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This paper introduces an application of wave propagation modeling to building response investigation. We have analyzed the computed responses of analytical building models and the obtained strong-motion records for forty-one actual buildings during five different earthquakes by using the normalized input output minimization (NIOM) method. This method can model wave propagation in multiple linear systems by considering the statistical correlation of the earthquake motions at different observation locations, and can reveal the arrival times of incident and reflected waves as well as their relative amplitudes. From these values, the fundamental period and the damping ratio of the building could be simply estimated. The estimated values were then compared with the values for the analytical building models, the values estimated in the previous studies for the actual buildings, and the building code formula. Keywords: strong motion records; buildings; wave propagation; fundamental period; damping 1. Introduction methods is the normalized input output minimization Records of building motion are a valuable source of (NIOM) method (Kawakami and Haddadi 1998, Haddadi experimental data containing all the uncertainties and Kawakami 1998), and we use it in our study. The inherent in the behavior of buildings during earthquakes. NIOM method models wave propagation in multiple However, the effects of ground motion on buildings linear systems by considering the statistical correlation cannot be directly seen from the records because they of the earthquake motions at different observation are much more complex than those during oscillator locations. experiments (Ohba 1981). Therefore, after getting the First, the computed response time histories of the records, they should be thoroughly analyzed. analytical building models are analyzed with the NIOM Analyzing the behavior of a structure during an method in order to check the applicability of the method earthquake involves two problems: vibration and wave to building records. Next, actual acceleration records propagation, because the vibration of a structure results recorded in forty-one buildings during several recent from seismic wave propagation in it. However, the most earthquakes are analyzed and compared with the results frequently used and conventional methodologies for for the analytical building models. Obtained wave arrival studying building response to earthquake motions are times and wave amplitude ratios are simply related to the system identification method and spectral ratio vibration properties like the fundamental period and analysis, which are vibration approaches. Vibration damping ratio of each building. Finally, conclusions are methods are well known and have been developed mainly drawn by comparing the NIOM results for the actual for structural engineering purposes. In contrast, wave and analytical buildings with those of previous propagation approaches have mainly been used to investigations. investigate ground motions during earthquakes, and researchers have used several methods (such as impulse 2. Data response and correlation functions) to simplify and 2.1 Computed Response of Assumed Analytical clarify the wave propagation in soil layers and to Building determine soil properties. The earthquake responses of six analytical buildings This paper introduces an application of wave with 2, 5, 10, 20, 30 and 40 stories were computed. The propagation modeling to the strong motion records for fundamental periods of these buildings are assumed to buildings. One of the wave propagation modeling be 0.5, 0.8, 1.6, 3.0, 4.0 and 5.0 s, respectively, based on previous research on the fundamental period of steel *Contact Author: Hideji Kawakami, Geosphere Research Institute moment-resisting frame buildings. The buildings used of Saitama University, 255 Shimo-ohkubo, Sakura-ku, in this paper are the same as in Chopra (1995), and all Saitama-shi, 338-8570 Japan the stories of these buildings have equal masses (45.3 t) Tel: +81-48-858-3543 Fax: +81-48-858-3543 and heights (4 m). Figure 1(a) shows a 10-story building e-mail: firstname.lastname@example.org and its properties. The story stiffnesses for the other (Received November 8, 2003 ; accepted April 6, 2004 ) buildings are given in Fig. 1(b). Journal of Asian Architecture and Building Engineering/May 2004/40 33 The