World Academy of Science, Engineering and Technology International Journal of Civil and Environmental Engineering Vol:4, No:10, 2010

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1 Evaluaon of he Dsplacemen-Based and he Force-Based Adapve Pushover Mehods n Sesmc Response Esmaon of Irregular Buldngs Consderng Torsonal Effecs R. Abbasna, F. Mohajer Nav, S. Zahedfar, and A. Tajk Absrac Recen years, adapve pushover mehods have been developed for sesmc analyss of srucures. Heren, he accuracy of he dsplacemen-based adapve pushover (DAP) mehod, whch s nroduced by Anonou and Pnho [004], s evaluaed for Irregular buldngs. The resuls are compared o he force-based procedure. Boh concree and seel frame srucures, asymmerc n plan and elevaon are analyzed and also orsonal effecs are akng no he accoun. These analyses are performed usng boh near faul and far faul records. In order o verfy he resuls, he Incremenal Dynamc Analyss (IDA) s performed. Keywords Pushover Analyss, DAP, IDA, Torson. A I. INTRODUCTION man challenge n performance-based desgn s o develop smple bu accurae mehods for esmang he sesmc response of srucures. In comparson wh meconsumng and complex mehods such as Incremenal Dynamc Analyss (IDA), non-lnear sac (pushover) analyses are more praccal and smple. Hence, pushover procedures have been exensvely developed durng he las decade. Convenonal pushover procedures, such as he Capacy Specrum Mehod [1] and The Dsplacemen Coeffcen Mehod [], only consder he domnan mode of srucure. Hence, hese mehods rely on a pushover analyss usng nvaran laeral load paerns o esmae deformaon demands under sesmc loadng. Based on hese assumpons, convenonal pushover mehods are known o have major drawbacks [3]. In order o solve he problems of convenonal mehods, several researchers have proposed advanced pushover procedures [4], [5], [6]. Some advanced mehods, such as MPA [4], rean he smplcy of nvaran load paerns whle R. Abbasna s wh he Iran Unversy of Scence and Technology, Narmak, Tehran, Iran (e-mal: abbasna@us.ac.r). F. M. Nav, s wh he Iran Unversy of Scence and Technology, Narmak, Tehran, Iran (correspondng auhor o provde phone: ; e-mal: foadmohajer@cvleng.us.ac.r). S. Zahedfar s wh he Iran Unversy of Scence and Technology, Narmak, Tehran, Iran (e-mal: saeedzahedfar@gmal.com). A. Tajk s wh he Iran Unversy of Scence and Technology, Narmak, Tehran, Iran (e-mal: A_ajk@cvleng.us.ac.r). some oher mehods, such as AMC [5], are usng varan load paern. In anoher aspec, advance pushover mehods could be caegorzed no wo man groups: dsplacemen-based loadng and force-based loadng adapve pushover mehods. Anonou and Pnho [004] evaluaed adapve and non-adapve pushover mehods based on force load vecors. They concluded ha force-based adapve pushover mehods canno reach o accepable resuls whle non-adapve mehods usng force-based loadng vecors could gve us more accurae resuls [7]. Based on hese observaons, hey proposed he dsplacemen-based adapve (DAP) analyss whch uses dsplacemen loadng vecor nsead of force loadng vecor [6]. Snce non-adapve pushover mehods, wh he rangular and unform dsrbuons, do no always provde curves ha consue a lower and an upper bound o he Incremenal Dynamc Analyss (IDA) response pons, seems necessary o develop adapve mehods nsead of non-adapve ones [7]. From anoher pon of vew, he sesmc response of asymmerc buldngs n he nelasc range s very complex. Alhough exensve research has been performed world-wde n he feld of nelasc orsonal response, general conclusons are lackng. Unforunaely, unl recenly lle aenon has been pad o he mos realsc bu mos complex case: mulsorey buldngs wh b-axal eccenrcy, subjeced o bdreconal ground moon. An overvew of recen research on orson was made by Ruenberg [00]. In hs paper, he effcency of he force-based and he dsplacemen-based adapve pushover procedures (FAP and DAP), nroduced by Anonou and Pnho [6], [7], s compared n rregular buldngs. In order o consder he orsonal effecs, dfferen srucures wh dfferen knd of rregulares n plan and elevaon are used. The resuls are compared wh convenonal procedure and IDA. Boh near faul and far faul records are also used. II. CONVENTIONAL PUSHOVER PROCEDURES Convenonal pushover analyss s he nonlnear ncremenal-erave soluon of he equlbrum equaon 34

