Preliminary Study on the Fragility Curves for Steel Structures in Taipei

Transcription

1 Earthquake Cherg: Egieerig Prelimiary ad Egieerig study o Seismology the fragility curves for steel structures i Taipei 35 1 Volume 3, Number 1, March 001, pp Prelimiary Study o the Fragility Curves for Steel Structures i Taipei Rwey-Hua Cherg 1) 1) Departmet of Costructio Egieerig, Natioal Taiwa Uiversity of Sciece ad Techology, Taipei, Taiwa, R.O.C. ABSTRACT The fragility curves are used to represet the probabilities that the structural damages, uder various levels of seismic excitatio, exceed specified damage states. This study first reviews differet methods i derivig fragility curves. A modified oliear static aalysis method is proposed ad the adopted i evaluatig the fragility curves for steel buildig structures i Taipei. Most steel buildigs i Taipei are high-rise ad were desiged to resist lateral loads by either special momet resistig frames (SMRF) or special momet resistig braced frames (SMRBF), which are targeted i this study. The capacity curves (ad spectra) for six SMRF ad six SMRBF, desiged accordig to the Taiwa seismic code prior to Chi-Chi earthquake, are computed by oliear static aalyses. The code-specified Taipei earthquake respose spectrum is trasformed ito the demad spectrum. After cosiderig the ucertaities i the buildig capacity spectrum, i the demad spectrum ad i the damage states, this study applies a Mote-Carlo simulatio method to compute the fragility curves. INTRODUTION Earthquakes may cause extesive direct/idirect losses i which structural damage plays a importat role. Oe of the most importat elemets i evaluatig the seismic damage to structures is the so-called fragility curve. The fragility curves for certai type of buildig structure are used to represet the probabilities that the structural damages, uder various levels of seismic excitatio, exceed specified damage states. I other words, each poit o the curve represets the probability that the spectral displacemet uder certai level of groud shakig is larger tha the displacemet associated with certai damage state. The damage states i this study are classified ito four categories, icludig slight damage (SD), moderate damage (MD), extesive damage (ED), ad complete damage (CD) [1]. This study first reviews differet

2 36 Earthquake Egieerig ad Egieerig Seismology, Vol. 3, No. 1 methods i derivig fragility curves, icludig empirical statistical methods, seismic code methods, oliear dyamic aalysis methods, ad oliear static aalysis methods. A modified oliear static aalysis method is proposed ad the adopted i estimatig the fragility curves for steel buildig structures i Taipei. The obtaied fragility curves ca be imported ito HAZ-TAIWAN for evaluatig the potetial seismic damage ad ecoomic loss, ad also serve as the basis for future seismic retrofittig ad seismic code revisio []. REVIEW OF METHODS Researchers i derivig the fragility curves geerally adopt four kids of methods, which are empirical statistical methods, seismic code methods, oliear static aalysis methods, ad oliear dyamic aalysis methods. Empirical statistical methods, or post-earthquake ivestigatio methods, provide the best estimates for fragility curves whe sufficiet ad reliable post-earthquake damage data are available, which is seldom realistic. Some researchers [3,4] collected, assorted, ad aalyzed the damage data ad the groud shakig data to give prelimiary estimates for the correspodig fragility curves. However, very few reliable data ca be used to coduct similar statistical aalyses i Taiwa, especially whe cosiderig the seismic fragility of steel structures i Taipei. Seismic code methods, adopted by HAZUS97 [1], estimate the capacity curve, demostratig the seismic capacity of certai type of structure, by the local code provisios. Specifically, the yieldig poit (yieldig stregth ad yieldig displacemet) ad the ultimate poit (ultimate stregth ad ultimate displacemet) for a give type of structure are determied based o a stadard code procedure; the correspodig capacity curve is simply a smooth curve coectig the origi, the yieldig poit, ad the ultimate poit. The fragility curves are the computed based o a capacity-demad spectrum method. This cocept is ot difficult to implemet, but the obtaied results may be highly iaccurate []. Noliear dyamic aalysis methods first defie ad calibrate the damage idices, fuctios of the maximum displacemet ad the cumulative eergy dissipatio, for structural compoets based o mootoic ad cyclic test results. Secodly, all sigificat ucertai seismic ad structural quatities are appropriately modeled by radom variables or radom processes. Fially, the simplified target structures are aalyzed by a sufficietly large umber of oliear time-history aalyses. The obtaied fragility curves are more accurate tha those obtaied by most methods, but require much more modelig ad computatioal effort. Noliear static aalysis methods compute the capacity curves by oliear static aalyses (or the so-called pushover aalyses) [5,6] ad evaluate the fragility curves by capacity-demad spectrum methods. Twelve seismic code-complyig steel structures are desiged i this study; the associated fragility curves are derived by a modified oliear static aalysis method, which is described i the ext sectio. The effort i dealig with the same problem by a oliear dyamic aalysis method is uder progress.

