Accuracy of nonlinear static procedures for estimating the seismic performance of steel frame structures

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1 Accuracy of nonlinar static procdurs for stimating th sismic prformanc of stl fram structurs M. Frraioli, A. Lavino, A.M. Avossa, A. Mandara Sconda Univrsità di Napoli, Dipartimnto di Inggnria Civil, via Roma 9 Avrsa (CE), Italy ABSTRACT: Th ffctivnss of nonlinar static procdurs for stimating th sismic prformanc of stl fram structurs was invstigatd. At this aim, th procdurs proposd in, ASCE/SEI 41-04, ATC40, FEMA440-ATC-55 wr applid for th displacmnt-basd prformanc assssmnt. Th rsults obtaind wr compard with incrmntal tim-history rspons. 1 INTRODUCTN Th validity and applicability of th static pushovr analysis hav bn xtnsivly studid in litratur. In particular, many rsarchrs hav compard th pushovr curvs with idalizd nvlops obtaind from incrmntal dynamic pushovr analyss of structurs subjctd to artificial and natural input ground motions. Furthrmor, th ffctivnss of th Capacity Spctrum Mthod (CSM) has bn validatd by comparison with xprimntal rsults from psudo-dynamic, cyclic and pushovr tsts. Th Capacity Spctrum Mthod - by mans of a graphical procdur - compars th capacity of th structur to rsist latral forcs to th dmands of arthquak rspons spctra. Diffrnt Nonlinar Static Procdurs (NSPs) basd on th CSM hav bn introducd in pr-standards rports and guidlins. Som of thm wr incorporatd in th nw gnration of sismic cods to dtrmin th dformation dmand imposd on a building xpctd to bhav inlastically. A simplifid procdur basd on th pushovr analysis and on th strngth rduction factor was introducd in th Eurocod 8 (003). Static Displacmnt-Basd Procdurs wr dvlopd by th Fdral Emrgncy Managmnt Agncy (FEMA). Th Cofficint Mthod (CM) of FEMA-356 (000) is basd on a displacmnt modification procdur in which svral mpirically drivd factors ar usd to modify th rspons of a linarly-lastic, singl-dgr-of-frdom (SDOF) modl of th structur. In othr words, th CM is a form of quivalnt linarization. Empirically drivd rlationships, that givs th ffctiv priod and damping as a function of ductility, ar usd to stimat th rspons of an quivalnt nonlinar SDOF systm. ATC-40 Rport (ATC-40, 1997) proposs thr nonlinar static procdurs (A,B,C) basd on th Capacity Spctrum Mthod and on th High Damping Elastic Rspons Spctra (HDERS) with an quivalnt viscous damping accounting for th nrgy dissipation du to yilding. Howvr svral dficincis wr found by som authors. In particular, no physical principl justifis th xistnc of a stabl rlationship btwn th hystrtic nrgy dissipation and quivalnt viscous damping, particularly for highly inlastic systms, and so th convrgnc of ATC itrativ procdurs is not assurd. Morovr, th ATC-40 procdurs rquir th calculation of th charactristic priods of arthquak spctrum which oftn ar not asy to b stimatd. Th N mthod proposd by Fajfar (001) combins togthr th visual rprsntation of th CSM and th suprior physical basis of Inlastic Dmand Rspons Spctra (IDRS) which ar xpctd to b mor accurat than HDERS, spcially in th short priod rang and in th cas of high ductility factors. Both CM and CSM wr found to provid substantially diffrnt stimats of targt displacmnt for th sam ground motion and th sam building (Aschhim t al., 1998; Akkar and Mtin, 007; Chopra and Gol, 00 Gol, 007; Miranda and Luiz-Garcia, 00) and hav proposd improvd procdurs for stimating th targt displacmnt. Rcntly, FEMA-440 documnt (ATC-55, 003) rxamind th xisting NSPs and proposd improvmnts to both th CM and CSM. Rcommndations in th FEMA-440 documnt hav bn adoptd in th ASCE/SEI standard (ASCE, 007).

