THE ROLE OF EPSILON FOR THE IDENTIFICATION OF GROUPS OF EARTHQUAKE INPUTS OF GIVEN HAZARD

Transcription

1 THE ROLE OF EPSILON FOR THE IDENTIFICATION OF GROUPS OF EARTHQUAKE INPUTS OF GIVEN HAZARD Tomaso TROMBETTI Stefano SILVESTRI * Giada GASPARINI University of Bologna, Italy

2 THE ISSUE 2

3 THE ISSUE 3 m 3 u 2 m 2 u m k 2 k 3 In any sound seismic engineering design, it is of prime importance the correct identification of the acceleration time histories to be used as inputs for non-linear dynamic analyses. reference design earthquake bin [Giovenale, Cornell, Esteva, EESD 24] k Typically, earthquake bins are identified by means of intensity measures (IMs) Bologna (Italy), P exceedance = % in 5 years Spectral acceleration has been widely proposed as optimal IM. S A S A 25 [cm/s 2 ] T 3 This research work points out the intrinsic limitations in the use of the spectral acceleration and proposes a possible way to overcome these limitations. 3

4 CURRENT METHODS FOR THE BIN IDENTIFICATION USING THE UHS Bologna (Italy), P exceedance = % in 5 years response spectra mean mean ± dev.std S A 3 25 [cm/s 2 ] 2 5 a single spectral ordinate at a time S A [g] single period The Uniform Hazard Spectrum (UHS) represents an ensemble of percentiles. Each piece of information collected in the UHS must be used one at a time and also the UHS does not allow to take into account the autocorrelation function between spectral ordinates at different periods. The current methods based upon the UHS leads to the identification of earthquake bins characterised by 4 the ensemble of spectral ordinates depicted in the figure on the right.

5 COMMENTS / LIMITATIONS..9.8 response spectra mean mean ± dev.std.7.6 S A [g] ) Loss of information (both in terms of spectral ordinates and autocorrelation function if synthetic records are used to create the bin) 2) Expected meaningfulness of the bin only for structures characterised by the selected period (with expected non-significance as soon as the non-linear behaviour is involved) 3) The earthquake bin is strongly structure-dependent instead of being only site-dependent. 5

6 RECENT PROPOSALS TO OVERCOME THESE LIMITATIONS 5 Bologna (Italy), P exceedance = % in 5 years S A 3 25 [cm/s 2 ] average of spectral ordinates at selected periods range of periods Recent research works [Baker & Cornell 25-27, Baker & Bianchini 28, Stanford University] proposed a new intensity measure based on an average of spectral accelerations (for the sake of precision, with the aim of identifying an optimal statistical predictor of the inelastic response of the structure). However, these proposals: still use single pieces of information (single values at different periods) still work with percentiles instead of the full distribution of the spectral ordinates of given hazard for a selected period (which, if coupled with the autocorrelation function, allows to treat the response spectra associated to a given hazard to a stochastic process) 6

7 THE SPECTRUM AS A STOCHASTIC PROCESS SA, SA,2 T T PDF S ( T ) PDF S ( T ) of A of A 2 T T SAn, T T T 2 T The autocorrelation function of the spectral ordinates is known [Baker and Cornell, BSSA 26] [Abrahamson and Silva s NGA report]. In this research work, we provide the probabilistic distribution functions of the spectral ordinates of given hazard which depend on the period. so that the statistical description of the process is complete and ready for: ) generating sample functions of the process to be used as spectra from which design seismic records can be obtained or, vice-versa: 2) verifying that groups of response spectra of historical records of earthquake ground motions (to be used as seismic input) satisfy, one the whole, the requirements of the process 7

8 THIS RESEARCH WORK 8

9 THE GROUND MOTION PREDICTION MODEL A central role in the PSHA procedure, is played by the ground motion prediction model. log GMP linear regression prediction actual observed values residuals.5 With reference to the spectral acceleration law by.5 Berge-Thierry et al. [23]: log R [km], M = ( = ) = R log S T.5s.4323 M log R residual A If we neglect the residual, we have a deterministic function: ( =.5 ) = ( = s) = g (, ) log SA T s.4323 M log R R log S T.5 M R A prediction log S ' T.5s A ( = ) i.e. we assimilate, for the moment, the ground motion parameter to its prediction: log SA ( T =. 5 s) = log SA '( T =.5s) 9

