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 T. Tromett, S. Slvestr and G. Gasparn Assocate Professor, DISTART Dept. of Structural Engneerng, Unversty of Bologna, Italy Ph.D Researcher, DISTART Dept. of Structural Engneerng, Unversty of Bologna, Italy Emal: ABSTRACT : Response spectra have een used n the years not only to otan estmatons of the sesmc response for gven dynamc system n response spectral analyss, ut also as a useful reference for the dentfcaton of desgn tme-hstory nputs (when tme-hstory analyss are requred. Ths second use s somehow mproper. Indeed, unform hazard response spectra are capale of provdng nformaton regardng the value of the spectral acceleraton (characterzed y a gven proalty of exceedance for a gven precse system perod, ut may prove to e msleadng when the overall shape of the desgn response spectra s used to dentfy groups of tme-hstory nputs. In ths paper, a tool whch can e used for the dentfcaton of groups of desgn tme-hstory earthquake nputs for Performance-Based Sesmc Desgn applcatons s developed. Frst, a pecular Proalstc Sesmc Hazard Analyss, whch allows to explctly handle (exclude/nclude and numercally quantfy the epstemc uncertanty due to the error of the ground-moton predcton (attenuaton model and to clearly dstngush t from the space-tme aleatory varalty, s developed. Second, grounded on ths Proalstc Sesmc Hazard Analyss, a relatonshp s otaned etween a gven hazard level and the statstcal characterstcs of the ensemle of response spectral ordnates, as computed at multple perods. Ths allows to dentfy the characterstcs that must e possessed y the groups of desgn earthquake nputs, y means of a newly ntroduced tool, heren defned as spectral cloud. KEYWORDS: ground moton predcton model, aleatory varalty, epstemc uncertanty, proalstc sesmc hazard analyss, spectral acceleraton

2 . INTRODUCTION In any sound sesmc engneerng desgn, t s of prme mportance the correct dentfcaton of the acceleraton tme-hstores to e used as nputs for the dynamc analyses. Wthn a Performance Based Sesmc Desgn framework [AOC Vson, 995], the choce of the reference desgn sesmc nput (commonly referred to as earthquake ns s deeply rooted upon ther proalstc dentfcaton. Typcally, the earthquake ns are dentfed y means of earthquake ntensty measures (IMs. IMs consstng of a scalar or vector-valued comnaton of selected ground moton parameters (GMPs assocated to a gven proalty. In recent years, many research works [Govenale et. al, 4, Tromett et al, 7] have focused on the dentfcaton of the optmal IM for earthquake n creaton. In ths paper, a specfc procedure s developed for the dentfcaton of the characterstcs that groups of earthquake nputs must posses n order to represent a sound engneerng choce of the desgn nput. In partcular, the procedure allows to dentfy the propertes whch must e satsfed y the desgn tme-hstores through the use of a newly ntroduced tool, heren defned as spectral cloud. The spectral cloud eng capale of provdng a proalstc condton upon the spectral ordnates at multple perods.. THE GROUND MOTION PREDICTION MODEL A central role, n the procedure for the dentfcaton of the spectral cloud, s played y the ground moton predcton model. The attenuaton law provdes an estmate of a selected ground moton parameter GMP gven the magntude M, the ste-epcentre dstance R and the local sol characterstcs. In general, attenuaton laws are otaned y means of a lnear regresson of the avalale data n logarthmc scale, so that they may e wrtten n the followng general form: log GMP predcton + resdual (. The predcton generally depends on the magntude M, the ste-epcentre dstance R and the local sol characterstcs (whch are fxed for a selected ste, as gven y: predcton g ( M, R (. where g s a determnstc functon of M and R. In common hazard calculatons, the magntude M and the dstance R are assumed as random varales whch nvolve the space- and tme-related sources of randomness (to whch the attenuaton law does not contrutes at all, so that the predcton may e seen as a determnstc functon of two random varales; the resdual s an addtonal random varale whch takes nto account the randomness of the error assocated wth the predcton provded y the attenuaton law or, usng the words of Berge-Therry et al. [3], the ntrnsc varalty of the data, and the error due to the attenuaton model. To etter explan the relatonshps etween these random varales, Equatons. and. can e rewrtten as Y Y' + Z g( M, R + Z, Y' g( M, R, where Y log GMP and Z resdual It s possle to dstngush two asc dfferent sources of randomness. The frst s related to the randomness n space and tme of occurrence of events and pertans to M and R. The second s related to the error n the attenuaton law n provdng a predcton for the ground moton parameter for a gven event (model-dependent source and pertans to Z. Accordng to the unamguous termnology suggested y Arahamson [988], t s possle to refer to the frst randomness as aleatory varalty (gven that t s the natural varalty of the physcal phenomenon, and to the second one as epstemc uncertanty (gven that t s the scentfc uncertanty n the smplfed model we use to descre the effects of the physcal phenomenon. Wthout wantng to get to the heart of the matter, the terms aleatory varalty and epstemc uncertanty, whch are commonly used n generc ways and often mxed up, are here used to underlne the separaton etween the two dstnct sources of randomness. To explctly take nto account ths separaton, t s here defned a new varale as the predcton,, of the ground moton parameter GMP: Y ' (.3 so that we can also express the predcton Y ' of the ase- logarthm of the ground moton parameter as Y' log GMP'. It s possle to wrte log GMP log + Z.