2 NON LINEAR STATIC PROCEDURES Th aformntiond nonlinar static procdurs (FEMA-356, ATC-40, FEMA-440, ASCE/SEI 41-06, CSM-N) usually rquir th pushovr analysis. Th structur is pushd statically to a targt displacmnt at th control nod to chck for th accptabl structural prformanc. Diffrnt hightwis distributions of latral loads ar usually considrd. In fact, th shar forcs vs. story drift rlationship may b vry snsitiv to th applid load pattrn. Gnrally, diffrnt distributions of latral load ar considrd. 1) Equivalnt latral forc distribution (ELFD). Th sismic forcs at ach floor lvl of th building ar distributd according to th distribution of floor mass m i and to th hight h i abov th bas. ) Fundamntal Mod Distribution (FMD). Th vrtical distribution is proportional to th floor masss and th fundamntal mod shap.. 3) Rspons Spctral Analysis distribution (RSA): th vctor of latral forcs is proportional to th story shar distribution calculatd by combining th modal rsponss of sufficint mods to captur at last 90% of th total mass. 4) Uniform Distribution (UD). Th latral load distribution is proportional to th floor masss m i. FEMA-356 rquirs dvlopmnt of th pushovr curv for two hight-wis distributions of latral forcs: UD and on slctd from ELFD, FMD and RSA. ATC-40, FEMA-440, and ASCE/SEI rquir only th fundamntal mod distribution..1 FEMA Cofficint Mthod Th Cofficint Mthod of (ASCE, 000) is basd on an quivalnt linarization. In particular, th maximum inlastic displacmnt (targt displacmnt) is obtaind from th linar lastic rspons of th quivalnt SDOF systm, as follows: T δ C C C C S g t = a (1) 4π S a is th rspons spctrum acclration at th ffctiv fundamntal vibration priod and damping ratio of th building undr considration. g is th acclration du to gravity. T is th ffctiv fundamntal priod of th building in th dirction undr considration, computd as follows: T = T K K () i i whr T i is th fundamntal vibration priod dfind from lastic ignvalu analysis; K i is th lastic stiffnss of th building; K is th ffctiv stiffnss of th building obtaind by idalizing th pushovr curv as a bilinar rlationship. Th cofficint C 0 is th modification factor that rlats th lastic rspons of a SDOF systm to th lastic displacmnt of th MDOF building. Th control nod of th building may b dfind from th first mod participation factor PF or slctd from tabulatd valus in FEMA-356. C 1 is th modification factor that rlats th maximum inlastic and lastic displacmnt of th SDOF systm computd from 1. T Ts C1 = ( ( R 1) Ts T ) R ; T < Ts (3) 1. 5; T < 0. 1s whr R is th ratio of lastic and yild strngths and T s is th cornr priod whr th rspons spctrum transitions from constant psudo-acclration to constant psudo-vlocity. Th modification factor C rprsnts th ffcts of pinchd hystrtic shap, stiffnss dgradation, and strngth dtrioration. C =1 for nonlinar analysis and tabulatd as a function of framing systm and prformanc lvl othrwis. Th modification factor C 3 accounts for th incrasd displacmnt du to P-dlta ffcts, as follows: 1. α 0 C 3 = (4) ; α < 0 ( α ( R 1) ) 3 T in which α is th ratio of th post-yild stiffnss to ffctiv lastic stiffnss.. ATC-40 - Capacity Spctrum Mthod Th ATC-40 procdurs ar basd on CSM Mthod. Th basic assumption is th quivalnt linarization by which th maximum inlastic dformation of a nonlinar SDOF systm (targt displacmnt) is approximatd from th maximum dformation of a linar lastic SDOF systm with quivalnt priod T q and quivalnt damping ratio ξ q, xprssd as follows: ( μ 1)( 1 α ) ( 1+ αμ α ) μ 1 Tq = T0 ; ξ q = ξ 0 + κ (5) 1+ αμ α π μ whr T 0 is th initial priod of vibration of th nonlinar systm; α is th post-yild stiffnss ratio; μ is th maximum displacmnt ductility ratio; κ is th adjustmnt factor to approximatly account for changs in hystrtic bhavior. In particular, th ATC-40 documnt dfins thr typs of hystrtic bhaviors: Typ A with stabl, rasonably full hystrtic loops. Typ C with svrly pinchd and/or dgradd loops. Typ B btwn Typs A and C. Th quivalnt linarization procdur rquirs prior knowldg of th displacmnt ductility ratio. ATC- 40 provids thr itrativ procdurs: (A, B, C). Th procdurs A and B ar th most transparnt and convnint for programming, whras th Procdur C is purly a graphical mthod that is not suitabl for programming.

