Three level performance based optimization method of steel frames

Transcription

1 Orgnal Artcle hree level performance based optmzaton method of steel frames Abstract hs paper descrbes a three level performance based optmzaton model and an estmate method of resdual top dsplacement for steel frames at three earthquae levels. he steel frames are supposed to be elastc at frequent earthquae, nelastc and hardenng at occasonal and rare earthquaes, respectvel. he estmate formula s derved and estmate procedure s gven n detal. he estmate method onl needs to use onl one pushover analss untl steel frames eld. he eld pont s obtaned automatcall n the proposed method. he estmate method s able to mae optmzaton process unnterrupted. Optmal desgn of a 3 stor 2 ba steel frame s demonstrated to valdate the proposed method. Kewords Earthquae resstant structures, Earthquae engneerng, Sesmc desgn, Steel frames, Optmzaton, Optmzaton models. a, b* a ABB Corporate Research Center, Vasteras, 72226, Sweden b Department of Cvl Engneerng, Aalto Unverst, Espoo, 02150, Fnland * qmao.lu@se.abb.com ht://dx.do.org/ / Receved: Aprl 01, 2016 In Revsed Form: December 15, 2017 Accepted: Februar 09, 2018 Avalable onlne: March 21, INRODUCION Performance based desgn refers to the methodolog n whch structural desgn crtera are expressed n terms of achevng a set of performance objectves Ghobarah, he structural sesmc desgn needs to be based on the defned mulle performance objectves and earthquae hazard levels. he performance objectves can be defned as three levels.e., servceablt, lfe safet, and collapse preventon assocated wth three earthquae hazard levels.e., frequent, occasonal, and rare earthquaes. he structure should has no damage at servceablt level when t meets the frequenc earthquae n Eurocode 8 European Commttee for Standardzaton, he structure s allowed to have moderate and severe damage at lfe safet and collapse preventon levels when t meets occasonal and rare earthquaes, respectvel n Eurocode 8 European Commttee for Standardzaton, herefore, at lfe safet and collapse preventon levels, t s necessar to explctl consder the nelastc behavor of the structures. he nonlnear tme hstor analss s beleved to be the most rgorous procedure to evaluate the nelastc behavor of structures. However, the nonlnear tme hstor analss methods are beleved not to be practcal for everda desgn because the nvolve computatonal and modelng effort, convergence problem and complext Lu et al., 2010; Genctur and Elnasha, he smplfed nonlnear analss methods are preferable to evaluate the nelastc behavor of structures n cvl engneerng practce Fajfar, 1999, he smplfed nonlnear analss methods are based on pushover analss to determne structural capact dagram and on desgn response spectra to represent demand dagram. he N2 method Fajfar, 1999, 2000, 2002; Kresln and Fajfar, 2012 and capact spectrum method Freeman, 1998; Zou and Chan, 2005; Genctur and Elnasha, 2008 are cal smplfed nonlnear analss methods. he N2 method has been mplemented n Eurocode 8 European Commttee for Standardzaton, he capact spectrum methods have been appled n AC 40 Appled echnolog Councl, 1996 and FEMA 356 Federal Emergenc Management Agenc, 2000 n dfferent teratve procedures. he N2 method can determne performance pont wth no need for teraton. he smplfed nonlnear analss methods have been brefl revewed n lterature Fajfar, he smplfed nonlnear analss methods can obtan sesmc demands top dsplacement, nter stor drfts, etc. at dfferent hazard levels. he sesmc demands are compared wth performance targets specfed lmts on top dsplacement, nter stor drfts etc. for the relevant performance levels. If the sesmc demands are equal to or less than performance targets, t means the structures perform well at dfferent hazard levels. If the sesmc demands are greater than performance targets, the structural performance s unsatsfed at dfferent hazard levels. he new structures must be modfed and the exstng structures must be strengthened untl the structures perform well at dfferent hazard levels. herefore, t s stll tedous to desgn a new structure whch can perform well at Latn Amercan Journal of Solds and Structures, 2018, 151, e07

