PROBABILISTIC SEISMIC DEMAND ANALYSIS CONSIDERING RANDOM SYSTEM PROPERTIES BY AN IMPROVED CLOUD METHOD

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1 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin PROBABILISTIC SEISMIC DEMAND ANALYSIS CONSIDERING RANDOM SYSTEM PROPERTIES BY AN IMPROVED CLOUD METHOD D-Gng Lu, Xio-Hui Yu, Feng Pn 3 nd Gung-Yun Wng 4 Professor, School of Civil Engineering, Hrbin Institute of Technology, Hrbin. Chin Grdute Student, School of Civil Engineering, Hrbin Institute of Technology, Hrbin. Chin 3 Civil Engineer, Shnghi Construction Group technology center, Shnghi. Chin 4 Professor, School of Civil Engineering, Hrbin Institute of Technology, Hrbin. Chin Emil: ludgng@hit.edu.cn, yuxiohui8358@63.com, pfhit@63.com, wnggy@hit.edu.cn ABSTRACT Probbilistic seismic demnd nlysis (PSDA) is n pproch for computing the men nnul frequency of exceeding specified seismic demnd for given structure t designted site, which combines the conventionl probbilistic seismic hzrd nlysis (PSHA) of the designted site with nonliner dynmic nlysis (NDA) of the given structure. Now PSDA hs lredy become n importnt prt of the new-genertion Performnce Bsed Erthquke Engineering (PBEE) proposed by PEER. The commonly used nonliner demnd estimtion method in PSDA is the so clled cloud nlysis procedure in which cloud of demnd vlues re generted when structure is subjected to suite of erthquke ground motion records. However, this method cn only tke into ccount the record-to-record (RTR) vrition in ground motions without considering the rndom of structures. In this pper, n improved cloud method bsed on Ltin Hypercube Smpling (LHS) method is proposed to build probbilistic seismic demnd models (PSDM) to consider inherent rndomness both in structures nd in ground motions. Furthermore, probbilistic seismic demnd frgility nlysis (PSDFA) nd probbilistic seismic demnd hzrd nlysis (PSDHA) re mde bsed on the improved probbilistic seismic demnd models. The methodology is pplied in three-by nd five-storey R.C. frme s n exmple. The probbilistic seismic demnd models with the conventionl cloud method nd the improved cloud method re compred, nd the probbilistic seismic demnd frgility curves nd hzrd curves re derived respectively. KEYWORDS: Seismic Demnd, Improved Cloud Method, Probbilistic Seismic Demnd Model, Seismic Demnd Frgility, Seismic Demnd Hzrd. INTODUCTION Developing from the conventionl probbilistic seismic hzrd nlysis (PSHA) of designted site, probbilistic seismic demnd nlysis (PSDA) is n pproch for computing the men nnul frequency (MAF) of exceeding specified seismic demnd for given structure t designted site, which couples ground motion hzrd curve for the designted site from PSHA with demnd results from nonliner dynmic nlysis (NDA) of the given structure under suite of erthquke ground motion records (Shome & Cornell, 998; Shome, 999). As one of centrl cores of probbilistic seismic performnce ssessment, seismic relibility nlysis, seismic frgility nlysis nd seismic risk nlysis, PSDA hs lredy become n importnt prt of the probbilistic decision-mking frmework of the new-genertion performnce-bsed erthquke engineering (PBEE) proposed by PEER (Cornell & Krwinkler, ; Moehle & Deierlein, 4). The commonly used nonliner demnd estimtion method in PSDA is the so clled cloud nlysis procedure in which cloud of demnd vlues re generted when structure is subjected to set of erthquke ground motion records. However, this method cn only tke into ccount the record-to-record (RTR) vrition in ground motions, or in other words, it generlly tkes the nlyzed structure s deterministic one without considering the inherent rndom properties of the system.

