AN ADAPTIVE CAPACITY SPECTRUM METHOD FOR ESTIMATING SEISMIC RESPONSE OF STEEL MOMENT-RESISTING FRAMES

Internatonal Journal of Earthquake Engneerng Num. Anno XXXIII Specale CTA 2015 Num. 1-2 AN ADAPTIVE CAPACITY SPECTRUM METHOD FOR ESTIMATING SEISMIC RESPONSE OF STEEL MOMENT-RESISTING FRAMES Massmlano Ferraol*, Angelo Lavno, Alberto Mandara Department of Cvl Engneerng, Desgn, Buldng and Envronment Second Unversty of Naples, Aversa (CE) SUMMARY: An adaptve verson of the capacty spectrum method s proposed to estmate deformaton demands of steel moment-resstng frames under sesmc loads. Its computatonal attractveness and capablty of provdng satsfactory predctons of sesmc demands n comparson wth those obtaned by other advanced nonlnear statc procedures n lterature are examned. Both effectveness and accuracy of these approxmated methods based on pushover analyss are verfed through an extensve comparatve study nvolvng both regular and rregular steel moment-resstng frames. The results obtaned by nonlnear statc procedures and nonlnear dynamc tme-hstory analyss under spectrum-compatble accelerograms are eventually compared. The proposed procedure generally gves a more accurate soluton than that obtaned from the other nonlnear statc procedures. KEYWORDS: steel moment-resstng frames, nonlnear analyss, adaptve pushover 1. INTRODUCTION The estmaton of lateral dsplacement demands s of prmary mportance n performance-based earthquake-resstant desgn. In fact, both structural and non-structural damage are prmarly related to lateral dsplacements. However, estmatng sesmc demands at hgh performance levels, such as lfe safety and collapse preventon, requres explct consderaton of the nelastc behavour of the structure. To ths purpose, the nonlnear response hstory analyss (NRHA) s the most rgorous method for the estmaton of sesmc demands. Nevertheless, the Nonlnear Statc Procedures (NSPs) are wdely used to calculate the deformaton demands wth satsfactory accuracy wthout the complex modellng and computatonal effort of NRHA. Some NSPs were ncorporated n the new generaton of sesmc codes n procedures based on Capacty Spectrum Method (CSM) or Dsplacement Coeffcent Method (DCM), such as n FEMA 273 [1997], ATC-40 [1997], FEMA 356 [2000], Eurocode 8 [2004], Italan Buldng Code [2008], FEMA-440 [ATC-55, 2005], ASCE/SEI 41-06 Standard [2005]. In general, applyng dsplacement rather than force loadng n pushover procedures would be the most sutable opton for nonlnear statc analyss of structures subjected to earthquake ground moton. However, due to the unvaryng nature of the appled dsplacement loadng vector, ths approach may neglect the strong varatons of the dsplacement pattern n post-yeld falure mechansm produced, for example, by strength rregulartes and soft storeys. Consequently, when an nvarant load pattern s used, the force-based pushover s to be preferred to the dsplacement- *Correspondng author: Massmlano Ferraol, Department of Cvl Engneerng, Desgn, Buldng and Envronment, Second Unversty of Naples, Aversa (CE), Italy. Emal: massmlano.ferraol@gmal.com

M. FERRAIOLI et. Al. based pushover. However, the accuracy of these conventonal force-based pushover analyses n predctng sesmc demands of structures and ther lmtatons especally for hgh-rse buldngs reman among the most controversal topcs. In fact, the tradtonal pushover analyss wth an nvarant lateral force pattern accurately estmates the sesmc response of low-rse and regular buldngs when the structural response s domnated by the frst mode. On the contrary, sgnfcant dfferences were found n hgh-rse buldngs, where the effects of the hgher modes cannot be neglected. The two major lmts of the conventonal pushover procedures are that, owng to the nvarant loadng pattern, the hgher mode effects are gnored and, therefore, the changes n the dynamc propertes of the structures are neglected. Ths s manly because nerta force dstrbuton changes contnuously under earthquake ground moton due to hgher mode contrbuton and stffness degradaton. Fgure 1. Cases studes: regular and rregular moment-resstng frames 48

