SEISMIC DESIGN OF STEEL CONCENTRIC BRACED FRAME STRUCTURES USING DIRECT DISPLACEMENT BASED DESIGN APPROACH

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1 SEISMIC DESIGN OF STEEL CONCENTRIC BRACED FRAME STRUCTURES USING DIRECT DISPLACEMENT BASED DESIGN APPROACH K.K. Wjesudara South Asa Isttute of Techology ad Medce P.O Box, Mlleum Drve,Malabe, Sr Laka. Tel. 7687, emal: Abstract The drect dsplacemet based desg (DDBD) procedure s well developed ad used for desgg reforced cocrete momet resstg frame structures, wall structures ad brdges. However, there s lmted umber of studes avalable o desgg steel cocetrc braced frame (CBF) structures usg DDBD approach. Therefore, t s ecessary to develop a DDBD procedure for CBF structures. O ths regards, ths paper proposes a DDBD procedure for steel cocetrc braced frame structures. The proposed procedure utlses yeld dsplacemet shape derved o the bass of tesle yeldg of the braces, ad equvalet vscous dampg equato of the system gve as a fucto of system ductlty ad o-dmesoal slederess rato by Wjesudara et al. () for cocetrc braced steel frame structures. Fally, the performace of four steel CBF structures desged accordg to the proposed DDBD procedure s studed usg olear dyamc respose of those structures. The results show that the performace of CBF structures s good agreemet wth the desg cosderatos. Keywords: Drect dsplacemet based desg, cocetrc braced steel frames, equvalet vscous dampg coeffcet, o dmesoal slederess rato, ductlty

2 . Itroducto The drect dsplacemet based desg (DDBD) was frst troduced by Prestley (99) ad t has bee subjected to cosderable research atteto Europe, New Zealad, ad North Amerca the terveg years (Prestley, ). The procedure s well developed for RC momet resstg frame, wall structures ad brdges over the last decade. However, DDBD procedure for steel cocetrc braced frame (CBF) has ot bee developed fully ad oly very lmted umber of studes has bee foud the lteratures. Medhekar ad Keedy () have developed a dsplacemet based desg procedure for (CBF) structures. However, that desg procedure, the equvalet vscous dampg (EVD) coeffcet of the equvalet sgle degree of freedom (SDOF) system s take as 5% of the crtcal dampg. More recetly, the DDBD procedure for CBF structures has bee developed by Della Corte ad Mazzola (8), but that procedure the referece s made to the Takeda-Th EVD expresso whch was developed for reforced cocrete (RC) structures. Goggs ad Sullva (9) revewed the apparet EVD of a umber of CBF structures, wth sleder braces, subject to shake table testg. They foud that the EVD coeffcet should be less tha that dcated by the Takeda- Th model, ad argued that there was a eed for EVD expressos specfc to CBF structures. Moreover, the DDBD procedure developed by Medhekar ad Keedy (), the yeld dsplacemet profle has bee developed o the bass of the yeldg of braces but eglectg the axal deformato of colums. Della Corte ad Mazzola (8) has derved the yeld dsplacemet profle of CBF structure usg the buckg state of braces. Ths approach mght be approprate oly for the CBF structures wth verted-v braced frames wth flexble beams where two cocetrc braces are coected at each storey level. As cosequeces, ths study proposed a ew DDBD procedure for steel CBF structures based o the yeld dsplacemet profle derved usg both the tesle yeldg of braces ad axal deformatos of colums, ad the EVD coeffcet equato developed by Wjesudara et al. (). The proposed procedure has bee valdated usg the performace of four steel CBF structures, whch were desged accordg to the proposed procedure. For the valdato purpose, the resultat dsplacemet, drft ad storey shear profles at peak storey dsplacemet from olear dyamc aalyses (NDAs) are compared wth the correspodg desg profles used DDBD procedure.. Drect Dsplacemet Based Procedure for Steel CBF Structures The complete DDBD procedure for steel CBF structures s summarsed here three steps: () evaluato of the yeld dsplacemet profle, () selecto of the desg dsplacemet profle, ad () the trasformato of a MDOF system to a equvalet SDOF system referrg the frst elastc mode of respose.

