SIMPLIFIED METHOD TO DETERMINE STRUCTURAL PERFORMANCE POINTS

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1 13 th Worl Confrnc on Earthquak Engnrng Vancouvr,.C., Canaa August 1-6, 2004 Papr No IMPLIFIED METHOD TO DETERMINE TRUCTURAL PERFORMANCE POINT Jang Hoon Km 1 an Dong Hoon Jwa 2 UMMARY Wth th mrgnc of th stat-of-th-practc sgn approach such as splacmnt bas an prformanc-bas mthos, th prcton of structural bhavor has bcom a part of th sgn procss. In ths noton, th ssmc mans n forc an splacmnt for sgn ar to b trmn by consrng th ntracton wth th structural capacty. Howvr, th computaton of th rsultant prformanc pont as pr th stat-of-th-practc approach rqurs a consrabl amount of work wth a gr of complxty. On bhalf of sgn ngnrs n practc, thrfor, an altrnatv approach, smpl but accurat nough, s propos n ths papr. For ths th AAHTO ssmc bas solaton sgn approach has bn rvw an mof to ft th nonlnar statc analyss procur for trmnaton of th prformanc pont of structurs n a smplr way. uch an aaptaton may b possbl for th fact that a structural systm subjct to th squntal formaton of amag u to arthquak loang kps softnng to rsult n pro shftng towar th longr s. Th suprorty of th propos mtho to th stat-of-th-practc approach s that th accptabl valu of prformanc pont can b appropratly obtan wthout constructng th so-call acclraton splacmnt rspons spctrum rqur n applyng th capacty spctrum mtho. Th valty of th propos approach was vrf by comparng th prct valus to th ons consr xact n th ltratur. INTRODUCTION In orr to conomcally attan th structural safty, th volvng sgn co provsons up to at hav rqur th matrals an sctons proprly proporton for th unrstmat capacty to xc th ovrstmat man. Although th structurs sgn n such a way may scur th safty margns approprat to th nvual mmbrs, thos co provsons ar rgar as a black box, snc thy o not nclu anythng about th post-yl bhavor. Ths mans that th prcton of th ssmc bhavor of a sgn structur may b out of sgnr s work scop, whl th aquat scton mnsons can b obtan by nputtng th loang conton, matral proprty an gomtrcal conton, or by smply 1 Assoc. Prof., Dpartmnt of Archtctur, Ajou Unvrsty, outh Kora, Emal: kmjh@ajou.ac.kr 2 tructural Engnr, -Tch Consultng Group, outh Kora, Emal: ngtoo@hotmal.com

2 followng th co provsons. Thrfor, a sparat analytcal stp s rqur to valuat th structural bhavor for an xpct arthquak loang. Th so-call splacmnt-bas sgn (DD) or th subsquntly volv prformanc-bas sgn (PD) approach, ntrouc n arly 1990 s, s th sgn mtho attmptng to contan th analytcal stp to prct th structural bhavor, n whch th ssmc man s calculat from th rlatonshp btwn th forc-formaton capacty curv an th rspons spctrum. That s, th assum or ntally calculat splacmnt man trggrs frst th calculaton of th hystrtc nrgy sspaton, an thn th nrgy-bas quvalnt amng rato from whch th splacmnt man on th rspons spctrum s back-calculat, an so on. Ths ntractv an tratv calculatng procss kps gong up to th convrgnc rsultng n th maxmum splacmnt man or altrnatvly th prformanc pont. It shoul b not hr that th structural capacty an th man ar rcognz as rathr mutually ntractng physcal quantts than thos to flatly compar. Ths nw sgn concpt has bn frst mpl n AAHTO [1], UC [2] an FEMA [3] for sgn of ssmc bas solaton rubbr barngs, an subsquntly ntrouc n ATC [4] an FEMA [5] for nonlnar nlastc statc analyss of th convntonally sgn structurs. Howvr, th gulns n ATC [4] to fn th prformanc ponts sm qut complcat nough for th ngnrs n practc to fl uncomfortabl. That s, th complxty s nvtabl for th fact that th DD rqurs bulng th splacmnt rspons spctrum an th PD rqurs th capacty spctrum n form of acclraton-splacmnt-rspons spctrum (ADR). Thrfor, th purpos of th prsnt papr s to suggst a practcal approach, smpl but accurat to an approprat gr, to fn th structural prformanc pont by slghtly mofyng th AAHTO sgn mtho of ssmc bas solaton rubbr barngs [7]. ELATIC REPONE PECTRUM Th quvalnt statc arthquak forc spcf as pr th co provsons for structural sgn s n form of bas shar, that s F = Cs W (1) whr C s nots th ssmc rspons coffcnt (that s, bas shar coffcnt) an W s th wght of th structur unr consraton. Th ssmc rspons coffcnt C s s varously fn n ach country co. In K [6], A I A I C = E E s / RT R (2) In UC [2], 1.25Z I Z I C s = (3) 2 / 3 RT1 R In AAHTO [7], A A C = 1.2 s / 3 RT R (4) whr A an Z ar th st coffcnt, I E an I ar th mportanc factor, s th sol factor an R s th rspons mofcaton factor n whch whn R = 1, C s bcoms th lastc ssmc rspons coffcnt. Ths quatons show that th ssmc sgn forc n th rgon of long pro gras at a

