Rotational Components of Strong-motion Earthquakes

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1 Rotatioal Compoets of Strog-motio Earthquakes Viet W Lee 1 ad Jiawe Liag 2 1 Professor, Civil & Evirometal Egieerig, Uiv. of Souther Califoria, Los Ageles, CA USA, vlee@us.edu 2 Professor, Shool of Civil Egieerig, Tiaji Uiversity, Tiaji , Chia, liag@tju.edu. ABSTRACT : The rotatioal ompoets assoiated with the dyami respose of elasti ad of poro-elasti half spae a be as importat as the atual deformatios, for omputatio of the respose of the half spae ad for the respose aalyses of egieerig strutures, ad must be iluded i all aalyses. The basi theory of elastiity shows that the Euleria strai tesor of deformatios is expressible i terms of the liear strai tesor ad the seod order terms of rotatio tesor. At preset, all strutural aalyses already deal with the geeralized displaemet of the 3 ompoets of traslatios ad the 3 ompoets of rotatios as the 6 idepedet ompoets. These six geeralized ompoets are ofte simultaeously estimated usig the kiemati equatios of fores ad momets Sie rotatios are defied i terms of derivatives of traslatios, with respet to spae oordiates, there always should exist aalyti expressios for rotatios i terms of the traslatios. This is ertaily the ase whe the traslatio ompoets are give i retagular oordiates. Usig surfae waves, the method of Trifua (1971) for geeratig artifiial, sytheti strog motio aelerograms a thus immediately be exteded to geerate both sytheti torsioal ad rokig aelerograms ad of their respose spetra (Lee ad Trifua, 1985, 1987, 2008). Those are ostruted from the horizotal ad vertial aeleratio ompoets. These rotatioal ompoets of strog groud motios are gaiig attetio as it is beomig evidet that they a otribute sigifiatly to the overall respose of strutures durig strog earthquake motios. This paper iteds to stress the importae of the rotatioal ompoets of aelerograms, to develop theories ad algorithms for geeratig rotatioal motios from the orrespodig available traslatioal motios, ad to hek suh validity from several atual, real rotatioal reordigs. KEYWORDS: Diffratio, Free-Field, Half-Spae, Elasti, Poroelasti 1. INTRODUCTION After 75 years of strog-motio reordig programs i the wester Uited States, ad durig the last 40 years, i may other parts of the world, there are ow well over several thousad strog motio aelerograms reorded, proessed ad available for distributio (Trifua 2007). The observatioal strog motio programs evolved from realizatio i early 1900s that it is essetial to reord strog groud motios ad strutural respose durig atual earthquakes (Hudso, 1983a, b; Trifua 2008a), to guide the developmet of the methods for aalysis ad preditio of strutural respose ad to provide a basis for their experimetal verifiatio. The strog motio database aumulated so far allows earthquake egieers to 1) study the strog groud motio i differet parts of the world, to uderstad its harateristis ad destrutive potetial i relatio to the tetoi harateristis of the areas where earthquakes our; 2) to aalyze ad iterpret the dyami respose of strutures durig earthquakes; 3) to idetify ad quatify the earthquake strog-motio parameters, assoiated with geeratio ad trasmissio of strog motio waves, ad 4) to provide a data bak

2 for future studies. While the reorded data is ivaluable for researh ad aalyzes of strog-motio, it still does ot over may differet reordig oditios, whih are eeded for egieerig desig. This is partiularly true i those regios, whih have low seismi ativity, ad have ot yet aumulated adequate strog motio database. It is also ofte eessary, to estimate future shakig at sites, whih have harateristis outside the rage of parameters for whih the reorded data is ow available. It is for these reasos that the developmet of moder empirial salig equatios for diret salig of Fourier ad respose spetral amplitudes stated to appear followig the suessful multi-statio reordig of Sa Ferado, Califoria, earthquake of 1971 (Trifua 1976; Lee 2002a,b, 2007). This work o diret empirial salig of spetral amplitudes also failitated the developmet of the algorithms for geeratio of sytheti strog-motio aelerograms of traslatioal ompoets of strog motio (Trifua 1971; Wog ad Trifua, 1979). It was also reogized that the torsioal ad rokig ompoets of strog motio, atig together with the traslatios, a otribute sigifiatly to the overall respose of strutures durig strog earthquakes (Kobori ad Shiozaki 1973; Luo 1976; Lee 1979). I absee of moder programs to develop strog-motio istrumets to reord the rotatioal ompoets of strog-motio, it beame eessary to estimate the rotatioal motios i terms of the orrespodig traslatioal motios. Trifua (1982) showed how the torsio ad rokig of groud surfae assoiated with iidet P, SV ad SH waves a be determied i terms of the orrespodig traslatioal amplitudes of motios. Few years later Lee ad Trifua (1985, 1987), showed how to geerate sytheti torsioal ad rokig aelrograms ad their Fourier ad respose spetra, startig from the amplitudes of horizotal motios. Durig the past two deades i spite of the fat that the egieerig studies have otiued to show the sigifiae of the rotatioal ompoets i strog motio exitatio for the respose of strutures (Trifua 2006, 2008b), the progress i developig ad deployig strog motio istrumets, whih a also reord rotatioal ompoets of earthquake waves has otiued to be slow. I this paper we adopted our approah for the geeratio of artifiial traslatioal ad rotatioal aelerogtrams, ad preset a physial, umerial method for its extesio to geeratio of rotatioal Fourier spetra from that of reorded traslatioal strog earthquake aelrograms, followed by examples of the estimated Foruier torsioal ad rokig spetra, ad of their orrespodig rotatioal time histories. We have the followig set of objetives i this paper: 1) To stress the importae of the rotatioal ompoets of aelerogtrams, ad promote the growth ad developmet of strog-motio istrumets for reordig, together with the traslatioal ompoets, the rotatioal ompoets of the aeleratios of earthquake motios. 2) With the availability of rotatioal ompoets of aeleratio beig so limited, we eed to develop theories ad algorithms for geeratig rotatioal motios from the orrespodig available traslatioal ompoets of motios at eah reordig sites. 3) To use existig istrumetatio to reord, ollet ad proess rotatioal time histories of aeleratio i future seismi evets if possible. This a the be used as a set of database to aalyze ad uderstad the rotatioal ompoets of aelerograms, ad to hek the validity of developed theories of geeratig the rotatioal ompoets of motios from traslatios. A ogoig projet, by oe of the preset authors of this paper, i Tiaji Uiversity of Chia, is the settig up of two six-ompoet (three traslatioal ad three rotatioal ompoets) sesors to reord rotatioal ompoets of free field ad strutures, respetively. The six-ompoet sesor is a ombiatio of EA-120 Fore Balaed Aelerometer ad R-1 Rotatioal Sesor. This will allow the reordig of both the rotatioal ad traslatioal ompoets. It is expeted that the istrumetatio may be of some help for the study of rotatioal groud motio.

3 2. FROM TRANSLATIONAL TO ROTATIONAL ACCELERATION SPECTRA A omprehesive review of the geeral method for the geeratio of sytheti aelerogram, developed by the Strog Motio Researh Group at the Uiversity of Souther Califoria, a be foud i Lee (2002). It geerates sytheti traslatioal aelerograms ad uses the existig theory of liear, elasti wave propagatio to geerate the other aompayig ompoets of strog motio, amely the rotatioal ompoets of torsio ad rokig, ad the strais ad urvatures. 2.1 Geeratio of Sytheti Torsioal ad Rokig Aelerograms The importae of torsioal ad rokig exitatios has bee idiated by the studies of Draviski ad Trifua (1979, 1980), Kobori ad Shiozaki (1975), Luo (1976), Bielak (1978), Lee (1979), Gupta ad Trifua (1987, 1988, 1990a, 1990b, 1990, 1991), Todorovska ad Lee (1989), ad Todorovska ad Trifua (1989, 1990a, 1990b, 1992, 1993). With the slow developmet of strog-motio istrumets that reord rotatioal ompoets of strog motios (Hudso, 1983; Trifua ad Todorovska, 2001), it beomes eessary to explore the possibility of estimatig those i terms of the orrespodig traslatioal ompoets of strog shakig. By osiderig the horizotal propagatio of plae waves with ostat veloity C, Newmark (1969) estimated the otributio to the displaemets of buildig foudatio, resultig