Damage Indices using Energy Criterion for Seismic Evaluation of Steel Frame Buildings

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1 amag Indcs usng Enrgy Crtron for Ssmc Evaluaton of Stl Fram Buldngs Prasad Prahlad Assstant Profssor, partmnt of Cvl Engnrng, N.I.T. Jamshdpur, Jharkhand, Inda Shrkhand Mansh & Agarwal Pankaj Assocat Profssor s, partmnt of Earthquak Engnrng, IIT Roork, Inda SMMARY Th contnt of th papr s basd on th prmss that ndcs n trms of damags ar ffctv mthods of valuaton of buldng structurs undr arthquak loadngs. Enrgy basd crtron has bn found ffctv tools n th prsnt stat of art bcaus nrgy bng th capacty of a structur to rsst any ssmc dmand n mor stabl mannr. Th man focus of ths papr rmans wth th formulaton of damag ndcs usng th nrgy basd valuaton usng som rfrnc stl buldng structurs undr varyng arthquak loadngs. amag mpls th rducton of rsstng capacty, whch may b loss of strngth or loss of nrgy absorpton du to cyclc loadng arsng durng arthquaks. Th analytcal xprssons drvd n trms of nrgy capacty hav bn valdatd through th data acqurd as rspons analyss n th prsnt work. Th concluson of th fndngs may b furthr nvstgatd through changng th structurs, typs of matrals and th loadng combnatons tc. Kywords: Enrgy basd ssmc valuaton, Stl fram buldngs, amag ndcs tc 1. INTROCTION Ssmc valuaton of stl buldng structurs has undrgon crtcal rapprasal n th rcnt past [SEAOC, 1995; ATC-4, 1996; FEMA-73, 1997; Gong, 3; Akyama, 1985]. Varous numrcal approachs hav bn rcntly mployd to trac out th smplcty for th valuaton of th rspons of stl buldng structurs undr ssmc loadng. Howvr, th approach of th nrgy basd balanc crtron as ffctv mthodology has bn addrssd [Park & Ang, 1985; Housnr, 1956; Bozorgna & Brtro, 4] n th rcnt past. In ordr to quantfy th damag ndcs rfrrd n ths papr, stl buldng frams hav bn modld for lnar and nonlnar analyss procdurs wth th gudlns of modlng of softwar s: ETABS, SAP-, RAM PERFORM 3. Wth th optmum condtons of th buldng prformanc crtrons, th loadng has bn assgnd undr th ssmc actons. Programs wr run undr batch mods. Th databas of th rspons paramtrs hav bn ncorporatd for th valdaton of th damags ndcs dvlopd n th contnt of ths papr. Th objctvs of th papr rmand wth th dvlopmnt of troubl-fr approach n trms of normalzaton of nrgy wth th stran nrgy for bttr ssmc valuaton tools. u to fast nonlnar analyss procdurs, local to global bhavor of structural and non structural mmbrs hav bn found a bttr tool wth promsng opportunty and n ths drcton, th us of fast spd and hgh mmory computr systms hav mad possbl to thnk and go ahad for furthr nvstgatons.. PROBLEM FORMLATION Chang of ntrnal nrgy capacty of a structur s rsponsbl for th dsturbanc rsultng nto partal or total damag; hnc t s mportant for dntfcaton and quantfcaton of varous typs of nrgy n ordr to pacfy smpl mthodology for ssmc rsstant dsgn. In ordr to achv such objctvs as th contnt of ths papr, th followng approach has bn mad

