DESIGN SPECTRA REDUCTION COEFFICIENTS FOR SYSTEMS WITH SEISMIC ENERGY DISSIPATING DEVICES LOCATED ON FIRM GROUND

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1 Th 14 th Worl Confrn on Earthquak Enginring DESIGN SPECTRA REDUCTION COEFFICIENTS FOR SYSTEMS WITH SEISMIC ENERGY DISSIPATING DEVICES LOCATED ON FIRM GROUND S. E. Ruiz 1, J. P. H. Toxqui 2 an J. L. Rivra 3 1 Profssor, Instituto Ingniría, Univrsia Naional Autónoma Méxio, Méxio City 2 Grauat Stunt, Instituto Ingniría, Univrsia Naional Autónoma Méxio, Méxio City 3 Ph.D., formrly at Instituto Ingniría, Univrsia Naional Autónoma Méxio, Méxio City sruizg@iingn.unam.mx, jhialgot@iingn.unam.mx ABSTRACT: A probabilisti man sismi analysis is prform to alulat amping offiints that tak into aount th fft of nrgy issipation on th sign sptral orinats. Two typs of EDDs ar stui: a) linar visous vis, an b) yiling amping systms. Th systms analyz ar loat on firm groun. KEYWORDS: Damping fators, rution offiints, sptral orinats, nrgy issipating vis 1. INTRODUCTION Svral simplifi sign approahs hav bn propos for th sismi sign of struturs with nrgy issipating vis (Collins t al 1995, Ramirz t al 2000, Hanson an Soong 2001, FEMA-450, Whittakr t al 2003 Thos approahs ar ommonly bas on th rution of th alration sign sptrum by mans of amping offiints that tak into aount th fft prou by th nrgy issipating vis (EDDs Th rution fators hav bn obtain, in gnral, from trministi non-linar rspons-history analysis of singl-gr-of-from (SDOF) systms with EDDs, subjt to ror or simulat groun motions. Th iffrn btwn th prsnt stuy an thos foun in th litratur is that hr w tak into aount th fft of all th possibl groun motion intnsitis xpt at th sit, by mans of th orrsponing sismi hazar urvs (Cornll 1968, Estva OBJECTIVE Th objtiv of this stuy is to prsnt a prour for alulating amping fators n to ru th sign psuo-alration sptral orinats, u to th prsn of nrgy issipating vis (EDDs) on th strutural systm. In orr to rah this objtiv a probabilisti man sismi analysis (PDSA) is prform to singl-gr-of-from (SDOF) systms with EDDS, loat on firm groun. Two typs of EDDs ar onsir: a) linar visous systms an b) yiling amping vis. 3. METHODOLOGY First, th uniform annual failur rat (UAFR) sptra of th systm without EDDs (onvntional systm) an, altrnativly, with EDDs (ombin systm) ar alulat using th algorithm prsnt in th nxt stion. Bas on this information, th ratio btwn th sptrum for th onvntional systm an that orrsponing to a systm