3 2 rat of 1/ T 1/ 3 or 1/ T 2 /, mor slowly than that of th groun rspons spctrum known to b 1 / T n gnral. Ths s a lbrat tratmnt to guarant th stablty of flxbl structurs. Howvr, for ssmc valuaton, any consrvatsm contan n th sgn shoul b lmnat for th prcton of ralstc bhavor of structurs unr arthquak loang. Thrfor, th lastc rspons spctrum wth 5% amng rato n Fg. 1 rcommn n ATC [4] may b approprat for ths purpos. In th fgur, th ffctv pak acclraton (EPA) C A s trmn for 2.5C A to b th avrag maxmum rspons of a structural systm wth 5% amng rato n th acclraton govrnng (short pro) rgon. Th psuo-acclraton rspons of a structural systm n th vlocty govrnng (m to long pro) rgon s trmn by C V / T, n whch C V s th avrag maxmum rspons of th onscon-pro structural systm wth 5% amng rato. Th structural analyss usng th lastc rspons spctrum along wth th quvalnt amng rato maks th ynamc analyss smpl, an bcoms th bass of DD an PD. a 2.5CA CV/T EPA=CA Fg. 1 ATC-40 5% amp lastc rspons spctrum [4] T AAHTO DEIGN METHOD OF EIMIC AE IOLATION RUER EARING [7] Th prmary purpos of ssmc bas solaton lastomrc barngs s to shft th natural pro of a structural systm to a longr s as shown n Fg. 2(a) to accommoat th ssmc forc wthn th lastc rang. Howvr, th flxbl natur of th structural systm u to rubbr barng accompans th ncras n th splacmnt man n rturn as shown n Fg. 2(b). nc th rspons spctrum of a long-pro structural systm for analyss follows th lastc groun rspons spctrum, th ssmc rspons coffcnt n Eq. (4) can b xprss as, consrng 5% amng rato, A Cs = 2. 5 A (5) T whr T s th ffctv natural pro of a structural systm nclung rubbr barngs for ssmc solaton an ncrass wth th splacmnt. In a gnral form, T s xprss as W T = 2 π (6) Σ k g ff

4 a Incrasng Damng Incrasng Damng T (a) Acclraton pctrum T (b) Dsplacmnt pctrum Fg. 2 Rspons of ssmc solaton rubbr barng systm whr g = 9810 mm/s 2 s th gravty acclraton. Th ovrall stffnss Σ kff nots th sum of ffctv stffnss of ssmc bas solaton rubbr barngs an th ffctv stffnss of an nvual rubbr barng can b obtan from th hystrtc forc-splacmnt rlatonshp at vry loang cycl as shown n Fg. 3. That s + F F k ff = (7) whr F an F ar rspctvly th postv an ngatv maxmum latral forc an an ar + + th corrsponng splacmnts. F, F, an ar xprmntally or analytcally obtan by constructng th capacty curv. Eq. (5) n consraton of amng ffct bcoms A Cs = (8) T whr s th amng coffcnt an pnnt upon th quvalnt amng rato. Tabl 1 prsnts Forc + Fmax F y - k ff + Dsplacmnt - FF mn Fg. 3 lnar bhavoral mol of ssmc bas solaton rubbr barngs