3 .3 FEMA Cofficint Mthod FEMA 440 introducs som improvmnts in Cofficint Mthod of FEMA-356. In particular, th cofficint C 1 is givn by R ; T < 0. s 0. 04a C1 = 1. T > 1. 0s (6) R ; 0. s < T < 1. 0s at whr a=130 for sit class A and B, 90 for sit class C, a=60 for sit classs D, E, and F. Th cofficint C is givn by 1. T > 0. 7s C = 1 R 1 (7) ; T 0. 7s 800 T Finally, th cofficint C 3 is liminatd and substitutd by a limitation on strngth to avoid dynamic instability. This limitation on strngth is spcifid by imposing a maximum limit on R givn by h Δ α d Rmax = + ; h = Δ y 4 ln ( T ) (8) In q.8 Δ d is th dformation corrsponding to pak strngth, Δ y is th yild dformation, and α is th ffctiv ngativ post-yild slop givn by ( α α ) α (9) = α P Δ + λ P Δ Whr α is th ngativ post-yild slop ratio; α P- Δ is th ngativ slop ratio causd by P-Δ ffcts; λ is th nar-fild ffct givn: λ=0.8 for S and λ=0. for S 1 < 0.6 (S 1 is dfind as th 1-scond spctral acclration for th Maximum Considrd Earthquak). Th α slop includs P-Δ ffcts, incycl dgradation, and cyclic dgradation..4 FEMA Capacity Spctrum Mthod Th improvd FEMA-440 CSM includs nw xprssions to dtrmin th ffctiv priod and ffctiv damping dvlopd by Guyadr and Iwan (006). Consistnt with th original ATC-40 procdur, thr itrativ procdurs for stimating th targt displacmnt ar also outlind. Finally, a limitation on th strngth is imposd to avoid dynamic instability. Improvd formulas for th ffctiv priod and viscous damping ar proposd as a function of svral cofficints tabulatd in Tabl 6-1 and Tabl 6- of FEMA-440 documnt. Th quivalnt linarization procdurs applid in practic normally rquir th us of spctral rduction factors to adjust an initial rspons spctrum to th appropriat lvl of ffctiv damping β ff. This factor B(β ff ) is a function of th ffctiv damping and is usd to adjust th spctral acclration ordinats as follows: ( S ) ( S ) B( β ) = (10) a β a β ff whr th damping cofficint B(β ff ) is givn by: 4 B = (11) 5.6 ln β ff.5 Capacity Spctrum Mthod -N Th CSM-N procdur is basd on th Capacity Spctrum Mthod and th Inlastic Dmand Spctra. Th inlastic sismic dmand is not rprsntd by High Damping Elastic Rspons Spctra (HDERS), but it is rprsntd by mans of th Inlastic Dmand Rspons Spctra (IDRS). Th IDRS is not dirctly obtaind through th nonlinar tim-history analysis of th quivalnt bilinar SDOF systm, but it is indirctly computd scaling th Elastic Dmand Rspons Spctra (EDRS). Th IDRS is obtaind by scaling th EDRS (with viscous damping ratio ξ=5 pr cnt) by mans a ductility rduction factor R μ. In particular, th inlastic psudo-acclration S a and displacmnt S d - which rprsnts th coordinats of th IDRS in ADRS format - ar charactrizd from th coordinats [S d ;S a ] of th EDRS (ξ=5 pr cnt) as follows: S S a d a = Sd R μ R μ μ S = (1) In q.1 μ is th ductility ratio and th ductility rduction factor R μ is dfind by th following sprssions drivd from statistical data analysis (Vidic t l. 1994): R ( μ 1) T = T μ 0 T T (13) 0.3 R μ = μ for T T0 with T0 = 0. 65μ T c Tc (14) whr T c is th cornr priod of th arthquak ground motion btwn th constant acclration rgion and th constant vlocity rgion of Nwmark- Hall rspons spctrum. 3 ASSESSMENT OF CURRENT NONLINEAR STATIC PROCEDURES FOR PERFORMANCE EVALUATN OF BUILDINGS 3.1 Study cass A 9-story stl fram dsignd according to th Italian Cod (008) and Plastic Dsign (Mazzolani t al., 1997) ar considrd in th analyss. Th dsign sismic action is dfind with soil class A, damping ratio ξ=5%, pak ground acclration PGA=g, bhavior factor q=6.5. Stl mmbrs ar mad from Italian S75 stl (f y =75 MPa). Th intrstory hight is 3.50 m for th first floor and