2 Th 14 th Worl Confrn on Earthquak Enginring with linar visous EDDs is obtain. Thos ratios ( Q ) ar th rution offiints of th sptrum assoiat with th onvntional strutur. Th visous EDDs onsir in this stuy ar suppos to hav amping offiints qual to 5, 10, 15, 20 an 25% of th ritial. Nxt, w obtain th UAFR sptra for iffrnt valus of SDOF systms with hystrti amprs. Th mhanial haratristis of th EDDs ar givn by mans of th paramtrs α an γ, fin as follows: α = K / K an γ = F y / Fy, whr K is th stiffnss of th, K is th stiffnss of th main SDOF systm, Fy is th yil for of th EDDs, an illustrat in Figur 1. Fy is th yil for of th main SDOF systm. Ths paramtrs ar Figur 1. Main strutural systm with hystrti EDDs Nxt, w plott on th sam graph th UAFR sptra orrsponing to systms with hystrti amprs an th UAFR sptra assoiat with systms with visous amprs. In this mannr w foun th intrstion points btwn both sptra an, bas on th oinins, w stablish quivalns btwn th two typs of ombin systms. Partiularly, w foun th visous amping valu of a onvntional systm with an annual probability of failur qual to that orrsponing to a systm having hystrti EDDs with α an γ paramtrs. 4. UNIFORM ANNUAL FAILURE RATE SPECTRA In orr to obtain th amping fators it is first nssary to onstrut th uniform annual failur rat (UAFR) sptra orrsponing to th SDOF systms with EDDs. In th following w rprou th algorithm propos by th authors (Rivra an Ruiz 2007) for systms with hystrti vis. 1. As a first stp, valus of th following paramtrs orrsponing to th ombin systm ar propos: sismi offiint (C y ), nominal strutural vibration prio T an mass M, as wll as valus of th ratios α = K / K an γ = F y / Fy Nxt, th nominal valu of th latral stiffnss of th ombin systm is alulat ( K T = 4π M / T ). Th nominal stiffnss valus ( K, K ) assoiat rsptivly with th onvntional an with th issipating systms ar obtain ( K = K T /( 1+ α) an K = αk 3. Th yil isplamnt valu of th ombin systm ( yt ) is alulat: yt = CyW /( K + K

3 Th 14 th Worl Confrn on Earthquak Enginring 4. Using th rlations mntion abov, th yil isplamnt valus of th onvntional an of th issipating systms ar alulat ( y = C yw /( K + γk ) an y = γ y 5. Th valus of th stiffnss an of th yil isplamnts of th onvntional systm ( K an y, rsptivly) ar us to alulat th paramtrs Γ 4 an Γ 5 (that orrspon to Babr an Wn 1981 mol In th as of a onvntional rinfor onrt strutur, ths paramtrs ar givn by Γ6 1 K Γ 5 = 2Γ 4 =, whr F υ F y = K y. y 6. In a similar way, th valus of th stiffnss an of th yil isplamnts orrsponing to th issipating systm ( K an, rsptivly) ar us to alulat th valus of th paramtrs Γ an Γ. For stl y Γ6 1 K ma issipating lmnts: Γ5 = Γ 4 =. 2υ F y 7. Eah ombin SDOF systm is subjt to a iffrnt alrogram. Hr w gnrat artifiial groun motions. Eah of ths is sal so that th sptral alration assoiat with th funamntal prio of th systm unr stuy has th sam rturn intrval ( T R ) (Shom an Cornll 1999 Th ratio btwn th sptral alration valu an th invrs of th rturn intrval is givn by th sit sismi hazar urv, whih is assum to b known. 8. Th pak systm isplamnt is obtain stp by stp in tim. Thn, th pak strutural utility man ( μ i ) orrsponing to th i-th simulat ror is alulat. 9. A nominal valu of th utility apaity of th ombin systm is propos ( μ a 10. Th strutural failur of th SDOF systm ours whn th utility man is gratr than th availabl utility (apaity); that is, whn μ i / μ a = Q i 1. Th annual strutural failur rat is valuat by mans of (Estva an Ruiz 1989): ν ν F = P( Q 1Sa ) Sa (1) S a whr ν / y it is th absolut valu of th rivativ of th sit sismi hazar urv (whih is assum to b known), an P( Q 1 y) is th onitional probability that th strutural failur ours, givn a sismi intnsity S a. 11. Th intgral is valuat numrially for iffrnt valus of C y, α, γ an μ a. With th rsults, th man hazar urvs assoiat with SDOF ombin systms with iffrnt vibration prios ar onstrut. In this stuy th strutural man is takn as th lasti for offiint ( S a / g ), so th man hazar urv is a S a / g - vrsus -ν F graph, whr g = gravity. 12. Th UAFR sptra ar rawn on th basis of th man hazar urvs assoiat with iffrnt strutural vibration prios. Th sam algorithm was appli to systms with linar visous vis, xpt that in this as th algorithm boms simplr than that srib abov. 4 5

