrbuton provded by te brace n copreon decreae. Te ncreae n te u of te orzontal coponent of te axal force n te brace poble becaue te ncreae n te axal force of te brace n tenon larger tan te decreae n te axal force of te brace n copreon. Te axu torey ear ( ax ) attaned wen te brace n tenon yeld. For t value of te axu torey ear, te brace n copreon ubjected to an axal force wc lower tan te bucklng retance and ger tan te pot-bucklng retance N u,rd. Wen te top dplaceent ncreaed up to te dplaceent u l correpondng to te ultate ductlty of te brace, te u of te orzontal coponent of te axal force n te brace decreae becaue te reducton of te axal force of te eber n copreon not balanced by te ncreae of te axal force of te brace n tenon. Te torey ear correpondng to te ultate trengt naed u,rd and evaluated by ean of te relaton ( N + ) co! u, Rd = pl,rd Nu,Rd () were N pl,rd te yeld trengt of te brace. In t paper te lateral torey trengt Rd aued equal to te ultate ear force u,rd utaned by te brace. T aupton underetate te axu torey ear ax by a axu of % (for! = 1.3) and content wt te degn etod tpulated by te Japanee ec code [3-3] or tat propoed by Marno and Nakaa [37-39]. Furter, t alo content n prncple wt EC []. T code conder te lateral trengt Rd equal to te ear force correpondng to te yeldng of te brace n tenon and to a null axal force n te brace n copreon. Once te torey lateral trengt Rd a been defned, te torey overtrengt can be evaluated a follow u,rd! = (7) Ed DAMAGE DISTRIBUTION CAPACITY FACTOR Te daage dtrbuton capacty factor DDC evaluate te tendency of concentrcally braced tructure to experence concentraton of daage on a few torey f te yeldng of a brace occur preaturely. In partcular, te DDC factor calculated uppong tat te brace n tenon of

3 1 Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 Boco et al. [kn] 1 5 u pl 5 1 u l 15 u [c] u buck ax b,rd u,rd contrbuton of te brace n tenon contrbuton of te brace n copreon = 1.3 [kn] 1 5 ax b,rd =. u,rd u l u buck u pl u [c] Fg. (1). Sear force dplaceent relatonp for dagonal brace wt noraled lenderne equal to 1.3 and.. te ngle torey ave yelded wle te brace of te oter torey are wtn te elatc range of beavour. To evaluate te DDC factor of te -t torey, a odal repone pectru analy carred out on te tructural odel n wc te brace of te -t torey are not ncluded o a to ulate te yeldng of te brace n tenon and te bucklng of te brace n copreon. On te contrary, an elatc beavour aued for te brace of all te oter torey. However, a reduced tffne condered for tee brace n order to ulate te extent of tffne degradaton occurrng after brace n copreon ave buckled. Te ntertorey dplaceent deand of all torey are coputed and noraled to te ultate value of te platc ntertorey dplaceent of te ae torey,.e. to te platc ntertorey dplaceent correpondng to te brace falure. Noralaton carred out wt te a of evaluatng te portance of te ntertorey dplaceent deand wt repect to te platc deforatve capacty of te ae torey. T analytcally expreed by te relaton & n 1 ' u # j & ' # $! u DDC = $! ( * () l l n 1% j= 1 ' u j " % ' u " j) were n te nuber of torey, Δu te requred ntertorey dplaceent and Δu l te ultate platc ntertorey dplaceent,.e. te dfference between te ntertorey dplaceent correpondng to te ultate ductlty of te brace (u l ) and te ntertorey dplaceent correpondng to te end of te elatc beavour (u pl ). All tee dplaceent lgtly depend on te dtrbuton of force appled to te tructure. Defnton of te Ultate Ductlty of te Brace Te avalable ductlty µ f of te brace largely related to te fracture of te cro-ecton followng local bucklng, provded tat brace connecton are adequately degned and detaled [-]. T paraeter evaluated ere by ean of te relaton propoed by Treblay [3] µ =. + 3! (9) f. Equaton (9) wa deterned by Treblay on te ba of te experental reult of cyclc loadng tet; n partcular, te ductlty of te brace wa calculated at eac cycle a te u of te abolute value of te ortenng and elongaton of te brace (δ - and δ + ) dvded by te axal elongaton of te brace at yeld δ y. Evaluaton of te Ultate Platc Intertorey Dplaceent Te ntertorey dplaceent correpondng to te ultate ductlty of te brace are calculated ere by addng te contrbuton relatve to tree dfferent range of beavour of te braced frae. Te procedure appled to braced frae n wc only brace are allowed to enter te nelatc range of beavour. Te aboveentoned dplaceent contrbuton are evaluated by elatc tructural odel wc ulate te ecancal properte of te frae wtn te range of beavour under conderaton. Te frt range of beavour of te braced frae caractered by axal force of brace not ger tan te bucklng