acceleration time history recorded in the basement two modes (Chopra 1995). The ground motion and the of 3710 Wilshire Boulevard during the San Fernando computed response acceleration time histories at the fifth earthquake of February 9, 1971 (National Geophysical floor and roof are shown in Fig. 2(a). In order to check Data Center 1996) was used as a motion at the ground the applicability of the wave propagation modeling surface (Fig. 2(a)). The responses of the assumed method to buildings, the computed responses were buildings were computed by modal analysis, in which a analyzed by the NIOM method. Rayleigh damping ratio of 5% was assumed in the first Fig.1. (a) Configuration and Properties of Assumed 10-story Building and (b) Distribution of Story Stiffness in Assumed 2-, 5-, 20-, 30-, and 40-story Buildings (Chopra 1995) Data Center 1996, CSMIP 1994, Architectural Institute of Japan 1996). However, due to the limited space, we have selected the four buildings shown in Table 1 to illustrate how the wave propagates in the building. Table 1 lists the building and earthquake names, epicentral distance, structural type, number of stories, and observed maximum accelerations at the basement and top floor of the four buildings. Table 1. Building Specifications Fig.2. (a) Computed Response Acceleration of Assumed 10- story Building with Damping Ratio of 0.05 and (b) NIOM Method Analysis Result 2.2 Records for Actual Buildings We analyzed strong-motion records of forty-one buildings: twenty-one buildings during the February 9, 1971 San Fernando earthquake, one building during the October 1, 1987 Whittier Narrow earthquake, four buildings during the October 19, 1989 Loma Prieta earthquake, thirteen buildings during the January 17, 1994 Northridge earthquake (all in California, USA), and two buildings during the January 17, 1995 Hyogo- ken Nanbu earthquake in Japan (National Geophysical 34 JAABE vol.3 no.1 May. 2004 Hideji Kawakami 3. Wave Propagation Modeling Method amplitude of the input model at the building’s roof at The normalized input output minimization (NIOM) time t = 0 is defined to be normalized to unity. The method used in this study can model wave propagation Lagrange multipliers method gives the following in multiple linear systems by considering the statistical equation: correlation of earthquake motions at different observation locations. It can simplify the complex waveforms observed, extract several wave components such as incident and reflected waves, and give arrival times of incident and reflected waves as well as their relative amplitudes. Since the wave propagation velocity depends greatly on the characteristics of materials and structures through which the wave propagates, we have (3) tried to apply such a method to building record analysis in addition to ground motion analysis. A brief review of where λ = Lagrange multiplier; c to c = weighting 0 M the method is given below (Kawakami and Haddadi constants of the squared Fourier amplitude spectra of 1998, Haddadi and Kawakami 1998, Oyunchimeg and the input and outputs; and k to k = those of their time 0 M Kawakami 2003). derivatives. When a time-invariant linear system is subjected to When the ratios of the two weighting constants (k and earthquake motion, the input and outputs of the system c) are chosen to be the same for the input and outputs in the frequency domain can be related by means of the (k /c = k /c = = k /c ), the simplified input and output 0 0 1 1 M M transfer functions H (ω) (l=1,2,…,M). In the case of models are determined by the following equations. observed earthquake motions, the outputs at each frequency are given by (1) (4) where ∆t = the sampling rate in the time domain; M = the number of output motions; N = the number of samples; and F(ω ) and G (ω ) = the Fourier transforms i l i of the observed earthquake input and output motions, respectively. It should be noted that input couldn’t be separated (5) from output in the analysis of a feedback system. In fact, all the observed motions can be considered as different outputs subjected to a common excitation. In this paper, the input motion means neither the incident-wave motion nor the excitation, but the motion observed at one The inverse Fourier transforms of Eqs. (4) and (5) give arbitrary location (the building’s roof in this paper). simplified input and output models in the time domain. Transfer functions depend only on the physical