ON THE GENERATION OF THE DESIGN EARTHQUAKE GROUND MOTION TIME HISTORY

Transcription

1 ON THE GENERATION OF THE DESIGN EARTHQUAKE GROUND MOTION TIME HISTORY by Izuru Okawa 1), Toshhde Kashma 1), Haruyuk Ktamura 2), Masanobu Tohdo 3), Shgek Saka 4), Masaharu Tangak ), Kunak Yamagsh ), Kohj Naraoka 6) ABSTRACT In desgn, t s common that the desgn moton s provded n the form of response spectrum. It s possble to calculate the maxmum response wth the desgn spectra usng the response spectral method such as the SRSS method, f the buldng s assumed lnear durng the earthquake. However, for such types of severe ground motons, the buldng wll exceed ts elastc lmt. The approxmate method lke SRSS s not suffcent for the relable response values. In such case we need to generate the desgn moton tme hstores. There s an ssue there, what knd of tme hstores should we make use. There may be two optons. (1) The past strong moton records (2) The smulated ground moton consderng the earthquake actvty n the area. It s known that n Japan, specfc buldngs such as the hgh rse buldngs or base-solated buldngs that necesstate the verfcaton of the realstc behavor durng earthquake for the reason that there are stll not suffcent desgn experences for the types of buldngs. In ths paper, the varatons of smulated ground motons wth respect to ther propertes n ground moton characterstcs and structural responses. KEY WORDS: Smulated motons Phase angles, Spectrum-compatble motons, Envelope functon, Duraton tme, Varatons 1. INTRODUCTION The conventonally used varous standard desgn spectrum, s a result of statstcal analyss of collected ground moton records for varous condtons. There are two types of desgn spectra. One s an envelope or ntegraton of many expected ground motons that are possble to occur n certan perod of tme. The other s one that s constructed based on the earthquake magntude, dstance and./or sol condton that s also expected to occur. From the vewpont of tme hstory generaton, the latter assumpton s sutable because the tme hstory s prmarly based on the mage of the expected earthquake event. The duraton tme, for example, depends on the earthquake magntude. However, the former type s, n realty, frequently found as the standard spectrum n engneerng practce. The tme hstory ncludes many factors related to varaton representng the accdental occurrence or rreproducblty of earthquake event. As a fundamental rule, several tme hstores are used for analyses. However, lmted number of tme hstores s used n realty due to the lmt n tme and cost. In recent revson of the notfcaton n the Japan Buldng Standard Law, t s suggested that several statstcally ndependent tme hstores should be used for the dynamc analyss. Fgure 1 shows the varaton of story drft for seven smulated nput motons that are compatble to the desgn spectrum. It s clearly seen that the varaton of story drft among 7 generated motons are rather larger than expected. In ths paper, some notes and consderatons of the 1) Buldng Research Insttute 2) Nkken Sekke Co. Ltd 3) Toda Corporaton 4) Hazama Corporaton ) Mtsu Corporaton 6) Shmzu Corporaton

