A GENERATION METHOD OF SIMULATED EARTHQUAKE GROUND MOTION CONSIDERING PHASE DIFFERENCE CHARACTERISTICS

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1 Otober 1-17, 8, Beijing, China A GENERATION METHOD OF SIMULATED EARTHQUAKE GROUND MOTION CONSIDERING PHASE DIFFERENCE CHARACTERISTICS T. Yamane 1 and S. Nagahashi 1 Senior Strutural Engineer, Strutural Engineering Group, Nien Seei Ltd., Toyo, Japan Professor, Dept. of Arhiteture and Civil Engineering, Chiba Institute of Tehnology, Chiba, Japan yamanet@nien.o.jp, nagahashi.sumio@it-hiba.a.jp ABSTRACT: A method of simulating earthquae ground motions for testing the strutural design of a building is proposed. In this method, a seismi wave is synthesized by ombining the Fourier amplitude presribed by the ω model with the Fourier phase from observation reords seleted on the basis of the standard deviation of the phase differenes, whih reflets the soure and propagation harateristis of an earthquae. The proposed method an simulate seismi waves for different types of earthquaes. The dynami behavior of a high-rise building subjeted to earthquae ground motions having a long-period omponent was found to differ from that of a building subjeted to a rustal earthquae in the viinity of a onstrution site. This finding suggests the importane of onsidering the earthquae environment around onstrution sites and using simulated earthquae ground motions to test the strutural designs of buildings. KEYWORDS: Waveform generation method, Simulated earthquae ground motion, Phase differene 1. INTRODUCTION Performane-based design has been employed reently for the strutural design of buildings in Japan. The basi onepts of performane-based design an be summarized as (i) examine the performane of eah building, and (ii) learly explain the results to the publi. To verify resistane to earthquaes, the earthquae environment, soil onditions at the onstrution site, et., must be onsidered and the dynami behavior of eah building evaluated. Sine step-by-step integration is performed when evaluating the dynami behavior of high-rise or base-isolated buildings, in order to satisfy the basi onepts of performane-based design, the dynami analyses must inlude the time histories of the aeleration of seismi waves, whih reflet the magnitude of the earthquae, the hypoentral distane, and the soil onditions. Thus, performane-based design requires a pratial method of simulating earthquae ground motions that an be easily applied to a variety of projets. The purpose of this paper is to propose a method of simulating earthquae ground motions for testing the strutural design of buildings. In this method, the Fourier amplitude is presribed by the ω model, and the Fourier phase is seleted from observation reords. The phase harateristis are quantitatively evaluated by adopting the standard deviation of the as an index of the properties of the hypoentral region, inluding the rupture proess, and the dispersion during wave propagation. Relational expressions of the standard deviation of the against the hypoentral distane, whih are obtained by analyzing the observation reords, are employed as riteria for seleting the Fourier phase. The seismi waves of inter-plate and rustal earthquaes synthesized by the proposed method differ in the shape of their wave envelopes and their frequeny harateristis. Furthermore, numerial analyses of high-rise buildings using these waves show that the dynami behavior of a high-rise building subjeted to an inter-plate earthquae with a long-period omponent differs from that of a building subjeted to a rustal earthquae in the viinity of the onstrution site. This finding suggests the importane of using simulated earthquae ground motions to evaluate the dynami behavior of buildings in performane-based design. Although this paper fouses on a method for generating a body wave, a method for generating a surfae wave (Love wave) in a sedimentary basin also has been developed using the same framewor. These synthesized Love waves have been applied to the design of a tower about 6 m in height that urrently is being onstruted in Toyo.