This procedure is called the NIOM method. properties of the system. Therefore, the same transfer Simplified models may not always represent what is function as the one that defines the relationship between happening physically in detail. However, because these the observed input F(ω ) and output G (ω ) should satisfy models satisfy the statistical correlations between the i l i the relationship between the simplified input model X(ω ) observed motions, they help us to simplify and extract and the simplified output model Y (ω ). the physical properties of the system. The NIOM method l i offers the advantage of being able to investigate building (2) responses using only observed earthquake motions The procedure leading to the simplified input and without introducing any structural information like story output models is shown schematically in Fig. 3. Here, stiffness or damping distributions. The capabilities and the observed input F(ω ) and output G (ω ) are used to i l i applicability of the method were presented previously compute the transfer function H (ω ). Minimizing the l i by the developers (Kawakami and Haddadi 1998, summation of the squared values of Fourier amplitude Haddadi and Kawakami 1998). spectra of the input and outputs would result in the simplified input model, X(ω ), and the simplified output models, Y (ω ), (l=1,2,…,M). However, if there is not l i any constraint, the minimization procedure gives zero input and zero outputs at all times, and it does not give any useful result. Therefore, we assume that the JAABE vol.3 no.1 May. 2004 Hideji Kawakami 35 Fig.3. Schematic Procedure of NIOM Method 4. Numerical Analysis models showed two peaks, the same as in the case of the 4.1 Analysis of Computed Response of Assumed analytical building in Figs. 2(b) and 4, and arrows (1) Building and (2) indicate incident and reflected waves, We applied the NIOM method to the computed respectively. Arrival times of the incident and reflected response accelerations of a 10-story building, such as waves for each floor are mostly the same. In Fig. 5, shown in Fig. 2(a), and obtained the simplified input simplified output models for the actual building records and outputs as shown in Fig. 2(b). Here, the response show a smaller amplitude for the reflected wave than acceleration at the roof was considered as the input, and for the incident wave in all cases, the same as in the the acceleration responses at the other nine floors were assumed building case in Fig. 4. considered as outputs. Figure 2(b) shows analysis results when the sampling rate of the time series was 0.02 s and weighting constants were k = 0.01, c = 1 and c = … = c 0 0 1 9 =1. As shown in Fig. 2(b), the input was modeled such that its amplitude at time t = 0 was unity and the amplitudes at the other times and output approached zero unless correlation existed between them. One may understand simply that the amplitude at the top story is assumed to be 1.0, and relative amplitude compared to the top story is obtained for the other stories. One can see two clear peaks in the simplified output models corresponding to the incident and reflected waves, which are indicated by arrows (1) and (2) in the figure, respectively. The incident wave propagates from the basement to the roof, whereas the reflected wave propagates from the roof to the basement. Arrival times of the incident and reflected waves were the same: 0.28 s and 0.48 s at the fifth floor and basement, respectively. The waveforms of the simplified models changed as we changed the value of k , which determines the contribution of high or low frequencies (Haddadi and Kawakami 1998), but the obtained wave arrival times were similar. However, due to the limited space, only figures showing NIOM analysis results for k = 0.01 are Fig.4. NIOM Analysis Results Using Computed Response of Assumed 10-story Building with Different Damping Ratios given in this paper. Also, Fig. 2(b) shows that the reflected wave amplitude was smaller than the incident wave amplitude in the 5. Discussion of Analysis Results output models. The NIOM results for the assumed Simplified models obtained from the NIOM method building were obtained by changing the damping ratio give the arrival time and relative amplitude of incident from 1% to 30%: cases of 1%, 5%, 10% and 20% and reflected waves at each level. Some interpretation damping are shown in Fig. 4. One can see that the of