INGEGNERIA SISMICA INTERNATIONAL JOURNAL OF EARTHQUAKE ENGINEERING Table 1. Man Characterstcs of accelerograms No. Input Staton PGA Date Tme Dr. ID (m/s 2 ) M w 1 Frul ST20 06/05/1976 20:00 NS 3.499 6.5 2 Montenegro ST64 15/04/1979 06:11 EW 2.199 6.9 3 Campano Lucano ST93 23/11/1980 18:34 NS 1.363 6.9 4 Izmt ST575 17/08/1999 00:01 NS 1.576 7.6 5 South Iceland ST108 17/06/2000 15:40 NS 1.238 6.0 6 Olfus ST101 29/05/2008 15:45 EW 0.439 6.3 7 LomaPreta ST47379 18/10/1989 00:04 NS 4.029 6.9 In order to overcome these drawbacks, some researchers proposed nvarant loadng patterns takng nto account the hgher mode effects [Chopra et al., 2002-2004], [Fajfar et al., 2005], [Janmeng et al., 2008], [Poursha et al., 2009], [Reyes et al., 2011], [Kresln et al., 2012]. Other researchers developed adaptve pushover procedures accountng for hgher mode effects and progressve damage accumulaton [Gupta et al., 2000], [Antonou et al., 2004], [Kalkan et al., 2006], [Pnho et al., 2008], [Ferraol et al., 2008], n order to overcome the most mportant lmtatons of tradtonal methods especally for estmatng sesmc demands of tall buldngs. However, some of these nonlnear statc procedures requre very complex analyses and they fal the target of usng procedures smpler than NRHA. Nevertheless, these adaptve pushover methods may represent an attractve dsplacement-based tool for structural assessment, fully complyng wth the recently ntroduced deformaton and performance orented trends n the feld of earthquake engneerng. 2. ADAPTIVE AND MULTIMODAL NONLINEAR STATIC PROCEDURES Generally, usng modal propertes of the structure n nonlnear statc analyss s the most accessble approach to take nto account the dynamc characterstcs of the system. However, the conventonal nonlnear statc procedures (NSPs) are based on the assumpton that the structure vbrates predomnantly n a sngle mode and that the dynamc propertes of the structure reman unchanged. Fgure 2. Pseudo-velocty spectra: a) recorded accelerograms b) adjusted accelerograms 49

M. FERRAIOLI et. Al. The frst assumpton s not always fulflled, especally n the case of hgh-rse buldngs and/or torsonally flexble plan-asymmetrc buldngs [Chopra et al., 2004], [Erduran et al., 2008], [Kresln et al., 2012], [Ferraol, 2010-2015]. Furthermore, the progressve changes n the modal propertes due to structural yeldng are generally neglected. In order to nclude the effects of hgher modes some advanced modal pushover procedures based on the elastc modal decomposton concepts were developed n lterature. Many of these procedures consder hgher modes n lateral load pattern n order to take nto account hgher mode effects both n plan and n elevaton. In partcular, n the wellknown modal pushover analyss (MPA) proposed by Chopra et al. [2004] hgher mode effects are consdered by analysng each mode as an equvalent sngle-degree-of-freedom system ncludng nonlnear propertes related to that mode. Kunnath [2004] nvestgated smple modal combnaton schemes to ndrectly account for hgher-mode effects. In order to take nto account hgher-mode effects, Poursha et al. [2009] proposed the consecutve modal pushover procedure that employs mult-stage and sngle-stage pushover analyses. The man problem of conventonal nonlnear statc procedures s that they are based on the assumpton that the mode shape remans un-changed after the structure yelds. In order to overcome ths drawback n recent years some adaptve pushover methods were proposed to nclude the effects of hgher modes and the changes n vbraton characterstcs due to the nelastc response. Specfcally, n order to nclude the changes n the dynamc propertes of the structure Gupta et al. [2000] proposed an adaptve pushover procedure based on an elastc demand spectrum. Kalkan et al. [2006] suggested a new pushover analyss procedure derved through adaptve modal combnatons (AMC) that accounts for the effects of both hgher modes and varyng dynamc characterstcs due to nelastc response. Antonou et al. [2004] proposed an nnovatve dsplacement-based adaptve pushover (DAP) procedure, n whch an ncremental updatng wth ncrement of ds-placement calculated accordng to the spectrum scalng s appled at each analyss step. Two shortcomngs of the modal combnaton rules can be ponted out: the frst one s that the result obtaned does not fulfl equlbrum; the second lmtaton s that sgns are lost durng the combnaton process elmnatng the contrbuton of negatve quanttes and consderng an always-addtve contrbuton of hgher modes. In addton, no soluton s provded to determne the target dsplacement n the adaptve nonlnear statc analyss. Summng up, n spte of ther deep theoretcal background, many of the aforementoned methods suffer from the quadratc modal combnaton rules, n whch the change n the sgn of storey components at hgher modes s not consdered as the sgn reversals of load vectors n hgher modes are neglected. Thus, the magntudes of the appled loads n all storey levels are postve. Ths napproprate always-addtve ncluson of hgher mode contrbuton through a non-weghted SRSS combnaton rule represents a further weakness of modal pushover procedures. Fnally, no mult-run methods are able to reflect the nteracton between modes n the nonlnear range. 3. PROPOSED ADAPTIVE CAPACITY SPECTRUM METHOD In ths study, a dsplacement-based Adaptve Capacty Spectrum Method (ACSM) based on the Inelastc Demand Response Spectra (IDRS) s proposed. At frst, the dsplacement-based adaptve pushover (DAP) procedure proposed by Antonou et al. [2004] s used to defne the capacty of the structure. Ths results n a varaton of the lateral force pattern durng pushover analyss. Thus, also the equvalent Sngle-Degree-of-Freedom system (SDOF), whch s representatve of the MDOF 3D model of the buldng n the Capacty Spectrum Method (CSM), 50

INGEGNERIA SISMICA INTERNATIONAL JOURNAL OF EARTHQUAKE ENGINEERING changes durng pushover analyss. In order to consder such effect, an adaptve verson of the Capacty Spectrum Method (ACSM) s proposed. At each step of the pushover analyss, a dfferent equvalent SDOF system s defned as a functon of the actual lateral dsplacement pattern. In partcular, the equvalent mass Meq and stffness Keq of the SDOF system at the -th step of the pushover analyss s expressed as a functon of the j-th storey dsplacement j and force Fj, as follows [Ferraol et al., 2008]: M K eq N m 2 j1 j j N j 1 m j 2 j N m 2 j1 j j N eq F N 2 1 j j j m j1 j j The global force-dsplacement capacty curve of the structure (base shear V versus top dsplacement TOP) s transformed step by step n the capacty spectrum (spectral acceleraton Sa versus spectral dsplacement Sd) n ADRS format (Acceleraton-Dsplacement Response Spectra), as follows: S S a V N j1 N j1 N j m j m j 2 j 2 j 2 1 m 1 j j d TOP N N m j1 j j where and are the base shear and the correspondng top dsplacement ncrements at the -th step of pushover analyss, respectvely, s the mass of the j-th storey, s the lateral dsplacement of the j-th storey at the -th step of pushover analyss and N s the number of storeys. The sesmc demand s represented through Inelastc Demand Response Spectra (IDRS) that are ndrectly computed by scalng the 5% damped Elastc Demand Response Spectra (EDRS) accordng to the R--T relatons avalable n lterature for the strength reducton factor [Vdc et al., 1994]. Specfcally, the nelastc pseudo-acceleraton Sa and dsplacement Sd, whch are the coordnates of the IDRS n ADRS format, are characterzed by the coordnates [Sde; Sae] of the EDRS (=5%) as follows: S S ae a (5) R Sde Sd (6) R where s the ductlty rato and s the ductlty reducton factor defned by Vdc et al. [1994], as follows: * * T T T0 R 1 1, (7) T0 * T T R. (8) 0 (1) (2) (3) (4) 51

M. FERRAIOLI et. Al. In Eqs. (7) and (8) depends on the ductlty and, therefore, on the lateral dsplacement of the equvalent SDOF system. Thus, some teratons are requred n order to match the values of ductlty obtaned from the demand spectrum and the capacty dagram, respectvely, wthn a gven tolerance. Fgure 3. Scatter plot of estmated target nterstorey drft for regular frames 52