10 THE PSHA PROCEDURE: OVERVIEW Assimilates the occurrence of seismic events to Poisson Processes Adopts the Gutenberg-Richter recurrence law Divides each r Seismo-Genetic Zone min in J sub-areas of circular shape, annular shape, or sectors r max PDF of the magnitude M: qˆi mi t pˆi exp qˆi mi ˆ ˆ M i = i i f m p q t e i f PDF of the distance R: R ( r) = 2 2 r max 2r r min Attenuation law: ( = ) = log S ' T.5 s g M, R given g = g(m,r) with M,R = random variables, it is possible to obtain PDF of g A PDF of SA ' T =.5s due to the contribution of all J sub-areas of the i-th seismic source zone: J ( K i + ) K S, ', ', 2,, ' exp 2,, ' Ai a i = α ij i ij a i ij a i j= f s K K s K s,, i

11 THE PSHA PROCEDURE: STEPS Step : choice of earthquake catalogue and seismic source zones Step 2: the recurrence law: Gutenberg-Richter relationship Step 3: the occurrence law: Poisson arrival process Step 4: the CDF and the PDF of the magnitude for each seismic source zone Step 5: choice of the ground motion prediction model Step 6: the CDF and the PDF of the prediction S A (T j ) of the spectral acceleration due to the seismic action of a single seismic source zone Step 7: the CDF and the PDF of the prediction S A (T j ) of the spectral acceleration due to the seismic action of more than one seismic source zone log λ M = p q M [ ] x ( λ ( mi ) t) λ P X = x = e x! FM m i i = e f ˆ ˆ M m i i = pi qi t e ( qˆ m) tpˆi exp i i m t i qˆi mi t pˆi exp qˆi mi log S ' = b T + b T M + b T log D+ b T D+ b T, s b A 2 3 b 4 5 F s ' = α exp K s ', i SAi, ' a, i ij 2, ij a, i j= J j= J ( K, i + ) K, i fs, ' s, ', 2,, ' exp 2,, ' Ai a i = αijk ik ijs a i K ijs a i I F s F s FSA ' a fs ' ( s ') A a = sa ' ( ') = ( ') A, i SA' a S ' a, i i= ( s ') K

12 RESULTS OF THE PSHA IN TERMS OF MEDIAN Up to here, the PSHA allows to associate: a given hazard level (specific probability of occurrence, for a given site, over a given observation time, T ) P with the median (5-percentile, or also prediction) of the spectral acceleration 5 Bologna (Italy), P exceedance = % in 5 years P = % in 5 years If we do this, for all periods, we obtain the median response spectrum of given hazard: S A [cm/s 2 ]

13 AND THE DISPERSION? The attenuation law estimates the dispersion of the S A (T j ) around its 5 percentile by means of the standard error (of the natural logarithm of the observed values with respect to the natural logarithm of the prediction in the case of spectral acceleration at a specified period T j ): SEln S A ( T j ) Epsilon is defined by engineering seismologists studying ground motion [Abrahamson, BSSA 988] as the number of standard deviations by which an observed logarithmic spectral acceleration differs from the mean logarithmic spectral acceleration of a ground-motion prediction (attenuation) equation [Baker & Cornell, EESD 25]. ε i ( Tj) ln s T ln s ' T = SEln SA( Tj) ai, j a j The statistical distribution of epsilon is generally considered to be well represented by the standard normal distribution. Ε = N (,) For many attenuation laws, Ε SE + log s log a ' T ',, S j A Tj = ',, = ( λζ, ) S T s T m r A j a j S T s T m r LN A j a j λ = ln sa' ζ = SE ln SA ( Tj) ( Tj ) SE is independent from both M and R ln SA( Tj) f s f s ' a = ',, a S T s T S T s T m r A j a j A j a j ' S T s T A j a j has a lognormal distribution 3

14 THE LOGNORMAL DISTRIBUTION From the numerical value of the standard error associated to the attenuation law, it is possible after few mathematical passages, to obtain the numerical values of the mean, standard deviation and cov of the lognormal distribution: ' = ( λ, ζ ) = ln s '( T ) S T s T LN λ A j a j j j j a j ζ = SE ( j) j ln SA T j j sa Tj SE SE 2 2 SA Tj 2 SA Tj λ + ζ ln ' + ln μ = e = e = s ' SA( Tj) sa '( Tj ) a Tj e σ ln 2 2 SE SE = = ' 2 ζ j 2 ln S ( T ) ln S ( T ) μ e ' ' s A j A j a Tj e e SA Tj sa Tj SA Tj sa Tj ζ 2 SE j ln S T cov ' A j = e = e SA Tj sa Tj 2 5 Bologna (Italy), P exceedance = % in 5 years 45 4 To sum up, for each value of T j, it is possible to characterise the conditioned distribution of S A (T j ) given the prediction associated to a given hazard as: S A [cm/s 2 ] 2 f s ' a( Tj) = e SA Tj sa Tj SE s T 2π ln S A ( T ) a( j) j 5 2 s a Tj sa ( T j) ln ln ' 2 SE ln S ( T ) 5 A j