3 . Evaluato of Yeld Dsplacemet Profle The yeld dsplacemet profle s developed o the bass of followg two assumptos: () bucklg of the compresso braces ad yeldg of the teso braces at all the storey levels occur smultaeously; ad () the force-deformato curve of a par of cocetrc braces at ay storey level of a steel CBF structure s approxmated to be b-lear. As stated the frst assumpto, achevg the brace yeldg ad buckg teso ad compresso smultaeously s relatvely easy practce for termedate or stocky braces, sce ther axal deformatos at yeldg ad bucklg are ot sgfcatly dfferet. However, ths assumpto may ot be applcable for sleder braces whch show sgfcat varato betwee the axal deformato at yeldg ad bucklg (Archambault et al.,995 ad Shaback, ). O the bass of the assumptos, the lateral dsplacemet at each storey level s bascally duced due to storey sway mechasm resultg the brace elogato teso ad shorteg compresso ad the rgd body rotato of the storey resultg the axal deformatos of the outer colums the braced bay as show Fgure (a) ad (b), respectvely. Fgure : (a) Sway mechasm (b) Rgd rotato of the storey of th storey Cosderg the th storey, the axal deformato (δ b ) of the teso brace ca be expressed as: b () L d, Lud, ylud, where L ud, ad L d, are the udeformed ad deformed legths of the brace at th storey, ad ε y s the yeld stra of the brace steel materal. Based o the deformed geometry show Fgure (a), the deformed legth of the brace ca be expressed as: L d, L L B sy, h y ud, ud, ()

4 where Δ sy, s the lateral dsplacemet duced by the sway mechasm at yeldg of the th storey teso brace, B s the bay wdth ad h s the storey heght. Sce the terms (Δ sy, ) ad (ε y L ud, ) are eglgble compared to other terms, Eq. () ca be rewrtte as: L y ud, y sy, h () B s cos where α s the agle of the brace to the horzotal le. Cosderg the rgd rotato at th storey as show Fgure (b), the rgd rotato at yeldg of the teso brace, θ y, ca be expressed as: ( h, h, ) ry, y, () B h Teso ad compresso forces developed outer colums the braced bay, resultg brace bucklg compresso ad yeldg teso, are sgfcatly dfferet to each other for the termedate ad sleder braces. However, the gravty loads dmsh the dfferece by decreasg the teso force ad coversely crease the compresso force. As cosequece of that, t s reasoable to assume that the axal elogato ad shortg of the outer colum teso ad compresso, respectvely are approxmately equal. Thus, Eq. () ca be rearraged the followg form Eq. (5). h ta h ry, h yc (5) B where ε yc s the yeld stra of the colum steel materal ad β s the rato of the desg axal force to the yeldg force of the colum secto at th storey. Fally, the total terstorey yeld dsplacemet at the th storey Δ y s: y, sy, ry, y y ta, h ych (6) s cos. Selecto of Desg Dsplacemet Profle The desg dsplacemet profle proposed to RC momet resstg frame structures by Prestley et al. (7) s used ths study. As hghlghted the study by Khatb et al. (988) ad Trembley ad Pocet () that the frst elastc mode of vbrato of cocetrcally braces frames (CBFs) dcates hgher ter-storey drft cocetrato at the frst storey level smlar to what s observed steel momet resstg frames (SMRFs) whe they subject to a severe shakg, eve though the drft cocetrato results both types of structures due to two dfferet structural actos. I CBFs, the drft cocetrato occurs due to elastc bucklg whle t s due to the elastc rotato of plastc hges SMRFs. Thus, t s reasoable to use Prestley et al. (7) desg dsplacemet profle to steel CBFs. The desg dsplacemet