retance. Conequently, n te frt odel te brace of all torey are elatc. Te tructure ubjected to tatc (1) lateral force F wc, takng nto account alo te effect of gravty load, are able to caue te brace of all torey to buckle (Fg. a) and tu lead te frae to te begnnng of te ucceve range of beavour n! (1) (1) F = " F (1) b,rd, j= + 1 j At te end of te decrbed elatc range of beavour te ntertorey dplaceent u buck ; te axal elongaton of te brace n tenon equal but oppote n gn to te axal ortenng of te brace n copreon and ndcated a δ b (1). Te ucceve range of beavour of te frae caractered by elatc brace n tenon and by brace n copreon were bucklng a already been aceved. Te odel ulatng t econd range of beavour of te tructure conder brace n tenon but not brace n copreon. () Te tatc lateral force F lead to an ncreae of te axal force n te brace n tenon equal to!n pl,rd = N pl,rd " (N b,rd " N q ) and to a reducton of te axal force n te brace n copreon equal to!n u,rd = N u,rd " ("N b,rd + N q ) (Fg. b). Tee lateral force are gven by te followng relaton

4 Predcton of te Repone of Concentrcally Braced Frae Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 11 n ( u,rd, " b,rd,) "! () () F = F (11) j j= + 1 Te reducton! N u, Rd of te axal force n te brace n copreon not oberved n te tatc analy of te tructure becaue te brace n copreon are not odelled n t tage of te tructural beavour. Owng to t, tee force are appled to te tructure at te nterecton of brace wt colun. Te ntertorey dplaceent aceved at te end of t range of beavour are added to u buck to evaluate toe correpondng to te end of te elatc beavour (u pl ). Te addtonal axal elongaton and ortenng of te brace obtaned n t range of beavour are ndcated a δ () b. Fnally, n te trd range of beavour (Fg. c) no ncreent of te axal force n brace poble becaue te brace n tenon are yelded and toe n copreon are n te pot-bucklng range of beavour. Te torey tffne only provded by te flexural tffne of te colun but t contrbuton neglected. Te ntertorey dplaceent can ncreae untl te axu ortenng n te brace δ ax (wc aued to be equal to te axu elongaton)!! µ (1) ax =. 5 y f Tu, te ntertorey dplaceent correpondng to te ultate ductlty of te brace (1) () pl (" # " # " ) co! u l u ax b b + = (13) and te ultate platc ntertorey dplaceent : (1) () (" # " # " )! l l pl $ u = u # u = co (1) ax Evaluaton of te Intertorey Dplaceent Deand b b Te ntertorey dplaceent Δu wc appear n Equaton () are evaluated by ean of a odal repone pectru analy. Te tructural odel nclude te concentrcally braced frae a well a te colun degned for gravty load (or te oent retng frae n te cae of dual yte). Te brace of te -t torey (.e. were te DDC factor calculated) are not ncluded n te nuercal odel. In fact, at t torey, te brace n copreon condered to ave buckled and tat n tenon condered to ave yelded. Te brace of all te oter torey are n te elatc range of beavour. However, te area A eq, j of te cro ecton of tee brace lower tan te nonal area A br, j to take nto account tat te lateral torey tffne reduce fro tat of te frt range of beavour to tat of te econd range (Fg. d) before te attanent of te ntertorey dplaceent u pl. Te equvalent area of te brace cro-ecton A eq, j defned o tat te lateral tffne of te j-t torey equal te lope of te elatc branc of te blnear dealzaton of te onotonc beavour repreented n (Fg. d). Baed on t aupton, te equvalent area of te brace cro-ecton A eq, j can be calculated by te followng relaton A u,rd, j eq, j = A (15) br, j u,rd, j! b,rd, j ANALYSED STRUCTURES A et of buldng tructure wt concentrc dagonal brace analyed n t paper. Te tructure defned by te nterecton of four fve-pan plane frae orented along te x-drecton and x tree-pan plane frae orented along te y-drecton. Te plan layout (Fg. 3) a well a te geoetrc and a properte are equal at all floor. Te lengt L of eac pan equal to wle te ntertorey egt equal to 3.3. Te dagonal brace are located along te pereter of te buldng. Caractertc value of dead and lve load are equal to. and. kn/², repectvely. Storey a calculated on te ba of a ean value of te gravty load equal to 5. kn/² []. Te buldng (a) (1) F 1 Frt range of beavour (b) () F n Second range of beavour (1) F 1 N b,rd,n-n q -N b,rd,n +N q () F 1 ΔN pl,rd,n ΔN u,rd,n N b, Rd, 1-N q -N b,rd,1+n q ΔN pl,rd,1 ΔN u,rd,1 (c) Trd range of beavour (d) 3 rd range pl = u, R d nd range buck = b,rd 1 t range blnear dealzaton u buck u pl u f u Fg. (). Idealed onotonc beavour of te braced frae: (a) elatc beavour, (b) beavour after te bucklng of te copreon brace, (c) beavour after te yeldng of te tenon brace, (d) torey ear force ntertorey dplaceent relatonp.