5 th valus of amng coffcnt for varous quvalnt amng ratos n whch th ntrpolaton s ma for th valus othr than shown. In th tabl, an L not th amng coffcnts appl to th short an long pro rangs, rspctvly. Comparng th valus of amng coffcnt to th ynamc amplfcaton factors normalz by that of ζ = wth 15.9% probablty of xcnc as suggst by Nwmark an Hall [8], th closnss can b obsrv at th rang of rlatvly lowr quvalnt amng rato. Th AAHTO amng coffcnt s qut clos to th valus of L of ATC ovr all rang of th quvalnt amng rato. It s not that th amng coffcnts n th vlocty govrnng rang only ar prsnt n AAHTO ssmc solaton rubbr barng sgn n th tabl, snc th pro of th structural systm wth rubbr barngs has alray bn shft to th rang of m to long pro. Tabl 1 Comparson of amng coffcnts btwn varous approachs Damng Coffcnt Equvalnt AAHTO Nwmark an ATC [4] Damng [1] Hall [8] havor Typ A havor Typ havor Typ C Rato ζ L L L L nc th ssmc solaton rubbr barngs hav th low hystrtc amng, thr s no ffrnc btwn th vsco-lastc amng rato an th ffctv amng rato. Thrfor, th ffctv amng rato can b fn as th loss factor usng th blnar mol n Fg. 3. That s 1 TotalAra ζ = (9) 2 2 π Σ kff whr TotalAra s th sum of th ara surroun by th forc-splacmnt hystrtc loop of ssmc bas solaton rubbr barngs an s th maxmum latral splacmnt of rubbr barngs at th cycl. Th pak psuo-acclraton rspons of an lastc systm a s rlat to th pak splacmnt rspons n a = ω 2 = Cs g (10) whr ω = 2π /T s th angular frquncy of a structural systm. ubsttutng Eq. (8) nto Eq. (10), th maxmum splacmnt rspons can b obtan as 250 A T = = (11) ummarzng th prvous quatons, bas shar F rqurs th ffctv natural pro T an amng coffcnt whch subsquntly rqur th ffctv stffnss Σ kff an th maxmum latral splacmnt. Howvr, snc T an ar rqur for Σ kff an agan, th whol sgn procss of th ssmc bas solaton rubbr barngs rqurs th tratv calculaton. Th sgn procur of AAHTO ssmc solaton lastomrc barngs can b summarz as follows.

6 tp 1 Dtrmn th plan sz, thcknss an numbr of rubbr an stl plats n consraton of combn loang, an construct th blnar forc-splacmnt capacty hystrtc loop. tp 2 Dtrmn th st coffcnt A an th sol factor. tp 3 Assum th ntal latral splacmnt 1. tp 4 Calculat T, Σ kff, ζ an usng Eqs. (6), (7), (9) an Tabl 1. tp 5 Calculat usng Eq. (11). If 1 s suffcntly small, thn s th maxmum latral splacmnt an th sgn procss ns. If not, rturn to tp 4 wth th calculat n th prsnt stp. tp 6 If th maxmum latral splacmnt shoul ncras or cras, rturn to tp 1 to chang aquatly th numbr an/or th thcknss of rubbr plats an go ovr all stps aftrwars. ADAPTATION TO DETERMINATION OF PERFORMANCE POINT Th ssmc valuaton of structurs nclus th prcton of th maxmum formatons for th xpct arthquak xctaton, an whthr th structural systm can accommoat thm locally an globally. ATC [4] an FEMA [5] suggst a mthoology to fn th so-call Prformanc Pont, th maxmum formaton at th top of th structural systm, usng th rlatonshp btwn th capacty curv an man rspons spctrum, an an accptabl lmtaton of story rft as pr th prformanc objctvs. In orr to aapt th sgn mtho of th ssmc solaton rubbr barng systm to th ssmc valuaton of th convntonal structural systm n whch th nonlnar nlastc bhavor s xpct, t s ncssary to tak a look at th analogy an ffrnc btwn th two systms. Analogy an Dffrnc Th prmary analogy btwn th ssmcally solat an convntonal structural systms s that th latral splacmnt man s locally concntrat on th crtcal scton. In th bas-solat structurs, almost all latral splacmnt man s accommoat by th shar formaton of rubbr barngs. In th sam tokn, th nlastc latral splacmnt man mpos on th convntonal structural systm s accommoat by th rotatonal capacty of plastc hngs. Anothr analogy s that th ffctv stffnss crass as th latral splacmnt man ncrass as ncat n Fg. 4 whr th scant stffnss s rgar as th ffctv stffnss. It shoul b not that vry pont on th forc-splacmnt capacty nvlop has ts own partcular ffctv stffnss. Forc Forc kff Dsplacmnt kff Dsplacmnt (a) Rubbr barng (b) tructural lmnts Fg. 4 Charactrstcs of hystrtc loops