3 Cherg: Prelimiary study o the fragility curves for steel structures i Taipei 37 PROPOSED METHOD The proposed oliear static aalysis method is discussed i details as follows: Classificatio of Model Structures Thirty-seve model structures, distiguished by their costructio materials, lateral force resistig systems, ad heights, are cosidered i HAZ- TAIWAN. It is foud i a detailed survey [] that most steel buildigs i Taipei are high-rise ad were desiged to resist lateral loads by either special momet resistig frames (SMRF) or special momet resistig braced frames (SMRBF), accordig to the Taiwa seismic code prior to Chi-Chi earthquake. It is therefore determied that these two types of structures are the targeted model structures. Computatio of Capacity Curves A capacity curve is used to demostrate the seismic capacity of a structure; its ordiate is usually the base shear V ad its abscissa is usually the maximum displacemet δ. The capacity curves associated with twelve codecomplyig target structures, belogig to the cosidered model structures, are computed i pushover aalyses, i which a suitable oliear aalysis program (e.g., DRAIN-DX) is ecessary. The lateral displacemets due to the progressively icreased lateral forces are computed i order. The m-th modal lateral force actig o the i-th story, Fi,m, is give by [5] F i, m = PFm φi, m Sa, m wi (1) where w i is the mass of the i-th story; Sa,m is the m-th modal spectral acceleratio; φi,m is the i-th story amplitude of the m-th mode shape, satisfyig ( w i φi, m )/ g = 1; is the total umber of stories; g is the gravitatioal acceleratio; PFm is the m-th modal participatio factor determied by PF m = ( w φ i ( w φ i i, m i, m )/ g )/ g () The total lateral force applied at the i-th story, Fi, is thus k F i = F i, m (3) m= 1 where the determiatio of k is related to the cumulative effective mass coefficiet, i which the m-th modal effective mass coefficiet EMm is EM m = ( wi φi, m )/ g wi / g ( wi φi, m )/ g Trasformatio ito Capacity Spectra (4) To facilitate the subsequet solutio process, the obtaied capacity curves eed to be trasformed ito their correspodig capacity spectra i which the ordiate is the spectral acceleratio Sa ad the abscissa is the spectral displacemet Sd. Sd ad Sa are related to δ ad V as follows: S δ d = (5) PF 1 φ,1 V / W S a = (6) EM 1 where W is the total mass.

4 38 Earthquake Egieerig ad Egieerig Seismology, Vol. 3, No. 1 Ucertaity i Capacity Spectra The capacity spectra computed for target structures, belogig to a model structure, will be differet oe aother; the ucertaity ca be quatified by statistical approaches. A oliear regressio with o-costat variace aalysis [7] is coducted i this study to derive the mea capacity spectrum E ( Sa Sd = x) ad the stadard deviatio σ ( Sa Sd = x ) for a model structure. The associated probability desity fuctio is verified by goodess-of-fit tests. Therefore, the actual capacity spectrum for a model structure ca be expressed as S ( S = x ) = R( x ) E( S S x ) (7) a d a d = where R(x) deotes a series of radom variables with a mea of 1 ad stadard deviatios of σ(sa Sd = x). Determiatio of Demad Spectrum The ielastic respose spectrum may be determied by modifyig the elastic spectrum accordig to the ductility ratio µ *, which is defied as the ratio betwee the maximum displacemet ad the yieldig displacemet. The yieldig displacemet is take as the mea value of the yieldig displacemets computed for differet target structures belogig to a model structure. The media demad spectrum i this study is derived by applyig Newmark ad Hall approach [8] i modifyig the code-specified Taipei acceleratio spectrum. Figure 1 demostrates the variatio of the demad spectra for differet ductility ratios whe the peak groud acceleratio (PGA) equals 1g. To accout for the ucertaity i the demad spectrum, the actual demad spectrum is modeled as the media Fig. 1 Demad spectra for various ductility ratios demad spectrum times a radom variable S that is assumed to be logormally distributed with a media of 1 ad a coefficiet of variatio of [1]. Quatificatio of Damage States To obtai the fragility curves, differet damage states eed to be quatified i terms of spectral displacemets. The relatioship betwee damage states ad displacemets is, however, difficult to establish; it requires a large amout of local damage data ad soud egieerig judgmet. Based o the discussio with the practicig egieers ad the recommedatios i HAZUS97 ad VISION000, the mea iterstory drifts associated with slight damage state, moderate damage state, serious damage state, ad complete damage state are set to be 0.5%, 1.5%,.5%, ad 3% respectively. The trasformatios from the iterstory drifts to the spectral displacemets are established through careful examiatios o the output of oliear aalyses. The statistical iformatio o the spectral displacemets correspodig to differet damage states is also obtaied.