5 1 Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 Boco et al. are founded on oft ol (ol C accordng to EC) n a gly ec area wt a reference ground acceleraton a g,r equal to.35 g.. C C1 C3 C Fg. (3). Plan layout of te condered tructure. C C Y. Te nuber of torey of te buldng n vared fro to 1 (n tep of ). Two ec-retant confguraton are condered for eac tructure: te ec acton reted only by te braced frae n te frt confguraton wle t utaned by te braced frae and oent retng frae (MRF) n te econd. Tee two confguraton wll be erenafter referred to a ple concentrcally braced yte and dual yte, repectvely. Te colun are contnuou along te egt of te buldng. Bea-to-colun connecton and colun-bae connecton are alway pnned n te plane of te braced frae. All te oter connecton are aued to be rgd and full trengt n dual yte and pnned n ple concentrcally braced yte. For eac of te obtaned tructure, brace are degned accordng to two dfferent degn etod a tpulated n EC [] and propoed by Marno and Nakaa [37-39]. Furterore, te effect of te ec force on dpatve eber (brace and alo bea n te cae of dual yte) evaluated by eter te odal repone pectru analy or te lateral force etod of analy. Te total nuber of frae analyed n t paper (3 egt x ec-retant confguraton x degn etod of brace x etod of analy). Eac frae dentfed by te nuber of torey (, or 1), te degn procedure (EC for Eurocode or MN for Marno and Nakaa) and te etod of analy ( for odal repone pectru analy or for lateral force etod of analy). Te ybol D added at te begnnng of te label of te frae to ndcate dual yte. X bracng Te degn ec acton deterned by te elatc pectru propoed by EC (1993) reduced by te beavour factor q. Te pectru repreentatve of ol type C and caractered by a g,r =.35 g. Te value of q aued equal to. for brace degned accordng to EC, wle t 3.5 wen te degn etod by Marno and Nakaa ued. Tee value are adopted for bot te ple concentrcally braced yte and te dual yte. Becaue of te yetry of te tructure, te nuercal analye for te evaluaton of te nternal force of te dpatve eber are carred out ung a two-denonal odel wc repreent alf te tructure of te buldng. T nuercal odel nclude only te braced frae n cae of ple concentrcally braced yte and bot te braced frae and oent retng frae for dual yte. Dagonal brace are generally lender and are expected to buckle at a low level of ec force. Tu, accordng to a plfcaton allowed by EC, only te brace n tenon are ncluded n te nuercal odel adopted for te evaluaton of te degn nternal force. Brace are ade of quare ollow ecton and teel grade S35. Wde-flange ape and teel grade S35, S75 and S355 are ued for bea and colun. Te ae ecton ued for te colun of two conecutve torey. Sple Concentrcally Braced Syte Brace are degned by te etod uggeted n EC [] and by tat propoed by Marno and Nakaa [37-39]. Te cro-ecton ze of te brace deterned at eac torey by equatng te degn torey ear force to te ear trengt. Accordng to EC, te lateral trengt provded by a par of dagonal brace aued to equal te orzontal reacton of te tenon brace, wc uppoed to carry te axal force correpondng to t yeldng N pl,rd. Terefore, te area of te cro-ecton of te brace of te -t torey Ed, A br, = (1) f co! y Accordng to Marno and Nakaa [37-39], te ear trengt provded by a par of dagonal brace equal to te ear u,rd calculated by Equaton () and te area of te cro-ecton of te brace of te -t torey Ed, A = (17) br, ( + " ) f co! 1 u y were χ u te rato of N u,rd to N pl,rd. In t paper, χ u evaluated by an equaton of te noralzed lenderne!. T equaton a been nuercally deterned and reported n [1]. Regardle of te degn etod adopted, te oogenety trengt condton of EC atfed,.e. te rato between te axu and nu overtrengt of te brace Ω,ax /Ω,n never greater tan 1.5. After te cro-ecton ze of te brace a been agned, te degn nternal force of bea and colun of te braced frae are deterned by te capacty degn crtera decrbed n EC [, 5]. In general, te degn axal force N Ed, of bea and colun at te -t torey calculated by te followng equaton N = 1 "! N (1) Ed, NEd, G, +. 1 ov,n Ed,E, were N Ed,G, and N Ed,E, are te axal force provded by gravty load and ec force, and te coeffcent γ ov te teel overtrengt factor aued equal to 1. a uggeted n [9]. Bea utan only te axal force 1.1 γ ov Ω,n N Ed,E, (N Ed,G, = ) and te bendng oent caued by te gravty load. For colun, N Ed,G, calculated accordng to te trbutary area concept and N Ed,E, gven by te degn ec analy. Bendng oent on colun ave been neglected. Te cro-ecton ze of te bea deterned by two degn condton: te bendng oent ut be aller tan te platc oent retance reduced becaue of te

6 Predcton of te Repone of Concentrcally Braced Frae Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 13 axal force and te axal force ut be aller tan te bucklng retance reduced becaue of te bendng oent. Te bucklng retance of te bea calculated aung tat te bucklng about te weak ax retraned by te deck. Te cro-ecton ze of te colun obtaned by equatng ter axal force to ter bucklng retance. Bot oent and bucklng retance are deterned accordng to te trengt crtera tpulated n Eurocode 3 (EC3) aung te partal afety coeffcent γ M and γ M1 are equal to unty []. Te bea-to-colun connecton of te colun tat do not belong to te braced frae are pnned. Tee colun utan only te axal force provded by gravty load (gravty colun). Te degn axal force evaluated accordng to te trbutary area concept and conderng te load per quare eter of te non-ec tuaton equal to 9. kn/. Te cro-ecton ze of tee colun