4 Th 14 th Worl Confrn on Earthquak Enginring 5. GROUND MOTION AND SEISMIC HAZARD CURVES Th SDOF ombin systms analyz ar loat on firm soil. On hunr simulat groun motions wr us as xitations (s stp 7 of th abov algorithm Th simulations wr bas on th ror obtain in Filo Caballo station uring th Sptmbr 19, 1985 arthquak. Th ror is shown in Figur 2a, an its fitt sptral nsity, S (ω), is prsnt in Figur 2b. Th fftiv uration was takn qual to 25s. Figur 2a Bas ror Figur 2b Ajust sptral nsity of th bas ror Th orrsponing sit sismi hazar urvs, for iffrnt strutural prios, ar shown in Figur 3. Figur 3. Sismi hazar urvs orrsponing to th SCT sit

5 Th 14 th Worl Confrn on Earthquak Enginring 6. UAFR SPECTRA FOR STRUCTURES WITH LINEAR VISCOUS DEVICES Th UAFR sptrum was alulat (with th algorithm mntion abov) for iffrnt valus of linar visous amping a to th main systm. In this stuy, a man failur strutural rat qual to was us. Th UAFR ' sptra orrsponing to six iffrnt ratios of ritial amping ( ς = 5, 10, 15, 20, 25 an 30%) ar shown in Figur 4a. From ths sptra, th ratio btwn ah of thm an that orrsponing to ς = 5% was obtain. That ratio ( Q ), prsnt in figur 4b, is th rution offiint of th strngth sptrum that taks into aount th prsn of th visous amprs. Figur 4b orrspon to valus of Q for iffrnt strutural prios an for ' rlations of ritial amping ratios qual to ς ς = 5/30, 5/25, 5/20, 5/15 an 5%/10%. / Figur 4a. UAFR sptra for iffrnt prntags of visous amping Figur 4b. Rution fators. Visous amping Bas on Figur 4b it is possibl to stablish ruls for th onstrution of sign sptra that onsir th fft of xtra amping a to th main strutural systm. For xampl, th authors hav propos to th thnial ommitt in harg of formulating th Sismi Dsign Comision Fral Eltriia (CFE) Manual (now unr rvision) to inorporat th following amping fator xprssion: β ( T ) 1 Q 0.05 ς = = ' λ Whr β is th amping fator that multiplis th sign alration sptrum for 0.05 amping, in orr to tak into aount th prsn of th xtra amping of th strutur is th strutural prio of intrst T ' ς is th fration of ritial amping of th strutur plus EDDs

6 Th 14 th Worl Confrn on Earthquak Enginring 0.45 ; if T < T (2a) λ = T 0.45 T 0.6 ; if T T (2b) T is th prio whr th form of th sptrum hangs 7. UAFR SPECTRA FOR STRUCTURES WITH HYSTERETIC DEVICES Following th algorithm list in stion 4, w onstrut a numbr of UAFR sptra for SDOF systms with hystrti vis. Thos sptra orrspon to SDOF systms having EDDs with iffrnt valus of th paramtrs α an γ (thos wr fin in stion 3, figur 1 Eah of th UAFR sptra of SDOF with hystrti issipating vis was plott on th sam Figur 4a (whih orrspon to sptra for systms with visous amprs) in orr to fin oinin points btwn thir orinats an, in this way, to fin th quivalnt visous amping for ah as. In th following w will try to xplain this prour by mans of an xampl. Figur 5 shows (with isontinuous lins) th sptra that appar in Figur 4a (thy orrspon to SDOF with visous amprs) as wll as th sptrum (with blak full lin) that orrspons to SDOF with hystrti amprs. For this xampl w hav slt th following paramtrs: α = 1 an γ = In Figur 5 w also iniat svral full r irls that orrspon to six points whr th isontinuous urvs (visous amping as) intrst th blak ontinuous urv (hystrti amping as Th orrsponing quivalnt visous amping for th strutural prios ( ) iniat in Figur 5 ar prsnt in th thir olumn of Tabl 1, whr th first olumn rprsnts th strutural prio an th son is th sismi offiint (vrtial axis of Figur 5 Th pairs of valus of th first an th thir olumn in Tabl 1 ar prsnt graphially in Figur 6. This shows that th quivalnt fftiv ritial amping ratio ς pns on th strutural prio, an prsnts its maximum valu los to th ominant prio of th sptrum (in this as, qual to 0.15s Th form of th fitt urv in Figur 6 (for prios longr than th ominant prio) oul b, for xampl, th following xprssion: T = 6 ς ( T + A) + B (3) whr A an B ar onstants that pn on th α an γ valus.