2 KU=P n a fne elemen formulaon, where K s he nonlnear sffness marx, U s he dsplacemen vecor and P s a predefned load vecor appled laerally over he hegh of he srucure n relavely small load ncremen. Ths laeral load can be a se of forces or dsplacemens ha have a necessarly consan rao hroughou he analyss (fxed paern). A he end of each eraon, he reacon vecor of he srucure s calculaed from he assemblage of all fne elemen conrbuons. The ou-of-balance forces are eravely reappled unl convergence o a specfed olerance s reached [8]. Fg. 1 Yeldng sequence hrough convenonal pushover analyss. The procedure connues eher unl a predefned lm sae s reached or unl srucural collapse s deeced. Ths arge lm sae may be he deformaon expeced for he desgn earhquake n case of desgnng a new srucure, or he drf correspondng o srucural collapse for assessmen purposes. Furhermore, s presumed ha he fne elemen code has been suffcenly verfed, so ha numercal collapse, as opposed o srucural, s no operave. Generally, hs procedure allows racng he sequence of yeldng and falure on he member and srucure level, as well as he progress of he overall capacy curve of he srucure. Ths process s shown n Fg. 1 [8]. The Dsplacemen Coeffcen Mehod (DCM-[]) and he Capacy Specrum Mehod (CSM-[1]) are he mos wdely known convenonal mehods. In he dsplacemen coeffcen mehod, op s maxmum expeced dsplacemen s consdered as srucural performance pon. The modfed dsplacemen of elasc response specrum s used for esmang he maxmum dsplacemen of he equvalen nonlnear sngle degree of freedom sysem. The dsplacemen demand of he mehod s deermned from he elasc one by usng a number of correcon facors based on sascal analyses. Accordng o FEMA 356, he arge dsplacemen, whch s he maxmum dsplacemen occurrng a he op of srucures durng a chosen earhquake, can be deermned as Te C0C1CC3S a g (1) 4 where C 0 s he dfferences of dsplacemens beween he conrol node of MDOF (mul degree of freedom) buldngs and equvalen SDOF sysems, C 1 s he modfcaon facor for esmang he maxmum nelasc deformaon of SDOF sysems from her maxmum elasc deformaon, C s he response o possble degradaon of sffness and energy dsspaon capacy for srucural members durng earhquakes, C 3 s he modfcaon facor for ncludng he P effecs, T e s he effecve perods of evaluaed srucures, S a s he specral value of acceleraon response correspondng o T e, and g s he acceleraon of gravy []. The capacy specrum mehod (CSM) was frs nroduced by Freeman [9], [10] as a rapd evaluaed procedure for assessng he sesmc vulnerably of buldngs. Aferwards, ATC-40 [1] nvesgaed CSM procedure n deals. Ths procedure compares he srucural capacy n he form of a pushover curve wh demands on he srucure n he form of an elasc response specrum. The graphcal nersecon of he wo curves approxmaes he response of he srucure [9] [11]. In order o accoun for he effecs of nonlnear behavor of srucures, equvalen vscous dampng has been mplemened o modfy he elasc response specrum. Impled n he capacy specrum mehod s ha he maxmum nelasc deformaon demand of a non-lnear sngle-degree-of-freedom (SDOF) sysem can be approxmaely esmaed by an erave procedure of a seres of lnear secan represenaon sysems. Therefore, avods dynamc analyss of nelasc sysems []. III. ADVANCED PUSHOVER PROCEDURES In order o accoun for hgher modes effecs, advanced pushover procedures have been developed. As a pon of vew, hese advanced mehods can be dvded no wo man groups: he procedures wh nvaran load paerns and he procedures wh varan load paerns. Pare e al. [1] was he frs one who suggesed he Mul-Modal Pushover (MMP) procedure. MMP was ryng o accoun for hgher modes effecs regardng o consan load paern. MMP was hen refned by Moghadam and Tso [13]. Chopra and Geol [4] have developed hese mehods and proposed a Modal Pushover Analyss (MPA) procedure. Invaran load paern canno consder he changes n he dynamc characerscs