5 Cherg: Prelimiary study o the fragility curves for steel structures i Taipei 39 Evaluatio of Exceedace Probability A fragility curve is a probabilistic measure for the fragility of a model structure uder various levels of groud shakig; the exceedace probability over a damage state ds is defied as P ( ds Sd ) = P ( X > Sd, d Sd ) (8) where Sd,ds is the spectral displacemet correspodig to ds; Sd is the mea spectral displacemet uder certai level of groud shakig; X is the spectral displacemet correspodig to the itersectio of the capacity spectrum ad the demad spectrum. It is oted that both X ad Sd,ds i Eq. (8) are radom variables. HAZUS97 trasforms Eq. (8) ito the followig equatio: 1 S d P ( d s S d ) = Φ l (9) β ds S d, d s where Φ( ) is the cumulative distributio fuctio of a stadard ormal radom variable; S d, d s is the media of Sd,ds; βds is expressed as β = (10) ds ( CONV[ βc, βd ] ) + ( βm, ds ) where βc, βd, ad βm,ds are, respectively, the logormal stadard deviatios of the capacity spectrum, the demad spectrum, ad ds. The term CONV i Eq. (10) deotes the process of combiig differet sources of ucertaity, but is ot explaied i further details. This formulatio is based o the assumptio that S d,d s is logormally distributed ad P ( X > S d, d Sd ) ca be modeled by a s ormal cumulative distributio fuctio. However, the latter assumptio may be far from the truth accordig to the study i []. s To solve Eq. (8), a Mote-Carlo simulatio method, rather tha a umerical itegratio method [], is adopted i this study. The procedure for computig a poit i the fragility curve, i.e., the exceedace probability over certai damage state for a give spectral displacemet Sd *, is explaied as follows. The correspodig spectral acceleratio Sa * is first determied by examiig the mea capacity spectrum. The mea demad spectrum is the costructed based o the calculated ductility ratio ad Sd * (or Sa * ); it is oted that Sd * is the abscissa of the itersectio of the mea capacity spectrum ad the mea demad spectrum. After itroducig the ucertaity i the capacity spectrum, i the demad spectrum, ad i the damage state, a Mote-Carlo simulatio procedure is adopted for evaluatig Eq. (8). All poits o a set of fragility curves correspodig to a model structure ca be obtaied by repeatig the aforemetioed procedure with differet Sd * ad damage states FRAGILITY CURVES FOR TAIPEI STEEL STRUCTURES The oliear static aalysis method proposed i the previous sectio is used to derive the fragility curves for Taipei high-rise steel structures. As metioed earlier, SMRF ad SMRBF, desiged accordig to the Taiwa seismic code prior to Chi-Chi earthquake, are the two model structures cosidered here. Six target structures are desiged for each model structure; their heights rage from 69 meters to 103 meters. The capacity curve for each target structure is computed by the procedure described i the previous sectio. Dead loads ad

6 40 Earthquake Egieerig ad Egieerig Seismology, Vol. 3, No. 1 partial live loads are icluded i the aalyses. The member oliear behaviors are modeled by respective biliear curves with a hardeig ratio of %. May experimetal results idicate that brittle failures occur frequetly i the steel beam-colum coectio area due to improper weldig. Based o [9] ad the result statistics of the experimets coducted i Taiwa [10], it is assumed hereiafter that the stregth of a steel beam-colum coectio reduces to 0% of its origial stregth after the associated plastic rotatio exceeds 0.01 radia. Noliear aalyses with this assumptio pluggig ito a biliear model yield more realistic capacity curves tha those without this assumptio; the compariso for target structure 1 is show i Fig.. It is apparet that the actual seismic capacity is less tha the predictio based o a ideal biliear model. The compariso betwee the capacity spectra computed i this study ad those estimated by a stadard code procedure [] idicates that the applicatio of a stadard code procedure may iduce a cosiderable error. I additio, it is also observed, after a parameter study, that the cotributio from higher vibratio modes is ot egligible. The capacity curves computed for six target structures, belogig to a model structure, are trasformed ito the correspodig capacity spectra, i which a oliear regressio with o-costat variace aalysis is performed. It is foud that a logormal probability distributio yields the best fit for either SMRF or SMRBF. The mea capacity spectrum E ( Sa Sd = x) is best fitted by a third order polyomial; its coefficiets ad the coditioal stadard deviatio σ ( Sa Sd = x) are also obtaied. The statistical iformatio o the spectral displacemets correspodig to differet damage states for SMRF ad SMRBF ca be obtaied by the procedure described i the previous sectio. To be coservative, the mea values are take as 90% of their origial calculated values while the stadard deviatios are take as 110% of their origial calculated values. Fially, the computed fragility curves for SMRF ad SMRBF by Mote-Carlo simulatio aalyses are show i Fig. 3 ad Fig. 4. It is apparet that SMRBF possesses higher seismic capacity tha SMRF; this observatio is aticipated sice SMRBF is of dual lateral resistig systems. A effort i derivig these fragility curves by a oliear dyamic aalysis method is uder progress; its Fig. Compariso of capacity curves for target structure 1 Fig. 3 Fragility curves for SMRF