2 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin In order to overcome the shortcomings of the trditionl cloud method in PSDA, n improved pproch combining cloud method (Mckie & Stojdinovic, ) with Ltin Hypercube Smple (LHS) method (Hwng, et l., ) is firstly proposed in this pper. The PSDA is run on three levels: probbilistic seismic demnd model (PSDM) building, probbilistic seismic demnd frgility nlysis (PSDFA), nd probbilistic seismic demnd hzrd nlysis (PSDHA). PSDM is the core of PSDA, nd lso the bsis for PSDFA nd PSDHA, which represents the functionl reltionships between ground motion intensity mesures (IMs) nd engineering demnd prmeters (EDPs) of structures. PSDFA is used to provide the conditionl filure probbilities of exceeding demnd limit sttes of structures given spectrum of IM levels, nd genertes series of seismic demnd frgility curves of structure corresponding to the demnd limit sttes. On the other hnd, PSDHA is done to estimte the men nnul frequency of exceeding given demnd level, nd results in seismic demnd hzrd curve of the structure in conjectured seismic hzrd environment. As cse study, the developed methodology is pplied in three-by nd five-storey R.C. frme structure designed ccording to Chinese seismic design code of buildings (GB5-). The spectrl ccelertion corresponding to the fundmentl period of structures, i.e. S (5%,T ), is selected s IM. The mximum inter-storey displcement ngle (ISDA) is chosen s the globl EDP of structures. The clssicl cloud method nd the improved cloud method re both employed to run the PSDA, nd the PSDMs from the two methods re compred. The PSDFA nd PSDHA re crried on using the different PSDMs. It is demonstrted tht the PSDMs without considering rndom system properties in structures will under-estimte the potentil hzrd probbility of structures.. CLOUD METHOD AND ITS IMPROVEMENT.. Cloud Method The commonly used nonliner demnd estimtion method in PSDA is the so clled cloud nlysis pproch in which cloud of demnd vlues re generted when structure is subjected to suite of erthquke ground motion records. The min ingredients of cloud method to formulte the PSDMs of interest re s follows: () Assemble suite of N ground motions which re pplicble to the re of interest. This suite should represent brod rnge of vlues of the chosen intensity mesure. Actully, the cloud describes the selection of erthqukes with vrible IMs. The rtificil or synthetic site-specific ground motions cn be used just like in Monte Crlo simultion. However, the most commonly used method is the bin pproch which subdivides ground motions into hypotheticl bins bsed on mgnitude (M w ), epicentrl distnce (R), nd locl soil type (Shome et l., 998), s shown in Figure (). As result, the record-to-record rndom properties in ground motions cn be considered thoroughly. In this pper, the ground motion records re selected ccording to the four bins shown in Figure (), nd the spectrl ccelertion S (T ) corresponding to the fundmentl period of the structure is selected s n intensity mesure. () Select the clss of structures to be investigted. Associted with this clss re set of engineering demnd prmeters tht cn be mesured during nlysis to ssess structurl performnce under the considered motions. In this pper, the mximum inter-storey displcement ngle (ISDA) θ mx of structures is chosen s n engineering demnd prmeter. (3) Build the nonliner finite element model to chrcter the clss of structures selected, with provisions to vry designs of the clss through the use of design prmeters. Thus, portfolio of structures is generted by different reliztions of the design prmeters. The finite element model should cpture the mteril nd geometric nonliner chrcteristics in the structure. (4) Perform full nonliner time history nlysis of the structure subjected to the selected ground motion record until ll ground motions nd structurl model combintions hve been exhusted. Key responses or demnd prmeters should be monitored throughout the nlysis. (5) Record nd plot the pek vlues of the selected engineering demnd prmeter (EDP) versus the vlue of intensity mesure (IM) for tht ground motion, s shown in Figure (b). A regression of this cloud of dt is then used to derive the PSDMs between ground motion intensity mesures nd structurl engineering demnd prmeters.. Improved Cloud Method Erthquke is low-probbility, high-consequences, nd lrge-uncertinty hzrd event. The bove-mentioned cloud method cn only consider the inherent rndomness in ground motions in the form of record-to-record vrition. It cnnot tke the inherent rndomness in structurl systems nd the epistemic uncertinty in modeling into ccount, lthough it does consider clss of structures through vrying the design prmeters.