7 Th 14 th Worl Confrn on Earthquak Enginring Figur 5. Intrstion of th sptrum orrsponing to th systms with hystrti EDDs (blak ontinuous lin) with th sptra orrsponing to th systms with visous amprs (isontinuous lins) Tabl 1 Equivalnt visous amping for iffrnt prios Prio (s) T C y ς Figur 6. Efftiv ritial amping ratios orrsponing to a systm with α = 1, γ = 0.3 an ritial amping ratio qual to 5%.

8 Th 14 th Worl Confrn on Earthquak Enginring 6. CONCLUSIONS W hav shown a gnral prour for th alulation of rution fators bas on th ratio btwn th UAFR sptrum orrsponing to th onvntional fram an that of th systms with EDDs, both assoiat with th sam annual failur rat. This approah is bing follow at th National Univrsity of Mxio for th alulation of amping fators. Thos will b submitt for thir possibl inlusion in th CFE Sismi Dsign Manual. ACKNOWLEDEGEMENTS Thanks ar givn to L. Estva for his valuabl ommnts. This rsarh was sponsor by Instituto Invstigaions Elétrias (IIE), by Comisión Fral Eltriia (CFE), an by DGAPA-UNAM-IN REFERENCES Babr T.T. an Wn Y.K. (1981 Ranom vibration of hystrti graing systms. Journal of th Enginring Mhanis Division 107:EM6, Collins, K.R., Wn, Y.K. an Fouth, D.A. (1995), Invstigation of Altrnativ Sismi Dsign Prours for Stanar Builing, Rport No. UILU-ENG , Univrsity of Illinois at Urbana-Champaign, Illinois, USA Cornll, C.A. (1968 Enginring sismi risk analysis. Bulltin of Sismologial Soity of Amria 58:5, Estva, L. (1968), Bass para la Formulaión Disions Disño Sísmio, UNAM, DF, Méxio Estva, L. an Ruiz, S.E. (1989 Sismi failur rats of multistory frams. Journal of Strutural Enginring 115:2, FEMA450 (2003), NEHRP Rommn Provisions for Sismi Rgulations for Nw Builings an Othr Struturs, Builing Sismi Safty Counil National Institut of Builing Sins, Washington D.C., USA Hanson, R.D. an Soong, T.T. (2001), Sismi Dsign with Supplmntal Enrgy Dissipation Dvis, Earthquak Enginring Rsarh Institut, California, USA Ramirz, O., Constantinou, M., Kirhr, C., Whittakr, A., Johnson, M. an Gomz, J. (2000), Dvlopmnt an Evaluation of Simplifi Prours of Analysis an Dsign of Builings with Passiv Enrgy Dissipation Systms, Multiisiplinary Cntr for Earthquak Enginring Rsarh, NY, USA Rivra, J.L. an Ruiz S.E. (2007 Dsign approah bas on UAFR sptra for struturs with isplamnt- pnnt issipating lmnts. Earthquak Sptra 23:2, Shom, N. an Cornll, C. A. (1999), Probabilisti Sismi Dman Analysis of Nonlinar Struturs, Rport No. RMS-35, Dpartmnt of Civil Enginring of th Stanfor Univrsity, California, USA Whittakr, A., Constantinou, M., Ramirz, O., Johnson, M. an Chrysostomou, C. (2003 Equivalnt latral for an moal analysis prours of th 2000 NEHRP provisions for builings with amping systms. Earthquak Sptra 19:4,