2 dfferent hazard levels usng the smplfed nonlnear analss methods, although t ma be relatvel smple to evaluate performance of the exstng structures at dfferent hazard levels. he author ever proposed an optmzaton procedure for sesmc desgn of steel frames for mult performance and mult hazard levels Lu and Paavola However, the nter stor drfts are drectl treated n the constrants, for a buldng wth man stores, t s not convenent to mplement. In ths paper, a novel method to estmate resdual top dsplacements of steel frames at dfferent hazard levels s developed. A three level performance based optmzaton model s proposed. he paper s arranged as follows. In secton 2, three performance levels and assocated three hazard levels are defned. In secton 3, elastc and nelastc demand spectra n AD format are obtaned based on elastc acceleraton response spectrum n EC8. In secton 4, a proposed method to obtan capact spectrum dagram s developed based on the N2 method. Compared wth the N2 method, the proposed method use onl one pushover analss untl steel frames eld. It also doesn t need an engneerng judgements to get the eld pont. In secton 5, reducton factor and ductlt factor are determned. In secton 6, an estmate method of resdual top dsplacement s developed for three level performance based desgn. In secton 7, computatonal procedure of resdual top dsplacement s gven n detal. In secton 8, a three level performance based optmzaton model s formulated based on the resdual top dsplacements. Fnall, as the llustraton of the developed approach, optmal desgn of a 3 stor 2 ba steel frame s demonstrated usng ANSYS software. 2 DEFINIION OF PERFORMANCE AND HAZARD LEVELS he performance targets are specfed lmts on an response parameters,.e., top dsplacement, nter stor drfts, stresses, strans, etc., at relevant performance levels. he performance objectves requre that the structural sesmc demands are equal to or less than performance targets at relevant performance levels. he top dsplacement of the MDOF sstem s a good ndcator of the global deformaton of the buldngs subjected to earthquae loadng. he top dsplacement s often taen as the global sesmc demand n performance based desgn Kresln and Fajfar, 2012; Fajfar, 1999, 2000, 2002; Ghobarah, 2001; Zou and Chan, 2005; Genctur and Elnasha, However, the top dsplacement can not reveal the global damage degree of the buldngs subjected to severe ground motons. he resdual top dsplacement s not onl a good ndcator of the global deformaton of the buldngs subjected to earthquae loadng, but also can reveal the global damage degree of the buldngs subjected to severe ground motons. In mult level performance based desgn, the resdual deformaton of the buldngs s present at the lfe safet and collapse preventon levels. In ths paper, the sesmc demand of steel frame s the resdual top dsplacement. he lmts on resdual top dsplacement are performance targets. he performance objectves requre that the resdual top dsplacement of steel frame s equal to or less than the lmts on resdual top dsplacement at relevant performance levels. he three performance levels, correspondng damage states and lmts on resdual top dsplacement are defned n able 1. he performance levels are assocated wth earthquae hazard and desgn levels. hree earthquae hazard levels assocated wth the three performance levels are proposed n able 2. 3 SEISMIC DEMAND DIAGRAM Accordng to the precedng defnton, the sesmc demand dagram s the elastc acceleraton response spectrum at servceablt level, and nelastc acceleraton response spectra at lfe safet protecton and collapse preventon levels. able 1: hree performance levels, correspondng damage states and lmts on resdual dsplacement. Performance levels Damage states Lmts on resdual top dsplacement Servceablt No damage 1 D 0 Lfe safet Moderate damage and reparable 2 D 5 cm Collapse preventon Severe damage 3 D 15 cm Earthquae frequenc able 2: Proposed earthquae hazard levels. Return perod for ears Probablt of exceedance Frequent 43 50% n 30 ears Occasonal 72 50% n 50 ears Rare % n 50 ears Pea ground acceleratons n ths paper 1 ag 0.15g 2 ag 0.40g 3 ag 0.60g Latn Amercan Journal of Solds and Structures, 2018, 151, e07 2/18