4 0 m for th othr floors. Th bay lngth is 5.00 m. Th dsign sctions of th two stl frams ar rportd in tab.1. In tab. th modal proprtis ar shown. A distributd plasticity-fibr lmnt modl implmntd in Sismostruct (SismoSoft, 004) is considrd in th analyss. Sourcs of gomtrical nonlinarity considrd ar both local and global. Th sprad of plasticity along th lmnt drivs from an inlastic cubic formulation with two Gauss points to us for numrical intgration of th quilibrium quations. A bilinar modl with kinmatic strain-hardning of % is usd for stl. Th plastic rotation and th rsidual strngth ar dfind according to. Th following currnt nonlinar static procdurs ar considrd in th analysis: 1) ATC 40 undr first mod distribution (FMD); ) ATC 40 undr uniform distribution (UD); 3) Cofficint Mthod (CM); 4) FEMA 440 Cofficint Mthod (CM); 5) Capacity Spctrum Mthod (CSM-N) undr first mod distribution; 6) Capacity Spctrum Mthod (CSM N) undr uniform distribution; 7) FEMA 440 (CSM) undr first mod distribution; 8) FEMA 440 (CSM) undr uniform distribution. Th NSPs ar applid for th prformanc-basd assssmnt of IC08 and PD- Stl frams. Thr diffrnt lvls of prformanc ar considrd: Immdiat Occupancy ; Lif Safty LS; Collaps Prvntion. Th accptanc critria ar basd on th following prformanc paramtrs: 1) intrstory drift damag indx (IDI); ) plastic rotations in columns and bams. (dfind by tab.5.6 of )..5.0 IDA-01 CSM-N (FMD) CSM-N (UD).5.0 IDA-0 CSM-N (FMD) CSM-N (UD) IDA-03 CSM-N (FMD) CSM-N (UD) IDA-04 CSM-N (FMD) CSM-N (UD) IDA-05 CSM-N (FMD) CSM-N (UD).5.0 IDA-06 CSM-N (FMD) CSM-N (UD) Figur 1 Pak ground acclration vrsus roof displacmnt Nonlinar Static Procdurs and Incrmntal Dynamic Analysis (Italian Cod 008 dsign)

5 Tabl 1. Dsign Sctions of Stl Frams 9 Storys 3 Bays - IC08 9 Storys 3 Bays PD-SLS N BEAMS EXT. INT. EXT. INT. BEAMS COUMNS COUMNS COUMNS COUMNS 1 IPE70 HE0B HE80B IPE70 HE500B HE500B IPE70 HE0B HE80B IPE70 HE450B HE400B 3 IPE70 HE0B HE80B IPE70 HE450B HE400B 4 IPE70 HE0B HE80B IPE70 HE450B HE400B 5 IPE70 HE0B HE80B IPE70 HE400B HE400B 6 IPE70 HE0B HE80B IPE70 HE400B HE360B 7 IPE70 HE0B HE80B IPE70 HE400B HE340B 8 IPE70 HE0B HE80B IPE70 HE30B HE300B 9 IPE70 HE0B HE80B IPE70 HE60B HE40B Tabl. Fundamntal priods and modal mass ratio 1 mod mod 3 mod T(s) α(%) T(s) α(%) T(s) α(%) IC PD SLS Th limit valus for th intrstory drift damag indx ar: 1) IDI=1 for limit stat; ) IDI= for LS limit stat; 3) IDI=4 for limit stat. Th assssmnt of currnt nonlinar static procdurs is carrid out by comparison against dynamic analysis. At this aim, six ground motions ar slctd to b consistnt to EC8 8 lastic spctra for diffrnt soil classs (tab.3). In fig.3 th nvlop valus of th spctral acclration ar compard to EC8 lastic rspons spctra. In figs.1- th roof displacmnt for th limit stats (Immdiat Occupancy ; Lif Safty for non-structural lmnts ; Lif Safty for structural mmbrs, Collaps Prvntion - ) ar rlatd to th corrsponding pak ground acclration lvls (PGA). Furthrmor, also th pak roof displacmnt obtaind from th incrmntal dynamic analysis is rportd..5.0 IDA-01 CSM-N (FMD) CSM-N (UD).5.0 IDA-0 CSM-N (FMD) CSM-N (UD) IDA-03 CSM-N (FMD) CSM-N (UD) IDA-04 CSM-N (FMD) CSM-N (UD) NSPs out of rang IDA IDA-06 CSM-N (FMD) CSM-N (UD) Figur. Pak ground acclration vrsus roof displacmnt Nonlinar Static Procdurs and Incrmntal Dynamic Analysis (PD-SLS dsign)