INGEGNERIA SISMICA INTERNATIONAL JOURNAL OF EARTHQUAKE ENGINEERING Fgure 4. Scatter plot of estmated target nterstorey drft for rregular frames 4. VALIDATION AND COMPARISON WITH OTHER NSPS A number of steel moment-resstng frames desgned to meet sesmc requrements of Italan Buldng Code [2008] have been consdered n the numercal analyses (Fg. 1). The desgn sesmc acton has been defned assumng sol class A, dampng rato =5%, peak ground 53

M. FERRAIOLI et. Al. acceleraton PGA=0.25g, behavour factor q=6.5 for regular frames and q=6.5 0.80=5.2 for rregular frames. Steel members are made of S275 steel grade (fy=275 MPa). The nterstorey heght s 3.5m for the frst floor and 3.0m for the upper floors. The bay length s 5.0m. A dstrbuted plastcty-fbre element model mplemented n the SesmoStruct code [2004] has been used n nonlnear analyses. Sources of geometrcal nonlnearty taken nto account are both local and global. A blnear knematc hardenng materal model has been used for steel. More detals about the case studes can be found n Ferraol et al. [2014a, 2014b, 2014c]. The effectveness of the proposed adaptve capacty spectrum method (ACSM) s evaluated by comparng ts predctons wth estmates obtaned from a comprehensve set of nonlnear response hstory analyses (NRHA). Fgure 5. a) Interstorey drft profle. b) Error profle. Regular frames 54

INGEGNERIA SISMICA INTERNATIONAL JOURNAL OF EARTHQUAKE ENGINEERING To ths am, a set of seven nput ground motons have been carefully chosen n such a way to be consstent wth the selected target response spectrum (5%-damped Eurocode 8 type 1 elastc spectrum for sol class A). The SesmoMatch software [2004] has been used to enforce the spectrum-compatblty of the selected accelerograms. Table 1 shows the parameters of the ground moton records. In Fg. 2, the pseudo-velocty response spectra of both recorded and adjusted accelerograms are plotted. The accuracy of ACSM s compared wth other more advanced nonlnear statc procedures: 1) MP: Mult-mode Pushover analyss [Sucuoğlu et al., 2011]; 2) MMPA: Modfed Modal Pushover Analyss procedure assumng hgher modes as elastc [Chopra et al., 2004]; 3) N2-EXT: Extended N2 method consderng hgher mode effects both n plan and n elevaton [Kresln et al., 2012]. The nterstorey drft rato of each storey (defned as the storey drft dvded by the storey heght) plays an mportant role n the amount of damage nduced n the structure durng earthquake ground moton. Consequently, ths quantty has been used for comparng the results of Nonlnear Statc Procedures (NSPs) and Nonlnear Response Hstory Analyss (NRHA). Specfcally, the mean value of the seven nterstorey drfts resulted from NSPs has been compared wth the mean value of the peak nterstorey drfts resulted from NRHA. In Fg. 3-4, the nterstorey drft ratos obtaned wth Nonlnear Response Hstory Analyss (NRHA) are assumed as reference values (X-axs) and the correspondng values obtaned wth the Nonlnear Statc Procedures (NSPs) are shown on the Y-axs. The bsector represents the cases when the NSP and the NRHA gve the same results. The straght lnes numbered -10%, -20% and -30% are representatve of error rates (defned n subsequent Eq. (9)) equal to -10%, -20% and -30%, respectvely.,, (%) 100 jnsp jnrha Error. (9) jnrha, In Eq. (9), s the mean value of the seven peak nterstorey drfts n the j-th storey resultng from NRHA,, s the mean value of the seven nterstorey drfts n the j-th storey resultng from NSP. The scatter plot of Fgs. 3-4 evaluates the coherence between NSP and NRHA. When all the ponts representng the pars of values are clustered around the bsector, both methods tend to gve the same results. On the contrary, when some values le below the straght lnes numbered -10%, t means that the error of the mean nterstorey drft comng from the NSP wth respect to the mean nterstorey drft of the NRHA s greater than 10%. Fnally, t s to be observed that the ponts over the bsector are stuatons where the value estmated from NSP procedure s greater than the correspondng value from NRHA. In these cases, the NSP gves conservatves estmates of the nonlnear dynamc response. Fgs. 3-4 show that all the procedures tend to underestmate the nterstorey drft demand for PGA=0.20g and underestmaton s even greater than 30%. For PGA=0.20g, all framed structures respond elastcally, and the dfferences between NSP and NRHA are not dependent on the nelastcty n the structure, but entrely arse due to the dfference n the analyss method. For values of PGA rangng from 0.40g to 0.80g the amount of nelastcty n the structure ncreases, and the accuracy depends on the NSP used to compute the sesmc demand. Specfcally, both N2-EXT and MMPA tend to overestmate the nterstorey drfts when compared to NRHA. The dscrepancy ncreases wth both the number of storeys and rregularty n elevaton. On the other sde, the MP procedure underestmates the nterstorey drft demand of rregular frames, whle the proposed ACSM tends to gve the most accurate solutons. 55