3 3.1 Horzontal elastc acceleraton response spectrum n AD format he elastc response acceleraton spectrum Sae for the horzontal components of sesmc acton s defned n Eurocode 8 European Commttee for Standardzaton, 2004 as 0 B : SaeagS B B C : Sae2.5agS 1 C C D : Sae2.5agS C 2 4 s: S 2.5a S D ae g where S ae D s the elastc response spectrum; s the vbraton perod of a lnear sngle degree of freedom sstem; a g s the desgn ground acceleraton on e A ground; s the lower lmt of the perod of the constant B spectral acceleraton branch; s the upper lmt of the perod of the constant spectral acceleraton branch; C D s the value defnng the begnnng of the constant dsplacement response range of the spectrum; S s the sol factor and s the dampng correcton factor wth a reference value of 1 for 5% vscous dampng rato. For ground e A, S 1.0, B 0.15 s, C 0.4 s, D 2.0 s. For an elastc SDOF sstem, the dsplacement response spectrum s 2 Sde S 2 ae 2 4 where S de s the value n the dsplacement spectrum correspondng to the perod and a fxed vscous dampng rato. Elastc response spectrum n AD format s obtaned b Eqs. 1 and 2. For the horzontal elastc acceleraton response spectrum defned n EC8, the cut off perod s obvousl 2 s n AD format. 3.2 Horzontal nelastc acceleraton response spectrum n AD format he relatonshp between nelastc response spectrum and elastc response spectrum Vdc et al., 1994 s S a d ae R S 3 S S 4 R de where Sd and a S are the values n the dsplacement and acceleraton spectra, respectvel, correspondng to the perod and a fxed vscous dampng rato 5%. R s the reducton factor and s ductlt factor. B substtutng Eqs. 3 and 4 nto Eq. 2, we obtan the nelastc response spectrum functon n AD format, 2 Sd S 2 a 5 4 For a blnear spectrum, R ( 1) 1 C C 6 Latn Amercan Journal of Solds and Structures, 2018, 151, e07 3/18

4 R C 7 he reducton factor R and the ductlt factor are related to both the structural response and the elastc acceleraton response spectrum. 4 SRUCURAL CAPACIY DIAGRAM 4.1 Base shear op dsplacement dagram MDOF sstem he nonlnear statc analss also called pushover analss s used to obtan the base shear force V top dsplacement D dagram for MDOF sstem. A monotoncall ncreasng pattern of lateral forces s appled to t structures n pushover analss. A planar steel frame shown n Fgure 1 s assumed where n s the number of the stor, the heght of the th stor s h and the mass of the th stor s m. he nverted trangular load pattern wth maxmum loadng at top and zero loadng at the ground level s emploed n ths paper. It s assumed that the lateral force at the th stor shown n Fgure 1 s proportonal to the component of the assumed dsplacement shape weghted b the stor mass m Fajfar and Gaspersc, 1996, P pm 8 where the component of the assumed dsplacement shape s 1 n 1 h h 9 where p controls the magntude of the lateral loads of steel frames. he dsplacement shape follows the frst vbraton mode of the buldng. herefore, the base shear force V s n V P p m pm 1 1 n 10 where m n m s the equvalent mass of the SDOF sstem. 1 Latn Amercan Journal of Solds and Structures, 2018, 151, e07 4/18

5 Fgure 1: n stor steel frame. he equal loadng steps are appled to steel frame n pushover analss. he loadng step sze stor s p P m pm 11 c p where p, and c s a constant nteger. c he ncrement of base shear force s V n P 12 1 P at the th In the base shear force V top dsplacement D dagram shown n Fgure 2, V t j and D tj are the ncrement of base shear force and ncrement of top dsplacement at the jth load step, respectvel. Fgure 2: V -D dagram. t Latn Amercan Journal of Solds and Structures, 2018, 151, e07 5/18

6 4.2 he F D dagram, eld pont and elastc perod equvalent SDOF sstem he dsplacement of the equvalent SDOF sstem s D D t he base shear force of the equvalent SDOF sstem s V F where s a constant and calculated as n m m 1 n n 2 2 m m Provded that both shear force V and top dsplacement D are dvded b, the force dsplacement relatonshp determned for the MDOF sstem,.e., V D dagram, becomes the shear force and dsplacement relatonshp for the equvalent SDOF sstem,.e., the shear force F and dsplacement D dagram shown n Fgure 3. t t Fgure 3: F D dagram. In Fgure 3, F j and D j are shear force ncrement and dsplacement ncrement of the equvalent SDOF sstem at the jth load step, respectvel. j j 16 F V tj j 17 D D In ths paper, the steel frame s assumed to eld f the followng nequaltes are true at the th load step, Latn Amercan Journal of Solds and Structures, 2018, 151, e07 6/18

7 F D 1 1 F D 18 OR D D 19 1 where s a constant and greater than 1. For structure steel, the modulus of elastct s much greater than the tangent modulus, n ths paper, 2 or bgger can be enough to judge that the eldng happens. herefore, the dsplacement at the eld pont n F D dagram s D F D 20 1 he shear force at the eld pont n F D dagram s F 21 1 In ths paper, the post eld stffness s dealzed to be zero Aschhem and Blac he dealzed F D dagram s shown n Fgure 4. he elastc perod of the dealzed blnear sstem can be calculated as md 2 22 F Fgure 4: Idealzed F D dagram thc lne. 4.3 Capact dagram n AD format he capact dagram n AD format shown n Fgure 5 s obtaned b dvdng the forces n the force deformaton dagram,.e., the dealzed F D dagram, b the equvalent mass m. S a F 23 m Latn Amercan Journal of Solds and Structures, 2018, 151, e07 7/18