from torsioal earthquake groud motios. Tso ad Hsu (1978) used a similar approah, i additio to assumig that the motios also iluded plae o-dispersive waves. Natha ad MaKezie (1975) disussed the possible averagig effets of foudatio sizes o the resultig torsioal exitatios of buildigs. These ivestigatios, however, failed to osider the depedee of the phase veloity o the frequeies of the iomig Love waves. Their assumptio that the iomig waves are of ostat phase veloity at all frequeies, makes their results useless. Lee ad Trifua (1985) iluded the effets of wave dispersio ad trasiet arrivals i the estimatio of torsioal aelerograms i a elasti layered half-spae. The earlier work of Trifua (1982) i alulatig the torsioal ompoet of iidet plae SH-waves was exteded to eable the alulatio of torsio from surfae Love waves. Lee ad Trifua (1985) exteded the work of Wog ad Trifua (1978, 1979) i geeratig sytheti ompoets of traslatioal motios to that of Torsioal motios. It was show that the torioal motio ψ ( x, yt, ) at the groud surfae (y = 0) is related to the orrespodig horizotal trasverse displaemet wxyt (,, ) by iω ψ w y = = (1) 0 2 x where ω ad x are respetively the frequey ad phrase veloity i the horizotal x- diretio of the waves. This is applied i the frequey domai to the Foruier Trasform of the sytheti ompoets of aeleratio. Lee ad Trifua (1987) later further exteded their work of the geeratio of sytheti Torsioal motios above to that of geeratig sytheti rokig motios.. It was show i a similar way that the rokig motio φ ( x, yt, ) at the groud surfae (y = 0) is related to the orrespodig vertial displaemet vxyt (,, ) by iω φ v y = = (2) 0 x y= 0 y= The Ratios of Rotatioal to Traslatioal Fourier Spetra The work of Lee ad Trifua (1985, 1987) allows the sytheti, artifiial Rotatioal Fourier Spetra to be ostruted together with the Traslatioal Fourier Spetra usig Eq (1) for Torsioal Spetra ad Eq (2)

4 for Rokig Spetra. From the Rotatioal Fourier Spetra, Iverse Fourier Trasform will geerate the orrespodig sytheti Torsioal ad Rokig aeleratio time histories. This proedure a the be adopted ad modified to geerate estimated Rotatioal Fourier Spetra from atual traslatioal Fourier Spetra omputed from atual reordigs Equatio (1), for example, shows that the ratio of the Fourier spetra of Torsioal motio, Ψ ( ω), to that of the orrespodig trasverse horizotal motio, H ( ω ) is Torsio FS Ψ( ω) iω Ψ( ω) ω = =, with amplitude = (3) Horizotal FS H( ω) 2 ( ω) H( ω) 2 ( ω) where ( ω ) is the horizotal phase veloity. Table 1 Layered Medium ear El Cetro, Califoria. Figure 1 Layer # Depth (km) α β Desity (gm/) , : P, S-Wave Veloities, (km/se) α β As a illustratio we osider a 7 Layer model (Table 1). It orrespods to a site i El Cetro, Califoria. Dispersio urves for this site were omputed usig the Haskell-Thomso matrix method (Haskell, 1953; Thomso, 1950), whih geerated the phrase ad group veloity urves for Rayleigh ad Love waves. The 5 solid urves i Fig. 1 from the right are for the first five modes of the Rayleigh waves. Mode 1 is the rightmost solid urve, followed by Modes 2 to 5 to the left. The same holds for the five modes of Love waves, whih are plotted with dashed lies. At short periods, all urves approah the miimum shear S - wave veloity (i this example 0.30 km/s). The two leftmost urves are the orrespodig body waves, with the dashed S wave below the solid P wave o top. The body S-wave starts at the miimum, 0.3 km/s at period ear 0.1se ad quikly rises to the maximum ad quikly rises up to the maximum as the 6 th ad 7 th modes of waves. β, 3.7 km/s. The body P-wave similarly starts at the miimum β α, 0.6 km/s α, 6.4km/s. They will be iluded with the surfae waves respetively Next we osider the trasverse ompoets of Love waves ad the orrespodig torsioal waves. Lee th ad Trifua (2008) oted, the waves orrespodig to their m mode, defied (oly) withi the frequey bad, ω ±Δ ω, are Ψ m ( ω) iω Ψ m( ω) ω =, with amplitude = (4) H ω 2 H ( ω) 2 ( ) m m m m