5 profle s obtaed from a ormalsed elastc mode shape δ, ad the dsplacemet of the lowest floor Δ as gve Eq. (7): (7) H (8) H H H (9) H H where the ormalsed elastc mode shape depeds o the heght H, ad roof heght H.. Trasformato of MDOF System to SDOF System I the very frst stage of the DDBD desg procedure, t s essetal to represet a MDOF system by a equvalet SDOF system based o the frst elastc mode of respose ad the trasformato of a MDOF system to a equvalet SDOF system s also based o the followg assumptos: () MDOF system respods harmocally the assumed shape; () the base shears developed by a MDOF system ad ts equvalet SDOF system are same; ad () the work doe by the lateral earthquake force o both systems s same. The equvalet SDOF system propertes such as effectve mass (m eff ), secat stffess (k eff ), EVD coeffcet (ξ CBF ), equvalet desg dsplacemet (Δ d ), ad the base shear (V)ca be expressed as descrbed below o the bass of the above metoed assumptos. The desg dsplacemet (Δ d ) ad yeld dsplacemet (Δ y,eff ) of equvalet SDOF system foud from the assumpto by equatg the work doe by the lateral forces o the each system, ca be expressed as gve Eq. () ad Eq. (), respectvely. d m m where m s the mass at the heght H assocate wth dsplacemet Δ. () m y, y, eff () m Cosderg the assumptos ad, the effectve mass of the SDOF system ca be defed as: y,

6 m eff m eff () The effectve heght of the equvalet SDOF structure s gve by: H e ( m H ) ( m ) The desg dsplacemet ductlty factor of a equvalet SDOF system s related to the equvalet yeld dsplacemet Δ y,eff, ad the desg dsplacemet Δ d as: () d () y, eff The EVD coeffcet of the equvalet SDOF system proposed by Wjesudara et al. () for CBFs, whch ca be related to the desg dsplacemet ductlty demad ad the odmesoal slederess rato (λ), s used. The study by Wjesudara et al. () frstly aalyses fftee dfferet pre-determed sgle storey CBFs order to evaluate the EVD coeffcet through a area based approach (Jacobse 96) whch s the corrected for the results of elastc tme hstory aalyss (ITHA) coducted usg real accelerograms. I geeral, the area based hysteretc EVD s greatly a fucto of the o-dmesoal slederess ad the ductlty, but does ot vary sgfcatly for dfferet dagoal bracgs. The corrected hysteretc EVD values also exhbt a sgfcat depedecy o the o-dmesoal slederess. Therefore, Wjesudara et al. () proposed blear dampg expressos for desg of CBFs as a fucto of the ductlty ad the o dmesoal slederess as show Eq. (5.a) ad Eq. (5.b). CBF.. 5 (5.a) CBF.. 5 (5.b) It s mportat to ote that the o-dmesoal slederess rato of the equvalet SDOF system s assumed to be equal to the average value of o-dmesoal slederess ratos of the MDOF system. Sce, o-dmesoal slederess rato of the equvalet SDOF system s ot avalable at the begg of the desg procedure, t s suggested to assume a value ad verfy t after szg the braces. If the dfferece betwee the assumed o-dmesoal slederess rato ad average o-dmesoal slederess rato s sgfcat, the the desg procedure has to be repeated wth the ew average value of o-dmesoal slederess rato utl the dfferece s sgfcat.