obtaned by equatng ter axal force to ter bucklng retance evaluated accordng to EC3 aung γ M1 = 1.. Dual Syte Accordng to EC, te lateral force appled on te dual yte are dtrbuted between te CBF and MRF accordng to ter lateral tffne. Furter, EC recoend tat te MRF and te CBF of te dual yte are degned n accordance wt te provon tpulated for te two tructural type ndependently condered. Tu, te degn of te braced frae perfored n coplance wt te prevou ecton. For oent retng frae, te dpatve zone are located at te end of all te bea, at te bae of te frt torey colun and at te upper end of te top torey colun [7-51]. Te nu requred value of te bea platc odulu obtaned by equatng te degn bendng oent M Ed to te platc bendng oent M pl,rd. Degn bendng oent are calculated a te u of te effect of te gravty load and ec acton condered n te ec degn tuaton. A recoended n EC, bea ecton are elected o tat te degn axal force and te ear force do not decreae te full platc oent and te rotaton capacty at te platc nge. Te bea degned are alo verfed to utan te gravty load of te non-ec degn tuaton and lt te deflecton to te reference value reported n EC3 for te ervceablty lt tate. Te overtrengt factor Ω MRF of te dpatve zone of te MRF are calculated a te rato of te full platc retance n bendng to te degn bendng oent. Te degn nternal force of te non-dpatve zone of colun are obtaned, accordng to te capacty degn crtera, by te followng relaton N = 1 "! N (19a) Ed, NEd, G, +. 1 Ed, M Ed, G, +. 1 ov ov MRF n MRF n Ed,E, M = 1 "! M (19b) Ed, Ed, G, +. 1 ov MRF n Ed,E, = 1 "! (19c) Ed,E, Ten, te colun cro-ecton are elected o tat two condton are verfed: () te degn bendng oent lower tan te flexural trengt reduced becaue of te axal force; () te degn axal force lower tan te bucklng retance reduced becaue of te degn bendng oent. Bot flexural and bucklng retance are calculated accordng to EC3 aung γ M = γ M1 = 1. []. NUMERICAL ANALYSES Te ec beavour of te condered tructure evaluated by ncreental nonlnear dynac analy. Te ngle nonlnear dynac analy carred out by ean of te OpenSEES progra [5]. Te peak ground acceleraton a g caled n tep of.g n order to etate te peak ground acceleraton correpondng to te attanent of te ultate ductlty n brace. Modellng of Structure Te nuercal analye are carred out on twodenonal odel rater tan on tree-denonal odel (a an exaple ee Fg. for dual yte). T plfcaton poble becaue of te yetry of te tructure [53-5] and becaue no n-plan varaton of te nonal a, tffne and trengt dtrbuton condered [57-59]. Te yetry n te dynac and ecancal caractertc of te yte allow te adopton of a odel wc repreent alf te tructure of te buldng. Te brace are odelled by four nonlnearbeacolun eleent. Te cro-ecton dvded nto fbre. Te yteretc beavour of teel repreented by te odel of Menegotto-Pnto [, 1] wt kneatc and otropc ardenng. An ntal caber dplaceent of.1% of te brace lengt appled at brace d-lengt [-]. Te corotatonal teory ued to repreent te oderate to large deforaton effect of nelatc bucklng of te brace [5]. Bea and colun of te CBF are expected to rean elatc and, terefore, are odelled by ean of elatcbeacolun eleent. Wen dual yte are condered, te platc beavour of bea and colun of te oent retng frae odelled by ean of bea- WtHnge eleent. Te dept of te platc nge aued to be equal to te egt of te cro-ecton of te condered eber. Te vcou dapng force are obtaned troug te forulaton propoed by Rayleg. In partcular, te tffne coeffcent appled to te ntal tffne atrx of te eleent. A vcou dapng rato equal to.5 fxed for perod equal to toe of te frt and trd ode of vbraton of te - and 1-torey tructure, and for perod equal to toe of te frt and econd ode of vbraton of te -torey tructure. For eac tructure te analy perfored conderng te P-Δ effect. Sec Input Te ec nput conttuted by ten artfcal accelerogra. Te artfcal accelerogra are generated by ean of te oftware SIMQKE []. Te ngle artfcal accelerogra defned by a tatonary rando proce odulated by ean of a copound ntenty functon [7]. Te total duraton of eac accelerogra fxed equal to 3.5. Te eartquake re te ; te paraeter IPOW of te frt branc of te ntenty functon and te paraeter ALFA of te trd one are aued equal to and.5, repectvely (Fg. 5a). Te duraton of te tatonary part of te accelerogra equal to 7. and, terefore, lower tan te nu value uggeted by EC,.e. 1. Te

7 1 Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 Boco et al. COLUMN TYPE C1 C C C C C1 C C3 C C C3 C (a) C1 C C C1 C C3 C3 C C C C C L L L L L L L L L Fg. (). Nuercal odel for ec acton actng along (a) x-drecton, (b) y-drecton. adopted value a reulted fro a prevou nvetgaton n wc natural and artfcal accelerogra were copared n ter of nput energy pectra, Ara ntenty, frequency content and nuber of equvalent cycle []. Te coparon between te elatc repone pectru of eac accelerogra and te elatc pectru propoed by EC for ol C own n (Fg. 5b). Repone Paraeter For eac nonlnear dynac analy, te repone of te frae expreed n ter of te requred ductlty of te brace. To be content wt te approac followed by Treblay n dervng te ultate ductlty µ f, te brace ductlty deand µ d calculated ere a te u of te axu abolute value of te ortenng δ - and elongaton δ + dvded by δ y. Ten, te daage ndex calculated by Equaton (1). Te peak ground acceleraton ncreaed n tep of.g untl te axu daage ndex along te egt of te buldng equal to 1. ALIDATION OF THE PREDICTION OF THE DAM- AGE DISTRIBUTION Te daage dtrbuton frt predcted wt reference to te ngle accelerogra. Conequently, n