5 th where m = m ( ω) is the Love wave phrase veloity of m mode at frequey ω = ω, H m ( ω) ad Ψ m ( ω) are respetively the m th mode of the Traslatioal ad Rotatioal Fourier Spetra from the surfae Love waves. Figure 2 Figure 2 shows a plot of these ratios, ω 2 m ( ω ), for the five modes of Love waves plus body S-waves as the 6 th mode. The urves for the 6 modes all have similar slopes, espeially at high frequey, alog the diagoal gree lie, a straight lie joiig the limits of ω 2βmax at the low frequey ed ad ω 2βmi at the high frequey ed. This is a importat liear tred, whih will be used. A simple way to haraterize the ratio, Ψ ( ω) H ( ω), with all the modes of Love waves ombied, withi ω ±Δ ω, would be to write (Lee ad Trifua, 2008): Ψ( ω) Ψ( ω) Ψ( ω) iω = = = H( ω) ω= ω H( ω) H( ω) 2 ω= ω (5) where = ( ω), ad 1 represet weighted average of the phrase veloities of all the modes of Love waves at frequey ω. Note that this ratio is depedet oly o the frequey ad the loal phase veloity spetra of the surfae Love waves at the site. a be viewed as the weighted average of the phrase veloities = ( ω ) at frequey m m ω = ω of the 5 modes of surfae Love waves plus body S-wave. The dashed lie as ratio of amplitudes ω 2 ( ω ) will thus be ruig very lose alog the dashed lie from ω 2βmax at low frequeies to ω 2β at high frequeies. mi I exatly the same way, the spetra of rokig, Φ ( ω), a ext be expressed i terms of the spetra of the vertial traslatio, V ( ω ). The vertial traslatioal Fourier spetrum is ow omposed of modes of Rayleigh surfae waves. Now Rokig FS Φ( ω) Φ( ω) Φ( ω) iω = = = =, (6) Vertial FS V( ω) V( ω ) V ( ω) ω ω ω ω ω ω = = = with = ( ω ), represetig the weighted average of the M modes of Rayleigh waves. Reall that these ratios of Rotatioal to Traslatioal Fourier Spetra (Equatio (5) for Torsio, ad Equatio (6) for Rokig) are depedet oly o the phrase veloity spetra, = ( ω), at the site ad are idepedet of the iput traslatioal aeleratio. I other words, for the same site, the ratios of Rotatioal to Traslatioal Fourier Spetra are always give by Equatios (5) ad (6), depedet oly o the loal geology at

6 the site. Suh ratios a be estimated at eah site by geeratig the sytheti Rotatioal ad Traslatioal aeleratio (Lee ad Trifua, 1985, 1987) Spetra ad takig their ratios. With these ratios available at a give site, ad if a atual aelerogram reordig is available at the same site, they a ext be applied to alulate the rotatioal Torsioal ad Rokig Fourier Spetra from the atual Traslatioal Fourier Spetra. The orrespodig rotatioal time histories a the be alulated usig Iverse Foruier Trasform. 3. EXAMPLE: ROTATIONAL SPECTRA AT PACOIMA DAM, CALIFORNIA Figure 3 Figure 4 Here s a example of the estimatio of rotatioal spetra from the traslatioal ompoets of the Fourier spetra of reord strog motio. We illustrate this for aelerogram reorded at Paoima Dam, i Califoria, durig the Feb 9, 1971 Sa Ferado Earthquake. Its S74W (horizotal) ompoet had a peak 2 2 aeleratio of m / se, while the vertial ompoet had a peak aeleratio of m / se. The Fourier spetra of these two horizotal (as the trasverse) ad vertial ompoets a be used to estimate the orrespodig torsioal ad rokig ompoets of motio. Figure 3 shows a plot of the Fourier Spetrum amplitudes of the horizotal ompoet S74W i uits of i/se ad the orrespodig torsioal ompoet i uits 3 of mrad /se ( 10 rad / se ). Figure 4 shows the plot of the Fourier Spetrum amplitudes of the vertial ompoet of aeleratio, ad of the orrespodig rokig ompoet. The geologial veloity profile of Table I is used, where the miimum ad maximum shear wave veloities, β mi ad β max, rage from 0.30 km/se to 3.70 km/se. Figures 5 ad 6 show the orrespodig rotatioal torsioal ad rokig time histories obtaied from the iverse FFT of the orrespodig rotatioal spetra i Figures 3 ad 4. The rotatioal aeleratio, veloity ad

7 displaemet are respetively give i uits of mrad/s/s, mrad/s ad mrad, where 1 mrad = rad. The peak rotatios of 0.3 mrad for torsio ad 0.6 mrad for rokig are obtaied from the alulatios. Figure 5 Figure 6 It is see that oe a use this proedure to geerate estimated time histories for torsioal ad rokig motios for every reorded traslatioal aelerogram for whih the site geologial date are available. Further work ad approah a be foud i Lee ad Trifua, Next we ll wait for the atual reordig of traslatioal ad rotatioal data to be ompared with the estimated data by this proedure.

8 REFERENCES Bouho, M., ad Aki, K. (1982). Strai ad rotatio assoiated with strog groud motio i the viiity of earthquake faults. Bull. Seism. So. Am. 72(5), Graizer, V.M. (2006). Tilts i strog groud motio. Bull. Seism. So. Am., 96(6), Haskell, N.A. (1953). The Dispersio of Surfae Waves o Multilayered media, Bull. Seism. So. Am., 43: Hudso, D. E. (1983a) History of Aelerograph Developmet, Pro. of the Golde Aiversary Workshop o Strog Motio Seismology, U.S.C., Hudso, D. E. (1983b). Strog Motio Istrumetatio Systems, Pro. of the Golde Aiversary Workshop o Strog Motio Seismology, U.S.C., Kobori, T. ad Shiozaki, I. Torsioal (1973). Vibratio of Struture Due to Obliquely Iidet SH Waves, Pro. Fifth W orld Cof Earthquake Eg. 1(22). Lee, V. W. (1979). Ivestigatio of Three-Dimesioal Soil-Struture Iteratio, Dept. of Civil Eg. CE 79-11, Uiversity of Souther Califoria, Los Ageles, 1979 Lee, V.W. (2002a). Empirial Salig of Strog Earthquake Groud Motio: Part I: Atteuatio ad Salig of Respose Spetra, ISET J. Earthquake Tehology, 39(4) Lee, V.W. (2002b) Empirial Salig of Strog Earthquake Groud Motio: Part III: Sytheti Strog Motio, ISET J. Earthquake Tehology, 39(4), Lee,V.W. (2007). Empirial Salig of Strog-Motio Respose Spetral Amplitudes - a Review ISET J. Earthquake Tehology, 44(1) Speial Issue V. W. Lee ad M.D. Trifua (1985). Torsioal Aelerograms, It. J. Soil Dyamis & Earth quake Eg., 4(3), Lee, V. W. ad M.D. Trifua (1987) Rokig Strog Earthquake Aeleratios, It. J. Soil Dyamis & Earthquake Eg., 6(2), Lee, V.W. ad Trifua, M.D. (1995). Frequey Depedet Atteuatio Futio ad Fourier Amplitude Spetra of Strog Earthquake Groud Motio i Califoria, Report CE 95-03, Dept. of Civil Eg., Uiv. of Souther Califoria, Los Ageles, CA. Lee, V.W. ad Trifua, M.D. (2008). Empirial Salig of Rotatioal Spetra of Strog Earthquake Groud Motio, BSSA Speial Issue o "Rotatioal Seismology ad Egieerig Appliatios (i review). Luo, J. E. (1976). Torsioal Respose of Strutures for SH Waves The Case of Hemispherial Foudatio, Bull. Seism. So. Amer. 66, Takeo, M. (2006). Groud rotatioal motios reorded i ear-soure regio of earthquakes, Ch. 12 i the book Earthquake Soure Asymmetry, Strutural Media ad Rotatio Effets, R. Teisseyre, M. Takeo, ad E. Majewski (eds.). Spriger, Heidelberg, Germay, Takeo, M., ad Ito, H.I. (1997). What a be leared from rotatioal motios exited by earthquakes?, Geophys. Jour. It., 129, Thomso, W.T. (1950). Trasmissio of Elasti Waves through a Stratified Solid Medium, J. Appl. Phys, 21: Trifua, M.D. (1971) A Method for Sythesizig Realisti Strog Groud Motio, Bull. Seism. So. Amer., 61, Trifua, M.D. (1972). Stress Estimates for