Calculation of Response Spectrum Directly from Bedrock to Free Field using Equivalent Linear Method

Transcription

1 alculatio of Respose pectrum Directly from Bedrock to Free Field usig Equivalet Liear Method H. L. Xig Research Istitute of tructural Egieerig ad Disaster Reductio, Togji Uiversity, haghai, P.R. hia Urba Mass Trasit Railway Research Istitute, Togji Uiversity, haghai, P.R. hia T. Jiag &.T. Xue Research Istitute of tructural Egieerig ad Disaster Reductio, Togji Uiversity, haghai, P.R. hia.z. Geg Urba Mass Trasit Railway Research Istitute, Togji Uiversity, haghai, P.R. hia UMMARY: A stable relatioship betwee the peak groud velocity (PGV) ad the maximum value of %-damped pseudo-spectral velocity (PV) is preseted based o aalysis of 88 far-field horizotal records selected from large earthquakes i EMD ad PEER strog motio database. The validity ad applicability of this relatioship is ispected usig dow-hole array records. As for the ear-field groud motios, this relatioship also shows good agreemet by chagig the regressio parameter. A ew approach to calculate the free field respose spectrum of horizotal layered site directly from that of the bed rock is preseted with the followig iterative procedure: ) Usig the stadard equivalet liear method model ad the equivalet strai level (with assumed value for the first iteratio) at the middle of each layer, determiig the trasfer fuctios for a uit iput specified at the bedrock. ) Treatig the earthquake motios as a statioary radom process with the power spectral desity (PD) fuctio which is coverted from the bed rock respose spectrum, calculatig the PD fuctios of every layer usig the trasfer fuctios. 3) By covertig the PD fuctios to PV, evaluatig the PGV from the PV, ad calculatig strai usig the relatioship betwee the shear strai ad particle velocity of homogeeous medium, obtai the equivalet strai of every layer, which is used to get ew shear modulus ad dampig to determie ew trasfer fuctios. After the iteratio is coverged, free field respose spectrum is derived from PD fuctio of the surface layer. The usefuless ad validity of the proposed method is demostrated by comparig the results with those of covetioal methods for differet project sites. Keywords: Free field respose spectrum; peak groud velocity; equivalet liear method; horizotal layered site. INTRODUTION There are maily two methods to derive free field respose spectrum (spectral acceleratio (A) is used i this paper) from the bedrock A. Oe is the covetioal method () that several artificial accelerograms, geerated i accord with the give bedrock R, serves as bedrock iput. The average free field A is obtaied with a site respose aalysis procedure such as HAKE(chabel et al.,97). This is the most commoly way i practice(gb 77-, ) but it is very cumbersome. The other oe is to calculate the free field A directly from the bedrock A. tudies o this method are firstly reviewed at the begiig of this paper. The the relatioship betwee PGV ad maximum value of PV is cofirmed by regressio of the strog earthquake motio. Based o this relatioship ad the theory of equivalet liear method, a ew method for calculatio of free field A directly from bedrock A is proposed. To demostrate the validity of the ew method, several examples are preseted icludig five sites used i the former research. u had proposed a method usig the assumptio that the iput motio is statioary Gaussia process ad the soil profile ca be modeled as multi-degree-of-freedom(mdof) system (u et al.,99). With the relatioship betwee power spectral desity (PD) fuctio ad the spectral acceleratio (A), the free field A is calculated from the bedrock A. Furthermore, equivalet liear theory is used to