of srucures. Hence, adapve pushover procedures have been developed. The frs aemp whch ulzes fully adapve paerns was nroduced by Bracc e al. [14]. Lefor [15] developed hs mehod. A dfferen adapve mehodology was proposed by Gupa and Kunnah [16], n whch he appled load s consanly updaed durng he analyss. Ths concep has been developed and used n dfferen versons of advanced pushover mehods. As menoned before, Anonou and Pnho [003], [004] nvesgaed adapve and non-adapve pushover mehods and concluded ha force-based adapve procedures canno predc he sesmc responses accuraely. Based on hese observaons, hey nroduced he dsplacemen-based adapve pushover (DAP) procedure. Aferwards, so many oher advanced pushover procedures usng he dsplacemen-based loadng vecor are proposed [5] and seems ha he nroducon of he dsplacemen-based loadng could be a urnng pon n developmen of pushover mehods. 343

3 IV. THE DISPLACEMENT-BACED ADAPTIVE PUSHOVER PROCEDURE The dsplacemen-based adapve pushover (DAP) procedure, whch s nroduced by Anonou and Pnho [6], s based on usng dsplacemens nsead of forces n order o make he load paerns. The DAP procedure can be mplemened n four man sages; (a) defnon of nomnal load vecor and nera mass, (b) compuaon of load facor, (c) calculaon of normalsed scalng vecor and (d) updae of loadng dsplacemen vecor. Whle he frs sep s mplemened only once, a he sar of he analyss, he hree remanng pars are repeaed a every sage durng he analyss [6]. Snce n adapve pushover, n conras o convenonal one, he loadng vecor shape s auomacally defned and updaed a each analyss sep, he nomnal load vecor, U 0, mus have a unform (recangular) dsrbuon shape n hegh, n order o preven from he dsoron of he load vecor confguraon a any analyss sep. Knowng he U 0, he magnude of he loadng vecor, U, a each sep s calculaed by he produc of s nomnal counerpar, U 0, and he load facor (). The load facor s auomacally ncreased, by means of a load conrol or response conrol ncremenal sraegy, unl a predefned analyss arge, or numercal falure, s reached. U.U 0 () In order o deermne he shape of he load vecor (or load ncremen vecor) a each sep, he normalsed modal scalng vecor, D s used. Ths normalsed modal scalng facor s compued a he sar of every load ncremen. In order o compue D, frsly he scalng dsplacemen vecor, D, should be deermned. The scalng dsplacemen vecors, whch reflec he acual sffness sae of he srucure, are obaned drecly from he egen vecors, as descrbed n (3), where s he sorey number, j s he mode number, s he modal parcpaon facor for he jh mode, j s he mass normalsed mode shape value for he h sorey and he jh mode, and N sands for he oal number of modes, calculaed hrough an egenvalue analyss. In he egenvalue analyss, frsly modal shapes and modal parcpaon facors and fnally modal loads are calculaed and hen, SRSS or CQC combnaon rules are used o combne hem. N D D j j1 N j1 ( ) (3) j j The maxmum dsplacemen of a parcular floor level (he relave maxmum dsplacemen beween ha floor and he ground), canno be a good ndcaon of he acual level of damage ncurred by buldngs subjeced o earhquake loadng. j On he conrary, nersorey drfs, obaned as he dfference beween floor dsplacemens a wo consecuve levels, feaure a much clearer and drec relaonshp o horzonal deformaon demand on buldngs. Hence, based on nersorey drfs, Equaon (3) could be wren as: D where k 1 N j1 k j N, j 1, j S j j1 (4) j s he nersorey drfs for each mode and D s he dsplacemen paern a he h sorey whch s obaned hrough he summaon of he modal-combned nersorey drfs of he soreys below ha level. Equaon (4) also ncludes an addonal parameer S d,j ha represens he dsplacemen response specrum ordnae correspondng o he perod of vbraon of he jh mode, whch s called specral amplfcaon. In oher words, he modal nersorey drfs are weghed by he S d value a he nsananeous perod of ha mode, so as o ake no accoun he effecs ha he frequency conen of a parcular npu me-hsory or specrum have n he response of he srucure beng analyzed. Alhough usng he relave dsplacemen beween floors n order o deermne he floor dsplacemen