15 IF WE WANT TO GO ON (THE STANDARD APPROACH) f S s ' ' ' ' A a = fsa s s a a fs s A a dsa and, making reference to a selected probability of occurrence, the uniform hazard spectrum can be obtained (which roughly corresponds to the 7-percentile of the lognormal distribution): 5 Bologna (Italy), P exceedance = % in 5 years S A 3 25 [cm/s 2 ] BUT IT IS NOT NECESSARY. 5

16 NUMERICAL EXAMPLE: UNIFORM HAZARD SPECTRUM AND UNIFORM HAZARD SPECTRAL CLOUD spectral cloud S A [g]

17 NUMERICAL EXAMPLE 7

18 for the city of Bologna (ITALY), over an observation time of t = 5 years, for a rare seismic hazard level. P = % in 5 years %.9 in 5 years CDF NUMERICAL EXAMPLE: PSHA RESULTS Bologna (Italy), t =5 years, bedrock, ZS9 subdivision, Berge-Thierry et al. 23 attenuation law P exceedance = Bologna (Italy), t =5 years, bedrock, ZS9 subdivision, Berge-Thierry et al. 23 attenuation law P exceedance = %.9 in 5 years.8.7 prediction of the spectral acceleration: s '.6 a Tj spectral acceleration: s a ( T j ) CDF CPTI2 earthquake catalogue of the Italian territory and ZS9 seismic subdivision for the Italian territory (4 area source zones) spectral attenuation law specifically developed for moderate European earthquakes by Berge-Therry et al. [23].2. F PRED (pred) F GMP (gmp) Spectral Acceleration S A (T=.2s) [cm/s 2 ].2. F PRED (pred) F 3 8 GMP (gmp) Spectral Acceleration S A (T=.4s) [cm/s 2 ] Bologna (Italy), t =5 years, bedrock, ZS9 subdivision, Berge-Thierry et al. 23 attenuation law P exceedance = %.9 in 5 years CDF F PRED (pred) F 69.6 GMP (gmp) Spectral Acceleration S A (T=.6s) [cm/s 2 ] Bologna (Italy), t =5 years, bedrock, ZS9 subdivision, Berge-Thierry et al. 23 attenuation law P exceedance = %.9 in 5 years CDF F PRED (pred) F GMP (gmp) Spectral Acceleration S A (T=.8s) [cm/s 2 ] 8

19 NUMERICAL EXAMPLE: UNIFORM HAZARD SPECTRUM AND UNIFORM HAZARD SPECTRAL CLOUD spectral cloud S A [g]

20 FULL CONDITION ON THE SAMPLE OF SPECTRAL ORDINATES { s } ' (, ) a, Tj sa,2 Tj sa, i Tj sa, n Tj sa Tj = LN λ j ζ j response spectra mean mean ± dev.std.9 spectral cloud S [g] A S A [g]

21 SIMPLIFIED CONDITIONS ON THE MEAN AND THE STD OF THE SAMPLE T j mean s, ai T j std s, ai T j cov s ai, ( T j ) [s] [cm/s 2 ] [cm/s 2 ] cov The spectral ordinates used for the identification of the seismic inputs for design purposes are largely affected by the epistemic error of the ground motion prediction model. 2

22 NUMERICAL EXAMPLE: RESPONSE SPECTRA OF EARTHQUAKE BINS EARTHQUAKE BIN FROM UNIFORM HAZARD SPECTRUM seismic records all scaled to the same S A (T =.5s) value EARTHQUAKE BIN FROM UNIFORM HAZARD SPECTRAL CLOUD seismic records which roughly satisfy, on the whole, the lognormal distribution at each T j..9.8 response spectra mean mean ± dev.std..9.8 response spectra mean mean ± dev.std S A [g].5 S A [g]

23 CONCLUSIONS The research work (through the development of a peculiar PSHA procedure) identifies the statistical characteristics of the ensemble of the spectral ordinates, as computed at multiple periods, for groups of earthquake inputs characterized by given hazard. The statistical characterisation of the spectral cloud as here proposed allows to: overcome the problems deriving from the use of the uniform hazard spectrum; identify earthquake inputs which retain their significance independently from the period range considered; obtain groups of design earthquake inputs which can be used for different structures and for structures with substantial variations in vibration periods; link the identification of the seismic hazard strictly to the site, without involving the structure. A numerical application allows also to recognise that the spectral ordinates used for the identification of the seismic inputs for design purposes are largely affected by the epistemic error/uncertainties of the ground motion prediction model (cov = ). 23

24 THANK YOU!!! 24