2 varaton n the smulated ground moton wll be dscussed. 2. TYPES OF TIME HISTORIES There are three types of motons consdered n the current desgn of specfc buldngs. (1) Standard type ncludes the well-known El Centro/194 NS component, Taft/192 EW Component and Hachnohe harbor for Tokach-ok earthquake/1968 EW component. These motons become a knd of standard n Japan. The moton of ths type has no varaton and s manly used to examne the overall valdty of the desgn. Because those standard motons are used n most of the desgn practce. Therefore, the desgner can check the level of sesmc performance n reference to other desgn examples. (2) Ste-specfc type ncludes the modfed motons recorded at the ste or nearby stes, and the smulated motons based on the event that has the largest nfluence to the buldng concerned. The ground moton of ths type can be evaluated as the moton due to the selected specfc earthquake. The moton has varaton based on the rupture process uncertanty of the causatve fault that s specfed as of the largest nfluence n the area. (3) Desgn spectrum-compatble type ncludes two optons. One s the moton that has clear defnton of correspondng to an earthquake event. The other s not necessarly related wth some specfc event, snce the spectrum s determned from the statstcal analyss of varety of motons. Here, we need a strategy to fx the ground moton parameters to generate tme hstores. The moton of ths type had been frequently used n the past. The technque for generatng ground moton of the type s avalable. (Ref.1) We wll manly focus on type (3) n ths paper. 3. VARIATION IN DESIGN MOTIONS The Prmary factors causng the ground moton varaton mght be as follows; a) Sesmologcal factors It s relevant to earthquake occurrence or fault rupture. In engneerng these factors are nvolved n phase scatter all together. b) Varaton of tme hstory propertes It ncludes response spectra, peak ampltudes, energy spectra, etc. c) Varaton of the structural response It nvolves non-lnear response, responses of mult-story buldngs 4. GENERATION OF SPECTRUM COMPATIBLE MOTIONS 4.1 Wave forms The commonly used waveform for the tme hstory of ground moton s as follows; at () = Et () Acos( ωt+ φ) (1) where, at () s acceleraton tme hstory wth unform tme nterval, Et () s the envelope functon to make wave non-statonary. A, ω, φ are ampltude, crcular frequency and phase angle of the -th component, respectvely The tme hstory can be generated by the nverse Fourer transform. The response spectra of the generated moton are calculated. If the convergence to the target (desgn) spectra s not good, the ampltude of each component s modfed n accordance wth the correspondng spectral value. Ths process s repeated untl a good agreement s attaned. The fnal tme hstory wll be served as the desgn moton for the verfcaton n dynamc analyss. The tme hstory generaton scheme usng unformly random phase angles and envelope functon does not always promse the convergence to the target response spectrum for a larger number of cyclc computaton. The poor convergence s sometmes mproved by changng the ntal set of phase angles. It s known that the usage of the phase angles of the recorded moton generates a tme hstory smlar

3 waveform appearance wth the orgnal recorded moton. In ths case, the envelope functon s not necessary. The expresson (1) becomes as follows removng the envelope functon. follows..31m.774 T d = 1 (3) Where, M s earthquake magntude at A t (2) * () = cos( ω + φ ) * where, φ s the phase angle of the recorded moton. The target response spectrum s generally gven for % of crtcal dampng. It s necessary for the generated moton to be compatble to only the % dampng response spectrum. It s not generally requred, however, that the generated moton s compatble to the spectral values wth dampng other than % of crtcal. Ths condton seems too loose, turns out to result n actually restrcted by the fx of duraton. 4.2 Duraton tme and envelope functon The duraton tme and envelope functon are necessary n the smulaton process of ground moton. In some cases, the velocty and dsplacement tme hstores have drft n baselnes. One should be careful for t n computng peak ampltudes. There are number of defntons of duraton tme. The three plausble defntons of duraton tme T d are ; (1) The cumulatve tme n whch the ampltude exceeds certan level of threshold. (2) The elapse tme from moment the ampltude exceeds a specfed level frst to the last tme t falls below the level. (3) The tmes correspondng to % and 9% of cumulatve energy. The cumulatve energy s the proportonal to the ntegraton of squared ampltude. Physcally, the duraton tme s related wth the tme requred for the fault to complete the rupture. In addton, the delays and advances of sesmc waves make the waveform longer, f there are many paths for the sesmc waves. An example of the equaton for duraton tme s as The most essental value n desgn s the maxmum response. In addton, strong moton records are dgtzed wth tme nterval of. to.2 second. The spectrum-compatble moton can be defned wth any of the tme nterval; t should be specfed as smaller when one needs to evaluate the response wth prevalng hgher modes. The property of the desgn ground moton specfed n the notfcaton No.1461 n the revsed Japanese Buldng Standard Law should be as follows; The smulated moton should be compatble n acceleraton response spectrum wth % crtcal dampng specfed n the notfcaton No The compatblty to the target spectrum (desgn spectrum) should be checked n the perod range between.2 and second. However, under some specfc condtons such as the deep sedment n the ground surface (here wthn the scope as deep as up to layer wth shear wave velocty of 3,m/s), much longer component than -second should be consdered, snce the deep subsurface condtons sometmes generate surface waves that domnates n longer perod component. The typcal areas n Japan nclude the plan areas (Kanto plan, Osaka plan, Nohb plan etc., please refer the BCJ gudelne, 1992) 4.3 Compatblty to Target There are several ndces proposed to check f the moton s compatble to the target spectrum. Here, the recommended values for each ndex s added. (1) Smallest Spectral Rato, ε mn ε mn Sa( T, h) =.8 DSa( T, h) mn where, S ( T, h) s the acceleraton response a