order to evaluate te overtrengt factor, te torey ear Ed of Equaton (7) calculated by odal repone pectru analy conderng te elatc repone pectru correpondng to te accelerogra under exanaton. Te ae pectru adopted to evaluate te requred ntertorey dplaceent of Equaton (). Te expected daage ndex DI e ten calculated wt reference to eac accelerogra by Equaton (). Fnally, te ean value (wt repect to te 1 accelerogra) of te expected daage ndex are calculated. Tee value are caled o tat te axu daage ndex equal to 1 and are copared to te actual value DI dyn obtaned by nonlnear dynac analye. Overtrengt Factor of te Condered Frae Fg. () ow te egtwe dtrbuton of te noraled overtrengt factor (Ω /Ω,n ) for oe ple and dual dagonal braced frae degned accordng to EC. Specfcally, at eac torey te axu, nu and ean value of te overtrengt factor obtaned wt reference to te condered accelerogra are reported. In 1-torey ple concentrcally braced frae degned by te lateral force etod of analy (1EC), te egtwe dtrbuton of te overtrengt factor qute cattered. Te overtrengt generally low at te upper torey becaue te lateral force etod of analy gnfcantly underetate te degn ec acton at te ger torey of te buldng. Te egtwe dtrbuton of te overtrengt factor ore unfor n 1-torey ple concentrcally braced frae degned by odal repone pectru analy. However, n tee buldng, te oogenety trengt condton not rgorouly atfed. T apect anly due to te dfference between te degn pectru (were a non-zero nu value of te peudo-acceleraton uually condered) and te repone pectru correpondng to te accelerogra under exanaton. Wen dual yte are condered, te ec force at te upper torey are anly utaned by te oent retng frae and te cro-ecton of te brace are generally overzed n order to atfy te upper lt on te noraled lenderne condered n EC. For t reaon n 1- torey dual yte te overtrengt factor generally ger at te upper torey.

8 Predcton of te Repone of Concentrcally Braced Frae Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 15 (a) I (t) I(t) = (t / ) t < (b) S1. a /g 1. I(t) = 1 I(t) = e -.15 (t-11) <t<11 t> Eurocode pectru (ol C) t () T. () Fg. (5). Artfcal accelerogra: (a) Copound ntenty functon; (b) coparon between elatc repone pectra of te accelerogra and EC elatc pectru. 1 n 1 n 1 n 1 n 1 1 axu 1 1 ean nu 1EC 1EC D1EC D1EC 1.. Ω /Ω,n Ω /Ω,n Ω /Ω,n Ω /Ω,n 3. n n n n EC EC DEC DEC 1.. Ω /Ω,n Ω /Ω,n Ω /Ω,n Ω /Ω,n 3. Fg. (). Hegtwe dtrbuton of te overtrengt factor. In -torey yte te oogenety trengt condton alway atfed, regardle of te adopted etod of analy. Daage Dtrbuton Capacty Factor of te Condered Frae Fg. (7) ow te egtwe dtrbuton of te daage dtrbuton capacty factor for te yte decrbed n te prevou ecton. Te evaluaton of te DDC factor alot ndependent of te condered accelerogra. In fact, te nu, axu and ean value of te DDC factor at eac torey are alot concdent. For ple teel frae wt dagonal brace te nu value of te DDC factor along te egt of te frae qute low (cloe to.5). Furter, n 1-torey frae t nu value reaced at te upper torey,.e. were te preature yeldng of te brace n tenon occur. Tu, g value of te daage ndex are expected at te upper torey of tee yte. Intead, wen dual yte are condered, te DDC factor qute unfor and te value DDC n larger tan. and.3 n - and 1- torey yte, repectvely. Note tat a value of DDC n equal to.3 wa uggeted n a prevou tudy for te degn eccentrcally braced dual yte [- 7] n order to enure a collape ecan caractered by g value of te daage ndex along te egt of te buldng. Effectvene of te Propoed Forulaton Fg. () ow te coparon between te daage ndex evaluated by ean of te nonlnear dynac analy (black dot), aued a te actual value, and tat provded by te propoed etod (wte dot). Te coparon retrcted to oe of te analyed - and 1-torey frae. Te propoed expreon are able to predct accurately te repone of yte n wc te daage anly retrcted to a few torey. In te oter cae, te etod tll able to predct an alot unfor dtrbuton of te daage, altoug oe not neglgble catterng between actual and expected value can be found at oe torey. Te coparon between expected and actual value of te daage ndex repreented for all te condered frae n (Fg. 9a). Te x- and y-coordnate of te dot repreented n t fgure are te actual and predcted value of te daage ndex DI at eac torey. Dot lyng along te bector denote tat te propoed etod predct a value of te daage ndex equal to tat obtaned by te nonlnear dynac

9 1 Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 Boco et al. 1 n 1 n 1 n 1 n EC 1EC D1EC D1EC DDC DDC DDC DDC 1. n n n n EC EC DEC DEC DDC DDC DDC DDC 1. Fg. (7). Hegtwe of te daage dtrbuton capacty factor. 1 n 1 n 1 n 1 n EC expected value actual value 1EC D1EC D1EC DI DI DI DI n n n n EC EC DEC DEC DI DI DI DI Fg. (). Coparon between te actual and expected value of te daage ndex. 1. DI e DI e =DI dyn +..5 DI e =DI dyn DI DI e DI e =DI dyn +..5 DI e =DI dyn DI 1. dyn D-EC torey torey 1 torey EC D-MN MN torey torey 1 torey Fg. (9). Coparon between predcted and actual value of: (a) te daage ndex of brace of ngle torey (b) te average daage ndex. analy. T fgure ow tat te abolute value of te dfference between actual and expected daage ndexe generally lower tan.. Better reult are obtaned n ter of te ean value of te daage ndex (DI ) of all te brace of te analyed frae (Fg. 9b). CONCLUSION In t paper a relaton prevouly propoed to predct te egtwe daage dtrbuton of eccentrcally braced frae at collape extended to frae wt concentrc dagonal bracng.