6 and FEMA 440 (Cofficint Mthod) giv vry similar prformanc points. This rsult drivs from th valu of th cofficint C 3 that is approximatly qual to in both cass. Th procdurs basd on th High Damping Elastic Dmand Rspons Spctra (HDERS) such as ATC 40 and FEMA 440 (ATC 55) giv highr pak ground acclrations whn compard to th othr NSPs. Furthrmor, th rsults ar mor snsitiv to th latral forc distribution of th load pattrn. In particular, th uniform distribution may giv a nonconsrvativ ovrstimation of PGA. This rsult drivs from th diffrncs btwn HDERS and IDRS dfind by qs In ordr to show this ffct in fig-4 th HDERS and th IDRS at collaps ar compard. Th PGA rquird to intrsct th dmand and th capacity spctrum in th prformanc point at collaps is.14 for HDERS and 1. for IDRS. For th El-Cntro arthquak ground motion th nonlinar static procdurs giv valus of PGA that ar ovr th maximum valu of th graphic. This rsult is a consqunc of th acclration rspons spctrum of El-Cntro that quickly dcrass for priods vry clos to th fundamntal priod of th framd structurs considrd. Tabl 3. Earthquak ground motion charactristics. N. Input Dat Dir. Ms PGA/g t R (s) T C (s) 1 Bvagna NS Sturno EW Gubbio NS San Frnando NS El Cntro NS M. S. Svrino EW Psudo-Vlocity Sv (cm/sc) EC8-A EC8-B EC8-C EC8-D EC8-E Maximum Minimum Natural Priod (sc) Figur 3. Envlop of ground motion rspons spctra and comparison with lastic rspons spctra of Eurocod 8. Sa/g IDRS PGA/g=1. Capacity Curv Sd (cm) HDERS PGA/g=.14 PP Figur 4. Comparison btwn ATC40 High Damping Elastic Dmand Rspons Spctrum (HDERS) and CSM-N Inlastic Dmand Rspons Spctrum at collaps. PD-SLS fram. 4 CONCLUSNS Th accuracy of currnt nonlinar static procdurs for stimating th displacmnt rspons of stl framd structurs was invstigatd. Th procdurs basd on th on th Capacity Spctrum Mthod and on th High Damping Elastic Rspons Spctra (ATC-40, FEMA 440) giv dformation much diffrnt than th nonlinar rspons history analysis. On th contrary, th suprior physical basis of Inlastic Dmand Rspons Spctra (CSM-N) provd to giv mor accurat stimation of th sismic prformanc, spcially in th cas of stl framd structurs that usually xhibit high ductility factors. REFERENCES ASCE, 000. Prstandard and Commntary for th Sismic Rhabilitation of Buildings. Rport No. FEMA-356, Building Sismic Safty Council, Fdral Emrgncy Managmnt Agncy, Washington, D.C. ASCE, 007. Sismic Rhabilitation and Evaluation of Existing Buildings, ASCE Standard No. ASCE/SEI 41-06, Amrican Socity of Civil Enginrs, Rston, VA. Aschhim, M.A., Maffi, J. & Black, E Nonlinar Static Procdurs and Earthquak Displacmnt Dmands. Procdings of 6th U.S. National Confrnc on Earthquak Enginring, Earthquak Enginring Rsarch Institut, Sattl, WA. Akkar, S. & Mtin, A Assssmnt of Improvd Nonlinar Static Procdurs in FEMA-440. Journal of Structural Enginring, 133(9): ATC Sismic Evaluation and Rtrofit of Con-crt Buildings. Rport No. ATC-40, Applid Tchnology Council, Rdwood City, CA. ATC Improvmnt of Inlastic Sismic Analysis Procdurs. Rport No. FEMA-440, Applid Tchnology Council, Rdwood City, CA. Chopra, A.K. & Gol, R.K Evaluation of NSP to Estimat Sismic Dformation: SDF Systms. Journal of Structural Enginring, 16(4): CEN Eurocod 8. Dsign of structurs for arthquak rsistanc. Europan Committ for Standardization. Frraioli, M., Lavino, A., Avossa, A.M., Mandara, A. Displacmnt-basd sismic assssmnt of stl momnt rsisting fram structurs. 14th World Confrnc on Earthquak Enginring. Octobr Bijing, China Gol, R.K Evaluation of Modal and FEMA Pushovr Procdurs Using Strong-Motion Rcords of Buildings. Earthquak Spctra, 1(3): Gol, R.K. & Chadwll, C Evaluation of Currnt Nonlinar Static Procdurs for Concrt Buildings Using Rcordd Strong-Motion Data, Final Rport, California Strong Motion Instrumntation Program, CDMG, Sacramnto, CA. Guyadr, A.C. & Iwan, W.D Dtrmining Equivalnt Linar Paramtrs for Us in a Capacity Spctrum Mthod. Journal of Structural Enginring, 13(1): Miranda, E. & Ruiz-Garcia, J. 00. Evaluation of Approximat Mthods to Estimat Maximum Inlastic Displacmnt Dmands. Earthquak Enginring and Structural Dynamics, 31(3): SismoSoft, 008. SismoStruct. A computr program for static and dynamic analysis for framd structurs. (onlin) avaliabl from URL: