DRAFT: US-JAPAN PBEE PAPER BY CORDOVA ... - PEER
DRAFT: US-JAPAN PBEE PAPER BY CORDOVA ... - PEER DRAFT: US-JAPAN PBEE PAPER BY CORDOVA ... - PEER
A simple extension to current practice that can help capture the period shift effect is to introducea second intensity parameter that reflects spectral shape. The proposed parameter to do this is aratio of spectral accelerations at two periods,Sa( Tf)RS a= (1)S ( )aT 1where T 1 is the first mode period and T f is a longer period that represents the inelastic (damaged)structure. This ratio can then be combined with the first mode spectral acceleration, S a (T 1 ), togive the following new two-parameter hazard intensity measureαa( T 1R SaS * = S )(2)where α and the ratio T f /T 1 are determined by calibration to optimize the intensity index byminimizing the variability in computed results.Attenuation Functions for Two-Parameter IndexGiven the prevalence of linear spectral acceleration in codes and practice, most hazardassessment techniques and data are geared toward this predicting this quantity. For example,national hazard maps available from USGS define earthquake hazard in terms of spectralacceleration at two periods (roughly T = 0.2 second and 1 second) representative of short andlong period structures. In devising new intensity measures, it is convenient if they can bederived by manipulating existing models and hazard data.Since the proposed intensity measure, S * , is simply a function of the spectral acceleration at twodifferent periods (T 1 and T f ), it is relatively straightforward to modify existing attenuationfunction to accommodate this index. Equations 3 and 4 show the transformation of a singleparameter attenuation function, E[ln S a (T x )], to the modified function, E[ln S * ], where E[ln …] isread as the “expected value of the natural log of the given parameter” and other variables are asdefined previously:*ln S = (1 −α )ln Sa( T1) + α ln Sa( Tf)(3)*[ S ] (1 −α ) E[ ln S ( T )] + αE[ ln S ( T )]E ln =a 1a f(4)In addition to the expected value of S*, the standard deviation, σln S*, must also be defined. Thisin turn requires the correlation between spectral accelerations at the two periods, S aT ) and( 14
S a( T f) . Inoue (1990) provides the following empirical correlation coefficient, ρln Sa1ln Sa f, thatfills this need:ln Sa1ln Sa f( T ) − ln( 1/ T )ρ = 1−0.33ln 1/1f(5)Given this correlation expression, the standard deviation of S* can be defined as follows:σ2 22 2( 1 − α ) σln Sa+ α σln Sa+ 2ρ( )1 f ln Sa1ln Sa1 − α ασfln Saσln Sa f= (6)2ln S*1Most spectral attenuation relationships define empirical coefficients as a function of frequency orperiod that can be manipulated to calculate S* according to Eq. 4. For example, Abrahamson &Silva (1997) define an attenuation function as follows:n[ S ] a + a (m-m ) + a ( 8.5-m)+ [ a a (m-m )] ln(R)E ln =1 4 1 123+13 1(7)awhere the a-coefficients are tabulated by Abrahamsom & Silva, m is the earthquake magnitude,m 1 is a given base magnitude, and R is the distance from the epicenter to the site. SubstitutingEq. 7 into Eq. 4, one obtains the following relationship for modified coefficients that can beapplied in the otherwise standard attenuation relationship to obtain S*:a*x= a a(8)( 1−α )xT1+ αxT 2These new relationships can then be applied in a standard probabilistic site hazard analysis wherethe required performance is evaluated on the basis of this new intensity, S*.3. BUILDING TESTBEDSIn related research (Mehanny et al., 2000, 2001) several moment frame structures have beendeveloped and analyzed to exercise seismic assessment and design provisions for compositeconstruction. These frames are utilized here to provide the basis for calibrating the new intensitymeasure parameters, α and T f /T 1 , and illustrate their application in a probabilistic performanceassessment. The case study structures consist of three six-story frames and one twelve-storyframe, all of which are designed according to provisions of the International Building Code (ICC2000) and AISC Seismic Provisions (1997) for a site in a high seismic region of California. Dueto space limitations the frames are only briefly introduced here. For further details the reader isreferred to Mehanny et al. (2000, 2001).5
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- Page 9 and 10: While the two-parameter index reduc
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A simple extension to current practice that can help capture the period shift effect is to introducea second intensity parameter that reflects spectral shape. The proposed parameter to do this is aratio of spectral accelerations at two periods,Sa( Tf)RS a= (1)S ( )aT 1where T 1 is the first mode period and T f is a longer period that represents the inelastic (damaged)structure. This ratio can then be combined with the first mode spectral acceleration, S a (T 1 ), togive the following new two-parameter hazard intensity measureαa( T 1R SaS * = S )(2)where α and the ratio T f /T 1 are determined by calibration to optimize the intensity index byminimizing the variability in computed results.Attenuation Functions for Two-Parameter IndexGiven the prevalence of linear spectral acceleration in codes and practice, most hazardassessment techniques and data are geared toward this predicting this quantity. For example,national hazard maps available from <strong>US</strong>GS define earthquake hazard in terms of spectralacceleration at two periods (roughly T = 0.2 second and 1 second) representative of short andlong period structures. In devising new intensity measures, it is convenient if they can bederived by manipulating existing models and hazard data.Since the proposed intensity measure, S * , is simply a function of the spectral acceleration at twodifferent periods (T 1 and T f ), it is relatively straightforward to modify existing attenuationfunction to accommodate this index. Equations 3 and 4 show the transformation of a singleparameter attenuation function, E[ln S a (T x )], to the modified function, E[ln S * ], where E[ln …] isread as the “expected value of the natural log of the given parameter” and other variables are asdefined previously:*ln S = (1 −α )ln Sa( T1) + α ln Sa( Tf)(3)*[ S ] (1 −α ) E[ ln S ( T )] + αE[ ln S ( T )]E ln =a 1a f(4)In addition to the expected value of S*, the standard deviation, σln S*, must also be defined. Thisin turn requires the correlation between spectral accelerations at the two periods, S aT ) and( 14
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