2 .1. strbuton of Input Ssmc Enrgy Lft sd of th nrgy balanc quaton (.1) s nput ssmc nrgy and rght sd rprsnt th nrgy dstrbutd ag nrgy componnts. A componnt s capabl to absorb a sgnfcant amount of nrgy as lastc stran nrgy and dsspats nrgy through dampng n th lastc rgon. Rst of th nrgy s dsspatd through yldng n th nlastc rgon. Thrfor, dstrbuton of nrgy ag ts componnts s th frst task for furthr procdng to dsgn. E = E S + E K + E + E hξ (.1) vdng quaton (.1) by th nput ssmc nrgy to both sds of th quaton 1 = E S /E + E K /E +E /E + E hξ /E (.) Equaton (.) s n th normalzd form, whr th normalzng paramtr s nput ssmc nrgy tslf. Expandng th varous nrgy paramtrs on th rght sd of th quaton (.), th followng smpl normalzd valus, w gt..1.1 Normalzd Stran Enrgy = ku 1 ms pv u = = s pv u s d (.3) Whr u s th rlatv dsplacmnt and S d s th spctral dsplacmnt...1. Normalzd Kntc Enrgy = u s pv (.4) Whr th numrator s rlatv vlocty and th dnomnator s th psudo vlocty Normalzd ampng Enrgy = u (.5) n s pv Equatons (.5) and (.4) rval that normalzd dampng and kntc nrgy rato s n.1.4 Normalzd Hystrtc Enrgy = 1 u s d (.6) Ths mpls that normalzd stran nrgy s rlatd wth th normalzd hystrtc nrgy by a factor qual to cumulatv ductlty mnus 1.As dscussd abov dstrbuton of nput ssmc nrgy ag ts componnts s rqurd for makng th dsgn dcson. Valdaton of normalzd nrgy componnts from quatons (.3) to (.6) rqurs tm hstory analyss analyss and spctral vlocty of th concrnd ground motons. Rspons paramtrs n trms of nrgy aftr nonlnar analyss of rprsntatv stl buldng frams undr varyng arthquak ground motons wll b th contnts of th nxt chaptr: modlng and analyss. Som mor nvstgatons basd on th structur and ground moton ntracton n trms of nrgy nput and mchancal charactrstcs has bn formulatd as followng.

3 .. Rlaton btwn Hystrtc Enrgy (E ) and Stran Enrgy (E s ) Forc F y Δ y Δ max Fgur.1. Hystrtc loop for lasto-plastc systm Hystrtc nrgy through yldng s th outcom of th svr ground moton, whn a structur yld and taks th advantag of ductlty. Elasto-plastc bhavor of stl frams has bn consdrd snc such a bhavor s closly rlatd wth th stl fram actual bhavor. E h = 4 k y 1 = 8E s 1 (.7) urng succssv loop undr varyng arthquak ground motons, th total nrgy dsspatd for dsplacmnt ductlty =, 3, 4, 5, 6, 7 tc s followng: n E 8E s +16E s +4E s +3E s +4E s (.8) E h 1 Th abov quaton (.8) forms th arthmtc progrsson wth th rsultant valus. n 1.., E = 8n E s (.9) Whr E s th total nrgy dsspatd, n s th total numbr of loops and E s s th stran nrgy. For damag valuaton, th quaton (.9) s an mportant quaton. Th valus of th succssv hystrtc nrgy ar normalzd by th total hystrtc nrgy, whch has bn rcognzd an ffctv damag ndx. A damag ndx through normalzng th succssv hystrtc nrgy wth th sum of hystrtc nrgy s th proposd damag ndx as th contnt of prsnt study. Th normalzd valus provd a trnd of contnuous damag spctrum.3. Smplfcaton of Park and Ang amag Modl Park and Ang damag modl s smplfd for ovrall damag ndx n ordr to corrlat th local damag ndx to th global prformanc as on of th drvatv of th prsnt study. amag modls accountng for th combnaton of maxmum dformaton and dsspatd nrgy hav bn ntroducd [Park & Ang, 1995]. Th modl proposd by Park and Ang modl uss damag ndx dfns by max u de R y H y u = ultmat dformaton undr otonc loadng y = yld dformaton, de H =cumulatv hystrtc nrgy β = non-ngatv paramtr =.5 for stl structurs (.1)

4 Forc dyn = max mum dformaton n dynamc loadng max mum dformaton n otonc loadng A structur undr quvalnt loadng wll yld mor than th actual dynamc loadng dyn Fgur. Forc vs. dformaton rlaton for otonc and dynamc loadng [Akyama, 1985] A structur undr quvalnt loadng wll yld mor than th actual dynamc loadng Extnson of quaton (.1) for a gnralzd form, whr th damag at componnt lvls may drctly b ncorporatd for th ovrall damag ndx. Rarrangng th qn. (.1), w gt th followng xprsson dyn R y dyn de H R y, sng and Mg Mg de (.11) H Puttng dyn =.6 as suggstd by rsarchrs Park and Ang [Park & Ang, 1995] th maxmum damag undr undrctonal dynamc loadng normalzd wth th max dformaton undr otonc loadng. ymg(5 3) deh EH 5 E H ymg y (.1) (.13) 1E E 1H 1H E H E H Ovrall amag Indx Th gnralzd xprsson for ovrall damag ndx ( Tn ) Tn = (.14) Th xprsson (.14) of damag ndx s th xtnson of Park and Ang damag ndx for ovrall damag, whch ncluds th damag ndx at th lmnt lvl to th global lvl. s th damag ndx at th lmnt lvl MOELING AN ANALYSIS Stl buldng frams hav bn modld for lnar and nonlnar analyss n th nvronmnt of RAM PERFORM 3. Buldng frams hav bn subjctd latral loads du to ssmc acton n addton to ts gravty loadng. Tm hstory analyss wr also prformd on th rspctv buldng frams.

5 4. RESLT AN ISCSSIONS Rsults hav bn tabulatd aftr analyss of th rspctv buldng frams and dscussons wr mad n stp ws as follows: 4.1. Normalzd Hystrtc Enrgy Rsult: Th valu of hystrtc nrgy and stran nrgy ar known from th rspons hstory for nrgy for th buldng frams n th study for a st of arthquaks. Tabls 4. to 4.3 contan th hystrtc nrgy normalzd wth th stran nrgy for th buldng framworks. Tabl 4.1 gvs dtals of normalzd hystrtc nrgy for th varyng ground motons wth hghr pak ground acclratons Rsult scussons Th major task of ssmc dsgn s to ncorporat stffnss, strngth and ductlty dgradaton undr th svr arthquak ground motons n analytcal xprssons. Hystrtc nrgy n th normalzd form conssts of ths ntr factor nto a sngl quaton. As found th data for normalzd hystrtc nrgy n th tabl. 4.1, th valu s narly constant for varyng ground motons. Th nxt attrbut of ssmc dsgn s to consdr th stabl paramtr or such attrbut whch can b stabl whn any drastc chang taks plac. Normalzd hystrtc nrgy whr th bas s stran nrgy and th numrator s hystrtc nrgy s usful for such dsgn dvlopmnt. On of th problm formulatons s normalzd hystrtc nrgy n trms of hystrtc nrgy and normalzd yld strngth as a contnt of ths program. Tabls 4. to 4.3 hav bn usd for th ntrprtaton of normalzd hystrtc nrgy for thos frams whch hav dsspatd sgnfcant amount of nput ssmc nrgy through yldng of bams/columns at varous floors rval th pattrn of nrgy consumd by th structurs as stran, kntc and hystrtc nrgy. Hystrtc nrgy as th major sourc of consumpton of nput ssmc nrgy undr svrs arthquak ground motons provds stabl and promsng rspons whl a structur has no othr altrnatvs for fal saf dsgn. Tabl 4.1. Enrgy dstrbutons (%) for thr story 3 buldng framwork. Input Stran Kntc Hystrtc % E S % E K % E h Earthquak Enrgy Enrgy Enrgy Enrgy ground motons (knm) (knm) (knm) (knm) Northrdg, (.5165g) Northrdg(x.5165g) Northrdg(3x.5165g) Northrdg(4x.5165g) Tabl 4.. Normalzd hystrtc nrgy for thr story buldng fram. Sl. No Elastc stran nrgy Hystrtc Enrgy Rato of hystrtc and lastc stran nrgy Earthquak ground motons Northrdg E-W,(.5165g) Northrdg E-W,(x.5165g) Northrdg N-S,(x.4158g) El Cntro E-W,(.148g) El Cntro E-W,(x.148g)

6 Tabl 4.3. Normalzd hystrtc nrgy for thr story 3 buldng fram. Sl. Elastc stran Hystrtc Rato of hystrtc Earthquak ground motons No nrgy Enrgy nrgy and lastc stran nrgy Northrdg E-W, (.5165g) Northrdg E-W,(x.5165g) Northrdg N-S,(x.4158g) El Cntro E-W,(.148g) El Cntro E-W,(x.148g) 4.. Hystrtc Enrgy (E ) and Stran Enrgy (E s ) Equaton (.9) rlats th total hystrc (E ) and stran nrgy (E S ).., E = 8 n 1 n E s, Whr E s th total nrgy dsspatd, n s th total numbr of loops and E s s th lastc stran nrgy. E rprsnts damag, whch has th rlaton wth lastc stran nrgy through th numbr of loops, whn a componnt ylds and dsspats nrgy durng rvrsal of strsss arsng du to svr arthquak loadngs. For sngl hystrtc loop, hystrtc nrgy s rlatd wth th stran nrgy through cumulatv ductlty. Snc dsplacmnt approach for prformanc valuaton s poorly ratd bcaus th cumulatv ductlty s not asly accssbl. Howvr, for th known valus of succssv loop, th hystrtc nrgy s rlatd wth th cumulatv ductlty and can b stmatd through th smpl rlaton. Furthr, th sam xprsson has bn usd for fndng total nrgy dsspatd (H ) durng rvrsal of strsss du to varyng arthquak ground motons. Tabl 4.4. Hystrtc nrgy for th succssv loop Sl. No. μ E h = Es E s E s 3E s 4E s 5E s 6E s H = E h E s 3E s 6E s 1E s 15E s 1E s Tabl 4.5. amag ndx Sl. μ amag Indx 1 n E 1 (I)= H h 1/1 =.67 Immdat 3/1. 6/1.4 1/ /1.71 Lf 1/1 1. Collaps Occupancy amag Control Safty Tabl 4.6. Hystrtc nrgy for lasto-plasto loop for succssv dsplacmnt ductlty Sl. splacmnt (μ-1) 4(μ-1)/(μ) E h =..Eso No. uctlty (μ) E so =μ x E s Rmarks 1 1 Algbrac sum of th hystrtc 1 4 nrgy for succssv cycls may b 3 3 8/3 8 conomcally usd for prdcton of nrgy to b dsspatd durng contnuous spctrum of damags / /6

7 amag rato amag nrgy rato Wth th known valu of stran nrgy usd for Opratonal occupancy, th rlaton of varous prformanc objctvs on th contnuous spctrum of damag spctrum to b prdctd. Such a knd of rlaton n btwn th hystrtc nrgy and th stran nrgy s unqu n ts charactrstcs and can b usd for th smplst formulaton of prformanc ndx. Tabl 4.7. Thortcal and xprmntal damag rato for thr story 3 fram: Northrdg Earthquak (.6g) Sl splacmnt Cumulatv Thortcal damag Exprmntal damag Tabl 4.8. Thortcal and xprmntal damag rato for thr story 3 fram: El Cntro Earthquak (.6g) Sl.No. splacmnt Cumulatv Thortcal damag Exprmntal 1 uctlty 1 uctlty 1 rato rato damag 1, 1,8,6,4, Cumulatv ductlty Ex damag Th damag ndx Fgur 4.1. Thortcal and xprmntal damag rato: Cumulatv ductlty 1,5 1,5 1 Cumulatv ductlty Thortcal damag rato Fgur 4.. Thortcal and xprmntal damag rato vs. Cumulatv ductlty

8 Momnt (knm) Momnt (knm) Acclraton (g) Tm (sc) Fgur 4.3. Acclrogram of Northrdg E-W (.5165g) wth scal factor ,6 -,4 -,, Rotaton (rad),,4 Fgur 4.4. Tm hstory of bam two on ground floor of nn story fram Tm (sc) Fgur 4.5. Tm hstory of bam on on ground floor of nn story fram

9 Momnt (knm) Momnt (knm) Momnt (knm) ,6 -,4 -,, Rotaton (rad),,4 Fgur 4.6. Hystrtc loop of bam fourth on ground floor of nn story fram Tm (sc) Fgur 4.7. Tm hstory of ffth bam on ground floor of nn story fram ,4 -,,,,4 Rotaton (rad) Fgur 4.8. Hystrtc loop of ffth bam on ground floor of nn story fram