2 cosider dyamic soil properties. But the complicated parameters hider the wide applicatio of this method. Based o the elastic assumptio ad the relatioship betwee A ad PD of the earthquake records, Afra ad Pecker(Afra ad Pecker,)proposed a method with the followig process: calculatig the PD from the give bedrock A as the iput; calculatig the trasfer fuctio of the soil which is or layers at the top of the rock base; multiplyig the iput PD ad the trasfer fuctio ad covertig it ito the A. But this method will be ufeasible whe the oliear properties of the soil could ot be egligible durig strog earthquakes. Miura (Miura et al.,) covert the bedrock A ito Fourier amplitude spectrum usig the relatioship betwee A ad the -dampig velocity spectrum which is assumed to approximate the Fourier amplitude spectrum. The site is simplified as a sigle layer ad the respose of the free field is calculated with the harmoic iput motio. The the site is idealized as a lumped-mass system with the mass at the cetral of each layer ad the trasfer fuctios of the first ad secod model are determied. The strai of each layer is calculated usig the lumped mass model by supposig the respose of the field is idetical ad the site respose is cotrolled by the first mode. New dyamic soil properties are refied usig the strai as well as the fuctio of dyamic shear moduli ad dampig ratio versus strai amplitude. The the stiffess ad dampig of the lumped mass model are also reewed based o the ew dyamic soil properties. The iteratio is eded whe the period of the first mode is coverged. This method is proposed i the ew versio of buildig stadard law of Japa. But there are two mai shortcomigs of this method. Oe is the assumptio that the basic model of the MDOF domiates the site respose ad the respose of soil strata is equivalet to a sigle layer o half space. The other oe is that oly two specific values of A correspodig to the first ad secod periods are worked out while the mai part of the A is costructed with empirical methods.. RELATIONHIP BETWEEN PGV AND PETRA Like the peak groud acceleratio (PGA), PGV is a key parameter idicatig the characteristics of the groud motios. A widely used relatioship, proposed by Newmark(Newmark ad Hall,98), is give as: PGV ca be calculated from the %-damped PV at secod( PV(T=.)), PGV PV T.. (.) Based o aalysis of the published groud-motio predictio equatios, stochastic simulatios ad strog motio data, Bommer(Bommer ad Alarco,) educed empirical correlatios betwee PGV ad A at. secod, PGV A T. (.) To derive a ew predictive equatio betwee the PGV ad spectrum for the purpose of accuracy ad stability, this paper performs regressio aalysis of the far-field strog motio records selected from the database of the Pacific Earthquake Egieerig Research eter (PEER)( ad eter for Egieerig trog Motio Data(EMD)( Oly oe horizotal compoet amog the three compoets at a statio is used. The statios of all records selected are either free field sites or groud ad basemet of oe or two stories buildigs i order to reduce the ifluece of soil-structure iteractio. Totally 88 samples from large earthquakes listed i table are aalyzed. With regressio aalysis, a kid of stable liear correlatios betwee PGV ad the maximum PV of the far-field strog motios is, PGV max PV R (.3)

3 The parameter R ca be idetified with the simple least-squares optimal fit ad R =3. for far-field records. Fig. shows the compariso of the results calculated with differet equatios ad those of the records. The results of the proposed equatio are i good agreemet with the records tha the other two. Table.. Parameters of Earthquake Records elected for Predictio of PGV No Earthquake Time Mag. Epiceter Locatio Far-field records Near-field records Name (mm/dd/yy) (M L ) (Log., Lat.) Epicetral um Epicetral um. Dist.(km) Dist.(km) oaliga // N,.3W 3.7~ Morga Hill // N,.8W 3~ Palm prigs 7/8/ N,.W 9~83.~. Whittier Narrows // N,8.7W 9.3~.9 3.9~9. Loma Prieta /8/ N,.88W 8~.8.8~. 8 Laders /8/ N,.3W 3~ Bigbear /8/99. 3.N,.83W 37.9~ Northridge /7/ N,8.W 3~7..~. 9 Hector Mie // N,.7W 8.~ Aza /3/. 33.N,.W 9~ Big Bear ity //3. 3.3N,.8W 3~ 8.~7. 8 a imeo //3. 3.7N,.W 8~. 3 Parkfield 9/8/. 3.8N,.37W 3~.~9. Aza //. 33.3N,.7W 9~.~9. 8 Yucaipa //.3 3.N,7.W 8~8.9~ 8 hi-hi Taiwa 9// N,.8E 9.3~8 8.~ PGV calculated (cm/s). alculated=recorded alculated by Eq.(3) alculated=recorded alculated by Eq.() alculated=recorded alculated by Eq.() Figure. ompariso of PGV with differet equatios for the far-field records Near-field records with epicetral distace less tha km are selected from those large earthquakes i Table... Two horizotal compoets of the same statio are regarded as two idepedet radom processes ad totally 3 records are used. Regressio result shows that the relatioship betwee PGV ad the maximum value of PV ca also be expressed with Eq.(.3) with parameter R., as illustrated i Fig.. Furthermore, compariso i Fig. idicates that with icrease of PGV, the deviatio of Eq.(.3) will icrease. There is a suggestio that the predictive equatio should be carefully hadled for sites close to active faults where ear-source directivity effects may be expected. It is recommeded that R i predictive Eq.(.3) equal to 3. for far-field earthquake sites while for ear-source sites a smaller value of. is proposed i this paper. Aother compariso is made of the above-metioed records with data recorded by dow-hole arrays which provide detailed iformatio for studies of local site effects. The acceleratio time series whose