8 In ths paper, the pushover analss s used untl the steel frames eld and the capact dagram s obtaned. It s relatvel easer to succeed n performng pushover analss untl a steel frame elds than untl the frame collapses. It doesn t need an engneerng judgements to get the eld pont. herefore, t can mae the optmzaton process unnterrupted. Fgure 5: Capact dagram n AD format. 5 DEERMINAION OF REDUCION FACOR AND DUCILIY FACOR Elastc response spectrum n AD format, nelastc spectrum n AD format and capact dagram n AD format are plotted n the same pcture shown n Fgure 6 C or Fgure 7 C. he reducton factor R s defned as the rato between the acceleratons correspondng to the elastc and nelastc u sstems, R u ( S ) ae 24 S a S s the eld acceleraton shown n Fgure 6 or Fgure 7. If S S ( ),.e., R 1, the steel frame response s elastc. herefore, the ductlt factor s where Sae( ) s the acceleraton value of the elastc spectrum dagram at the perod and a a ae u 1 when R 1 25 follows: If If S S ( ) a C u,.e., R 1, the steel frame response s nelastc. he ductlt factor can be calculated as ae shown n Fgure 6, u Sd Sde( ) Sae( ) R 26 u D D S a where Sde( ) s the dsplacement value of the elastc spectrum at the perod and d S s dsplacement value at the ntersecton of the nelastc spectrum dagram and capact dagram shown n Fgure 6 and Fgure 7. If shown n Fgure 7, C C R Latn Amercan Journal of Solds and Structures, 2018, 151, e07 8/18

9 Fgure 6: Elastc spectrum AD format, nelastc spectrum AD format and capact dagram C. Fgure 7: Elastc spectrum AD format, nelastc spectrum AD format and capact dagram C. 6 ESIMAE OF RESIDUAL DEFORMAION For the case of three performance levels.e., servceablt, lfe safet, and collapse preventon, the three correspondng structural characterstcs.e., stffness, strength and deformaton capact domnate the performances as shown n Fgure 8. he cal performance curve of the steel frames shown n Fgure 8 ndcates that no resdual top dsplacement s present at servceablt level, moderate and tolerable resdual top dsplacement s present at lfe safet level, and large resdual top dsplacement s present at collapse preventon level. Although the permanent deformaton exsts at lfe safet and collapse preventon levels, the steel frame s n strong hardenng phase at servceablt and n wea hardenng phase at collapse preventon, respectvel. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 9/18

10 S d Fgure 8: pcal performance curve of steel frames. he dsplacement demand of the equvalent SDOF sstem can be determned from the defnton of ductlt as D 28 * he dsplacement demand of the equvalent SDOF sstem s transformed bac to the top dsplacement of the MDOF sstem, D S D 29 * t d If the steel frame s at elastc phase the ductlt factor 1, No resdual top dsplacement s present after earthquae,.e., D 0. If the steel frame s at nelastc and hardenng phase the ductlt factor 1 as shown n Fgure 9, accordng to the dealzed F D dagram, the resdual dsplacement demand of the equvalent SDOF sstem can be estmated as D D D D D F 30 1 p e F1 where D e s the elastc dsplacement. Fgure 9: Real and dealzed F D dagram. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 10/18

11 If the steel frame s stll at hardenng phase as shown n Fgure 9, the estmated value of the resdual top dsplacement of the steel frame D p s greater than the real resdual top dsplacement D h. herefore, the estmated value of the resdual top dsplacement s conservatve n Eq. 30. he resdual dsplacement of the equvalent SDOF sstem s transformed bac to the top resdual dsplacement of the MDOF sstem, D D D D F 1 p F1 As shown n Fgure 8, the steel frame s n elastc phase at servceablt, n strong hardenng phase and n wea hardenng phase at lfe safet and collapse preventon levels, respectvel. herefore, the resdual top dsplacement of steel frame at servceablt, lfe safet and collapse preventon levels, can be estmated as: f 1, D 0, f 1, the resdual top dsplacement s calculated b usng Eq COMPUAIONAL PROCEDURE FOR RESIDUAL OP DISPLACEMEN he Eq. 31 s used to estmate the resdual top dsplacement of the MDOF sstem when the steel frame response s at elastc phase or nelastc and hardenng phase. he detal of computatonal procedure s: Step 1 Perform pushover analss wth the equal loadng step sze controlled b Eq. 11 untl the steel frame elds Inequaltes Eq. 18 or Eq. 19 s true. Step 2 Obtan the eld pont D, F usng Eq. 20 and Eq. 21. Step 3 Calculate the elastc perod of the equvalent SDOF sstem usng Eq. 22. Step 4 Calculate the reducton factor R wth Eq. 24. u Step 5 If Ru 1, then 1, the resdual top dsplacement s D 0. If Ru 1, calculate the reducton factor wth Eq. 26 f C or Eq. 27 f C. Estmate the resdual top dsplacement D usng Eq HREE LEVEL PERFORMANCE OPIMIZAION MODEL he e A ground moton s defned wth the elastc acceleraton response spectrum accordng to Eq. 1, whch has been normalzed to pea ground acceleraton a g equal to a g, a g and a g at frequent, occasonal and rare earthquae, respectvel. he desgn ground acceleratons for the three level performances depend on dfferent desgn codes and can be chosen b structural engneers. he resdual top dsplacements of steel frames can be 1 2 estmated b the computatonal procedure n secton 7. he resdual top dsplacements are denoted as D, D 3 and D at frequent, occasonal and rare earthquae, respectvel. D 1, 2 D and 3 D are the lmts on resdual top dsplacements related to servceablt, lfe safet and collapse preventon levels, respectvel. he three level performance optmzaton model s proposed as Fnd d Mn M d d d d 1 1 S.. D D 2 2 D D 3 3 D D d d d 1,2,, N 32 Latn Amercan Journal of Solds and Structures, 2018, 151, e07 11/18

12 where d and M d are the vector of desgn varables and mass of steel frame, respectvel. d, d, and d are the th desgn varable, ts lower and upper boundar, respectvel. N s the number of desgn varables. he resdual top dsplacements D, D, and D are dependent on the vector of desgn varables d. In ths paper, the frst order optmzaton method of ANSYS s emploed to solve the optmzaton model of Eq. 32. In the frst order optmzaton method, the senstvt analss technque s used to construct optmzaton algorthm and determne the searchng drecton. ANSY uses the fnte dfference methods to perform senstvt analss. he frst order method converts the constraned optmzaton problem nto a seres of unconstraned optmzaton problems b usng penalt functon methods. Accordng to the lateral load pattern of Eq. 8, the relaton between the resdual top dsplacements D j D j 1, 2,3 and resdual nter stor drfts dr n j dr , 2,, n s Assumed that the th stor of the structure s wea/soft stor, the resdual nter stor drft of the th stor s the domnant component n the contrbuton to the resdual top dsplacement. D d j dr 34 he desgn varable of columns at the th stor s, we have d. Dfferentatng Eq. 34 wth respect to desgn varable D d j dr d 35 Snce the th stor of the structure s wea/soft stor, the frst order senstvt Eq. 35, the frst order senstvt of resdual top dsplacement,.e., D d j dr d wll be ver large. In, wll become ver large. he desgn wth wea/soft stor s mpossble to be chosen as the optmum desgn. herefore, although there are no explct constrants on nter stor drfts n three level performance optmzaton model of Eq. 32, the nter stor drfts local sesmc demand are ndrectl constraned b usng resdual top dsplacement constrants, lateral load pattern nverted trangle and senstvt analss technque. 9 EXAMPLE 9.1 Optmal desgn of a 3 stor 2 ba steel frame A three stor two ba steel frame shown n Fgure 10 s fxed at the ground. he heght of the frst, second and thrd stores are h1 5.4m, h2 3.6m and h3 3.6m, respectvel. he steel frame conssts of 4 groups ncludng B1, C1, C2, and C3 shown n Fgure 10. All the members are H shape secton shown n Fgure 10. he desgn varables are the sze of flanges and webs also shown n Fgure 10. he desgn varable vector s d h, t, h, t, h, t, h, t, h, t, h, t, h, t, h, t bf bf bw bw 1cf 1cf 1cw 1cw 2cf 2cf 2cw 2cw 3cf 3cf 3cw 3cw he Beam189 element s used to analze the steel frame. Beam189 s a 3 D 3 node element. hs element s well suted for lnear, large rotaton, and large stran nonlnear analss. he Beam element s developed b Smo and Vu Quoc 1986, and Ibrahmbegovc he pushover analss s carred out usng ANSYS software. he propertes of materal are defned as: Modulus of elastct, Yeld stress E 200 GPa, Posson s rato MPa and Secant modulus of plastct E MPa, Denst 7857 g/m. he mass of ever s floor s assumed to be 1000 g. he desgn space,.e., lower and upper lmts, s shown n able 3. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 12/18

