Multivaiate distibution models fo design spectal acceleations based on unifom hazad specta Shun-Hao Ni, De-Yi Zhang, Wei-Chau Xie, & Mahesh D. Pandey Depatment of Civil and Envionmental Engineeing Univesity of Wateloo, Wateloo, Ontaio, Canada ABSTRACT The site-specific design specta, Unifom Hazad Specta (UHS, have been adopted as input design gound motions in seveal seismic design codes. Howeve, the UHS, which ae deived fom the conventional Pobabilistic Seismic Hazad Analysis (PSHA, can only povide the pobability of exceeding a specified value of spectal acceleation at individual vibation peiod. To pefom eliability-based seismic design of stuctues, the analytical multivaiate distibution of design spectal acceleations at multiple peiods fo a given site is desied. In this pape, a multivaiate distibution model of design spectal acceleations at multiple peiods ae deived based on the maginal distibution model, knowledge of coelations among the spectal acceleations, and the existing unifom hazad specta. RÉSUMÉ Les spectes de conception spécifique au site, les Spectes Unifome de Risque Sismique (SURS, ont été adoptés comme motions du sol dans plusieus codes de conception paasismique. Toutefois, SURS, qui sont déivées de la Méthode d'analyse Pobabiliste du Dange Sismique (MAPDS, ne peuvent founi que la pobabilité de dépasse une cetaine valeu de l'accéléation spectale à la péiode individuelle de vibation. Pou effectue une conception paasismique basée su la fiabilité des stuctues, la distibution analytique multi vaiée des accéléations spectale de conception à des péiodes multiples pou un site donné, est désie. Dans cet aticle, un modèle de distibution multi vaiée des accéléations spectale de conception à de multiples péiodes est déivées basée su le modèle de distibution maginale, la connaissance des coélations ente les accéléations spectales, et les spectes du isque unifome existantes. 1 INTRODUCTION Unifom Hazad Specta (UHS have been used as input design spectal acceleations (acceleation seismic design specta in a numbe of seismic design codes (e.g., ASCE, 2005, NBCC, 2010. The UHS, howeve, ae deived fom the conventional Pobabilistic Seismic Hazad Analysis (PSHA (McGuie, 2004, which computes the pobability of exceeding a theshold value of spectal acceleation at individual vibation peiod sepaately. As esults fom a suite of maginal hazad analyses fo spectal acceleations at diffeent peiods, the UHS ae thus unable to povide any knowledge about the simultaneous occuence of spectal acceleations at multiple vibation peiods (Bake and Conell, 2006. The concept of a single design eathquake is then lost on an UHS. To obtain knowledge about the simultaneous occuence of spectal acceleations at multiple peiods, a Vecto-valued Pobabilistic Seismic Hazad Analysis (VPSHA was pefomed by Bazzuo and Conell (2002. This analysis is a diect extension of the conventional PSHA. The only additional infomation equied is the joint distibution of spectal acceleations fo a given magnitude and distance. Howeve, the esults of both the conventional and the vecto-valued PSHA can only povide numeical spectal acceleations associated with a small numbe of coesponding levels of pobability of exceedance, athe than analytical multivaiate distibution functions of spectal acceleations at multiple peiods. To pefom eliability-based seismic design of stuctues, an analytical multivaiate distibution function of design spectal acceleations at multiple vibation peiods fo a given site is desied. Fo seismic design and analysis of stuctues, the oiginal peception of a single design eathquake can also be achieved if the multivaiate distibution is used. The pobability of exceedance obtained fom the multivaiate distibution is fo the simultaneous occuence of the design spectal acceleations. In this pape, the conventional and vecto-valued PSHA ae fist descibed in Sections 2 and 3, espectively. The basic concept of spectal coelation is biefly intoduced in Section 4. In Section 5, the maginal distibution model fo spectal acceleation at individual peiod is veified empiically based on the esults of PSHA. The multivaiate distibution model of design spectal acceleations at multiple peiods ae then deived based on the maginal distibution model, knowledge of coelations among the spectal acceleations, and the UHS. The multivaiate distibution model is validated using the esults of the VPSHA. 2 PROBABILISTIC SEISMIC HAZARD ANALYSIS Pobabilistic Seismic Hazad Analysis (PSHA has been widely used in eathquake engineeing fo decades. It combines all possible and elevant eathquake scenaios