3 Note that, y defnton, the predcton ( GMP log log medan Y ' tres to capture oth the mean value, μ log GMP, and the medan,, of the ase- logarthms of the oserved values of the ground moton parameter, so that μ and log ( log GMP log GMP. medan From the theory of functons of random varales, and gven that the logarthm functon s monotonc, the medan of ( log the GMP, ( GMP medan, may e otaned as GMP medan GMP, that s ( GMP. Note medan medan that the same concluson does not apply to the mean value. As per the aove consderatons, the (whch does not account for random varale Z s assocated to the aleatory varalty only, whle the GMP s assocated to oth the aleatory varalty and the epstemc uncertanty. It s thus fundamental to clearly dstngush etween GMP and. To quanttatvely account for ths epstemc uncertanty, t s generally made reference to the standard error of estmate. 3. THE PSHA PROCEDURE In order to assocate gven values of the ground moton parameter GMP to correspondng proaltes of exceedance, a PSHA procedure s generally carred out. The PSHA procedure leads to the dentfcaton of the f gmp, and the Cumulatve Dstruton Functon (CDF, Proalty Densty Functon (PDF, GMP FGMP gmp, of the GMP, as computed over a gven oservaton tme t. The procedure (ased upon the consoldated approach suggested y Cornell [968], s composed of the 8 steps summarzed elow. Step : dentfcaton of the reference earthquake catalogue (same hypothess as per Cornell s approach. Step : the recurrence rate λ of sesmc events and the sesmc magntude M s assumed to follow λ s gven y: Gutenerg-Rchter relatonshp. For each -th sesmc source zone, the event rate ( m λ ( m pˆ exp( qˆ m wth pˆ exp( p ln and qˆ q ln. Step 3: the occurrence of sesmc events s modeled as a Posson arrval process. Consequently, the proalty, P[ X x], that exactly x events characterzed y magntude M strctly larger than m occur wthn the -th sesmc source zone over a gven oservaton tme t, s gven y: [ ] x ( λ ( m t λ m t P X x e (3. x! Step 4: the PDF of the magntude M for each sesmc source zone over a gven oservaton tme t s otaned as J q ˆ m t p ˆ exp q ˆ ˆ ˆ m fm m α p q t e (3. wth α S / S, S representng the area of the -th su-dvson of the -th sesmc source zone and representng the area of the -th sesmc source zone. Step 5: for a gven ste, the ground model predcton gves: log T + T M + T log D + T D + T, s ( Where D represents the dstance (ether epcentral or hypocentral etween the locaton of the event and the ste T, of nterest, s represents the local sol characterstcs (whch are fxed for a selected ste, 3 ( T, 4 ( T and 5 (, S T, T s are approprate coeffcents whch may depend on the structural perod of vraton T and on the local sol characterstcs s. f r are known, t s possle to otan: Step 6: once M f m and R

4 wth: J K, F ( gmp' α exp K gmp' (3.4 GMP', J ( K + K ( ',, ' exp, ' α,, (3.5 f gmp K K gmp K gmp K qˆ, ( T ln K ˆ, t p exp K, T + 3 T log R + h + 4 T R + h + 5 T, s ln (3.7 Step 7: assumng that the sesmc actvtes of all sesmc source zones are ndependent from each other: and: ( ' ( ' GMP' I (3.6 F gmp F gmp (3.8 f ( gmp' Step 8: Fnally the PDF of the GMP can e expressed as: where ' F ( gmp' gmp' ( ' GMP GMP gmp ' (3.9 f gmp f gmp f gmp dgmp' (3. f gmp s known, whle f GMP gmp ' gmp s to e determned as: (, f gmp f gmp f m r dm dr (3. GMP gmp ' GMP gmp ', m, r M, R gmp ' In general, most attenuaton laws provdes, together wth the predcton equaton, the standard error, log GMP, of the ase- logarthm of the oserved values of the ground moton parameter wth respect to the ase- logarthm of the predcton gmp ', as gven y: N ( log