3 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin M w M w =5.7 θ mx R= R(km).5 () Ground motion bins Figure Sketch digrm of cloud method S(g) (b) PSDM In order to consider the rndom of structures, the smples of structures should be generted by Monte Crlo simultion (MCS) to group into motion-structure pirs with the selected ground motions. However, the crude MCS usully requires lrge number of seismic response simultions in order to obtin ccurte sttistics of the structurl EDPs of interest. To reduce the number of crude MCS, mny vrince reduction techniques hve been invented, mong which, Ltin hypercube smpling (LHS) is comprtively efficient method (Hoshino & Tkemur, ). As strtified smpling pproch, LHS cn provide relible estimtes of the response sttistics with significnt smller number of smples tht required by crude MCS. It hs been proved by numerous reserchers tht for given number of rndom smples of structures, the vrince in the response sttistics obtined from LHS is much smller thn tht obtined from crude MCS. To combine Ltin hypercube smpling with the cloud method in PSDA in order to consider the rndom system properties of structures, in this pper, the rnge of ech rndom system property is divided into the sme number of segments with equl probbility s the number of input ground motions t the specified seismic hzrd level. From ech segment, one representtive smple, tken s the medin vlue of the segment, is selected. Then, in the nonliner time history simultions, the smples of ll vribles re rndomly mtched without replcement in sets contining one smple from ech input rndom vrible; these sets re finlly mtched rndomly to the input ground motions. As result, Ltin hypercube smpling cuses the entire rnge of ech rndom vrible to be represented in the set of vribles used in the simultion. The bsic procedure of the improved cloud method bsed on LHS is s follows: () Assemble suite of N ground motions ccording to the bin pproch shown in Figure (); () Select the clss of structures of interest nd the corresponding engineering demnd prmeters; (3) Build the nonliner finite element model of the subject structure; (4) Quntify the letory rndomness nd epistemic uncertinty in structurl systems; (5) Generte N sttisticl smples of the subject structure by LHS, these smples should be generted by smpling on vrious modeling prmeters which my be deemed significnt (e.g. mteril strength, geometric size, dmping rtio, etc.); (6) Estblish set of N ground motion-structure pirs ccording to the strtegy in LHS; (7) Perform full nonliner time history nlysis for ech ground motion-structure smple to simulte cloud of structurl response dt; (8) Regress this cloud of dt to derive the PSDMs between IMs nd EDPs. 3. PROBABILISTIC SEISMIC DEMAND ANALYSIS BY IMPROVED CLOUD METHOD 3.. Probbilistic Seismic Demnd Model (PSDM) The probbilistic seismic demnd models (PSDMs) re results of the improved cloud nlysis in PSDA, which

4 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin provide the reltionships between IMs nd EDPs. The rndomness in seismic demnd of structures comes both from the rndom input erthquke ground motions nd from the rndom. Let us denote the ground motion intensity mesure by S, the rndom engineering demnd prmeter by D. The demnd D cn be written in terms of the product of it medin vlue ˆD nd rndom error ε: D = ˆ Dε (3.) It is generlly ssumed tht ε cn be represented by lognorml distribution with medin nd stndrd devition of lnε: ˆ ε = (3.) σ ln ε = βd S where, β D S is clled dispersion, which is model error in nture depending to some degree on the level of S, we ssume tht it is constnt. The best estimte of the resulting demnd model ˆD, is defined s the medin, or the men of the nturl log of the N IM-EDP dt points: N ˆ ln( D) = ln( Di ) (3.3) N i = where, D i is the ith EDP dt. The dispersion is dependent on the number of prmeters (df) being estimted in liner regression on the EDP dt: N i= β DS = ( ln( D) ln( ˆ )) i D N df where, df =, since we only hve two prmeters D nd S. It