13 Beam B1 mm Column C1 mm Column C2 mm able 3: Desgn space, ntal and optmum desgns. Intal desgn Optmum desgn Desgn space Lower lmts Flange wdth Flange thcness Web depth Web thcness Upper lmts Flange wdth Flange thcness Web depth Web thcness Flange wdth Flange thcness Web depth Web thcness Flange wdth Column Flange thcness C3 mm Web depth Web thcness Structural mass g Perod of SDOF,, s Yeld dsplacement of SDOF, D, cm Ductlt factor, op dsplacement, D, cm t 1st level nd level rd level st level nd level rd level D D Note: t In ths example, the pea ground acceleratons are assumed as ag 0.15 g, ag 0.4 g and ag 0.6 g at frequent, occasonal and rare earthquae, respectvel. he lmts on resdual top dsplacements are assumed as 1 D 0, 2 D 5 cm, and 3 D 15 cm. he frst order optmzaton method of ANSYS s emploed to solve the optmzaton model of Eq. 32. he frst order method wll perform a maxmum of 30 teratons upon executon, usng a lne search step equal to 100% of the maxmum possble value, and a 0.200% dfference appled to the desgn varables to obtan the frst order senstvt. he tolerance of the objectve functon s 10 g,.e., the algorthm converges f the dfference between the two adjacent objectve values, acheved b the adjacent two lnear searchng, s less than 10 g. Startng wth the ntal desgn shown n able 3, the frst order optmzaton method converges at the sxth teraton to obtan the optmum desgn shown n able 3. In the optmzaton process, the frame mass, ductlt factor, and resdual top dsplacements at the servceablt, lfe safet and collapse preventon levels are shown n Fgures 11, 12, and 13, respectvel. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 13/18

14 Fgure 10: hree stor steel frame wth H shape sectons of members. 9.2 Observaton of optmum desgn able 3 shows that the resdual top dsplacement of the optmum desgn at collapse preventon level s cm and the upper lmt of the desgn space s 15 cm, and the resdual top dsplacement of the optmum desgn at lfe safet level s 1.40 cm and the upper lmt of the desgn space s 5 cm. herefore, the constrant of the resdual top dsplacement s actve at collapse preventon level, not actve at lfe safet level. If the constrant upper lmt of resdual top dsplacement at collapse preventon level n ths mathematcal model s changed, the dfferent optmum desgn wll be acheved. he optmum desgn s compared wth the ntal desgn shown n able 3, and n Fgures 11, 12 and 13. he mass of optmum desgn decreases. he perod and eld dsplacement of the equvalent SDOF ncrease, the ductlt factor ncreases, and the top dsplacements of the steel frame at servceablt, lfe safet, and collapse preventon levels ncrease. able 4: Inter stor drfts of optmum desgn unts: cm. 1st stor 2nd stor 3rd stor Servceablt level Lfe safet level Collapse preventon level able 5: Inter stor drft rato of optmum desgn. 1st stor 2nd stor 3rd stor Servceablt level Lfe safet level Collapse preventon level Latn Amercan Journal of Solds and Structures, 2018, 151, e07 14/18

15 Structural mass (g) Iteraton number Fgure 11: Objectve functon varaton n optmzaton process. In the desgn codes, the nter stor drfts must be verfed to satsf the lmtatons. herefore, the pushover analses of the optmum desgn should be used once agan. Accordng to the top dsplacements shown n able 3, the pushover analss stops f the top dsplacement s equal or greater than cm at collapse preventon level, equal or greater than cm at lfe safet level, equal or greater than 4.42 cm at servceablt level. he results show that the pushover analss stops at cm, cm, and cm, at collapse preventon, lfe safet, and servceablt levels, respectvel. he nter stor drfts of the optmum desgn at collapse preventon, lfe safet, and servceablt levels are lsted n able 4. he pushover analss process at collapse preventon level s shown n Fgure 14. he dagrams of top dsplacement, nter stor drfts shown n Fgure 14 ndcate that the steel frame s at the hardenng phase at collapse preventon level. herefore, the estmated method of the resdual top dsplacement proposed n ths paper s reasonable. he nter stor drft and top dsplacement dagrams of the optmum desgn at servceablt, lfe safet, and collapse preventon levels are shown n Fgure 15. he nter stor drft ratos of the optmum desgn n able 5 ndcate that the dstrbuton of nter stor drft ratos are relatvel unform at servceablt, lfe safet, and collapse preventon levels, respectvel. Fgure 12: Ductlt factor varaton n optmzaton process. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 15/18