and the levels of pobability of gound motion paametes in the analysis. The famewok of the PSHA was established by Conell (1968 40 yeas ago. The cuent well-known PSHA method is ecognized as the Conell- McGuie PSHA (McGuie, 2004. Fo the seismic hazad evaluation of a site of inteest having n potential seismic souces, the annual mean ate of exceedance of gound motion paamete, such as spectal acceleation at a specified peiod S a, is given by: λ n νi { P { s m, } fm, R ( m, dmd, [1] } = > s m i= 1 i associated with multiple gound motion paametes, such as spectal acceleations at multiple vibation peiods as will be discussed in Section 3. Fo a given annual pobability of exceeding s *, a plot of the theshold s * fo a ange of vibation peiods of inteest at the site gives the Unifom Hazad Spectum (UHS as shown in Figue 1. The spectal acceleations on a UHS ae sepaately deived by the PSHA. The plot of the pobability of exceedance against the spectal acceleation at a given peiod is called seismic hazad cuve. whee ν i is the annual mean ate of occuence of eathquakes on seismic souce i above a lowe bound of eathquake magnitude and s * is the theshold value of spectal acceleation S a. f M,R(m, is the joint pobability density function of magnitude M and souce-site distance R. In some cases, the magnitude and distance ae assumed to be statistically independent. f M(m can usually be assumed to be an exponential distibution, which is tuncated at lowe- and uppe bounds, namely, m min and m max. f R( can be deived based on the assumption that a seismic focus is equally likely to occu anywhee on the seismic souce given the occuence of an eathquake having cetain magnitude. P( in equation 1 is the complementay standad nomal cumulative distibution function: ln s µ m, >, = 1 Φ S (, [2] a { a s m } P S m, whee µ lnim, and lnim, ae the conditional mean and standad deviation of the natual logaithm of S a povided in any customay attenuation elation. Given a pai of magnitude m and distance, the natual logaithm of S a has been widely veified to be well epesented by the nomal distibution maginally (Jayaam and Bake, 2008. The tempoal uncetainties of the occuence of eathquakes ae defined using the Poisson pocess and the pobability of exceeding s * in a time peiod t is given by t { a } 1 exp ( λ P S > s = t. [3] In most stuctual engineeing pactices, the time peiod t is taken as one yea o 50 yeas. The use of the Poisson pocess can also be extended to descibe the tempoal uncetainties of the occuence of eathquakes s Figue 1. Concept of unifom hazad specta 3 VECTOR-VALUED PROBABILISTIC SEISMIC HAZARD ANALYSIS The conventional PSHA fo a specific site povides the annual pobability of exceeding a theshold value of a single gound motion paamete, such as spectal acceleation at a given peiod. In many pactical applications, howeve, the joint knowledge of a vecto of gound motion intensity measues can impove the accuacy in the pediction of stuctual esponse induced by the eathquake. To demonstate the pocedue of Vecto-valued Pobabilistic Seismic Hazad Analysis (VPSHA, fist poposed by Bazzuo and Conell (2002, two spectal acceleations at diffeent vibation peiods ae consideed as follows. In the two-dimensional case, the joint annual mean ate of exceedance of spectal acceleations at peiods T 1 and T 2, i.e., S a(t 1 and S a(t 2, is given by λ s1, s2 { a ( 1 > 1, a ( 2 > 2, } ( m, dmd n P S T s S T s m m = ν i. [4] i= 1 f M, R i
The joint complementay cumulative distibution function of S a(t 1 and S a(t 2 conditional on a pai of magnitude m and distance fo seismic souce i, in equation 4, can be obtained by integating the coesponding pobability density function, i.e., P{ > s1, > s2 m, } = f T1 T2 [5] s2 s1 ( s, s m, ds ds. (, ( 1 2 1 2 Given a pai of m and, a vecto of the natual logaithm of spectal acceleations at multiple peiods has been empiically tested to follow multivaiate nomal distibution (Jayaam and Bake, 2008. The joint pobability density function conditional on m and in equation 5 can be ewitten in the conditional fom: f ( s, s m, = f ( s m,, 1 2 1 f ( s s, m,. 2 1 Based on the veified assumption of multivaiate lognomal distibution fo spectal acceleations at multiple peiods given m and, the maginal distibution of S a(t 1 is given by f a 1 ( s m 1 ln S ( T m, 1 s1 m, m, [6] 1 ln s µ, = Φ (, [7] and the distibution of S a(t 2 conditional on S a(t 1=s 1 can be expessed as ( 2 1,, f s s m = s Φ 2 s1, m, 1 ln s2 µ s1, m, (. s1, m, The mean and standad deviation of the conditional distibution in equation 8 ae given by µ = µ s1, m, m, ( s m,, 1 m, m, 2 s1, m, m,, [8] [9] + ρ ln µ, = 1 ρ. In equations 7 and 9, the means and standad deviations fo lns a(t 1 and lns a(t 2 conditional on m and can be obtained fom the attenuation elations of S a(t 1 and S a(t 2, espectively. The coelation coefficient between lns a(t 1 and lns a(t 2 has been empiically deived (Bake and Jayaam, 2008 as will be discussed in Section 4. Theoetically, two-dimensional case can be easily extended to a vecto of gound motion paametes and the additional equiement is to eplace the coelation coefficient with the full matix of coelation coefficient. Equation 3 can then be applied fo descibing the tempoal uncetainties when the joint annual mean ate of exceedance of spectal acceleations at multiple peiods has been obtained in equation 4. 4 CORRELATION COEFFICIENTS OF SPECTRAL ACCELERATIONS The coelation coefficient between the natual logaithm of spectal acceleations at any two peiods, given a scenaio eathquake (m and, has been empiically obtained. Based on the coelation coefficient fo any two peiods, a symmetic positive semi-definite matix of the coelation coefficient can be detemined fom all the combinations of spectal acceleations at diffeent peiods. The most ecent matix of coelation coefficient fo a scenaio eathquake has been defined based on egession analysis of a lage stong motion database by Bake and Jayaam (2008. In the following numeical example, this matix of spectal coelation is adopted fo the detemination of multivaiate distibution model. It is noted that, due to the limited gound motion ecods, the coelation coefficient between spectal acceleations at any two peiods fom diffeent pais of m and is still unavailable. This limitation may pevent the application of the multivaiate distibution model on stuctues having long fundamental peiod as will be seen late. This poblem can be esolved with the incease of ecoded gound motions. 5 DISTRIBUTION MODELS OF SPECTRAL ACCELERATIONS In Sections 2 and 3, the conventional PSHA and vectovalued PSHA wee descibed. Having obtained the maginal and joint distibutions of spectal acceleations conditional on a pai of m and, the pobabilities of exceedance can be detemined by integating all possible magnitudes and distances fo the site of inteest based on equations 1, 3 and 4. The esulting pobabilities of exceedance associated with the coesponding theshold values of spectal acceleations, howeve, can only be plotted numeically. The analytical maginal and joint distibution functions, fo the numeical esults geneated by PSHA and VPSHA, ae still unknown. To pefom eliability-based seismic design of stuctues, the analytical multivaiate
distibution function of design spectal acceleations at multiple peiods fo a given site is desied. Moeove, except fo the UHS associated with a small numbe of coesponding levels of pobability of exceedance fo a given site, the full infomation equied fo pefoming a PSHA o VPSHA is usually unavailable fo stuctual analysts. A eliable analytical multivaiate distibution function of design spectal acceleations at multiple peiods, which can be obtained based on as little infomation as possible, such as only the existing UHS, is pefeable. 5.1 Maginal Distibution Model Accoding to the seismic hazad cuve fom the PSHA fo a site of inteest (i.e., UHS in this pape, given a level of pobability of exceedance, a spectal acceleation value at a specified peiod can be obtained. Fo this spectal acceleation value associated with the coesponding pobability of exceedance, a scenaio eathquake (o socalled beta eathquake, epesented by magnitude m β, distance β, and ε β, can be detemined by applying the Pobabilistic Seismic Hazad Deaggegation (PSHD (McGuie, 1995. ε β is defined as the numbe of logaithmic standad deviations by which the logaithmic spectal acceleation deviates fom the mean. The beta eathquake deived fo a given pobability level contibutes the most seismic hazad to the spectal acceleation at the selected peiod among all possible eathquakes consideed in the PSHA. The beta eathquake can thus