hs been observed by numerous reserchers tht the medin demnd nd the spectrl ccelertion follow the typicl power lw reltionship: Dˆ = ( S ) b (3.5) where, nd b re prmeters, which cn be determined from liner lest-squres regression technique. In evidence, the reltionship between the medin demnd nd the spectrl ccelertion in the log spce is liner: ln( Dˆ ) = ln( ) + bln( S ) (3.6) 3.. Probbilistic Seismic Demnd Frgility Anlysis (PSDFA) PSDM cn be used to provide probbilities of exceeding demnd limit sttes, given mesure of intensity. Such reltionship is termed demnd frgility curve. Therefore, the demnd frgility cn be defined s conditionl filure probbility tht drift demnd D exceeds certin drift limit d, given specific spectrl ccelertion S : FR( x) = P[ D> d S = x] (3.7) The frgility is modeled commonly by lognorml cumultive distribution function (CDF), choice tht hs been supported by numerous reserch progrms during the pst decde in disprte fields. Then it is described by FR( x) = Φ [ln( x/ mr)/ βr] (3.8) where, m R is the medin spectrl cpcity, β R is the logrithmic stndrd devition, nd Φ[] is the stndrd norml integrl. The prmeters m R nd β R mesure the inherent rndomness in seismic cpcity. With the ssumption tht the seismic demnd cn lso be described by lognorml distribution, with medin s defined in Eq. (3.5) nd dispersion s defined in Eq. (3.4), we cn determine the prmeters in Eq. (3.8). Note tht Eq. (3.7) cn be further rewritten s: ln( d) ln( Dˆ ) FR ( x) = Φ (3.9) βds Substituting Eq. (3.5) in Eq. (3.9), one hs (3.4)

5 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin ln( d) ln( ) b ln( d) ln ( x) ln( x) FR ( x) = Φ =Φ b (3.) β β DS DS b Compring Eq. (3.8) with Eq. (3.), we cn obtin the prmeters m R nd β R s follows: ln d ln mr = exp (3.) b βd IM β R = (3.) b The demnd limit sttes re multiple levels corresponding to the different seismic dmge sttes, or seismic performnce levels of structures. For exmple, we generlly clssify the seismic dmge sttes of structures into five levels: intct, minor dmge, moderte dmge, severe dmge, nd collpse. Therefore, the corresponding four levels of demnd limit sttes re: equl to or lrger thn minor dmge demnd, equl to or lrger thn moderte dmge demnd, equl to or lrger thn severe dmge demnd, nd equl to or lrger thn collpse demnd. In order to chrcterize the four performnce levels, we denote the four deterministic limit vlues of i drift demnd s d ( i =,,3,4). Then, we cn derive the four levels of seismic demnd frgilities: i FRi ( x) = P[ D> d S = x] =Φ [ln( x/ mri )/ βr ] (3.3) i ln d ln mri = exp (3.4) b 3.3. Probbilistic Seismic Demnd Hzrd Anlysis (PSDHA) PSDM cn be lso used to estimte the men nnul frequency of exceeding given demnd, nd results in structurl demnd hzrd curve in conjectured hzrd environment. Actully, demnd hzrd curve is the finl result of PSDHA, just like the seismic hzrd curve of the site in the conventionl PSHA. The demnd hzrd is defined s the probbility of filure of exceeding given demnd under spectrum of possible erthqukes, which is determined by convolving the demnd frgility curve of the structure nd the seismic hzrd curve of the designted site: HD( d) P[ D> d] = P[ D> d S = x] dhs ( x) = F ( ) ( ) R x dhs x (3.5) where, HS ( x ) is the seismic hzrd function derived from PSHA of the site. For specific building sites, it hs been shown by mny reserchers tht t moderte to lrge vlues of ground ccelertion, there is logrithmic liner reltion between first modl spectrl ccelertion S nd the exceednce probbility H S (x) (Cornell et l., ). This reltionship implies tht S cn be described by Type II distribution of lrgest vlues k k HS ( x) = p[ S ] exp[ ( / )] > x = x k kx (3.6) where k = chrcteristic extreme, k = shpe prmeter. The hzrd curve cn be pproximtely liner on log-log plot in the region of interest (Cornell et l., ): k HS ( x) k x (3.7) After integrtion of lognorml demnd frgility function defined in Eq. (3.) nd derivtive of seismic hzrd function presented in Eq. (3.7), the demnd hzrd in Eq. (3.5) cn be obtined s n explicit formultion (Cornell et l., ): d k HD( d) HS ( S )exp DS b β = (3.8) d where, S represents the spectrl ccelertion corresponding to specific drift limit d. Their reltionship is defined in Eq. (3.5), which cn be trnsformed to the following expression d / b S = ( d/ ) (3.9)

6 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin d Then, H( S ) is the seismic hzrd with spectrl ccelertion of d S, representing the probbility of ground motion intensity tht cuse drift d or more. Eq. (3.8) is the most importnt result of PSDHA, from creful inspection of it, we cn see tht the seismic hzrd curve for the drift demnd H D (d) is equl to the seismic hzrd function of the site evluted t the spectrl ccelertion corresponding to this drift demnd, times correction fctor relted to dispersion in the drift demnd estimtion for given spectrl ccelertion. 4. CASE STUDY: R.C. FRAME STRUCTURES 4.. Bsic Dt A three-by nd five-storey R.C. frme structure, s shown in Figure, is tken s the exmple in this cse study. It is designed ccording to Chinese seismic design code of buildings (GB5-). F5 F4 F3 F F.KN/m 3.3KN/m 3.3KN/m 3.3KN/m 3.3KN/m 3.3m 3.3m 3.3m 3.3m 4.4m 6.m.4m 6.m Figure Three-by nd five-storey R.C. frme 4.. PSDM Bsed on Cloud Method The S (5%,T ) is chosen s the IM, nd the mximum ISDA θ mx is chosen s the EDP. ground motion records re selected from the dtbse of PEER. We tke Ms = 5.7 nd D = 3(km) s the limit vlues to crve these ground motion records into four bins, s shown in Figure (). The nonliner finite element model of the structure is developed by OpenSees pltform. Without considering the rndom, the nonliner time history nlysis is done for the model subjected to the sets of ground motion records. The PSDM bsed on cloud method is obtined vi log-liner regression method s ln( ˆ θ mx ) =.7ln( S ) 3.685, β DS =.7 (4.) The PSDM in this cse is shown in Figure 3() PSDM Bsed on Improved Cloud Method In this cse, we consider four rndom mteril properties of the structure: the yield strength f y nd initil elstic modulus E of steel; the compressive strength f c nd crushing strength f cr of concrete. The sttistics of the four rndom vribles re shown in Tble 4.. All vribles re ssumed s norml distribution, nd independent on ech other. Tble 4. Sttistics of rndom vribles Distribution prmeters Distribution prmeters Vribles Vribles Men vlue Std Men vlue Std f y(n/mm ) f cr (N/mm ) E (N/mm ) 4 4 f (N/mm ) c

7 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin According to the strtegy in the improved cloud method, the 4 mtrix of rndom smples is firstly generted by using LHS method, nd smples of the finite element model re obtined. Then couples of ground motion records nd structurl smples re combined. The nonliner time history nlysis is exhusted for ll ground motion-structure pirs. From the recorded spectrl ccelertion-isda results, the PSDM considering the rndom is derived by using the improved cloud method: ln( ˆ θ mx ) =.ln( S ) 3.578, β DS =.3 (4.) The PSDM in this cse is shown in Figure 3(b) Time history nlysis results ln(isda)=.7ln(s) R =.946 σ = Time history nlysis results ln(isda)=.ln(s) R =.946 σ = ln(θ mx ) -5-6 ln(θ mx ) ln(s/g) () Cloud method ln(s/g) (b) Improved cloud method Figure 3 PSDM in two cses 4.4. Drift Demnd Frgility Curves In Chinese seismic design code of buildings (GB5-), the limit vlue of the mximum ISDA for R.C. frme structure in elstic servicebility limit stte is /55, while in elstoplstic collpse limit stte /5. In this 3 pper, the four levels of drift demnd limit stte re chosen s: d = / 55, d = / 55, d = 4 / 55, 4 nd d = /5. The drift demnd frgility curves bsed on the bove PSDMs re shown in Figure 4. From Figure 4, we cn see tht the drift demnd frgility curves bsed on the PSDMs without considering rndom tend to be less thn tht bsed on the PSDMs considering rndom. It mens tht drift demnd frgility curves without considering rndom under-estimtes the conditionl filure probbility given the occurrence of the erthquke ground motion. Exceedence probbility Exceedence probbility Considering rndom. Without considering rndom S (). Considering rndom. Without considering rndom S d = /55 (b) d = /55