10 Predcton of te Repone of Concentrcally Braced Frae Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 17 Te daage dtrbuton calculated a a functon of te brace overtrengt factor and te daage dtrbuton capacty factor. Te propoed expreon valdated wt reference to 1 ple braced frae and 1 dual braced yte degned accordng to dfferent etod and caractered by a dfferent nuber of torey. Te nuercal nvetgaton ow tat te propoed expreon able to predct accurately te nonlnear dynac repone of yte n wc te daage anly retrcted to a few torey. If t not te cae, te etod can provde notceable reult altoug oe not neglgble catterng between actual and expected value can be found at oe torey. CONFLICT OF INTEREST Te autor() confr tat t artcle content a no conflct of nteret. ACKNOWLEDGEMENT Declared none. REFERENCES [1] D.A. Foutc, Sec beavor of eccentrcally braced teel buldng, Journal of Structural Engneerng (ASCE), vol. 115, pp , 199. [] L.W. Lu, J.M. Rcle, and K. Kaa, Global Perforance: General Report, In: Proceedng of te Internatonal Workop and Senar on Beavour of Steel Structure n Sec Area, 1997, pp [3] D.A. Foutc, S.C. Goel, and C.. Roeder, Sec tetng of a full cale teel buldng Part I, Journal of Structural Dvon (ASCE), vol. 113, pp , 197. [] I.F. Katb, S.A. Man, and K.S. Pter, Sec Beavour of Concentrcally Braced Steel Frae, Report. No. UCB/EERC- /1, Eartquake Engneerng Reearc Center, Unverty of Calforna, Berkeley, 19. [5] R. Treblay, Sec beavor and degn of concentrcally braced frae, Engneerng Journal AISC, vol. 3, pp. 1-1, 1. [] R. Treblay, Acevng a table nelatc ec repone for ult-tory concentrcally braced teel frae, Engneerng Journal AISC, vol., pp , 3. [7] M. Lacerte, and R. Treblay, Makng ue of te brace overtrengt to prove te ec repone of ulttory plt-x concentrcally braced teel frae, Canadan Journal of Cvl Engneerng, vol. 33, pp ,. [] F.M. Mazzolan, G. Della Corte, and M. D Anello, Experental analy of teel dpatve bracng yte for ec upgradng, Journal of Cvl Engneerng and Manageent, vol. 15, pp. 7-19, 9. [9] P.P. Ro, and A. Lobardo, Influence of te lnk overtrengt factor on te ec beavour of eccentrcally braced frae, Journal of Contructonal Steel Reearc, vol. 3, pp , 7. [1] M. Boco, E.M. Marno, and P.P. Ro, Influence of te Brace Overtrengt on te Sec Beavor of Cevron Braced Frae, In: Proceedng of te t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea 9, pp [11] A. Zona, L. Ragn, and A. Dall Ata, Sentvty-baed tudy of te nfluence of brace over-trengt dtrbuton on te ec repone of teel frae wt BRB, Engneerng Structure, vol. 37, pp , 1. [1] T. Nonaka, An elatc-platc analy of a bar under repeated axal loadng, Internatonal Journal of Sold Structure, vol. 9, pp. 59-5, [13] E.P. Popov, K. Takana, and C.W. Roeder, Structural Steel Bracng Syte: Beavor under Cyclc Loadng, Report. No. UCB/EERC-7/17, Eartquake Engneerng Reearc Center, Unverty of Calforna, Berkeley, 197. [1] S. Kggn, and C.M. Uang, Reducng redual drft of bucklngretraned braced frae a a dual yte, Engneerng Structure, vol., pp ,. [15] C. Aryaratana, and L.A. Fanetock, Evaluaton of bucklngretraned braced frae ec perforance conderng reerve trengt, Engneerng Structure, vol. 33, pp. 77-9, 11. [1] G.A. MacRae, Y. Kura, and C. Roeder, Effect of colun tffne on braced frae ec beavour, Journal of Structural Engneerng (ASCE), vol. 13, pp ,. [17] Y. Kura, and G.A. MacRae, Effect of Colun Flexural Caractertc on Sec Beavor of Braced Frae wt Fxed Colun Bae, In: Proceedng of te 5t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea, pp [1] M. Boco, and P.P. Ro, A New Propoal of Sec Degn of Dual Eccentrcally Braced-Moent Retng Syte, In: Proceedng of te 5t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea, pp [19] Q. Xe, and L. Zao, Sec beavor of dual yte wt bucklng-retraned brace, Journal of Senyang Janzu Unverty, vol., pp. 1-5,. [] X. J, M. Kato, T. Wang, T. Htaka, and M. Nakaa, Effect of gravty colun on tgaton of drft concentraton for braced frae, Journal of Contructonal Steel Reearc, vol. 5, pp. 1-15, 9. [1] M. Boco, and E.M. Marno, Degn etod and beavor factor for teel frae wt bucklng retraned brace, Eartquake Engneerng and Structural Dynac, vol., pp , 13. [] Eurocode. Degn of tructure for eartquake retance. European Cottee for Standardaton 3, pren //3. 