2 Otober 1-17, 8, Beijing, China. PHASE DIFFERENCE CHARACTERISTICS OF EARTHQUAKE GROUND MOTIONS.1. Definition of Phase Differene Charateristis As is well nown, a disrete time history, f (t), is expressed by a finite Fourier series: f ( t) = N / = a os( ω t + φ ) (.1) where a, φ, ω, and N denote the Fourier amplitude, the Fourier phase, an angular frequeny of the th degree, and the number of disrete data points, respetively, and providing ω = φ = φ N / = based on Fourier expansion. The differene in the Fourier phase for any adjaent frequenies, whih are arranged so as to have the value of an interval [, -π ], is defined as the phase differene (Ohsai, 1979): φ = φ + φ ( = 1,, L, N / 1) (.) 1 Examples of a distribution of the that is represented as a histogram by dividing the interval [, -π ] into 3 are shown in Figure.1. These examples suggest that the onvergene of the orresponds to the envelope of the aeleration waveform. It also suggests that this feature an be used when onsidering the harateristis of a time history, i.e., the phase harateristis. This study proposes the adoption of the standard deviation of the as an index, whih will permit quantitative evaluations of the phase harateristis. For instane, the standard deviation of the normalized by π is.18 in the ase of the example in Figure.1(a) and 5 in the ase of the example in Figure.1(b), respetively. Analysis of a number of reords onfirmed that this index σ / π an be used to ontinuously evaluate the differenes in the phase harateristis of earthquae ground motions σ / π = π -1.π -1.5π -.π (a) EW omponent aelerogram and distribution of the reorded at Hahinohe Harbour during the 1968 Toahi-oi Earthquae Time (se) 1..5 σ / π = 5 -.5π -1.π -1.5π -.π (b) NS omponent aelerogram and distribution of the reorded at JMA Kobe during the 1995 South Hyogo Prefeture Earthquae Figure.1 Examples of aelerograms and distributions of the.. Standard Deviation of the Phase Differenes for Crustal Earthquaes Quantitative evaluations of the phase harateristis apply the standard deviation of the, σ / π. For rustal earthquaes, plots of σ / π against the hypoentral distane, r, for the reords from KiK-net (a system of sending strong-motion data on the Internet in Japan) seismi stations with respet to the Western Tottori Prefeture Earthquae are shown in Figure..

3 Otober 1-17, 8, Beijing, China Standard deviation of Hypoentral distane (m) Figure. Standard deviation of the against the hypoentral distane for the reords at the KiK-net observation sites with respet to the Western Tottori Prefeture Earthquae The plot in Figure. shows that the longer the hypoentral distane, the greater the standard deviation of the. In other words, the effet of the dispersion in the wave propagation is refleted in the index σ / π. A regression analysis of the data shown in Figure. gives the following relational expression. σ / π = r (.3) In the proposed method, Eqn..3 is used as the riteria for seleting the Fourier phase from a number of reords of earthquae ground motions in the ase of rustal earthquaes. The value of σ / π is based on the hypoentral distane under the given onditions. Thus, the appropriate Fourier phases having the speified value of σ / π are seleted from reords of earthquae ground motions..3. Standard Deviation of the Phase Differenes for Inter-plate Earthquaes For inter-plate earthquaes, the relational expressions are determined by analyzing the observation reords for the 3 Toahi-oi Earthquae. Figure.3(a) shows σ / π with respet to the NS omponent of the reords from K-NET (another system similar to KiK-net) seismi stations in Hoaido. (a) σ / π of the NS omponent N Standard deviation of Hypoentral distane (m) (b) Plots of σ / π of the NS and EW omponents in eastern Hoaido (In the diretion of rupture propagation) Standard deviation of Hypoentral distane (m) () Plots of σ / π of the NS and EW omponents in western Hoaido (In the orthogonal diretion of rupture propagation) Figure.3 Standard deviation of the at the K-NET observation sites in Hoaido with respet to the 3 Toahi-oi Earthquae Assuming the strie slip, i.e., the diretion of rupture toward the northeast on the fault plane of this earthquae, the data is analyzed in eah area divided by the solid line in Figure.3(a). σ / π of the observed data in the eastern and western areas are plotted against r in Figures.3(b) and (), respetively. An analysis of these data produes the following relational expressions. In the diretion of rupture propagation: σ / π = 5 + 3r (.4) In the orthogonal diretion of rupture propagation: σ / π = 8 + 3r (.5)