7 Cherg: Prelimiary study o the fragility curves for steel structures i Taipei 41 obtaied i this study are ow verified by those computed by a oliear dyamic aalysis method; the compariso results will be published i the ear future. ACKNOWLEDGEMENTS Fig. 4 Fragility curves for SMRBF compariso with the curret results will be published later. CONCLUSION This study first reviews four differet methods i derivig fragility curves; a modified oliear static aalysis method is the proposed ad adopted i evaluatig the fragility curves for high-rise SMRF ad SMRBF steel structures i Taipei. The target structures are desiged accordig to the Taiwa seismic code prior to Chi-Chi earthquake; their member oliear behaviors are modeled by respective biliear curves with possible stregth reductios due to brittle failures. The probabilistic iformatio o the capacity spectra ad o the damage states is ratioally determied. After cosiderig the ucertaities i the buildig capacity spectrum, i the demad spectrum ad i the damage states, this study applies a Mote-Carlo simulatio method i computig the probability that the spectral displacemet is greater tha the displacemet correspodig to a give damage state. Fragility curves ca the be costructed by cosiderig various levels of seismic excitatio ad buildig damage states. The prelimiary results The fiacial support from Architecture ad Buildig Research Istitute ad the help of former graduate studets Yu-Che Kuo, Chu-Tai Shih, ad We-Cheg Hwag are greatly appreciated. REFERENCES 1. Risk Maagemet Solutios, Ic. (1997). Earthquake Loss Estimatio Method-HAZUS97 Techical Maual, Natioal Istitute of Buildig Scieces, Washigto, D.C.. Cherg, R.H. (1999). Prelimiary study of fragility curves for steel structures i Taipei (I), Research Report, Architecture ad Buildig Research Istitute (i Chiese). 3. Ogawa, N. ad Yamazaki, F. (1999). Image iterpretatio of buildig damage due to the 1995 Hyogoke-Nabu Earthquake Usig aerial photographs, Asia-Pacific Symposium o Structural Reliability ad Its Applicatios, Taipei, Taiwa. 4. Murao, O. ad Yamazaki, F. (1999). Developmet of fragility curves for buildig based o damage data due to the Hyokoke-Nabu earthquake, Asia-Pacific Symposium o Structural Reliability ad Its Applicatios, Taipei, Taiwa. 5. Applied Techology Coucil (1996). Seismic Evaluatio ad Retrofit of Cocrete Buildigs. 6. Lawso, R.S., Vace, V. ad Krawikler, H. (1994). Noliear static pushover aalysis-why, whe,

8 4 Earthquake Egieerig ad Egieerig Seismology, Vol. 3, No. 1 ad how? Proceedigs, 5th US Coferece o Earthquake Egieerig, Chicago, USA. 7. Devore, J.L. (000). Probability ad Statistics for Egieerig ad the Scieces, Duxbury, USA. 8. Newmark, N.M. ad Hall, W.J. (198). Earthquake spectra ad desig, Earthquake Egieerig Research Istitute, USA. 9. Nakashima, M., Miami, T. ad Mitai, I. (000). Momet redistributio caused by beam fracture i steel momet frames, Joural of Structural Egieerig, ASCE, Vol. 16, No Li, T.F. (199). Seismic behavior of steel structures, Master Thesis, Departmet of Costructio Egieerig, Natioal Taiwa Uiversity of Sciece ad Techology, Taipei, Taiwa, ROC (i Chiese). el