leads o beer resuls, however, Equaon (4) s approxmae, because s assumed ha he relave maxmum dsplacemen beween floors n all soreys occurs a he same me. Snce only he relave values of sorey dsplacemens (D ) are of neress n he deermnaon of he normalsed modal scalng facor D, whch defnes he shape, no he magnude of he load or load ncremen vecor, he dsplacemens obaned by (4) are normalsed so ha he maxmum dsplacemen remans proporonal o he load facor: D D max D 1 Once he normalsed scalng vecor D has been deermned, knowng he value of he nal nomnal load vecor U 0, he loadng dsplacemen vecor U a a gven analyss sep should be updaed. Updang he loadng vecor could be done usng one of wo alernaves; oal or ncremenal updang. Wh oal updang, he load vecor U a a gven analyss sep s obaned hrough a full subsuon of he exsng balanced loads by a newly derved load vecor, compued as he produc beween he curren oal load vecor, he curren normalsed modal scalng vecor U and he nomnal load vecor U 0, as shown n (6) : (5) 344

4 U. D. U (6) 0 Wh ncremenal updang, he load vecor U a a gven analyss sep s obaned by addng he load vecor of prevous sep P -1 a newly derved load vecor ncremen, compued as he produc beween he curren load facor ncremen, he curren normalsed modal scalng vecor U and he nomnal load vecor U 0, as shown n (7) : ypcally employed a he me. A dealed descrpon of he srucure may be found n Carvalho e al. [1999]. The srucure has been exensvely suded analycally. The four-sorey frame consss of wo bays of 5.0m span and one bay of.5m span. The ner-sorey hegh s.7m, he slab hckness s 0.15m wh a wdh of 4.0m. Ths srucure has been modeled as a wo dmensonal frame. Durng he analyses, hs model s known as M1. U U. U (7) 1 D. 0 The procedure of he force-based adapve pushover (FAP) s he same bu he only dfference s he applcaon of forcebased loadng vecor nsead of dsplacemen. In hs paper, boh procedures wll be nvesgaed. V.STRUCTURAL MODELS AND GROUND MOTIONS A. Models In order o compare he accuracy of DAP and FAP mehods n sesmc response esmaon of rregular buldngs, regardng o orsonal effecs, dfferen ypes of rregular srucures has been chosen. Usng hese models, an exensve sudy has been carred ou, nvolvng sac and dynamc nonlnear analyss. In hs secon, he models wll be nroduced. The ICONS Frame: Ths srucure s esed n Ispra, Ialy n May 1999 (Fg. ). I s seleced for s srong rregulary n plan and s varyng n-plane sffness. The srucure was desgned by Carvalho e al. [1999] wh he objecve of represenng desgn and consrucon pracce n many Souhern European and Mederranean counres n he 1950 s and 60 s. The desgn procedure followed regulaon requremens ha were n use hen, and made use of maerals Fg. 3 The SPEAR frame, Plane. The SPEAR frame: A hree dmensonal based on a full scale srucure esed n 00 whn European nework Sesmc Performance Assessmen and Rehablaon (SPEAR). I feaures rregulares boh n plan and elevaon (Fg. 3 and Fg. 4) [17]. Fg. The ICONS frame Fg. 4 The SPEAR frame, Acual Srucure. 345

5 Fg. 5 The hree sorey b-dreconally eccenrc sysem M3 (a) Plan vew (b) Elevaon vew and secons properes The es buldng has been desgned for gravy loads alone, usng he concree desgn code appled n Greece beween 1954 and I was bul wh he consrucon pracce and maerals used n Greece n he early 70 s. The srucural confguraon s also ypcal of non earhquake-ressan consrucon of ha perod [17]. Durng he analyses, hs model s known as M. The hrd model s a 3 sorey concree frame buldng whch has an asymmercal srucural plan. The sysem has a bdreconal eccenrcy of 5% of he plan dmenson n boh drecons. The descrpon of hs model s shown n Fg. 5 (a) and (b). All dmensons are n mllmeer unless saed oherwse. Ths model, whch s adoped from Erduran [18], s a hree dmensonal model. Durng he analyses, hs model s known as M3. Fg. 6 The 1 sorey seel frame Nex model s a 1 sorey dmensonal seel frame buldng whch s vercally rregular. The hree dmensonal form of hs model s nroduced n FEMA 451 [19], bu here only he wo dmensonal model s nvesgaed. Columns range n sze from W4x146 a roof o W4x9 a ground level. Beams