4 spectrum for perod T, and dampng ξ. DS ( T, h ) s the target spectrum. a (2) Coeffcent of Varaton, ν 2 ( ε 1.) (4) ν =. N (3) Average Error, 1 εave 1 ε ave.2 () where, ε N ε ave = (6) Sa( T, h) ε = (7) DS ( T, h) a : Number of Perods where the error s examned The followng two values are recommended to confrm the damage potental of the moton. (4) Spectral Intensty Rato, SI rato Here the ntegraton wll be done for between 1 and second snce the moton s manly use n the desgn of long perod buldngs, although the Housner s proposal for the Spectral Intensty s the ntegraton between.1 and 2. seconds. SI rato 1 ps ( ) 1 v T dt = 1. DpS ( T) dt () Energy Spectrum V E v (8) 2E = (9) M. EVALUATION OF VARIATIONS OF THE GENERATED GROUND MOTIONS Tme hstores compatble to the desgn spectra are generated and ther varatons are examned..1 Nonstatonary tme hstory wth envelope functon and unformly random phase angles 1) Effect of duraton tme Four types of duraton tme,.e,.1, 2, 6, 12 seconds were used to generate the ground motons. The desgn spectra used here s an acceleraton response spectrum shown n Fg.3. For each duraton tme, thrty tme hstores were generated. The non-statonary property, gven wth three parts,.e., buld-up, constant, and decay, are dentcal of a proporton to total duraton tme. The wave forms are shown n Fg.4. The frequency dstrbuton of peak acceleraton and velocty are shown n Fg.. The left hand fgure shows the frequency dstrbuton of the peak acceleratons. It s seen that shorter duraton tme gves larger varatons. The varaton for the 12-second moton s much less than the shorter duraton cases. The rght hand fgure shows the same dstrbuton for peak veloctes. It s also seen that the longer duraton tme shows less varatons wth peak ampltudes. 2) Varaton of response spectra In Fg.6, the averages of the rato to the target spectra (%) are shown for duratons of 1 and 12 seconds for 1, 2, 1, 2 % dampng. The varaton of the acc spectral values wth dampng at the longer perod range becomes less for shorter duraton tme hstores. Ths s due to that the wave number s not enough for the response to grow up for shorter duraton and longer perod. As a result, the response value s less vared wth dampng. In Japan, the followng formula s commonly used to modfy the response spectrum wth dampng other than %. F h = h (1) Fg.7 shows concdence between equaton (1) and the rato computed from the generated ground motons. It s seen that for shorter perod the rato becomes nearly unty for 1% and 1 to 1.2 for 2% dampng, for longer perod the rato s much larger wth shorter duraton and less n longer duraton and almost constant for ntermedate perod. Fg.8 shows the rato between the response spectrum and the pseudo response spectrum for dsplacement. It s seen that for longer perod the dfference becomes