4 PGA are ot less tha.g are selected from MIP database of four dow-hole arrays: Treasure Islad, La ieega, Eureka ad El etro-melolad. Two horizotal compoets of each statio, regarded as idepedet samples, are all selected ad total records are. Detailed iformatio of the dow-hole arrays ad earthquakes are described by Graizer et al.(). Regressio results of the dow-hole array records are show i Fig.3, which shows that the results obtaied with Eq.(.3) whe R is equal to 3. are i good agreemet with the records. alculated=recorded alculated with Eq.(3) alculated=recorded alculated with Eq.(3) PGV calculated (cm/s) PGV calculated (cm/s).. Figure. Near-field records results.. Figure 3. Dow-hole array records results 3. PROPOED METHOD 3.. Derivatio of A from PD ad vice versa The correlatio betwee A ad PD is sigificat i the simulatio of strog groud motio. Park(99) ad Paskalov ad Reese(3) preseted simple approach to covert A from PD ad vice versa, respectively. Iasmuch as the approach of Park eeds o umerical iteratios, it is feasible for discrete A ad PD. Forward equatio of derivatio of A from PD is, m k j k, jk j A, p R,, k,,..., j Where pj G j j; j ad j k, k, j k, j, R (3.) G is the discrete PD ad is dampig ratio. Usig v R i Eq.(3.) ca be expressed as,.77 k, jk, j, lvjte l vt j e Where T e is the effective duratio of groud motio ad ca be determied from the give evelope fuctio. Backward calculatio of PD from A is obtaied by solvig the followig equatio, mi,,, k j p ; j,..., m m A k p jrk, jk j (3.) j j j

5 3.. Itroductio of Equivalet Liear Method The sigificace of the local site effect has log bee recogized by earthquake egieers ad seismologists which triggers the developmet of equivalet liear method for D seismic respose aalysis sice the later 9s(Idriss ad eed,98). HAKE is the most popular program o the list as it was amog the first to be developed, ad eve today it is still widely used ad studied. By idealizig the soil deposit as a stratified, viscoelastic medium ad the seismic excitatio as vertical propagatio shear waves, the equivalet liear method adjusts the stiffess ad dampig ratio of soil layers i each iteratio util they are compatible with the strai level iduced by the earthquake u z, t with frequecy ca be writte as loadig. I frequecy domai, harmoic displacemets uz, t Uzexpit U z E expikzf exp ikz, where k G i modulus ad mass desity, respectively. E ad. The amplitude of the icidet ad reflected wave at depth z is ; G ad are shear F ca be determied with the boudary coditios: displacemets ad shear stress must be cotiuous at the layer iterfaces as well as the stress must be zero at the free surface. Thus the trasfer fuctio HN, betwee the displacemets at layer ad the iput layer N ca be defied as, H N, E E N F F N (3.3) Where E ad F are amplitude of icidet ad reflected waves ca be determied with the boudary coditios. Dyamic shear modulus G ad dampig ratio at layer are defied as a fuctio of the equivalet shear strai as followig, G G (3.) Here,max,. ad,max is the maximum strai at layer. As strai is ukow beforehad, iteratio is eed Algorithm of the Proposed Method With the assumptio that the earthquake motio is a statioary Gaussia radom process ad the theory of the equivalet liear method, site respose uder arrow bad frequecy ca be determied by idealizig the soil strata as liear system. Accordig to the stochastic vibratio theory, the relatioship of PD fuctio betwee of layer ad iput layer N is give as, G H G (3.) The maximum strai, N N,max is eeded to determie the equivalet strai whe the trasfer fuctio is calculated with equivalet liear method. Whe plae shear waves propagate i a homogeeous soil, the shear strai at some arbitrary poit z is directly proportioal to the particle velocity u at that poit: u tz V u u (3.) z V t V s s s