vary from W30x108 a roof o W30x13 a ground level. Fg. 6 shows he schemac descrpon of he model. More deals abou he model could be found n [19]. Ths model s known as M4 durng he analyses. The las model, whch s orgnally adoped and developed from Ray Chaudhur and Vllaverde [0], s a four sorey seel frame buldng, whch n plan and elevaon s rregular. The buldng has a unform mass dsrbuon over he hegh and a non-unform sffness dsrbuon. Fg. 7 (a) shows he plan of he framng sysem and Fg. 7 (b) shows he elevaon of he frame. Ths model s referred as M5 durng he analyses. As menoned above, dfferen models wh dfferen knds of rregulares and also dfferen maerals are chosen o perform a comparave sudy on effcency of force-based and dsplacemen-based adapve procedures n sesmc response esmaon of buldngs wh consderable orsonal effecs. Table I shows he bref descrpons of models M1 o M5. TABLE I REVIEW OF MODELS Models Number of Irregulary Soreys Plan Ele. Modelng Maeral M1 3 D concree M 3 3D concree M3 3 3D concree M4 1 D seel M5 4 3D seel The analyses have been mplemened usng he open source fne elemen plaform, OpenSees []. B. Ground Moon Ensemble In order o develop a se of benchmark responses agans whch o compare he resuls of analyses, a se of records havng far-faul and near-faul characerscs were compled. 346

6 The characerscs of he records are summarzed n Table II [1]. VI. RESULTS For each one of he consdered srucures, he resuls are presened. Capacy curves for dfferen models are depced. In Fg. 8 and Fg 9, a seres of op dsplacemen versus base shear plos, comparavely llusrang sac and dynamc resuls obaned for dfferen models subjeced o equally dverse earhquake records, s gven. For each model, wo pushover curves are presened, one resuled from usng farfaul records and he oher one s resuled from usng near-faul records. In order o compare he resuls, for each model, pushover curves resuled from Adapve pushover procedures (force-based and dsplacemen-based), IDA and convenonal pushover mehods are depced n he same graph. I s observed ha for srucures wh rregulary n plan and elevaon, IDA pons were dfferen from pushover analyses resuls, adapve or non-adapve. In oher words, seems ha adapve pushover procedures, boh force-based and dsplacemen-based mehods, canno predc he sesmc Fg. 7 The four sorey seel frame M5 (a) Plan Vew (b) Elevaon responses of rregular srucures precsely. Jus for M4 whch was only vercally rregular, he resuls seem o be beer. In srucures wh asymmerc plans, by ncreasng he rregulary, he accuracy of resuls s decreasng, as shown n Fg. 8 and Fg. 9. For mos of hese analyses, resuls of he force-based and he dsplacemen-based adapve procedures were very close. Also here was no a sgnfcance dfference beween near faul and far faul resuls. Only n M1 (The ICONS Frame) whch n plan and elevaon was rregular, he resuls for far faul and near faul records were dfferen. Based on he resuls, seems ha for rregular srucures, especally for hree dmensonal rregular buldngs whch orsonal effecs play an mporan role n he srucure, he convenonal pushover analyss gves beer resuls raher han he dsplacemen-based or force-based adapve pushover procedures. In addon o above, n srucures wh plan rregulary, such as M and M5, for boh force-based and dsplacemenbased, he dfference beween IDA and adapve and convenonal pushover analyses were more han ohers. TABLE II CHARACTERISTICS OF THE GROUND MOTIONS Ground Moon Magnud Record Year Dsance a (km) PGA (g) PGV (cm/s) PGD (cm) Far-faul Near-faul e Kobe Kobe Norhrdge Norhrdge Imperal Valley Imperal Valley Tabas Tabas San Fernando San Fernando a 347