7 Th prmary ffrnc btwn th solat an th convntonal structural systms s th afforablty of r-cntrng proprty aftr th xtrnal sourc of xctaton s rmov. Th amag u to th nlastc formaton lavs th prmannt splacmnt n th convntonal structural systm, whl n th ssmcally solat structural systm th orgnal structural form can b ovrcom. Anothr ffrnc s that th hystrtc nrgy sspat by th ssmc solaton rubbr barngs s nglgbly small, compar to that by th convntonal structural systm. Evn though such ffrncs may caus th occurrnc of som rror n aaptaton of th ssmc bas solaton rubbr barng sgn to th ssmc valuaton of th convntonal structural systm, t may not b a srous problm, as long as such an rror s accptabl an th propos mtho can gvs a smpl but suffcntly accurat soluton. Calculaton of Prformanc Ponts In a way smlar to th ssmc bas solaton barng sgn, th ssmc rspons coffcnt for th convntonal structural systm s trmn by th lastc rspons spctrum as shown n Fg. 1. That s CV Cs = 2. 5C A (12) T whr T s th prmary natural pro of th systm. Consrng th ffct of quvalnt amng rato, C C = V s T (13) whr follows th AAHTO valus n Tabl 1. nc th natural pro of a structural systm bcoms longr as th structural amag ncrass an spras ovr, L s suppos to b us as suggst n ATC [4]. ut bcaus valus of L n ATC o not ffr btwn bhavor typs an ar smlar to AAHTO valus, n Tabl 1 s us nsta for smplfcaton. In orr to trmn th valu of, th quvalnt amng rato ζ shoul b calculat. Rfrrng to Fg. 5 [4], th quvalnt amng rato, th sum of th vsco-lastc amng rato ζ o an th hystrtc amng rato, can b trmn as [9] κ ED ζ = ζ o + (14) 4π E o Forc F lnar rprsntaton Capacty curv k ntal k ff F y E o y Dsplacmnt E D Fg. 5 Equvalnt amng rato

8 E D whr, th nrgy sspat by th hystrtc amng, s th ara surroun by th blnar hystrtc loop, an E = F / 2 s th stran nrgy at th maxmum splacmnt. A mult-lnar o hystrtc loop can concurrntly b convrt to a blnar hystrtc loop by nforcng to mantan th sam ara unr th curv at th maxmum splacmnt as shown n Fg. 6. Accorngly, th yl pont vars wth th maxmum splacmnt unr consraton. Th quvalnt amng mofcaton factor κ rflcts th mprfcton of hystrtc bhavor of a structural systm. Th consr blnar hystrtc mol n Fg. 5 to calculat th quvalnt amng rato as pr Eq. (14) s a prfct paralllogram. Howvr, n ralty, snc th rnforc concrt or th mtal structural systm occus a part of th paralllogram n th hystrtc loop as shown n Fg. 7, th nrgy sspaton capacty an th corrsponng quvalnt amng rato shoul b rajust. Expanng Eq. (14) n rfrnc to Fg. 5 bcoms [4] 0.637κ ( y F ) ζ = (15) F F F kntal A2 lnar Rsprsntaton Capacty Curv A1 Not: A1 = A2 y Fg. 6 Dtrmnaton of blnar bhavoral mol Fg. 7 Dffrnc btwn actual structur an alz mol

9 Thn th maxmum splacmnt at th top lvl, that s, th prformanc pont, on th blnar capacty curv can b trmn by substtutng Eq. (13) nto Eq. (10) n consraton of th rlatonshp btwn th psuo-acclraton rspons an th maxmum splacmnt rspons of th lastc systm. That s 250Cv T = (16) nc som varabls n ths quatons ar functons of thmslvs, th maxmum rspons shoul tratvly b calculat by assumng an ntal valu as n th ssmc solaton rubbr barng sgn. Th sgn procur can b summarz as follows. tp 1 Prform th nonlnar nlastc analyss of a structural systm for statcally appl quvalnt arthquak loang to construct th capacty curv (that s, forc-splacmnt nvlop). tp 2 Dtrmn C A an C V bas on th groun moton lvl rqur for th prformanc objctvs. tp 3 Assum th ntal prformanc pont p1 an construct th corrsponng blnar capacty curv. tp 4 Calculat T, Σ kff, ζ an usng Eqs. (6), (7), (15) an Tabl 1. tp 5 Calculat th maxmum splacmnt at th top of th structur usng Eq. (16). If p, 1 s suffcntly small, thn bcoms th maxmum splacmnt (prformanc pont) an go to tp 6. If not, rturn to tp 3 wth th calculat n th prsnt calculaton stp an contnu th subsqunt procur. tp 6 If th story rft an th jont formaton calculat from th maxmum splacmnt at th top of th structur cannot b accommoat by th prsnt form of th structural systm, practc th ssmc rtroft an rturn to tp 1 an go ovr all stps aftrwars. WORKED EXAMPLE FOR VERIFICATION Th valty of th propos smplf mtho of analyss s vrf by applyng t to an xampl n ATC-40 Appnx A [10]. Th xampl structur s a m-rs rnforc concrt bulng locat n Escano Vllag of tanfor n Calforna, U..A. Th plan an lvaton of th bulng ar shown n Fg. 8 an th tycal story hght s 2.769m, ovrall bulng hght from th groun lvl to th roof floor s 22.15m, an th wght s 52700kN. Th sol factor s D (rock) an th st coffcnt s Z = 0.4, an th corrsponng ssmc coffcnt s C A = an C V = In accoranc wth th xampl, th ssmc rtroft was practc, snc th orgnal bulng was consr not uctl. Th analyss was prform n th longtunal an th transvrs rctons. Usng th gvn capacty curvs n both rctons n th xampl, th calculaton procur to trmn th prformanc ponts as pr th propos mtho s summarz n Fg. 9. Th maxmum splacmntbas valu for quvalnt amng mofcaton factor κ as pr ATC [4] s us for consstncy n comparson. Howvr, a mor approprat valu coul b obtan by consrng th nrgy absorpton ffcncy. Th prformanc ponts calculat by th varous mthos ar compar n Tabl 2. Th valus obtan by th capacty spctrum mtho an th nonlnar nlastc tm-hstory analyss ar gvn n th xampl. Th propos mtho prcts th prformanc pont as 304mm an 357mm n th longtunal an transvrs rctons, rspctvly. Although ths valus qut ffr from th valus obtan by th capacty spctrum mtho [4,10], thy ar closr to th valus obtan by th nonlnar nlastc tmhstory analyss mtho. Ths ncats th capablty of th propos smpl mtho to prct mor accurat valus compar wth th capacty spctrum mtho whch rqurs mor complcat procur.

10 Howvr, t s rqur that mor structurs havng vrs capacty curvs wth th complcat natur b nvstgat for th full vrfcaton. Fg. 8 Plan an lvaton of th xampl bulng [10]

11 Longtunal Drcton Transvrs Drcton tp 2 C A = 0. 47, C V = C A = 0. 47, C V = tp 3 p1 =250mm assum p1 =300mm assum tp 4 Σ kff =35kN/mm, T =2.46s Σ k ff =25.3kN/mm, T =2.9s κ =0.54, ζ =0.273, κ =0.58, ζ =0.266, tp 5 (1)=284mm (1)=338mm N.G. p1 (1) p1 (1) as har (kn) F kntal kff Frst Tral =250mm 2000 y Roof Dsplacmnt (mm) as har (kn) F y kntal kff Frst Tral =300mm Roof Dsplacmnt (mm) tp 3 tp 4 tp 5 N.G. =284mm rassum =338mm rassum Σ kff =31.4kN/mm, T =2.60s Σ k ff =23.2kN/mm, T =3.02s κ =0.53, ζ =0.272, κ =0.59, ζ =0.268, (2)=284mm (2)=338mm (1) (2) (1) (2) as har (kn) F kntal kff con Tral =284mm 2000 y Roof Dsplacmnt (mm) as har (kn) F kntal y kff con Tral =338mm Roof Dsplacmnt (mm) tp 3 tp 4 tp 5 O.K. =300mm rassum =351mm rassum Σ kff =30.0kN/mm, T =2.66s Σ k ff =22.5kN/mm, T =3.07s κ =0.55, ζ =0.281, κ =0.59, ζ =0.267, (3)=304mm (3)=357mm (2) (3) (2) (3) as har (kn) F kntal kff Thr Tral =300mm 2000 y Roof Dsplacmnt (mm) as har (kn) kntal 8000 F 6000 Thr Tral =351mm 4000 kff 2000 y Roof Dsplacmnt (mm) Fg. 9 Analyss procur as pr th propos mtho

12 Tabl 2 Comparson of Prformanc Ponts Analyss Mtho Longtunal Drcton (mm) Normalz by TH Transvrs Drcton (mm) Normalz by TH Capacty pctrum Propos Mtho Nonlnar Tm-Hstory (TH) CONCLUION (1) Th AAHTO ssmc bas solaton rubbr barng sgn mtho was slghtly mof to aapt to th valuaton of th convntonal structural systm for trmnaton of th prformanc pont. Th propos mtho was vrf to b much smplr but accurat through an xampl, compar wth th ATC-40 capacty spctrum mtho n whch ADR spctrum shoul b construct. Howvr, t s rcognz that furthr vrfcaton procur s rqur for applcaton to th structural systms havng mor complcat natur n capacty curvs. (2) Th quvalnt amng rato shoul b rajust bas on th nrgy absorpton ffcncy to appropratly tak th ffct of cyclc loang an uraton of arthquaks nto account. ACKNOWLEDGMENT Ths stuy was grant by 2002 POCO rsarch fun. That partal fnancal support s gratfully acknowlg. REFERENCE 1. AAHTO. Gu pcfcatons for smc Isolaton Dsgn. Amrcan Assocaton of tat Hghway an Transportaton Offcals, Washngton, D.C., U..A., UC. Unform ulng Co. Intrnatonal Confrnc of ulng Offcals, Calforna, U..A., NEHRP. Rcommn Provsons for smc Rgulatons for Nw ulngs: 1994 Eton. FEMA 222A, Fral Emrgncy Managmnt Agncy, Washngton, D.C., U..A. 4. ATC-40. smc valuaton an rtroft of concrt bulngs: Volum 1. Appl Tchnology Councl, NEHRP. Gulns for th ssmc rhabltaton of bulngs. FEMA 273, Fral Emrgncy Managmnt Agncy, Washngton, D.C., U..A., Archtctural Insttut of Kora. tanar Dsgn Loas for ulngs AAHTO. tanar pcfcatons for Hghway rgs: Dvson I-A smc Dsgn. 15th Eton, Amrcan Assocaton of tat Hghway an Transportaton Offcals, Washngton, D.C., U..A., Nwmark NM, Hall WJ. Earthquak pctra an Dsgn. Earthquak Engnrng Rsarch Insttut, rkly, Calf., Chopra AK. Dynamcs of tructurs Thory an Applcatons to Earthquak Engnrng. Prntc Hall Intrnatonal, Inc., ATC-40. smc valuaton an rtroft of concrt bulngs: Volum 2 Appncs. Appl Tchnology Councl, 1996.