5 larger. In addton, the comparson was made for mean energy spectra for motons wth dfferent duraton tme n Fg.9. It s seen that the energy spectral value for 12 second duraton becomes approxmately double of the 1 second duraton, and the spectrum s senstve to the duraton tme. 3) Nonlnear response characterstcs Tme hstory analyss s manly utlzed n nonlnear response analyss. Here, we used the 1dof b-lnear model to compare the non-lnear response propertes. The parameters used are the skeleton curves, stffness rato, buldng perod, stffness rato. We nvestgated the nfluence of these parameters on the peak dsplacement response wth motons of 4 types of duraton tmes. Fgure 1 shows the nfluence of yeld strength and buldng perod on the relaton of peak dsplacement and duraton tme. It can be commonly sad that the dsplacement response depends on duraton tme. The longer buldng perod gves larger varaton. The smaller yeld strength gves larger varaton n peak dsplacement. For the cases of elastc buldng perod less than 1 second, the response dsplacement does not depend too much on the duraton tme. Fgure11 shows the relaton between the dsplacement response and duraton tme for several stffness ratos. Although the dsplacement response and varaton become larger when the stffness rato s very small, the response becomes smaller and almost same regardless of the stffness raton when the raton s greater than.1 and as the duraton becomes longer. It s also seen that the dsplacement response does not depend on the strength except for the duraton of 1 second..2 Nonstatonary tme hstory wth phase angle of recorded moton Fgure 12 shows the supermposed plot of the acceleraton response spectra for dampng of 2,, 2 % wth 16 generated spectrum-compatble motons. In ths smulaton, the duraton of 12 second was used. In addton, the phase angles of the recorded moton are used to gve the nonstatonarty. The name of the records, mnmum error, coeffcent of varaton, average error are lsted n Table 1 wth the resulted PGA, PGV and PGD. In each drawng of Fg.12, the spectra derved usng the equaton (1) normalzed to the % spectrum s also drawn. It s seen from the fgure, the expected average spectra for ten generated motons slp out of the spectra computed wth F h. It can be ponted out that the assumed wave form wth duraton does not promse the relaton F h. CONCLUSIONS The varaton of the average of the peak ampltude, PGA and PGV and the lnear and nonlnear response spectral values were computed wth numbers of smulated waves wth varous duraton tmes. The varaton should be related wth the randomness nvolved n earthquake occurrence, such as rupture process, complexty of wave propagaton and the surface sols. However, to ncorporate all these uncertantes n the earthquake smulaton s almost mpossble snce the geophyscal and/or sesmologcal nvestgaton on the earthquake wave formng process s stll under development. Therefore, t s necessary for engneers to ncorporate the uncertantes all together n evaluatng the future strong ground motons. The number of tme hstores appled to the desgn analyss s qute lmted. The mportance s to know the possblty of loads or responses under such total uncertantes. In addton, the rght poston of tme hstory analyss n desgn practce should be reestablshed. REFERENCE (1) Y.Ohsak, R.Iwasak, T.Masao, I.Okawa, Phase Characterstcs of Earthquake Accelerogram and Its Applcaton, Trans. of th Int. Conf. On SMRT, K1/4, Vol.K(a) (2) Buldng Research Insttute, Buldng Center of Japan (1992), A Technque for Evaluaton of Desgn Earthquake Ground Moton, Buldng Center of Japan (3) K.Yamagsh, M.Tangak, I.Okawa, Average and Devaton Response Evaluaton for Varous Smulated Sesmc Motons Usng Random Phase Angles (Part 1,2), Summary of AIJ Annual Meetng, 2

6 story No.1 No.2 No.3 No.4 No. No.6 No.7 No.8 No.9 story No.1 No.2 No.3 No.4 No. No.6 No.7 No story drft (rad.) story drft (rad.) Fg.1 The dstrbuton of computed story drft for the spectrum compatble motons (Left: 1 BCJ level 2 motons, Rght: 7 BSL Notfcaton No1461 motons) Table 1 Lst of recorded moton for use of phase angle and resulte peak ampltudes Symbol Mn. error COV Ave. error t n PGA (cm/s/s) PGV (cm/s) PGD (cm) ELC4NS TAF2EW HAC68EW HAC68NS KSR93NS KSR93EW HKD93NS HKD93EW KSR94NS KSR94EW HCN94NS HCN94EW HCN9NS HCN9EW KOB9NS KOB9EW PGA:Peak Ground Acceleraton, PGV:Peak Ground Velocty, PGD:Peak Ground Dsplacement