7 Fg. 8 Pushover Curves (a) M1, Far-faul record (b) M1, Near-faul record (c) M, Far-faul record (d) M, Near-faul record VII. CONCLUSION The accuracy of he dsplacemen-based and he force-based adapve pushover procedures (DAP and FAP) whch were nroduced by Anonou and Pnho [004], n sesmc response esmaon of rregular buldngs, are nvesgaed. Irregular srucures n plan an elevaon wh consderable orsonal effecs have been analyzed. Four dfferen srucures wh plan rregulary and also a srucure wh elevaon rregulary have been analyzed. Boh far-faul and near-faul records have been used. To verfy hese analyses, he Incremenal Dynamc Analyss (IDA) s used. A summary of he man observaons and general conclusons of he presen sudy s presened below: Adapve pushover mehods, boh he dsplacemen based and force-based mehods, canno predc he sesmc responses of rregular srucures, especally he srucures wh plan rregulary. I seems ha convenonal pushover analyss can predc he sesmc responses of rregular buldng more accurae han adapve mehods. By ncreasng he rregulary of he srucures, especally he rregulary of plan, he accuracy of resuls s decreasng. The dfference beween he dsplacemen-based and he force-based adapve procedures n hree dmensonal srucures wh consderable orsonal effecs s very low. In rregular srucures, he convenonal pushover analyss and adapve procedures provde close resuls. The above sudy ndcaes ha furher research work s requred o compare he adapve and non-adapve and also he dsplacemen-based and he force-based adapve pushover procedures n sesmc response esmaon of rregular buldngs. Dfferen oher rregular srucures should be consdered, especally buldngs wh consderable plan rregulary whch orsonal effecs are noceable. I seems ha adapve procedures need more mprovemens o for enhancng n rregular srucures. 348

8 Fg. 9 Pushover Curves (a) M3, Far-faul record (b) M3, Near-faul record (c) M4, Far-faul record (d) M4, Near-faul record (e) M5, Far-faul record (f) M5, Near-faul record ACKNOWLEDGMENT The auhors would lke o acknowledge he fnancal suppor provded by he cvl engneerng deparmen of Iran Unversy of Scence and Technology (IUST). REFERENCES [1] ATC-40, 1996, Sesmc evaluaon and rerof of concree buldngs, Appled Technology Councl, Volume 1, Redwood Cy, Calf. 349

9 [] FEMA 356, 000, Presandard and Commenary for he Sesmc Rehablaon of Buldngs, Federal Emergency Managemen Agency, Washngon, D.C. J. Earhquake Eng., 13(11), Anono, S., and Pnho, R., 004, Developmen and verfcaon of a dsplacemen-based adapve pushover procedure, J. Earhquake Eng., 8(5), Anono, S., and Pnho, R., 004, Advanages and Lmaons of Adapve and Non-adapve Force-Based Pushover Procedures J. Earhquake Eng., 8(4), Papankolaou, V. K., and Elnasha, A. S., and Pareja, J. F., Lms of Applcably of Convenonal and Adapve Pushover Analyss for Sesmc Response Assessmen, Md-Amerca Earhquake Cener, Cvl and Envronmenal Engneerng Deparmen Unversy of Illnos a Urbana-Champagn. Freeman S. A., 1998, Developmen and use of capacy specrum mehod, The 6h US Naonal Conference on Earhquake Engneerng=EERI, Seale, Washngon, Paper No. 69. Pare, T. F., Sasak, K. K., Elbeck, D. H. and Freeman, S. A., 1996, Approxmae nelasc procedures o denfy falure mechansms from hgher mode effecs, Proc. of he Elevenh World Conference on Earhquake Engneerng, Dsc, Paper No. 966 Moghadam, A. S. and Tso, W. K., 00, A pushover procedure for all buldngs, Proc. of he Twelfh European Conference on Earhquake Engneerng, London, Uned Kngdom, Paper No Bracc, J. M., Kunnah, S. K. and Renhorn, A. M., 1997, Sesmc performance and rerof evaluaon of renforced concree srucures, Journal of Srucural Engneerng 13(1), Lefor, T., 000, Advanced pushover analyss of RC mul-sorey buldngs, MSc Dsseraon, Deparmen of Cvl and Envronmenal Engneerng, Imperal College London, Uned Kngdom. Gupa, B. and Kunnah, S. K., 000, Adapve specra-based pushover procedure for sesmc evaluaon of srucures, Earhquake Specra 16(), Fards, M.N., 00, Desgn of an Irregular Buldng for he SPEAR Projec - Descrpon of he 3- Sorey Srucure, Unversy of Paras Erduran, E., Assessmen of curren nonlnear sac procedures on he esmaon oforsonal effecs n low-rse frame buldngs, Engneerng Srucures 30 (008) [19] FEMA 451, 006, NEHRP Recommended Provsons: Desgn Examples, Federal Emergency Managemen Agency, Washngon, D.C. Ray ChaudhurS, Vllaverde R. Effec of buldng nonlneary on sesmc response of nonsrucural componens: a paramerc sudy. ASCE Journal of Srucural Engneerng 008;134(4): Pacfc Earhquake Engneerng Research Cener, hp://peer.berkeley.edu/smca 350