these results is needed to obtain the dynamic difference between the incident and reflected wave characteristics of the structure, as discussed below. amplitudes increased with increasing damping ratio, and that the arrival times were similar for different damping 5.1 Wave Arrival Time and Fundamental Period ratios. Arrival times of incident and reflected waves at the basement/ground floor (which are equal to the wave 4.2 Analysis of Actual Building Records travel times through the height of the building) are plotted The analysis results obtained using horizontal in Fig. 6 for the analytical and actual buildings. One can components of the records in the selected buildings are see that these were very similar except for a few cases. shown in Fig. 5, when k = 0.01, c = 1, and c = 1 (l =1, 2, This indicates the reliability of the NIOM method. 0 0 l …). As seen from this figure, the simplified output The question is what is the relationship between the 36 JAABE vol.3 no.1 May. 2004 Hideji Kawakami fundamental period and wave travel time through the distribution, where the order of story stiffness is reversed building. As we know, the fundamental frequency of a from that in case 1, gave T = 4τ = 4 × 0.48 = 1.92 s, while the fundamental period obtained by modal analysis was uniform soil deposit is given by , and the T = 2.11 s. It is interesting that the wave travel times τ are equal corresponding period is , where H = to 0.48 s in all three cases, but the fundamental periods thickness of the soil deposit; ν = shear wave velocity; differ from case to case. This means that the arrival time and τ = wave travel time through the soil deposit. obtained from the NIOM method is precisely the wave Corresponding to the non-uniform building model shown travel time from the basement to the roof. The wave in Fig. 1(a) [case 1 in Fig. 7], we assumed a 10-story travel time is used in the wave propagation problem, and uniform building with the same stiffness of 230 kN/cm it has a different meaning from the fundamental period, in all stories (case 2 in Fig. 7). This value of K = 230 kN/ which is used in the vibration problem. We can say that cm was chosen by assuming a series spring system: four times the wave travel time τ , T = 4τ , gives a reliable value of the fundamental period for a fairly uniform or , where K = stiffness of each story in case 1 low-rise building. However, the value of T = 4τ is overestimated (underestimated) compared with modal (see Fig. 1(a)) and K = story stiffness (same in all stories) analysis when story stiffness is decreased (increased) in case 2. from basement to roof. Figure 8 shows the NIOM analysis results for cases Figure 9 shows relationships between building height 1-3 in Fig. 7. As shown in case 2 in Figs. 7 and 8, the and the value of T = 4τ (fundamental period) for the obtained value of T = 4τ = 4 × 0.48 = 1.92 s for this uniform assumed buildings in Fig. 1 and several actual buildings building model deviates by about 3% from the (NS components) in comparison with the empirical fundamental period T =1.86 s calculated from the formula from the Uniform Building Code. The empirical smallest eigenvalue in the modal analysis. However, in the case of the analytical building shown in Fig. 1(a), i.e., case 1 in Figs. 7 and 8, T = 4τ = 4 × 0.48 = 1.92 s, while the fundamental period obtained by modal analysis was 1.60 s. Case 3 of the story stiffness Fig.6. Relationships between Arrival Times of Incident and Reflected Waves (NS Component) Fig.5. NIOM Results for Actual Building Records Fig.7. Fundamental Periods Obtained by Two Methods for (Horizontal Component) Assumed 10-story Buildings JAABE vol.3 no.1 May. 2004 Hideji Kawakami 37 Fig.8. NIOM Analysis Results for Cases 1-3 of 10-story Building in Fig. 7 Fig.10. Comparisons of Fundamental Periods of Buildings Obtained by NIOM Method and Previous Studies the 20-story hotel (NS component), 0.88 (0.88) s for 1640 S. Marengo (N038E component), and 0.64 (0.64) s for 8639 Lincoln avenue (S045E component) [values in parentheses indicate the periods obtained from reflected Fig.9. Relationships between Building Height and wave arrival times]. The differences between periods Fundamental Period (T = 4t) (NS Component) obtained from incident and reflected waves are less than 10%, and they may be because of the wide peaks in Fig. formula for estimating the fundamental period of 5. Meanwhile, the fundamental periods