Sa Ferado, Califoria Earthquake of February 9, 1971: Mai Evet ad Thirtee Aftershoks, Bull. Seism. So. Amer., 62(3), Trifua, M.D. (1974). A Three-Dimesioal Disloatio Model for the Sa Ferado, Califoria, Earthquake of February 9, 1971, Bull. Seism. So. Amer., 64(1), Trifua, M.D. (1976). Prelimiary Empirial Model for Salig Fourier Amplitude Spetra of Strog Groud Aeleratio i Terms of Earthquake Magitude, Soure to Statio Distae ad Reordig Site Coditios, Bull. Seism. So. Amer., 66(4), Trifua, M. D. (1982) A Note o Rotatioal Compoets of Earthquake Motios o Groud Surfae for Iidet Body Waves, Soil Dyamis ad Earthquake Egieerig 1(1),

9 Trifua, M.D. (1989a). Depedee of Fourier Spetrum Amplitudes of Reorded Strog Earthquake Aeleratios o Magitude, Loal Soil Coditios ad O Depth of Sedimets, It. J. Earthquake Eg. ad Strutural Dy., 18(7), Trifua, M.D. (1989b).Empirial Salig of Fourier Spetrum Amplitudes of Reorded Strog Earthquake Aeleratios i Terms of Magitude ad Loal Soil ad Geologi Coditios, Earthquake Eg. ad Eg. Vibratio, 9(2), Trifua, M.D. (1990). How to Model Amplifiatio of Strog Earthquake Motios by Loal Soil ad Geologi Site Coditios, It. J. Earthquake Eg. ad Strutural Dyamis, 19(6), Trifua, M.D. (1991). Empirial Salig of Fourier Spetrum Amplitudes of Reorded Strog Earthquake Aeleratios i Terms of Modified Meralli Itesity, Loal Soil Coditios ad Depth of Sedimets, It. J. Soil Dyamis ad Earthquake Eg., 10(1), Trifua, M.D. (2006). Effets of Torsioal ad Rokig Exitatios o the Respose of Strutures, Ch. 39 i the book Earthquake Soure Asymmetry, Strutural Media ad Rotatio Effets, Edited by R. Teisseyre, M. Takeo, ad E. Majewski, Spriger, Heidelberg, Germay, Trifua, M.D. (2007). Reordig Strog Earthquake Motio Istrumets, Reordig Strategies ad Data Proessig, Dept. of Civil Egieerig, Report CE 07-03, Uiv. Souther Califoria, Los Ageles, Califoria. Trifua, M.D. (2008a). 75 th Aiversary of Strog Motio Observatio A Historial Review, (submitted for publiatio). Trifua, M.D. (2008b). Buildigs as Soures of Rotatioal Waves, Chapter I.5 i Physis of Asymmetri Cotiua: Extreme ad Frature Proesses, edited by R. Teisseyre, H. Nagahama, ad E. Majewski, Spriger, Heidelberg, Germay, (i press). Trifua, M.D. (2008). The Role of Strog Motio Rotatios i the Respose of Strutures ear Earthquake Faults, Soil Dyamis ad Earthquake Egieerig, (i press). Trifua, M.D., ad Hudso, D.E. (1971). Aalysis of the Paoima Dam aelerogram, Sa Ferado, Califoria earthquake of Bull. Seism. So. Amer., 61(5), Wog, H.L., ad M.D.Trifua (1979). Geeratio of Artifiial Strog Motio Aelerograms, Iteratioal Joural of Earthquake Egieerig ad Strutural Dyamis, 7,