M. FERRAIOLI et. Al. Fgure 6. a) Interstorey drft profle. b) Error profle. Irregular frames Practcally, the pushover procedures based on nvarant lateral force patterns tend to accurately estmate the sesmc response of low-rse and regular frames, where the response s domnated by the frst mode and the development of a collapse mechansm of the global type s possble. On the contrary, these procedures become naccurate when appled to hgh-rse and rregular frames, where the hgher modes effects are sgnfcant and undesred collapse mechansm typologes may be developed. In partcular, hgh-rse frames are more prone to develop partal mechansms nvolvng a lmted number of storeys compared to the total number. The occurrence of such storey mechansm confrms the mportance of desgn procedures focusng on plastc mechansm control of MR-frames [Montuor et al., 2012-2015] [Pluso et al., 2014]. 56

INGEGNERIA SISMICA INTERNATIONAL JOURNAL OF EARTHQUAKE ENGINEERING In Fg. 5 the nterstorey drft and the error profles of the regular frames are plotted. The results obtaned for PGA=0.20g show that all procedures underestmate the nterstorey drft of all storeys. Moreover, the error s slghtly varable along the heght. When the value of PGA rses to 0.40g, the dfferences ncrease both along the heght and between the NSPs. For PGA=0.60g and even more for PGA=0.80g both N2-EXT and MMPA overestmate the nterstorey drfts, especally n the upper and lower storeys of the taller frames. On the contrary, the proposed ACSM procedure gves errors that are lower and less varable along the heght even when compared to the MP procedure. In Fg. 6 the nterstorey drft profles and the error profles of the rregular frames are plotted. Also n ths case, all procedures underestmate the nterstorey drft of all storeys for PGA=0.20g. Greater dfferences, both along the heght and between the NSPs, are evdenced for PGA=0.40g and, even more, for PGA=0.60g and PGA=0.80g. Both N2-EXT and MMPA overestmate the nterstorey drft, especally n the lower storeys, also for shorter frames and for PGA=0.40g. In the case of rregular frames, also the MP procedure seems naccurate snce t overestmates the nterstorey drfts, especally n the mddle storeys for PGA rangng between 0.40g and 0.80g. The proposed ACSM procedure gves errors that turn to be lower and less varable along the heght than other methods. In order to estmate the global accuracy of results gven by the mentoned NSPs, the mean value of maxmum nterstorey drft obtaned from NRHA s assumed as the reference value and the total error has been evaluated as follows: 1,, (%) 100 N jnsp jnrha Total Error (10) j 1 N jnrha, In Fg. 7 the total error s plotted as a functon of the peak ground acceleraton. The applcaton of both N2-EXT and MMPA procedure can potentally result n very naccurate solutons for the hgh values of peak ground acceleraton. Fgure 7. Total error as a functon of the peak ground acceleraton 57