gmp, a. d. log gmp ' log GMP (3. N where gmp ad,.. represent the -th value of the ground moton parameter among the avalale data used to defne the predcton equaton and N s the numer of the avalale data. It s then common practce, especally for spectral acceleraton laws, to defne the normalsed logarthmc resdual (epslon, ε, as follows: log gmp log gmp' ε (3.3 where gmp s the -th oserved value of the ground moton parameter. If, n Equaton (3.3,, the magntude m and dstance r used to compute the predcton gmp ' are equal to those assocated to the gmp data entry (and f the log GMP s that assocated to the predcton equaton used to compute the gmp ', the normalsed logarthmc resdual (epslon s a random varale Ε charactersed y the standard normal dstruton: Ε N (, (3.4 From Equaton (3.3, t s then possle to express GMP gmp', m, r as a functon of the random varale Ε log GMP

5 (epslon only, as gven y: log log ' ',, GMP gmp GMP gmp m r Ε + (3.5 whch, together wth Equaton (3.4 leads to: GMP gmp', m, r LN ( λζ, (3.6 wth: λ ln gmp' (3.7 and ζ lngmp, (3.8 For many attenuaton laws, f gmp s ndependent from oth M and R. As a consequence, ln GMP GMP gmp ', m, r does not depend on M and R, so that Equaton (3. ecomes: fgmp gmp '( gmp fgmp gmp ', m, r ( gmp (3.9 whch gves the followng fundamental result: GMP gmp' LN ( λ ln gmp', ζ ln GMP (3. whch characterses the dstruton of the GMP (due to the epstemc error assocated wth a fxed proalty of exceedance of the predcton gmp '. Eq. (3. allows to otan the moments/parameters whch characterse GMP gmp ': 4. UNIFORM HAZARD INPUT GROUP σ ' ' μ GMP gmp gmp e ζ (3. ' μ ' (3. GMP gmp GMP gmp e ζ ' e ζ δ (3.3 GMP gmp For the development of practcal sesmc desgn, the structural engneer needs to have at hand groups of earthquake nputs charactersed y a gven sesmc hazard level. The dentfcaton of earthquake nputs of gven hazard s generally otaned through a two-step assocaton. The frst step encompasses the assocaton etween an exceedance proalty, P, and a threshold value, w, of a gven physcal quantty, w, whch dentfes the ntensty of the sesmc event (often called Intensty Measure IM: P w (4. Ths assocaton can e expressed n words as follows: P proalty that a gve physcal quantty w whch dentfes the ntensty of the sesmc event exceeds a specfed threshold value w, for a selected ste over a gven oservaton tme t. The second (and susequent step sees the dentfcaton of the desgn sesmc records on the ass of the threshold value w. In the lght of the oservatons drawn from the study of the resduals developed n the prevous sectons, the hazard assocaton prolem expressed y Equaton (4. may assume dfferent meanngs dependng on the way the quantty w s assocated to the proalty of exceedance: ndeed, ths assocaton may e otaned ether (A n terms of the ground moton parameter tself (gmp or (B n terms of ts predcton ( gmp ', as gven n detal elow. 4. Approach A The approach A assocates, to a gven proalty P, a specfc threshold value gmp of the gmp, as otaned from:

6 gmp P gmp as per P f gmp dgmp (4. The assocaton provded y Equaton (4. accounts smultaneously for oth the ntrnsc hazard of the physcal phenomenon (.e. the space-tme aleatory varalty and the error n the evaluaton of the parameter whch dentfes the hazard (.e. the epstemc uncertanty due to the error of the attenuaton model, as they are grouped n the gmp. 4. Approach B The approach B assocates, to a gven proalty P, a specfc threshold value from: gmp' GMP gmp ' of the gmp ', as otaned P gmp ' as per P f gmp ' dgmp ' (4.3 The assocaton provded y Equaton (4.3 accounts only for the ntrnsc hazard of the physcal phenomenon (.e. the space-tme aleatory varalty, as contaned n the gmp '. Note that Equaton (4.3 may e seen as the result of a PSHA whch does not take nto account the ground moton varalty. The epstemc uncertanty due to the error of the attenuaton model may e then convenently accounted for through the use of approprate tools (e.g. gven percentle values of the EDP to otan desred confdence levels capale of handlng the dstruton of the GMP gmp ': gmp ' GMP gmp ' as per f GMP gmp' ( gmp ( APPLICATION TO THE SPECTRAL ACCELERATION Let us here specalze the analyss of aove wth reference to spectral acceleraton, SA( T, n what follows, the notatons GMP, gmp, and gmp ' thus specalse nto SA ( T, sa( T, SA '( T and sa' ( T, respectvely. 5. Unform hazard spectrum Followng approach A for the dentfcaton of the sesmc hazard, Equaton (4. specalses as follows: sa( T P s T as per P f s T ds T (5. a SA T a a As far as the dentfcaton of groups of sesmc records of gven hazard level s concerned, approach A s the common approach used n current performance ased sesmc desgn wth ts lmtatons. It s known that the ensemle of the spectral ordnates thus otaned (often referred to as the unform hazard spectrum, defned as the locus of ponts such that the spectral acceleraton value at each perod has an exceedance proalty equal to the specfed target proalty does not represent the spectrum of any sngle earthquake, gven that each PSHA, for each specfed T, s ndependent from the others. Consequently, each pece of nformaton collected n the unform hazard spectrum must e used one at a tme. 5. The spectral cloud It s known that the response spectrum s the curve that connects the m spectral ordnates, computed at the m multple reference perods. When a group of n sesmc records s consdered, n response spectra are otaned. We here defne as the spectral cloud the ensemle of the n m spectral ordnates, computed at the m multple reference perods, of a group of n sesmc records. Note that, n addton to the complete knowledge of the characterstcs of the dstruton of the m random varales, were the autocorrelaton functons of the response

7 spectra known, the n response spectra could e seen as a set of n sgnals of a stochastc process. 5.3 Unform hazard spectral cloud Followng approach B for the dentfcaton of the sesmc hazard, Equatons (4.3 and (4.4 specalse as follows: sa '( T P s ' T as per P f s ' T ds ' T (5. a ( S ' A T a a sa' ( T SA( T sa' ( T as per f '( s S a( T A T sa T The sesmc records of the group are charactersed y values of the ground moton parameter a, ( the whole, ft the dstruton ( SA T sa' T a (5.3 s T whch, on f s T of Equaton (5.3. The advantages of approach B over approach A ecome clear when more than one reference perods are consdered. Indeed, oth Equatons (5. and (5.3 can e easly appled smultaneously for dfferent perods T and T k ( k. Note that the smultaneous applcaton of Equatons (5. and (5.3 at dfferent reference perods T mplctly determne a unform hazard spectral cloud. Introducng the unform hazard spectral cloud, approach B enales to overtake the unform hazard spectrum concept. The applcaton of Equaton (5.3 nvolves the explct dentfcaton of f ( s ' a( T. Ths can e done n the lght of the epslon parameter, commonly used SA T sa T to dentfy the dsperson of the spectral acceleraton attenuaton law. The epslon parameter evaluated, at a specfed perod T, for the -th sesmc record, ε ( T, s gven y: ln sa, ( T ln sa' ( T ε( T (5.4 ln SA( T where the numercal values of are generally provded along wth the formulatons of the spectral ln S A ( T acceleraton attenuaton laws. The statstcal dstruton of epslon