determined by 3/4 different types of buildings is T = ah , where a = 0.0731, previous investigations were 3.4 s for the Transamerica 0.0853, and 0.0488 for reinforced concrete moment- building (Safak and Celebi 1991), 2.62 s for the 20-story resisting frames and eccentrically braced frames, steel hotel (Goel and Chopra 1997), and 1.03 s for 1640 S. moment-resisting frames, and all other buildings, Marengo (Hart and Vasudevan 1975). respectively; and h = building height in meters (ICBO Figure 10 compares the fundamental periods obtained 1997). by the NIOM method with those obtained by previous Fundamental periods (T = 4τ ) obtained from incident researchers for both the NS and EW components. Other wave arrival times in Fig. 5 were 5.12 (5.28) s for the actual buildings marked (open circle plots) include two Transamerica building (NS component), 2.24 (2.4) s for buildings studied by Celebi (1992), five buildings by 38 JAABE vol.3 no.1 May. 2004 Hideji Kawakami Goel and Chopra (1997), and three buildings by Hart damping ratio estimation may be improved by and Vasudevan (1975); our results show relatively good considering the ratio of wave amplitude decay to the wave agreement with those of the previous studies. propagation distance. Further study on this issue should One may think that errors in estimating fundamental be conducted in the future. However, we should notice periods in Fig. 7 are sizable and that the story stiffness that the damping ratio has been difficult to estimate distribution can be used to modify the estimated values accurately and that the estimated value depended greatly (T = 4τ ). One may also think that from the differences in on the method used. the fundamental period shown in Fig. 7, the differences between the NIOM and the previous studies should be larger than those obtained in Fig. 10. This discrepancy may be explained by taking the soil-structure interaction into consideration. 5.2 Wave Amplitude and Damping Ratio Figure 11 shows the relationships between incident and reflected wave amplitudes at the basement in the analytical and actual buildings (see Figs. 2(b) and 5). The amplitude of the reflected wave is clearly smaller than that of the incident wave, and this result can be explained by the damping of the structure. In order to clarify the effect of damping ratio on the Fig.11. Relationships between Incident and Reflected Wave wave amplitude, we analyzed the analytical buildings Amplitudes (NS Component) for various damping ratios. The difference between incident and reflected wave amplitudes increased with increasing damping ratio, as shown in Fig. 4, which shows the case of the 10-story building. Figure 12 shows the relationship between damping ratio and wave amplitude ratio (reflected to incident) at the basement for the 2-, 5-, 10-, 20-, 30-, and 40-story analytical buildings, which reveals a clear relationship between the two ratios. Therefore, the wave amplitude ratio obtained by the NIOM method can be used to estimate the damping ratio of the building. Figure 13 shows the wave amplitude ratio versus building height for analytical buildings with damping ratios of 1, 2, 3, 4, 5, 10, 20 and 30%. The values for several actual buildings are also plotted. The damping Fig.12. Relationships between Wave Amplitude Ratio and ratio for each actual building can be estimated by Damping Ratio for Assumed Multistory Buildings comparing the plot with the curves for different damping ratios. One can see that most of the actual buildings fall between curves corresponding to 1% and 10% damping, which is consistent with the recommended values of damping ratio for various types of buildings. The figure also has plots for buildings with damping ratios from 0% to 5% used in previous studies (Safak and Celebi 1991, Goel and Chopra 1997, Hart and Vasudevan 1975, Celebi 1992) indicated by open circles, 5% to 10% indicated by open squares, 10% to 15% indicated by open rhombuses, and more than 15% indicated by open triangles. Crosses in this figure correspond to buildings whose damping ratios were not available from the previous studies. One might think that the damping ratio of 10 % or higher appears to be too large for ordinary buildings if the soil-structure interaction is not considered. One may also think that Fig. 13 shows that the damping ratios obtained by the NIOM method do not agree well with Fig.13. Relationships between Building Height, Wave Amplitude