6 Where V s represets shear velocity of the soil layer. Thus the maximum strai of the soil uder statioary process iput is, max u max Vs PGV Vs (3.7) The followig iteratio steps are preseted for the calculatio of free-field A from the bedrock A: A Discrete bedrock PD fuctio N G is derived from bedrock respose spectra N A ad G ). et with Eq. (3.) (here that discrete poits will be sufficiet for N iitial equivalet shear strais to each layer. Determie dyamic material properties G ad of each layer based o Eq. (3.) ad equivalet shear strai. Obtai the trasfer fuctio HN, with Eq.(3.3) ad the recursio formulas for amplitude E ad F. alculate PD fuctio G i each layer uder the iput GN with Eq.(3.) ad the covert it ito the respose spectrum A by Eq.(3.). Determie PV usig the relatioship betwee PV ad A( PV A ), the get the maximum value max PV. PGV is estimated with Eq.(.3). Defie the maximum strai,max by Eq.(3.7) i each layer. Multiply,max by factor to get the equivalet strai. ompare the equivalet strai with their values i the previous iteratio. Iterate as ecessary. N. AE TUDIE.. Example The dow-hole array records obtaied at hiba site of Idustrial Istitute of ciece, Tokyo Uiversity, durig the 987 hiba-ke Toho-oki earthquake (M=.7) ad hi-fuji trasformer statio durig 983 Kaagawake-Yamaashike-Kekyo earthquake (M=.) are used to testified the propose method. oil profiles, dyamic soil properties, accelerometer locatios ad the acceleratio records are give by Yoshida(Yoshida et al.,) ad Jiag(Jiag ad Xig,7). Respose spectra ca be directly calculated from the acceleratio time histories. The % damped A obtaied from the acceleratios of the bottom accelerometers (8m depth at hiba site ad 8m depth at hi-fuji respectively) are used as iput to evaluate the surface respose spectra with factor R.. The A of the proposed method ad the records are very similar, as illustrated i Fig.. Record Record A (m/s ) 8 3 T (sec) A (m/s ) T (sec) (a) hiba site (b) hi-fuji site

7 Figure. ompariso of free field A betwee the results of the proposed method( red lie deoted as ) ad those of the dow-hole array records(dash lie).. Example Five soil profiles with detailed descriptio of soil properties(fig.(a)), depths ad shear wave velocities(fig.(a)), had bee give by Miura et al(). Respose spectra were established for earthquake magitude M= with a ear-field distace km ad a far-field distace km usig atteuatio relatioships for North hia regio(gb 7-97,998), as illustrated i Fig.(b). The free field A was calculated with the proposed method ad the coversioal equivalet liear method. It was oted that factors for evaluatig PGV i Eq.(.3) were 3. for far-field earthquakes ad. for ear-field oes. omparig the free field respose spectra evaluated by the proposed method ad the covetioal way i Fig. (b) ad (c), it was observed that results for far-field earthquakes were i excellet agreemet while spectra predicted by the proposed method were a little higher at site ad 3 for ear-field earthquakes. A smaller value of factor R. will improve the results of the proposed method, as is show i Fig. (c). This example idicates that the scalig factor R i Eq.(.3) is a key parameter of the proposed method. It is recommeded that the value of R is 3. for small to moderate earthquakes ad is. or smaller for large earthquakes G / G max.., F,t dampig ratio (%) A (m/s).. E- E-3.. trai. A of km A of km. T(sec) (a) oil properties (b) Iput A Figure. oil properties ad iput A for five typical sites. DIUION AND ONLUION Empirical relatioship betwee PGV ad the maximum value of % damped PV is regressed from more tha 8 far-field strog motio records obtaied at West America ad Taiwa regio. The validity of this relatioship is testified with dow-hole array records, which idicates that it ca be used for site respose aalysis. A simple method by which the free surface respose spectra ca be directly calculated from that of the bedrock is proposed based o the theory of equivalet liear method ad the relatioship betwee PGV ad respose spectra. ase studies show that the results of the propose method are always i good agreemet with those of the covetioal way as well as those derived from the dow-hole array records. Two key factors that ifluece the accuracy of the proposed method are the correlatio betwee A ad PD ad the approximate relatioship betwee PGV ad the respose spectrum. The approach to covert the A ito PD ad vice versa, as propose i sectio 3., does ot ifluece validity of the propose method if the approach has sufficiet accuracy. PGV ca be estimated by PV or A with Eq.(.) ad (.) respectively provided that those equatios satisfy the required precisio for specified site coditios. Because the relatioship of Eq.(.3) is regressed with far-field strog motio records, a smaller value of the scalig factor R is recommeded for ear-field earthquakes to take