4 Otober 1-17, 8, Beijing, China It should be noted that Eqns..4 and.5 are appliable in the ase of inter-plate earthquaes with a fault length of about 1 m beause the fault plane for the 3 Toahi-oi Earthquae was about 1 m in length. The equality of the gradients among Eqns..3,.4, and.5 shows that the effets of dispersion in wave propagation are generally similar and not dependent on the type of earthquae or the diretion of propagation. On the ontrary, the differenes in the interepts in those equations orrespond to the soure harateristis of earthquaes, inluding the rupture proess. 3. METHOD OF SIMULATING EARTHQUAKE GROUND MOTIONS 3.1. Amplitude Charateristis The Fourier Amplitude with respet to the soure harateristis is given by the ω model (Ai, 1967) as Ω A ( f ) = M Ω ( f ) = M A (π f ) (π f ) ( f ( f > f f ) ) (3.1) where Ω A ( f ), M, and f denote the soure spetrum of aeleration, seismi moment, and orne r frequeny. The propagation harateristis are written as R 1 U & θφ ( r, f ) = Ω ( f ) 3 A 4π ρv r (3.) where U & ( r, f ), ρ, V, and S Rθφ denote the Fourier Amplitude of aeleration on the bedro, density and veloity of a shear wave in bedro, and the radiation pattern oeffiient (Kanamori and Anderson, 1975). In this study, R θφ =.63 (Ai and Rihards, 198). Although the published values in papers or from publi institutions are basially utilized in the estimation of M and f, f = 1/( πτ ) is also used, where the rise time, τ, is given by S τ = (3.3) M Eqn. 3.3 was formulated from the relationships of τ =.5,1.6, 5. se. as M = 1.6 1, 5. 1, and Nm, respetively (Sato, 1979). 3.. Phase Charateristis The seletion riteria for the phase harateristis are disussed here beause the appliation of the Fourier phase from the observation reords is presupposed in the proposed method. For rustal earthquaes, Eqn..3 is used for seleting the phase harateristis from a number of reords. In other words, the Fourier phases, whih have the value of σ / π as speified by Eqn..3 based on the hypoentral distane, are applied when simulating earthquae ground motions. On the other hand, the seletion riteria for the Fourier phase for inter-plate earthquaes that have faults 1 m in length are given by Eqn..4 in the diretion of rupture propagation and by Eqn..5 in the orthogonal diretion. Based on Eqns..4 and.5, the relationship of σ / π at the observation sites to a fault plane typially an be represented as shown in Figure 3.1, inluding the interpolated evaluation in the oblique diretions.

5 The 14 th World Conferene on Earthquae Engineering Otober 1-17, 8, Beijing, China r = m σ / π =.14 r = 1 m σ / π =.11 r = m σ / π = 8 Hypoenter r = 1 m σ / π =.1 Fault plane r = 1 m σ / π = 9 r = 1 m r = m r = 3 m σ / π = 8 σ / π =. 11 σ / π =. 14 Figure 3.1 Standard deviation of the at the observation sites As for the site harateristis of the, the shape of the distribution of the at the surfae of the ground has been onfirmed to be similar to that in seismi bedro, as shown in Figure 3.. In other words, the values of σ / π for those waves are almost the same. Thus, the property of the phase differenes is fairly independent of the site harateristis. Consequently, the Fourier phase of the time history reorded at the surfae of the ground an be used to synthesize a wave in seismi bedro Time (se) Time (se) π -1.π -1.5π -.π (a) Aelerogram and distribution of the at the surfae of the ground π -1.π -1.5π -.π (b) Aelerogram and distribution of the in seismi bedro (GL-.1m) Figure 3. Comparison of distributions of the at the surfae of the ground and in seismi bedro at Hinase, Oayama ( r = 13 m), during the Western Tottori Prefeture Earthquae σ / π = 84 σ / π = Flow Chart of the Proposed Method A seismi wave in the bedro below a onstrution site is synthesized by ombining the Fourier amplitude and Fourier phase, as desribed above. The simulated earthquae ground motion at the surfae of the ground an be obtained by applying a multiple refletion theory of the horizontally layered soil model to the seismi wave in the bedro. A flow hart of this method is shown in Figure 3.3. Fourier amplitude Soure harateristis: Eqn. 3.1 Propagation harateristis: Eqn. 3. Fourier phase (Inverse Fourier transform) Synthesized wave in seismi bedro Synthesized wave at surfae of ground Soure and propagation harateristis: Crustal earthquae Eqn..3 Inter-plate earthquae In the diretion of rupture propagation Eqn..4 In the orthogonal diretion of rupture propagationeqn..5 Figure 3.3 Flow hart of the proposed method Site harateristis: Multiple refletion theory