appoximately epesent the seismic hazad to the spectal acceleation at the given peiod at the site of inteest by the deived m β, β, and ε β, instead of all possible eathquake theats. This appoximation has been used fo educing the UHS by Bake and Conell (2006. It is also assumed that the beta eathquakes (epesented by magnitudes m β and distances β obtained using the PSHD fo the same hazad cuve (i.e., spectal acceleations at the same peiod but diffeent pobability levels ae identical. This assumption is made because this is not an unusual case fo the numeical esults of the PSHD. Only the ε β value changes with the change of pobability of exceedance to chaacteize the vaiation of spectal acceleation. Futhe investigation is needed to validate this assumption. Theefoe, the spectal acceleations on diffeent levels of pobability of exceedance at the same peiod ae geneally induced by the same beta eathquake. Since the spectal acceleation at a given peiod conditional on a scenaio eathquake follows lognomal distibution as discussed in Section 2 and illustated in Figue 1, the spectal acceleation at a given peiod can thus be assumed to be lognomally distibuted maginally conditional on the beta eathquake, i.e., the seismic hazad cuve fo spectal acceleation at a given peiod fo a site empiically matches the complementay lognomal cumulative distibution function. Since all the infomation of vaiation on the spectal acceleation, including magnitude, distance, activity ate of seismic souce, and any othe gound motion paametes, has been included in the hazad cuve fom the PSHA, the mean and standad deviation of the maginal lognomal distibution fo the spectal acceleation can be simply detemined by using the egession analysis on the hazad cuve. It should be mentioned that although the assumption of lognomal distibution is based on the concept of the PSHD, the numeical esults of the PSHD (i.e., beta eathquake epesented by m β, β, ε β is not actually needed. The only infomation, on which the maginal lognomal distibution depends, is at least two spectal acceleation values at a given peiod associated with two coesponding levels of pobability of exceedance as shown in Figue 1. 5.2 Multivaiate Distibution Model Having obtained the maginal lognomal distibutions (mean and standad deviation values fo the spectal acceleations at given peiods sepaately as pesented in Subsection 5.1, the multivaiate lognomal distibution function fo those spectal acceleations can be diectly detemined, if the coesponding matix of coelation coefficient is available, by using the Nataf joint distibution model (Liu and De Kiueghian, 1986. Thee is also a moe compelling eason. Accoding to the concept of the UHS, the long peiod potion of the UHS epesents lage fa-field eathquakes, wheeas the shot peiod potion of the UHS epesents small neafield eathquakes (Atkinson and Beesnev, 1998. In pactical applications, the esponse specta of one lage fa-field and one small nea-field gound motions ae fequently used to cove the long peiod and shot peiod segments of the UHS, espectively (McGuie, 1995, Atkinson and Beesnev, 1998. The spectal acceleations at shot peiods can thus be safely assumed to be induced by a small nea-field beta eathquake. Since the spectal acceleations at multiple peiods have been tested to follow multivaiate lognomal distibution conditional on a given scenaio eathquake, the assumption of the multivaiate lognomal distibution fo spectal acceleations within shot peiod ange is thus theoetically easonable. Given a scenaio eathquake, the matix of coelation coefficient between spectal acceleations at any combinations of diffeent peiods is available as discussed in Section 4. The multivaiate lognomal distibution function fo the spectal acceleations at shot peiods can then be easily detemined based on the means and standad deviations of the maginal distibutions deived in Subsection 5.1 and the empiical matix of coelation coefficient. In pactice, the poposed distibution model and the analysis pocedue ae suitable only fo stuctues having shot fundamental peiods (usually less than 0.3 sec. This multivaiate distibution model could benefit the nuclea enegy industy, since nuclea stuctues (exiting plants and new designs ae vey stiff with fundamental peiods in the ange of 0.06-0.3 sec (EPRI, 2007. As mentioned in Section 4, the pocedue descibed above cannot be eadily used fo long-peiod stuctues.