7 Past earthquakes, dentfcaton of the earthquake sources Whether or not to evaluate ndvdually NO YES Use of standard spectrum Indvdual Evaluaton Generaton of tme hstory holdng spectral values of notfcaton No.1461 Modfcaton of recorded motons Selectons of Sources and ground moton evaluaton Condtons of Conformty Compatble to Sa longer than 6 sec of duraton Tme nterval suffcently small to the hgher mode of the structure Number of waves to be examned Target earthquakes Plate boundary Intra-plate local actve faults Deep sedment structure longer perod component Acc. Response spectrum Modfy or correcton Buldng heght, wdth and form Tme hstory 2 horzontal components Spatal varaton Sol amplfcaton Vertcal moton Dynamc analyss Fg.2 The flow chart for evaluatng the desgn ground moton tme hstory

8 cm/s 2 1 h=. cm/s 1 h=. G1 GX GA GC GB 4 4 GD Fg.3 Target response spectrum (Left: Acceleraton, Rght: Velocty) T d =1s T d =2s T d =6s T d =12s Fg.4 Example of tme hstory (1, 2, 6, 12 seconds) Frequency (a)peak Acceleraton(cm/s 2 ) Frequency (b)peak Velocty(cm/s) 4 8 Fg. Frequency dstrbuton of peak acceleraton and peak velocty of generated tme hstores

9 4. Td1s;h1% Td12s;h1% 1. h=.1 Rato to Target Spectrum (h=%) Td1s;h2% Td12s;h2% Td1s;h1% Td12s;h1% Td1s;h2% Td12s;h2% Perod(sec) Rato to Fh Rato to Fh T=.1s T=1.s T=1s h=.2 T=.1s T=1.s T=1s G1 G2 G6 GX GA GB GC GD Fg.6 Rato of response spectrum to the % dampng spectrum Fg.7 Comparson of F h values at.1, 1, 1 seconds for varous duratons Sd/(Sa%/ω 2 ) G1 G2 G6 GX GA GB GC GD h= Perod(sec) V E (cm/s) h=.1 G1(Td=1s) G2(Td=2s) G6(Td=6s) GX(Td=12s) Perod(sec) Fg.8 (lower) Rato of the dsplacement spectrum to the pseudo dsplacement spectrum Fg.9 Comparson of energy spectra of the smulated motons

10 Peak Dsp. Response (cm) y =. y ; K 2 =.1K 1 Tb=.s(mean) Tb=.7s(mean.) Tb=1.s(mean) Tb=3.s(mean) Td(sec) + - Peak Dsp. Response (cm) y =1. y ; K 2 =.1K 1 Tb=.s(mean) Tb=.7s(mean) Tb=1.s(mean) Tb=3.s(mean) Td(sec) (a)b-lnear model(α y =.α y ) (b)b-lnear model(α y =1.α y ) Fg.1 Relaton between duraton tme and peak dsplacement response for b-lnear model (1) Peak Dsp. Response (cm) K2=.*K1(mean) K2=.1*K1(mean) K2=.2*K1(mean) K2=.*K1(mean) (a)α y =.α y ; T b =3.s Td(sec) Peak Dsp. Response (cm) K2=.*K1(mean) K2=.1*K1(mean) K2=.2*K1(mean) K2=.*K1(mean) (b)α y =1.α y ; T b =3.s Td(sec) Duraton tme and peak dsp. response for B-Lnear model Fg.11 Relaton between duraton tme and peak dsplacement response for b-lnear model (2)

11 14 ξ=.2 12 Acc. (cm/s/s) Perod (s) ξ=. (ftted) Acc. (cm/s/s) Perod (s) ξ=.2 Acc. (cm/s/s) Perod (s) Fg.12 Computed response spectra and expected mean values