M. FERRAIOLI et. Al. Ths concluson s true for both regular and rregular frames. On the other sde, the results of MP procedure may become very naccurate when the peak ground acceleraton and the rregularty n elevaton ncrease. On the contrary, the proposed Adaptve Capacty Spectrum Method gve the most accurate solutons for both regular and rregular frames, especally for hgh values of peak ground acceleraton. 5. CONCLUSIONS An adaptve capacty spectrum method for estmatng sesmc demands of steel momentresstng frames was presented n ths paper. The proposed procedure takes nto account the frequency content of response spectra, the hgher mode effects, the progressve changes n the modal propertes due to structural yeldng and the nteracton between modes n the nelastc range. The accuracy and effcency of the proposed approach was valdated by comparng ts results wth estmates obtaned from the nonlnear response hstory analyss. The results were eventually compared wth those provded by other methods suggested n lterature. The accuracy of nonlnear statc procedures based on nvarant load patterns proved to be nsuffcent at the lower storey levels, and ths dscrepancy ncreases wth both the number of storeys and the rregularty n elevaton. The multmodal pushover methods accountng for hgher mode effects along the elevaton provde a more accurate estmaton of sesmc demands, but they become very naccurate when the peak ground acceleraton and the rregularty n elevaton ncrease. The proposed Adaptve Capacty Spectrum Method generally gves a more accurate soluton for both regular and rregular frames, especally for the hgh values of peak ground acceleraton. Moreover, the estmated errors are lower and less varable along the heght than other methods suggested n lterature. 6. ACKNOWLEDGEMENTS The subject dealt wth n ths paper s a part of the research project RELUIS 2014-2018 Steel and composte steel-concrete structures (Coordnators Prof. R. Landolfo, Prof. R. Zandonn), ssued by the Italan Dpartmento della Protezone Cvle. The actvty was carred out by the members of Research Unt of the Second Unversty of Naples, coordnated by Prof. A. Mandara. 7. REFERENCES Antonou, S. and Pnho, R. [2004] Development and verfcaton of a dsplacement-based adaptve pushover procedure, Journal of Earthquake Engneerng 8(5), 643-661. ASCE [2005] Sesmc Rehabltaton of exstng buldngs, ASCE/SEI 7-05, Publshed by the Amercan Socety of Cvl Engneers. ATC [1997] Sesmc Evaluaton and Retroft of Concrete Buldngs, Report No. ATC-40, Appled Technology Councl, Redwood Cty, CA. CEN [2004] Eurocode 8: Desgn of structures for earthquake resstance, Part 1: general rules, sesmc actons and rules for buldngs, EN1998-1, European Commttee for Standardzaton. Chopra, A.K. and Goel, R.K. [2002] A modal pushover analyss procedure for estmatng sesmc demands for buldngs, Earthquake Engneerng and Structural Dynamcs 31(3), 561 582. 58

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Internatonal Journal of Earthquake Engneerng Num. Anno XXXIII Specale CTA 2015 Num. 1-2 UNA VERSIONE ADATTIVA DEL METODO DELLO SPETTRO DI CAPACITÀ PER LA VALUTAZIONE DELLA RISPOSTA SISMICA DI TELAI SISMORESISTENTI IN ACCIAIO Massmlano Ferraol*, Angelo Lavno, Alberto Mandara Department of Cvl Engneerng, Desgn, Buldng and Envronment Second Unversty of Naples, Aversa (CE) SOMMARIO: É stata proposta una varante adattva del metodo dello spettro d capactà per la valutazone della rsposta nelastca d tela ssmoresstent n accao soggett ad azon d tpo ssmco. Sono stat esamnat suo vantagg computazonal e la sua capactà d fornre prevson accurate dell effet-tva rsposta ssmca n rapporto a quelle ottenute mpegando altre procedure statche non lnear dsponbl n letteratura. L effcaca e l accuratezza de metod approssmat basat sull anals d pushover è stata verfcata attraverso un estesa ndagne parametrca condotta su tela ssmores-stent n accao sa regolar, sa rregolar. I rsultat ottenut sono stat confrontat con quell ottenut dall anals dnamca non lneare condotta utlzzando segnal accelerometrc spettrocompatbl. * Correspondng author: Massmlano Ferraol, Department of Cvl Engneerng, Desgn, Buldng and Envronment, Second Unversty of Naples, Aversa (CE), Italy. Emal: massmlano.ferraol@gmal.com