16 Servceablt 8 level ral 1900ral 1900ral 1900ral 1900ral 1900ral 1900ral 1900ral Iteraton number Resdual dsplacement (cm) Fgure 13: Resdual top dsplacement varaton n optmzaton process. Fgure 14: Pushover analss to top dsplacement cm cm for optmum desgn. 1900ral 1900ral Stor 1900ral 1st performance level, nterstor drft 1st performance level, top dsplacement 2nd performance level, nterstor drft 2nd performance level, top dsplacement 3rd performance level, nterstor drft 3rd performance level, top dsplacement 1900ral Drft (cm) Fgure 15: Inter stor drfts and top dsplacement of 1st, 2nd and 3rd performance levels 9.3 Dscussons It s generall accepted that the damage of structure s stran and dsplacement related. he resdual top dsplacement can drectl measure the global damage of structures subjected to the dfferent earthquae hazard levels. Latn Amercan Journal of Solds and Structures, 2018, 151, e07 16/18

17 he estmate method of resdual top dsplacement assumes that the steel frame s n hardenng phase before t collapses. In ths paper, the estmate method of resdual top dsplacement onl needs the dsplacement and base shear force at eld pont. herefore, the pushover analss s used untl steel frame elds Inequaltes Eq. 18 or Eq. 19 s true. he computatonal tme to get the eld pont s far less than that to use pushover analss untl the structure collapses. On the other hand, t s not eas to succeed n demonstratng pushover analss untl structures collapse because the dfferent structures wll be produced n the optmzaton process. However, t s eas to succeed n mplementng pushover analss untl structures eld. herefore, the optmzaton process wll not be nterrupted. he estmate method of resdual top dsplacement can be used n optmzaton desgn of steel frames at servceablt, lfe safet, and collapse preventon levels. Wth the ad of Eq. 30 and Fgure 9, t can be observed that the estmated value of resdual top dsplacement s greater than the real resdual top dsplacement. herefore, the estmate method of resdual top dsplacement s conservatve. he resdual top dsplacement can be calculated usng the computatonal procedure dscussed n Secton he nput pea ground acceleratons,.e., a g servceablt, a g lfe safet, and a g collapse preventon are often gven n the desgn codes. herefore, t s onl necessar to determne the lmts on resdual top dsplacements,.e., D 1, 2 D and 3 D, n the optmzaton model Eq. 32. hen the optmzaton desgn of steel frames can be mplemented. he example n ths paper just shows how to mplement a three level performancebased optmzaton method of steel frames. Structural engneers should set up structural parameters and the lmts on resdual top dsplacements n real practce accordng to real structure and local desgn code, respectvel. he nverted trangular load pattern s n prncple naccurate for structures where hgher mode effects are sgnfcant. It s accurate enough for structures where the frst mode s domnant. Because of the fact that the non lnear analss pushover analss s based on a tme ndependent dsplacement shape, t ma not detect the structural weanesses whch ma be generated when the structures dnamcs characterstcs change after the formaton of the frst local plastc mechansm. Man researches Fajfar, 1999, 2000, 2002; Kresln and Fajfar, 2012 ndcate that the results obtaned b usng the N2 method are reasonabl accurate provded that the structure oscllates predomnantl n the frst mode. herefore, the N2 method has been mplemented n Eurocode 8 European Commttee for Standardzaton, he estmate method of resdual top dsplacement proposed n ths paper s drect developed based on the N2 method. For most buldng structures, the frst mode alwas domnates the vbraton. he resdual top dsplacement obtaned b usng the estmate method proposed n ths paper s reasonabl accurate for performance based desgn. he restrcton of the proposed method n ths paper s the same as the N2 method Fajfar CONCLUSIONS In ths paper, an estmate method of resdual top dsplacement has been developed for steel frames subjected to three earthquae hazard levels. A three level performance based optmzaton model s proposed based on the estmate method of resdual top dsplacement. he man conclusons are (1) he estmate method of resdual top dsplacement needs to use pushover analss untl steel frames eld. he resdual top dsplacement s estmated b usng the eld dsplacement value and eld loadng value at the eld pont. he estmated value of the resdual top dsplacement obtaned b the proposed method s greater than the real resdual top dsplacement. he estmated value of resdual top dsplacement s conservatve. (2) he computatonal tme to get the eld pont s far less than to get the collapse pont. It s relatvel easer to succeed n mplementng pushover analss untl steel frame elds than untl the frame collapses. It also doesn t need an engneerng judgements to get the eld pont and the optmzaton process wll not be nterrupted n the proposed method. (3) A three-level performance-based optmzaton model s proposed n ths paper. In the optmzaton model, the pea ground acceleratons related to three earthquae hazard levels and the lmts on resdual top dsplacement related to the three performance levels are requred to start optmzaton desgn. he pea ground acceleratons are often gven n desgn codes. However, the lmts on resdual top dsplacement related to lfe safet and collapse preventon levels should be further nvestgated n the future. (4) he pushover analses of the optmum desgn ndcate that the steel frame s at elastc phase at servceablt level, at nelastc and hardenng phases at lfe safet and collapse preventon levels. It means that the assumpton of resdual top dsplacement s relable when t s used n three-level performance-based optmzaton model. References Appled echnolog Councl 1996 Sesmc evaluaton and retroft of concrete buldngs, Report AC 40. Aschhem M and Blac EF 2000 Yeld Pont Spectra for Sesmc Desgn and Rehabltaton. Earthquae Spectra16: Latn Amercan Journal of Solds and Structures, 2018, 151, e07 17/18