7 The effectve perod T eff, at the desg dsplacemet of equvalet SDOF system s read from the dsplacemet spectrum at the EVD coeffcet calculated usg ether Eq. (5.a ) or (5.b). The secat stffess correspodg to the desg dsplacemet of the equvalet SDOF structure s gve by: K eff me (6) T e The desg base shear force for the MDOF structure s obtaed from the equvalet of SDOF structure as: F (7) b K e All the steps the DDBD procedure for steel CBF structures are summarzed the flow chart show Fgure. d. Nolear Dyamc Aalyss (NDA) Four steel CBF structures are desged accordg to the DDBD procedure descrbed secto. Two CBF structures are four ad eght storeys wth verted-v bracg cofgurato ad cotuous mddle colum that lks the brace-to-beam tersecto pots at each floor level drectly to the foudato whle other two frames are four ad eght storeys wth X bracg cofgurato. From ths pot owards, the verted-v bracg cofgurato wth mddle colum s called as IVMC bracg cofgurato. The floor pla ad the elevato of the buldgs were predetermed, as show Fgure. The heght of each storey s.5m ad the bay wdth of each of braced ad ubraced bays s 7m. The locatos of the cocetrc braced frames are show by the bold les Fgure. Fgure : Pla vew ad elevato of CBF structures wth (a) IVMC (b) X bracg cofgurato Seve real accelerograms are selected from PEER data base order to carry out the olear dyamc aalyses ad scaled to match wth the desg dsplacemet spectrum of 5% dampg as specfed EC-8 (5), for the sol class A wth the PGA of.g. Fgure shows the 5% damped dsplacemet spectra of the dvdual accelerogram ad average of the dvdual

8 Dsplacemet (m) accelerogram together wth the desg dsplacemet spectrum, the perod rage of to s. The average dsplacemet spectrum of the dvdual accelerogram s matched well wth the desg dsplacemet spectrum..6.5 rage Desg Spectrum Perod (s) Fgure : Dsplacemet spectra from the scaled atural accelerogram at 5% dampg I order to vestgate the performace of the buldg models desged accordg to the DDBD procedure proposed, olear dyamc aalyses are performed usg the OpeSEES fte elemet computer program (McKea et al. 7). The steel CBFs are modelled -D rather tha -D to permt the braces to buckle the out-of-plae drecto of the frame sce all the braces are desged ad detaled to develop the out-of-plae bucklg. Tremblay et al. (995) ad Tremblay et al. (996) poted out that the out-of-plae bucklg respose of a brace cocetrc cofgurato ca occur durg a severe shakg. The behavour of all the frame elemets except the braces s lmted to -plae dsplacemet by restrag the traslatoal degree of freedom the perpedcular drecto to the plae of the frame ad the rotatoal degrees of freedom the out-plae drectos. The colum-to-base ad the beam-to-colum coectos are modelled as ped coectos whle the colums are modelled as cotuous members. All the braces are modelled usg the elastc beam-colum brace model proposed by Urz (5) ad Urz et al. (8). I ths model, each brace s modelled usg two olear beam-colum elemets wth fve tegrato pots. All the colums ad beams are also modelled usg olear beam-colum elemets avalable OpeSEES frame work. The corotatoal theory was used to represet the moderate to large deformato effects of elastc bucklg of braces. Newmark accelerato tme tegrato scheme wth beta ad gamma.5 ad.5, respectvely ad taget stffess proportoal dampg equal to % of crtcal dampg s adopted for the aalyses.. Results ad Dscusso The average profles of peak dsplacemets, ter-storey drft ratos ad storey shears at the peak storey dsplacemet of two dfferet cocetrc bracg cofguratos resultg from NDA are compared wth the correspodg desg profles.

9 Fgure ad 5 llustrate the average dsplacemet ad drft rato profles for ad 8 storeys CBFs wth IVMC cofgurato, respectvely. Fgure 5 shows clearly that the resultat average dsplacemet profle of storey frame s almost lear ad well matched to the desg dsplacemet profle. The average drft rato s % below the desg drft rato at the st storey whle % below at the top storey. I the case of 8 storey frame, the average dsplacemet profle s farly matched wth the desg dsplacemet profles esurg that average dsplacemets do ot exceed the desg dsplacemets correspodg to the presumed dsplacemet shape sgfcatly as show Fgure 5. The average drfts at storey levels 5, 6 ad 7 are slghtly hgher tha the desg drfts, ad the maxmum of 6% hgher average drft s observed at 7 th storey level Desg EQ EQ EQ EQ EQ5 EQ6 EQ7 Des g Dsplacemet (m) Iter-storey drft (%) Fgure : rage tme-hstory respose of storey braced frame wth IVMC cofgurato Desg 5 EQ EQ EQ EQ EQ5 EQ6 EQ7 Des g..... Dsplacemet (m) Iter-storey drft (%) Fgure 5: rage tme-hstory respose of 8 storey braced frame wth IVMC cofgurato Fgure 6 shows the average storey shear dstrbuto at the atcpated peak dsplacemets ad the desg storey shear dstrbuto at the desg dsplacemets of ad 8 storey frames wth IVMC bracg cofgurato. At the peak storey dsplacemet of ay storey level, compresso ad teso braces are yelded ad buckled teso ad compresso, respectvely. Therefore, the storey shear forces obtaed represet the storey shear capacty. From the comparso of desg storey shear ad the storey shear capacty, t ca be cocluded that a CBF structure desged accordg to the DDBD procedure has hgher storey shear capacty tha the desg