8 Depth (m) accout of evidet oliearity of soil properties. It is oted that usig smaller value of R i the proposed method is solely for the sake of accuracy, ad ot idicates the ivalidity of Eq.(.3). Furthermore, the proposed method has the same advatages ad deficiecies as the equivalet liear method. ite site site 3 site site 3 t (a) soil profiles (:sad;; F:fill;; :clay;;t:silt) F t A (m/s ) A (m/s ). site site (b) Free field A with a bedrock iput A i accord with epicetral distace km (M=) R =. R =. site 3 R =. R =. site 3 3 (c) Free field A with a bedrock iput A i accord with epicetral distace km (M=) Figure. ompariso of respose spectra calculated with HAKE() ad the proposed method() site 3 R =. R =. site 3 3. site R =. R =. site 3. site. 3 8 R =. R =. site 3 REFERENE hia Earthquake Admiistratio, GB 7-97, (998), ode for seismic desig of uclear power plats, Beijig: hia Plaig Press.(i hiese) Geeral Admiistratio of Quality upervisio, Ispectio ad Quaratie of the People's Republic of hia, GB 77-, (), Evaluatio of seismic safety for egieerig sites, Beijig: tadards Press of hia.(i hiese) Afra, H., ad Pecker, A.,(), alculatio of free respose spectrum of a o-homogeeous soil deposit from bed rock respose spectrum. oil Dyamics ad Earthquake Egieerig, : 7-.

9 Bommer, J. J., ad Alarco, J. E., (), The predictio ad use of peak groud motio. Joural of Earthquake Egieerig, :, -3. Graizer, V., hakel, A., ad Hipley, P., (), Recet data recorded from dow-hole geotechical arrays. Proceedigs of MIP emiar o Utilizatio of trog-motio Data, acrameto, aliforia, Idriss, I. M., ad eed, H. B., (98), eismic respose of horizotal soil layers, Joural of the oil Mechaics ad Foudatios Divisio, AE, 9:M,3-3. Jiag, T., ad Xig, H., (7), A equivalet liear method cosiderig frequecy-depedet soil properties for seismic respose aalysis. hiese Joural of Geotechical Egieerig, 9:, 8-. (i hiese) Miura, K., Koyamada, K., ad Iiba, M.,(), Respose spectrum method for evaluatig oliear amplificatio of surface strata, Joural of tructural Egieerig, AIJ, 39,7-. Newmark, N. M., ad Hall, W. J.,(98), Earthquake spectra ad desig, Earthquake Egieerig Research Istitute, Berkeley, aliforia. Park Y. J., (99), New coversio method from respose spectrum to PD fuctios. Joural of Egieerig Mechaics, AE, :, Paskalov, A., ad Reese.,(3), Determiistic ad probabilistic floor respose spectra, oil Dyamics ad Earthquake Egieerig, 3, -8. chabel, P. B., Lysmer, J., ad eed H. B., (97), HAKE: a computer program for earthquake respose aalysis of horizotally layered sites, Report No. EER7-, Uiversity of aliforia, Berkeley. u, J., Zhou, X., ad Fa,., (99), A practical method for calculatig groud respose spectrum directly from base rock respose spectrum, hiese Joural of Geotechical Egieerig, :, 7-3.(i hiese) Yoshida, N., Kobayashi,., uetomi, I., al. et., (), Equivalet liear method cosiderig frequecy depedet characteristics of stiffess ad dampig, oil Dyamics ad Earthquake Egieerig, :, -.