6 Otober 1-17, 8, Beijing, China 4. VERIFICATION OF THE PROPOSED METHOD The proposed method was validated by omparing the synthesized waves with observed waves. Figure 4.1 shows the observed waves at Yubara, Oayama Prefeture, during the Western Tottori Prefeture Earthquae, the synthesized waves generated by this method, and their response spetra. In this simulation, Nm was adopted as M (Sagiya et al., ) and.1 se. as τ (Satoh, ). For the phase harateristis, a Fourier phase with σ / π of 7, whih is evaluated by Eqn..3 based on r = 31m, was seleted from reords of the same earthquae taen at a different observation site. Figure 4.1 shows that the synthesized waves an appropriately simulate the observed waves in terms of the maximum values, the shapes of the wave envelopes, and the frequeny harateristis (a) Aeleration of observed wave (b) Aeleration of synthesized wave Vel. (m/s) Vel. (m/s) () Veloity of observed wave (d) Veloity of synthesized wave Pseudo-veloity response (m/s) Observed wave Synthesized wave Period (se) (e) Pseudo-veloity response spetra (h=5%) Figure 4.1 Comparison between observed waves (EW omponent) reorded at Yubara, Oayama and synthesized waves with respet to the Western Tottori Prefeture Earthquae The other verifiation is shown in Figure 4., whih shows a omparison of the waves at Sapporo, Hoaido, 1 during the 3 Toahi-oi Earthquae. In this simulation, 1. 1 Nm was adopted as M (Earthquae Researh Institute, University of Toyo, 3) and 4.3 se. as τ based on Eqn The Fourier phase with σ / π of.17, whih is evaluated by Eqn..5 based on r = 86 m, was seleted from observation reords for the Western Tottori Prefeture Earthquae. In other words, a reord for a different type of earthquae is applied. As shown in Figure 4., the proposed method an simulate not only the duration of a wave but also the long-period omponent, whih appears in the veloity waveform (Figures 4.() and (d)). Though an f max (Hans, 198) of 1Hz is used in this method, the f max in the ase of Sapporo is assumed to be Hz by onsidering the redution in the high-frequeny omponents due to the refletion of waves in the thi deposits of the sedimentary basin. This is not a signifiant problem beause dynami analysis of a high-rise or base-isolated building does not require an aurate estimate for frequenies higher than Hz (a) Aeleration of observed wave (b) Aeleration of synthesized wave Vel. (m/s) Vel. (m/s) () Veloity of observed wave (d) Veloity of synthesized wave Pseudo-veloity response (m/s) Period (se) Observed wave Synthesized wave (e) Pseudo-veloity response spetra (h=5%) Figure 4. Comparison between observed waves (NS omponent) reorded at Sapporo, Hoaido and synthesized waves with respet to the 3 Toahi-oi Earthquae

7 Otober 1-17, 8, Beijing, China 5. EXAMPLES OF SIMULATED EARTHQUAKE GROUND MOTIONS Examples of simulated earthquae ground motions for inter-plate and rustal earthquaes are desribed in this setion. The ground motion of an inter-plate earthquae at Chiyoda-u, Toyo, is synthesized after the Great Kanto Earthquae in 193, whih had a moment magnitude, seismi moment, and fault length of M w = 7. 9, M = 8 1 Nm, L = 13 m, respetively (Wald and Somerville, 1995). σ / π =. 9 in onsideration of the positional relationship of the site to the fault plane shown in Figure 3.1. On the other hand, for the example 18 of a rustal earthquae with a fault length of 17 m, M w = 6. 5 ( M = 6. 1 Nm). Assuming that this earthquae would our near the site, i.e., the hypoentral distane is m in Eqn..3, σ / π is presribed for 6. Furthermore, the proposed method provides that the lower bound of r in Eqn. 3. is the fault length, namely the upper bound of Fourier amplitude in this equation is given thus. Table 5.1 is a list of the soil onditions at Chiyoda-u, Toyo. Table 5.1 Soil onditions at Chiyoda-u, Toyo Depth (m) Thiness (m) Veloity of shear wave (m/s) Density (t/m 3 ) Damping fator (%) Figure 5.1 shows the synthesized waves of the senario earthquaes mentioned above, the response spetra of those waves, and the distribution of the of observed waves seleted in these simulations. The data in Figure 5.1 onfirms that these waves differ onsiderably in the shape of the wave envelope and frequeny harateristis Time (se) Time (se) 1..5 (a) Inter-plate earthquae -.5π -1.π -1.5π -.π 1..5 (b) Crustal earthquae -.5π -1.π -1.5π -.π σ / π = 9 σ / π = 6 Pseudo-veloity response (m/s) Inter-plate earthquae Crustal earthquae Period (se) () Pseudo-veloity response spetra (h=5%) Figure 5.1 Comparison of synthesized waves for inter-plate and rustal earthquaes Numerial analyses of high-rise buildings were performed by these synthesized waves. Time histories of the displaement at the top of a 6-story building having a height of 4 m, a fundamental period of 6. se., and damping fator of % are shown in Figure 5.. Refleting the differene in the shapes of the wave envelopes and the frequeny harateristis shown in Figure 5.1, the high-rise building was vigorously shaen for a long time by the inter-plate earthquae (Figure 5.(a)). This was beause the synthesized wave had the long-period omponent exited by the thi deposits of the sedimentary basin in the Kanto Plain (Figures 5.1(a) and ()). On the other hand, the vibration of the building subjeted to a rustal earthquae in the viinity of the onstrution site gradually attenuated in a manner similar to free osillation (Figure 5.(b)). This finding suggests that it is important for designers to onsider the differenes in the properties of various types of earthquae ground motions.

8 Otober 1-17, 8, Beijing, China Displaement at the top of a building (m) (a) Inter-plate earthquae Displaement at the top of a building (m) (b) Crustal earthquae Figure 5. Comparison of the displaements at the top of a 6-story building subjeted to inter-plate and rustal earthquaes 6. CONCLUSIONS Relational expressions of the standard deviation of the with respet to earthquae ground motions against the hypoentral distane exhibit the soure and propagation harateristis of seismi waves. On the basis of those expressions, a method of simulating earthquae ground motions for testing the strutural designs of buildings is proposed. In this method, a seismi wave is synthesized by ombining the Fourier phase seleted from observation reords by means of a quantitative evaluation using those expressions and the Fourier amplitude presribed by the ω model. The proposed method was validated by omparing the synthesized wave with observed earthquae ground motions. Examples of the simulated earthquae ground motions for inter-plate and rustal earthquaes indiate differenes in those waves in terms of the shape of their wave envelopes and their frequeny harateristis. Furthermore, time history analyses using these synthesized waves show that a high-rise building subjeted to an inter-plate earthquae with a long-period omponent exited in a sedimentary basin will shae vigorously for a long time. On the other hand, a high-rise building subjeted to a rustal earthquae in the viinity of a onstrution site will experiene vibrations that gradually attenuate in a manner similar to free osillation. In performane-based design, it is important to onsider the earthquae environment around the onstrution site and appropriately apply simulated earthquae ground motions to the dynami analyses of the buildings. REFERENCES Ai, K. (1967). Saling law of seismi spetrum. Journal of Geophysial Researh 7:4, Ai, K. and P. G. Rihards (198). Quantitative Seismology, Theory and Methods, W. H. Freeman and Company, San Franiso, California, U.S.A. Earthquae Researh Institute, University of Toyo (3). /EIC_News /396.html Hans, T. C. (198). f max. Bulletin of the Seismologial Soiety of Ameria 7:6, Kanamori, H. and D. L. Anderson (1975). Theoretial basis of some empirial relations in seismology. Bulletin of the Seismologial Soiety of Ameria 65:5, Ohsai, Y. (1979). On the signifiane of phase ontent in earthquae ground motions. Earthquae Engineering and Strutural Dynamis 7, Sagiya, T., T. Nishimura, Y. Hatanaa, E. Fuuyama and W. L. Ellsworth (). Crustal movements assoiated with the Western Tottori Earthquae and its fault models. Journal of the Seismologial Soiety of Japan 54:4, (in Japanese). Sato, R. (1979). Theoretial basis on relationships between foal parameters and earthquae magnitude. Journal of Physis of the Earth 7, Satoh, T. (). Radiation pattern and fmax of the Tottori-en Seibu Earthquae and the aftershos inferred from KiK-net strong motion reords. Journal of Strutural and Constrution Engineering (Transations of Arhitetural Institute of Japan) 556, 5-34 (in Japanese). Wald, D. J. and P. G. Somerville (1995). Variable-slip rupture model of the great 193 Kanto, Japan, earthquae: Geodeti and body-waveform analysis. Bulletin of the Seismologial Soiety of Ameria 85:1,