Ratio, and Damping Ratio those obtained by the previous methods and that the JAABE vol.3 no.1 May. 2004 Hideji Kawakami 39 6. Conclusions References 1) Architectural Institute of Japan (1996) Collected strong motion This paper introduced an application of the wave records from 1995 Hyogo-ken Nanbu earthquake. Tokyo, Japan. propagation modeling method, i.e., the NIOM method, 2) California Strong Motion Instrumentation Program (CSMIP) to clarify how seismic waves propagate through a (1994) Processed data from the Northridge Earthquake of 17 building during an earthquake and to relate the obtained January 1994. Report OSMS 94-11, California. results to the dynamic properties of the building. Our 3) Celebi, M. (1992) Highlights of Loma Prieta responses of four tall buildings. Proceedings of Earthquake Engineering, Tenth World main conclusions are as follows. Conference, Balkema, Rotterdam. 1) The NIOM method gave results with two clear peaks 4) Chopra, A. K. (1995) Dynamics of structures: Theory and in the outputs, corresponding to incident and reflected applications to earthquake engineering. Prentice Hall, Inc., NJ. waves propagating vertically in the building. Such 5) Goel, R. K. and Chopra, A. K. (1997) Vibration properties of results were similar for all the analyzed actual buildings determined from recorded earthquake motions. Rep. No. UCB/EERC-97/14, Earthquake Engineering Research Center, buildings, which were of various structural types and University of California at Berkeley. materials. 6) Haddadi, H. R. and Kawakami, H. (1998) Modeling wave 2) The arrival time τ at the basement was precisely the propagation by using normalized input-output minimization wave travel time from the basement to the roof. Arrival (NIOM) method for multiple linear systems. Journal of Structural Mechanics and Earthquake Engineering, JSCE 584/I-42, 29-39. times of the incident and reflected waves were mostly 7) Hart, G. C. and Vasudevan, R. (1975) Earthquake design of the same. buildings: Damping. Journal of the Structural Division, ASCE, 11- 3) The wave travel time as calculated by the NIOM method has a different meaning from the fundamental 8) International Conference of Building Officials (ICBO). 1997. period, which is used in the vibration problem. The Uniform building code. Whittier, California. 9) Kawakami, H. and Haddadi, H.R. (1998) Modeling wave value 4τ gives an approximate value of the propagation by using normalized input output minimization fundamental period for a uniform or low-rise building. (NIOM). Soil Dyn. and Earthquake Engrg., 17, 117-126. However, it overestimates (underestimates) the 10) Kramer, S.L. (1996) Geotechnical earthquake engineering. Prentice fundamental period when story stiffness is decreased Hall, Inc., NJ. (increased) from basement to roof. 11) National Geophysical Data Center (1996) Earthquake strong motion 3-volume CD-ROM collection. Boulder, Colorado. 4) Reflected wave amplitude was always smaller than 12) Newmark, N.M. and Hall, W.J. (1982) Earthquake spectra and incident wave amplitude, and the ratio of the design. Earthquake Engineering Research Institute, Berkeley, amplitudes can be used to estimate the damping ratio. California. 13) Ohba, S. (1981) Wave propagation in buildings during oscillator Acknowledgements experiment, AIJ Annual Meeting, 849-850 (in Japanese). 14) Osaki, M. (1990) Vibration theory of buildings. Syokoku-sha, Tokyo We are grateful to the National Geophysical Data (in Japanese). Center, Colorado; California Strong Motion 15) Oyunchimeg, M. and Kawakami, H. (2003) A new method for Instrumentation Program, California Department of propagation analysis of earthquake waves in damaged buildings: Conservation, Division of Mines and Geology of the evolutionary normalized input-output minimization (NIOM). Journal of Asian Architecture and Building Engineering, 2(1), 9- USA; and Architectural Institute of Japan for providing the earthquake strong-motion records used in our study. 16) Safak, E. and Celebi, M. (1991) Seismic response of Transamerica building. II: System Identification. Journal of Structural Engineering, ASCE, 117( 8), 2405-2425. 40 JAABE vol.3 no.1 May. 2004 Hideji Kawakami
Journal of Asian Architecture and Building Engineering – Taylor & Francis
Published: May 1, 2004
Keywords: strong motion records; buildings; wave propagation; fundamental period; damping
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