3. [3] M. Boco, E.M. Marno, and P.P. Ro, Propoal of odfcaton to te degn provon of eurocode for buldng wt plt k eccentrc brace, Engneerng Structure, 13 [In Pre]. [] P.P. Ro, A Degn procedure for ted braced frae, Eartquake Engneerng and Structural Dynac, vol. 3, pp. 7-, 7. [5] M. Boco, and P.P. Ro, Sec beavour of eccentrcally braced frae, Engneerng Structure, vol. 31, pp. -7, 9. [] M. Boco, and P.P. Ro, A degn procedure for dual eccentrcally braced yte: Analytcal forulaton, Journal of Contructonal Steel Reearc, vol., pp. -5, 13. [7] M. Boco, and P.P. Ro, A degn procedure for dual eccentrcally braced yte: Nuercal nvetgaton, Journal of Contructonal Steel Reearc, vol., pp. 53-, 13. [] S.A. Anagnotopoulo, C. Alexopoulou, and K.G. Statopoulo, An anwer to an portant controvery and te need for cauton wen ung ple odel to predct nelatc eartquake repone of buldng wt toron, Eartquake Engneerng and Structural Dynac, vol. 39, pp. 51-5, 1. [9] M. Boco, A. Ger, and E.M. Marno, Correctve eccentrcte for aeent by te nonlnear tatc etod of 3D tructure ubjected to bdrectonal ground oton, Eartquake Engneerng and Structural Dynac, vol. 1, pp , 1. [3] M. Kreln, and P. Fajfar, Te extended N etod conderng ger ode effect n bot plan and elevaton, Bulletn of Eartquake Engneerng, vol. 1, pp , 1. [31] M. Boco, A. Ger, E.M. Marno, and P.P. Ro, Coparon of nonlnear tatc etod for te aeent of ayetrc buldng, Bulletn of Eartquake Engneerng, DOI: 1.17/ , 13. [3] M. Boco, A. Ger, and E.M. Marno, On te evaluaton of ec repone of tructure by nonlnear tatc etod, Eartquake Engneerng and Structural Dynac, vol. 3, pp. 15-, 9. [33] G. Brandono, M. Toreno, E. Grande, E. Mele, and A. De Luca, Sec degn of concentrc braced frae, Journal of Contructonal Steel Reearc, vol. 7, pp. -37, 1. [3] AIJ. Degn tandard for teel tructure. Arctectural Inttute of Japan: Tokyo, (n Japanee)

11 1 Te Open Contructon and Buldng Tecnology Journal, 13, olue 7 Boco et al. [35] AIJ. Gudelne for lt tate degn of teel tructure. Arctectural Inttute of Japan: Tokyo, 199. (n Japanee) [3] E.M. Marno, M. Nakaa, and K.M. Moala, Coparon of European and Japanee ec degn of teel buldng tructure, Engneerng Structure, vol. 7, pp. 7-, 5. [37] E.M. Marno, and M. Nakaa, Sec perforance and new degn procedure for cevron-braced frae, Eartquake Engneerng and Structural Dynac, vol. 35, pp. 33-5,. [3] M. Boco, E.M. Marno, and P.P. Ro, A Untary Approac for te Degn of Concentrcally Braced Frae, In: Proceedng of te 7t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea 1, pp. 1-. [39] E.M. Marno, A unfed approac for te degn of g ductlty teel frae wt concentrc brace n te fraework of Eurocode, Eartquake Engneerng and Structural Dynac, DOI: 1.1/eqe.33, 13. [] B. Saback, and T. Brown, Beavour of Square Hollow Structural Steel Brace wt End Connecton under Revered Cyclc Axal Loadng, Canadan Journal of Cvl Engneerng, vol. 3, pp , 3. [1] B.. Fell, A.M. Kanvnde, A.M. ASCE, G.G. Deerlen, F. ASCE, and A.T. Myer, Experental nvetgaton of nelatc cyclc bucklng and fracture of teel brace, Journal of Structural Engneerng (ASCE), vol. 135, pp. 19-3, 9. [] C.W. Roeder, D.E. Lean, K. Clark, J. Powell, J.H. Yoo, K.C. Ta, C.H. Ln, and C.Y. We, Influence of guet plate connecton and brace on te ec perforance of X-braced frae, Eartquake Engneerng and Structural Dynac, vol., pp , 11. [3] R. Treblay, Inelatc ec repone of teel bracng eber, Journal of Contructonal Steel Reearc, vol. 5, pp. 5-71, Nort Aercan Specal Iue. [] M. Boco, A. Ger, and P.P. Ro, Sec Repone of Dual Eccentrcally Braced Syte Degned by Eurocode, In: Proceedng of te 7t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea 1, pp [5] G. Brandono, A. De Luca, E. Grande, and E. Mele, Non-lnear Repone of Concentrc Braced Frae, In: Proceedng of te 5t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea, pp [] CEN. EuroCode 3. Degn of teel tructure Part 1 1: General rule and rule for buldng, EN European Cottee for Standardzaton, Bruxelle, 