18 European Commttee for Standardzaton 2004 Eurocode 8: Desgn of structures for earthquae resstance Part 1: general rules, sesmc actons and rules for buldngs. European standard EN Englsh Edton. Fajfar P and Gaspersc P 1996 he N2 method for the sesmc damage analss of regular buldngs. Earthquae Engneerng and Structural Dnamcs 25: Fajfar P 1999 Capact spectrum method based on nelastc demand spectrum. Earthquae Engneerng and Structural Dnamcs 28: Fajfar P 2000 A nonlnear analss method for performance based sesmc desgn. Earthquae Spectra 16: Fajfar P 2002 Structural analss n earthquae engneerng a breathrough of smplfed non lnear methods. Paper Reference 843, Proceedngs of 12 th European Conference on Earthquae Engneerng, London. Freeman SA 1998 he capact spectrum method as a tool for sesmc desgn. Proceedngs of the 11 th European conference on earthquae engneerng, Pars. Genctur B and Elnasha AS 2008 Development and applcaton of an advanced capact spectrum method. Engneerng Structures 30: Ghobarah A 2001 Performance based desgn n earthquae engneerng: state of development. Engneerng Structures 23: Ibrahmbegovc A 1995 On fnte element mplementaton of geometrcall nonlnear Ressner s beam theor: three dmensonal curved beam elements. Computer Method n Appled Mechancs and Engneerng 26: Kresln M and Fajfar P 2012 he extended N2 method consderng hgher mode effects n both plan and elevaton. Bulletn of Earthquae Engneerng 10: Lu Q, Zhang J and Yan L 2010 An optmal method for sesmc drft desgn of concrete buldngs usng gradent and Hessan matrx calculatons. Archve of Appled Mechancs 80: Lu Q and Paavola J 2015 An Optmzaton Procedure for Sesmc Desgn of Steel Frames for Mult Performance and Mult Hazard Levels. Advances n Structural Engneerng Smo JC and Vu Quoc L 1986 A three dmensonal fnte stran rod model. Part II: computatonal aspects. Computer Method n Appled Mechancs and Engneerng 58: Vdc, Fajfar P and Fschnger M 1994 Constant nelastc desgn spectra: strength and dsplacement, Earthquae Engneerng and Structural Dnamcs Washngton DC: Federal Emergenc Management Agenc 2000 Prestandard and commentar for the sesmc rehabltaton of buldngs, Report FEMA 356. Zou XK and Chan CM 2005 Optmal sesmc performance based desgn of renforced concrete buldngs usng nonlnear pushover analss. Engneerng Structures 27: Latn Amercan Journal of Solds and Structures, 2018, 151, e07 18/18