10 storey shear. It further dcates that the CBF structure could eve behave satsfactorly for hgher level of shakg tha the desg level of shakg. Therefore, DDBD procedure results coservatve desg. Desg 8 7 Desg Storey shear (kn) 5 Storey shear (kn) Fgure 6: Storey shear dstrbutos of ad 8 storey frames wth IVMC cofgurato Fgure 7 ad 8 llustrate the average dsplacemet ad drft profles obtaed from ad 8 storey CBFs wth X cofgurato, respectvely. Fgure 7 exhbts that the resultat average dsplacemet profle slghtly devates from the presumed lear dsplacemet shape resultg a hgher drft cocetrato at the st storey. The average drft rato s 7% above the desg drft rato at the st storey whle t s 8% below at the top storey. However, exceedace of the drft at the st storey s so sgfcat. Smlar to the 8 storey frame wth IVMC cofguratos, the average dsplacemet profle of 8 storey frame wth X cofgurato s farly matched wth the desg dsplacemet profles esurg that average drfts do ot exceed the desg drfts correspodg to the presumed dsplacemet shape sgfcatly. Smlar to the IVMC cofgurato, Fgure 9 exhbts that the resultat average storey shear forces, at lower storeys are sgfcatly hgher tha the desg values whle they are more smlar to the desg values at the upper storeys. Desg EQ EQ EQ EQ EQ5 EQ6 EQ7 Des g Dsplacemet (m) Iter-storey drft (%) Fgure 7: rage tme-hstory respose of storey braced frame wth X cofgurato

11 Desg 5 5 EQ EQ EQ EQ EQ5 EQ6 EQ7 Des g Dsplacemet (m) Iter-storey drft (%) Fgure 8: rage tme-hstory respose of 8 storey braced frame wth X cofgurato Desg 8 7 Desg Storey shear (kn) 5 Storey shear (kn) Fgure 9: Storey shear dstrbutos of ad 8 storey frames wth X cofgurato Curret force based desg phlosophy mplemeted may leadg codes for CBFs caot take to accout the effects of the slederess rato the respose of the structure. It s because the force reducto factors specfy for CBFs are depedet of slederess rato. Therefore, ths study, a DDBD procedure s developed to desg steel CBF structures cludg the effects of slederess rato. The yeld dsplacemet profle s evaluated o the bass of tesle yeldg of braces. The procedure uses assumed frst mode dsplacemet shape proposed by Prestley et al. (7) for MRFs as the desg dsplacemet profle wth the EVD equato proposed by Wjesudara et al. (). The proposed procedure s valdated usg NDA results of steel CBFs. The results of NDA prove that presumed lear dsplacemet shape proposed by Prestley et al. (7) for low-rse MRFs s reasoably vald for the low-rse CBF structures, eve though t depeds slghtly o the type of cocetrc cofgurato. Furthermore, the NDA results of the medum-rse CBF structures also prove that the presumed elastc frst mode dsplacemet shape wth hgher drft cocetratos at lower storeys s a reasoably good estmato for the