8 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin.9.8 Considering rndom Without considering rndom.9.8 Considering rndom Without considering rndom Exceedence probbility Exceedence probbility S S 3 (c) d = 4/55 (d) d 4 = /5 Figure 4 Drift demnd frgility curves bsed on different PSDMs 4.5. Drift Demnd Hzrd Curves As shown in Figure 5, the seismic hzrd curve of PSHA of the site in this exmple is pproximted s.38 H ( ).38 ( ) S x = x (4.3) In Figure 5, S (63,%, 5yr), S (%, 5yr) nd S (%, 5yr) re the vlues of S (5%, T ) corresponding to different exceednce probbilities in 5 yers. The drift hzrd curves coupling the seismic hzrd curve nd the drift frgility curves bsed on different PSDMs re derived, s shown in Figure 6. ln(exceedence probbility) S(63.%,5yrs) S with different exceedence probbility in fifty yers ln(pf)=ln(.38)-.38xln(s) S(%,5yrs) Men nnul exceedence probbility considering rndom properties in stucture without considering rndom properties in structure -9 S(%,5yrs) ln(s/g) Figure 5 Seismic hzrd curve for the designted site θ mx Figure 6 Drift demnd hzrd curves bsed on different PSDMs From Figure 6 we cn see tht, the drift demnd hzrd curve bsed on the PSDM without considering rndom is lower thn tht bsed on the PSDM considering rndom. Tht is to sy, PSDFA without considering rndom cn under-estimte the potentil hzrd probbility of structures. It is similr to the conclusion obtined from Figure 4. Therefore, it is necessry to consider rndom of structures when crrying on PSDA. 5. CONCLUSIONS This pper proposes n improved cloud method in PSDA to consider rndom of structures,

9 The 4 th World Conference on Erthquke Engineering October -7, 8, Beijing, Chin which is coupling of the conventionl cloud method nd Ltin hypercube smpling (LHS) method. The probbilistic seismic demnd nlysis of structures is crried on three levels: PSDM building, PSDFA nd PSDHA. A three-by nd five-storey R.C. frme structure is tken s cse-study exmple. The PSDMs bsed on cloud method nd improved cloud method re derived nd compred. The demnd frgility nd hzrd curves re obtined by PSDFA nd PSDHA, respectively. The conclusions re summrized s follows. () The improved cloud method proposed in this pper cn efficiently consider rndom in PSDA. () Both PSDFA nd PSDHA without considering rndom tend to under-estimte the structurl potentil seismic hzrd. Therefore, it s necessry to mke PSDA considering rndom system properties. 6. ACKNOWLEDGEMENTS The support of Ntionl Science Foundtion of Chin through projects (Grnt No. 975, , 585) is gretly pprecited. Any opinions, findings, nd conclusions or recommendtions expressed in this mteril re those of the uthors nd do not necessrily reflect those of the Ntionl Science Foundtion of Chin. REFERENCES Cornell, C.A, nd Krwinkler, H. (). Progress nd chllenges in seismic performnce ssessment. PEER Ceneter News 3:, -4. Cornell, C.A, Jlyer, F, Hmburger, R.O, nd Foutch, D.A. (). Probbilistic Bsis for SAC Federl Emergency Mngement Agency Steel Moment Frme Guidelines. ASCE Journl of Structurl Engineering 4:4, Hoshino, N, nd Tkemur, A. (). On reduction of finite smple vrince by extended Ltin hypercube smple. Bernoulli, 6:6, Hwng, H, Liu, J-B, nd Chiu, Y-H. (). Seismic Frgility Anlysis of Highwy Bridges. Report. Center for Erthquke Reserch nd Informtion. Mckie K, nd Stojdinovic, B. (). Probbilistic seismic demnd model for Cliforni highwy bridges. ASCE Journl of Bridge Engineering 6:6, Moehle, J. nd Deierlein G.G. (4). A frmework methodology for performnce-bsed erthquke engineering. Proceedings of the 3th World Conference on Erthquke Engineering, Vncouver, Cnd, -6 (CD-ROM). Ntionl Stndrd of Chin P.R. (). Seismic Design Code of Buildings (GB5-). Building Industry Press of Chin, Beijing. Shome, N, Cornell, C.A, Bzzurro, P, nd Crbllo, J.E. (998). Erthqukes, records, nd nonliner responses. Erthquke Spectr 4:3,