10 5. CONCLSION uctlty s assocatd wth damag. In ths rgard, cumulatv ductlty corrspondng to th dsplacmnt ductlty as stmatd n ths program s rlvant to th damag dntfcaton and quantfcaton. Thortcal damag s th rato of cumulatv ductlty corrspondng to th dsplacmnt ductlty dvdd by th quvalnt largst avalabl cumulatv ductlty usng th analytcal rsult. Exprmntal damag rato s rato of hystrc nrgy corrspondng to th dsplacmnt ductlty to th hghst hystrtc nrgy capacty. Both valus hav bn valdatd usng thr story 3 & stl buldng frams undr varyng arthquak ground motons. Enrgy basd valuaton for damag valuaton has bn found to b a stabl approach for analyss undr arthquak ground motons. Th papr dscusss th charactrstc of damags undr varyng arthquak ground motons as thy hav bn addrssd wth smplcty onc formulatd and dsgnatd n trms of damag ndcs usng th nrgy balanc crtron. Th scop of th contnt of th papr rman focusd on th furthr possblty of smplcty through mor and mor nvstgatons usng varous typs of constructon matrals, typs of structurs and th most mportant s th natur of th arthquak ground motons. ACKNOWLEGEMENT Th author of th papr s thankful to N.I.T. Jamshdpur for provdng m th fnancal and moral support for carryng out tm hstory analyss and sttng out othr rlvant fndngs usng th computatonal faclty of th lab. n th partmnt of Cvl Engnrng. Fnancal support by CPA schm of th nsttut for attndng ths confrnc s also acknowldgd REFERENCES Akyama H. (1985), Earthquak-Rsstant Lmt Stat sgn for Buldngs, nvrsty of Tokyo Prss, Tokyo, Japan. ATC-4 (1996), Ssmc Evaluaton and Rtroft of Concrt Buldng. Volum 1, Calforna,.S.A. Blum J. R. (196), A rsrv nrgy tchnqu for th arthquak dsgn and ratng of structurs n th nlastc rang, Procdng, World Confrnc on Earthquak Engnrng, Tokyo, Vol. II. Bozorgna Y. and Brtro V.V. (4), Earthquak Engnrng from Engnrng Ssmology to Prformanc- Basd Engnrng, CRC Prss, Washngton C,.S.A. FEMA-73 (1997), Fdral Emrgncy Managmnt Agncy, NEHRP Gudln for th Ssmc Rhabltaton of Buldngs, Buldng Ssmc Safty Councl, Washngton C,.S.A. Gong Y. (3), Prformanc-basd dsgn of stl buldng framworks undr ssmc loadng. Ph.. Thss. partmnt of Cvl and Envronmntal Engnrng, nvrsty of Watrloo, Octaro, Canada. Housnr G. W. (1956), Lmt dsgn of structurs to rsst arthquaks, 1 World Confrnc on Earthquak Engnrng, Brkly, Calforna,.S.A., pp. 5-1 to Park Y. J. and Ang A.H. (1985), Mchanstc ssmc damag modl for rnforcd concrt, Journal of Structural Engnrng, ASCE; 111: Prahlad P., Shrkhand M. and Agarwal P. (7), Prformanc-basd ssmc dsgn usng normalzd hystrtc nrgy, 7 SEAOC CONVENTION, Sptmbr 6-9 Squaw Crk, Calforna, SA, 7; pp RAM Prform 3, Vrson 4 (6), Nonlnar Analyss and Prformanc Assssmnt for 3 Structurs, Brkly, Computr & Structurs Inc., CA,.S.A., SEAOC Vson (1995), Prformanc-Basd Ssmc Engnrng of Buldngs, Vols. I and II: Concptual Framwork, Structural Engnrs Assocaton of Calforna, Sacramnto CA,.S.A. Surahman A. (6), Earthquak rsstant structural dsgn through nrgy dmand and capacty, Earthquak Engnrng and Structural ynamcs; 36: st