12 dsplacemet shape for medum-rse CBF structures. Ulkely the low rse CBF structures, ther dsplacemet profles are less sestve to the type of cocetrc cofgurato. Refereces Amerca Isttute of Steel Costructos (AISC) (5) Sesmc provsos for structural steel buldgs. Chcago. CEN () ENV 99--, Eurocode (EC): Actos o structures - Part -: Geeral actos Destes, self-weght, mposed loads for buldgs, Europea Commttee for Stadardsato, Brussels. CEN (5) ENV 99--, Eurocode (EC): Desg of steel structures Part.: Geeral rules ad rules for buldgs, Europea Commttee for Stadardsato, Brussels. CEN (5) ENV 998-, Eurocode 8(EC8): Desg provsos for earthquake resstace of structures, Part : Geeral rules, sesmc actos ad rules for buldgs, Europea Commttee for Stadardsato, Brussels. Della Corte G,Mazzola FM (8) Theoretcal developmets ad umercal verfcato of a dsplacemetbased desg procedure for steel braced structures. I: Proceedgs of the th world coferece o earthquake egeerg, Bejg, 7 Oct. Goggs JG, Sullva TJ (9) Dsplacemet-based sesmc desg of SDOF cocetrcally braced frames. I: Mazzola, Rcles, Sause (eds) STESSA 9. Taylor & Fracs Group, pp Khatb I.F., Mah S.A., Pster K.S., (988) Sesmc behavor of cocetrcally braced steel frames. Report o. UCB/EERC-88/. Berkeley: Earthquake Egeerg Research Ceter, Uversty of Calfora. McKea F, Feves GL, Jeremc B, Scott MH (7) Ope System for Earthquake Egeerg Smulato, Medhekar, MS, Keedy DJL. () Dsplacemet-based sesmc desg of buldgs applcato, Egeerg Structures, pp.. Medhekar, M.S., Keedy, D.J.L. () Dsplacemet-based sesmc desg of buldgs theory, Egeerg Structures, pp. -9. PEER (6] OpeSees: Ope System for Earthquake Egeerg Smulato, Pacfc Earthquake Egeerg Research Ceter, Uversty of Calfora, Berkeley, CA.

13 Prestly, M. J. N. () Myths ad Fallaces Earthquake Egeerg, Revsted The Mallet Mle Lecture, IUSS Press, Pava, Italy. Prestly M. J. N., Calv G. M., Kowalsky M.J., (7) Dsplacemet-based desg of structures, IUSS Press, Pava. Tremblay R, Tmler P, Brueau M, Flatrault A. (995) Performace of steel structures durg the Jauary 7, 99 Northrdge earthquake, Ca J Cvl Egg,8 6. Tremblay, R., Brueau M., Nakashma M., Pro H. G. L., Flatrault A. (996) Sesmc desg of steel buldgs: lessos from the 995 Hyogo-ke Nabu earthquake, Caada Joural of Cvl Egeerg (), Tremblay R. () Ielastc sesmc respose of steel bracg members, Joural of Costructoal Steel Research 58, Tremblay R, Pocet L. () Sesmc performace of cocetrcally braced steel frames multstorey buldgs wth mass rrgularty, Joural of Structural Egeerg,6-75. Urz P. Flppou F.C. ad Mah S.A. (8) Model for cyclc elastc bucklg for steel member, Joural of Structural Egeerg ASCE (), Urz, P. (5) Towards earthquake resstat desg of cocetrcally braced steel buldgs. Ph.D. thess, Uv. of Calfora, Berkeley, Berkeley, Calf. Whtmore, R. E. (95) Expermetal vestgato of stresses gusset plates, Bullet No. 6, Egeerg Expermet Stato, Uv. of Teessee, Koxvlle, May. Wjesudara, K.K, Nascmbee R, Sullva T.J () Equvalet vscous dampg for steel cocetrcally braced frame structures, Bullet of Earthquake Egeerg, vol.9 (5), Wjesudara, K.K, Bolog D, Nascmbee R, Calv G.M (9) Revew of desg parameters of cocetrcally braced frames wth RHS shape braces, Joural of Earthquake Egeerg, (S): 9-.