The oscillatoy modes of a long-peiod stuctue may cove the long and shot peiod egions, contolled by lage fa-field and small nea-field beta eathquakes, espectively. The coelation coefficients between spectal acceleations at two peiods fom diffeent pais of m and ae thus needed. This type of spectal coelation will become available in futue with the incease of ecoded gound motions. It should be emphasized that the design spectal acceleations (theshold values fo diffeent peiods, obtained fom the multivaiate distibution model fo a given annual pobability of exceedance, occu simultaneously. Hence, the combination of these design spectal acceleations gives a single design eathquake conceptually. 5.3 Numeical example In this section, the pocedue fo obtaining the maginal and joint distibution functions fo the design spectal acceleations (i.e., appoximate esults is demonstated though a PSHA example and then numeically veified by the esults of the VPSHA (i.e., accuate esults fo the same hazad envionment. As illustated in Figue 2, a simple numeical PSHA example is pefomed to obtain the seismic hazad cuves fo spectal acceleations at shot peiods of 0.06 and 0.3 sec, espectively, using equations 1-3. The attenuation elation, poposed by Abahamson and Silva (1997, is used fo obtaining the paametes in equation 2. hazad cuve points. It is noted that moe hazad cuve points can be used fo estimating paametes of the distibution functions by using the egession analysis if applicable. Having obtained the maginal lognomal distibutions fo spectal acceleations at 0.06 and 0.3 sec, based on the poposed pocedue, the joint lognomal distibution fo these two spectal acceleations can then be detemined by using the matix of coelation coefficient descibed in Section 4. The contous of the joint annual pobability of exceedance fo spectal acceleations at 0.06 and 0.3 sec, deived fom the poposed pocedue, ae illustated in Figue 5. To veify the joint lognomal distibution model, a VPSHA, based on the same seismic hazad infomation as the PSHA, is pefomed as shown in Figue 6. Compaed to the VPSHA (incopoating all possible eathquake theats, the elative eos of the spectal acceleations of the joint lognomal distibution (appoximating all possible eathquake theats by a single beta eathquake ae less than 5 pecent fo a given joint pobability of exceedance. Fo example, as shown in Figues 5 and 6, fo the maginal pobability of exceedance of 0.0004, spectal acceleations at 0.3 sec fom the lognomal model and the PSHA ae 1.47 g and 1.45 g, espectively. Having detemined the spectal acceleations at 0.3 sec, to achieve the joint pobability of exceedance of 0.0002, spectal acceleations at 0.06 sec fom the lognomal model and the VPSHA ae 0.51 g and 0.53 g, espectively. The elative eos fo the maginal and joint pobabilities of exceedance ae only 1.38% and 3.77%, espectively. Theoetically, the bivaiate lognomal distibution model demonstated in the numeical example can be easily extended to multivaiate lognomal distibution with moe than two andom vaiables. Figue 2. Seismicity of the site fo PSHA and VPSHA The esulting seismic hazad cuves fo spectal acceleations at 0.06 and 0.3 sec ae shown in Figues 3 and 4, espectively, in squae symbols. To detemine the paametes of the logaithm of spectal acceleation fo the maginal lognomal distibution function, two esulting hazad cuve points at fequently used pobability of exceedance of 0.002 and 0.0004 ae adopted as shown in Figues 3 and 4 in shaded squae symbols. Most of the esulting maginal complementay lognomal distibution cuves go though the unused Figue 3. Seismic hazad cuve fo spectal acceleation at 0.06 sec fom PSHA
Figue 4. Seismic hazad cuve fo spectal acceleation at 0.3 sec fom PSHA As a esult, the seismic hazad cuves deived fom the PSHA can be well epesented by the complementay maginal lognomal distibution. It is also satisfactoy to epesent the esults of the VPSHA by multivaiate lognomal distibution. The paametes can be easily detemined fom the seismic hazad cuves using egession analysis. 6 CONCLUSION Figue 6. Contous of VPSHA Fo the pope use of the unifom hazad specta in eliability-based seismic design, a multivaiate (lognomal distibution model is poposed, which can appopiately epesent the vaiation of spectal acceleations at multiple peiods fo a site of inteest. The pocedue of analysis contains the following main steps: 1. Select at least two spectal acceleations associated with the coesponding annual pobabilities of exceedance at each vibation peiod of inteest fo the UHS at the site. 2. Estimate the paametes of the maginal lognomal distibution fo spectal acceleations at all inteested peiods by solving the equations (fo the cases with two points o using the egession analysis (fo the cases with moe than two points. 3. Establish the multivaiate lognomal distibution of the spectal acceleations at the peiods of inteest based on the paametes, obtained in Step 2, and the empiical matix of coelation coefficient of the spectal acceleations. Figue 5. Contous of joint lognomal distibution ACKNOWLEDGEMENTS The eseach fo this pape was suppoted, in pat, by the Natual Sciences and Engineeing Reseach Council of Canada (NSERC and Univesity Netwok of Excellence in Nuclea Engineeing (UNENE. We ae also thankful to D. Jack Bake of Stanfod Univesity fo poviding us with the compute pogams of attenuation elation and spectal coelation. REFERENCES Abahamson, N.A. and Silva, W.J. 1997. Empiical Response Spectal Attenuation Relations fo Shallow Custal Eathquakes. Seismological Reseach Lettes, 68, 94-127.
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