5. [7] F.M. Mazzolan, and. Pluo, Meber Beavoural Clae of Steel Bea and Bea Colun, In: Proceedng of Frt State of te ArtWorkop, COSTI, 199, pp [] A. Ger, E.M. Marno, and F. Ner, A Sec Degn Metod for Hg Ductlty Steel Frae, In: Proceedng of te 5t Internatonal Conference on Beavour of Steel Structure n Sec Area - Stea, pp [9] T. Wang, J. McCorck, N. Yotake, P. Pan, Y. Murata, and M. Nakaa, Collape ulaton of a four-tory teel oent frae by a dtrbuted onlne ybrd tet, Eartquake Engneerng and Structural Dynac, vol. 37, pp ,. [5] M. D Anello, R. Landolfo,. Pluo, and G. Rzzano, Ultate beavour of teel bea under non-unfor bendng, Journal of Contructonal Steel Reearc, vol. 7, pp. 1-15, 1. [51] E.M. Güney, M. D'Anello, R. Landolfo, and K. Mererdaş, Novel forulaton of te flexural overtrengt factor for teel bea, Journal of Contructonal Steel Reearc, vol. 9, pp. - 71, 13. [5] S. Mazzon, F. McKenna, M.H. Scott, and G.L. Fenve. OpenSee Coand Language Manual, 7. [53] B. Calderon, A. D Aven, A. Ger, and Z. Rnald, Statc v. odal analy of ayetrc buldng: Effectvene of dynac eccentrcty forulaton, Eartquake Spectra, vol. 1, pp ,. [5] E.M. Marno, and P.P. Ro, Exact evaluaton of te locaton of te optu toron ax, Te Structural Degn of Tall and Specal Buldng, vol. 13, pp. 77-9,. [55] A.M. Atanatopoulou, T. Makaro, and K. Anataad, Eartquake analy of otropc ayetrc ulttorey buldng. Te Structural Degn of Tall and Specal Buldng, vol. 15, pp. 17-3,. [5] M. Boco, E.M. Marno, and P.P. Ro, An analytcal etod for te evaluaton of te n-plan rregularty of non-regularly ayetrc buldng, Bulletn of Eartquake Engneerng, vol. 11, pp , 13. [57] K.G. Statopoulo, and S.A. Anagnotopoulo, Accdental degn eccentrcty: I t portant for te nelatc repone of buldng to trong eartquake?, Sol Dynac and Eartquake Engneerng, vol. 3, pp , 1. [5] M. De Stefano, M. Tanganell, and S. t. Effect of te varablty n plan of concrete ecancal properte on te ec repone of extng rc fraed tructure, Bulletn of Eartquake Engneerng, vol. 11, no., pp. 19-1, 13. [59] G. Zárate, and A.G. Ayala, aldaton of Sngle Storey Model for te Evaluaton of te Sec Perforance of Mult-Storey Ayetrc Buldng, In: Proceedng of te 13t World Conference on Eartquake Engneerng, paper no. 13, [] M. Menegotto, and P. Pnto, Metod of Analy for Cyclcally Loaded Renforced Concrete Plane Frae Includng Cange n Geoetry and Nonelatc Beavour of Eleent under Cobned Noral Force and Bendng, IABSE ypou on Retance and ultate deforablty of tructure Acted on by well defned repeated load, Fnal Report, [1] F.C. Flppou, E.P. Popov, and.. Bertero, Effect of Bond Deteroraton on Hyteretc Beavour of Renforced Concrete Jont, Report. No. UCB/EERC-3/19, Eartquake Engneerng Reearc Center, Unverty of Calforna, Berkeley, 193. [] M. Dclel, and E. E. Calk, Pycal teory yteretc odel for teel brace, Journal of Structural Engneerng ASCE, vol. 13, pp ,. [3] M. D Anello, F. Portol, and R. Landolfo, Modellng Iue of Steel Brace under Extree Cyclc Acton, In: Proceedng of te Fnal Conference COST ACTION C: Urban Habtat Contructon under Catatropc Event, 1, pp [] M. D Anello, G. La Manna Abrono, F. Portol, and R. Landolfo, Modellng apect of te ec repone of teel concentrc braced frae, Steel & Copote Structure, An Internatonal Journal, vol. 15, 13. [5] P. Urtz, F.C. Flppou, and S.A. Man, Model for cyclc nelatc bucklng of teel brace, Journal of Structural Engneerng, vol. 13, pp. 19-,. [] SIMQKE Uer Manual. NISEE Software Lbrary, Unverty of Calforna, Berkeley, USA, 197. [7] Jennng PC, Houner GW, and Ta NC. Sulated Eartquake Moton. EERL Calforna Inttute of Tecnology, Paadena, Calforna, 19. [] F. Aara, M. Boco, E.M. Marno, and P.P. Ro, An accurate trengt aplfcaton factor for te degn of SDOF yte wt P-Δ effect, Eartquake Engneerng and Structural Dynac, 13, [In Pre]. Receved: July 31, 13 Reved: Septeber 9, 13 Accepted: Septeber 11, 13 Boco et al.; Lcenee Benta Open. T an open acce artcle lcened under te ter of te Creatve Coon Attrbuton Non-Coercal Lcene (ttp://creatvecoon.org/- lcene/by-nc/3./) wc pert unretrcted, non-coercal ue, dtrbuton and reproducton n any edu, provded te work properly cted.