s generally consdered to e well represented y the standard normal dstruton, as t was assumed n the analytcal developments of prevous n sectons. Consequently, all conclusons drawn startng from Equaton (3.3 and elucdated throughout secton stll hold for the case of SA ( T assumed as GMP. Thus, the fundamental result provded y Equaton (3.9 leads to:.e.: In words, A ( a' ( f s ' a( T e SA T sa T s T π ln SA ( T a( ' (, A a ln sa T ln sa ' T ln SA( T, T (5.5 S T s T LN λ ζ, T (5.6 S T s T has a lognormal dstruton, wth the followng parameters and moments: λ a (5.7 ζ (5.8 ln S A ( T ln s '( T λ + ζ ln sa '( T + ln S ( T A ln SA( T μ e e s ' S ' a ( T e (5.9 A T sa T

8 ζ ln SA T ln SA( T σ μ e s ' ' ' a ( T e e SA T sa T SA T sa T ζ (5. ln S T ' A δ e e (5. SA T sa T The condton gven y Equaton (5.6, as appled smultaneously at any reference perod T, characterses the unform hazard spectral cloud and ndrectly dentfes the group of unform hazard earthquake nputs. 6. FULL CONDITION ON THE SAMPLE OF SPECTRAL ORDINATES The group can e dentfed mposng that, at each perod T, the ensemle of the n sample values {,,,, } a a a a n a (,,,, s T s T s T s T s T follows the lognormal dstruton, as gven y: 7. CONCLUSIONS { s } ' (, a T sa T sa T sa n T sa T LN λ ζ (6. The research work presented n ths paper dentfes the statstcal characterstcs of the ensemle of the spectral ordnates, as computed at multple perods, for groups of earthquake nputs characterzed y gven hazard level (here defned as the unform hazard spectral cloud. The statstcal charactersaton of the spectral cloud (whch can e assmlated to a lognormal random process as here proposed allows to: treat separately and ndependently the epstemc uncertanty due to the error of the attenuaton model from all other tme- and space-related sources of aleatory varalty; dentfy earthquake nputs whch retan ther sgnfcance ndependently from the perod range consdered; otan groups of desgn earthquake nputs whch can e used for dfferent structures and for structures wth sustantal varatons n vraton perods; lnk the dentfcaton of the sesmc hazard strctly to the ste, wthout nvolvng the structure. ACKNOWLEDGEMENTS Dr. J.W. Baker of Stanford Unversty (Calforna s thanked for the useful dscusson had durng the ICASP conference n Tokyo (August 7 whch nspred most of the second part of the paper and for provdng useful materal and references. Fnancal supports of Department of Cvl Protecton (Relus 5 Grant Task : Assessment and mtgaton of the vulneralty of r.c. exstng uldngs s gratefully acknowledged. REFERENCES AOC Vson Commttee. Performance-ased sesmc engneerng for uldngs. Report prepared y Structural Engneers Assocaton of Calforna, Sacramento, CA, 995. Govenale P, Cornell CA, Esteva L. Comparng the adequacy of alternatve ground motons ntensty measures for the estmaton of structural responses. Earthquake Engneerng and Structural Dynamcs 4; 33: Tromett T, Slvestr S, Malavolta D, Gasparn G. A methodology for determnaton of effcent earthquake ns for Performance Based Sesmc Desgn. Proceedngs of the Tenth Internatonal Conference on Applcatons of Statstcs and Proalty n Cvl Engneerng (ICASP, Tokyo, Japan, July 3 August 3, 7. Arahamson NA. Statstcal propertes of peak ground acceleratons recorded y the Smart array. Bulletn of the Sesmologcal Socety of Amerca 988; 78: 6 4. Berge-Therry C, Cotton F, Scott O, Grot-Pommera DA, Fukushma Y. New emprcal response spectral attenuaton laws for moderate European earthquakes. Journal of Earthquake Engneerng 3; 7(: 93-. Cornell CA. Engneerng sesmc rsk analyss. Bulletn of the Sesmologcal Socety of Amerca 968; 58: