Alternate Methods for Construction of Design Response Spectrum

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1 Proc. Natl. Sci. Counc. ROC(A) Vol. 22, No. 6, pp Alternate Methods for Construction of Design Response Spectrum R. YAUNCHAN TAN Department of Civil Engineering National Taiwan University Taipei, Taiwan, R.O.C. (Received October 16, 1997; Accepted April 8, 1998) ABSTRACT The evolutionary power spectrum (EPS) of any single seismic accelerogram is estimated using two techniques. An appropriate normalization factor is adopted to build the shape function of the expected EPS of a particular set of accelerograms. The relationship between this factor and the peak ground acceleration (PGA) is investigated. Therefore, under a design g-level, the expected EPS is obtained by multiplying the shape function by a suitable normalization factor. Based on the expected EPS, a set of simulated acceleration time histories can be generated to calculate the response spectral values. The available data include 159 strong-motion records in Taiwan with PGA greater than 0.05 g. The data are classified according to site condition and earthquake duration. It is shown that, within the range of the periods of interest, the spectrum of the hard site obtained using the conventional method is not conservative. The spectra obtained using the two proposed methods are preferable for the design of some important structures and tall buildings. Key Words: design spectrum, evolutionary power spectrum, subprocesses technique, multifilter technique I. Introduction The response spectrum method is widely employed in aseismic design analysis for many reasons. A design response spectrum usually incorporates the spectra of several earthquakes and represents a kind of average spectrum (Newmark et al.,197). Before performing statistical analysis, each response spectrum is generally normalized so that the maximum ground-motion acceleration is 1.0 g. Such normalization, although simple, is far from reasonable as the normalization factor is fixed for all the components having different frequencies in a particular earthquake. It is the objective of this paper to propose two alternate approaches in which more realistic normalization factors are used to construct a design response spectrum. Since both its frequency contents and mean square values vary with time, an earthquake acceleration time history can be regarded as a nonstationary random process. A useful stochastic model to describe such a process can be characterized by an evolutionary power spectrum (EPS) (Priestley, 1965). This spectrum is a time dependent function and has a physical interpretation as a local energy distribution over frequency. Estimation of the EPS of an individual accelerogram can be carried out using several tech- niques (Kameda,1975; Schüeller and Pradlwarter,1988). Each EPS is definitely different in pattern and intensity, depending on several factors, i.e., the magnitude, epicentral distance, focal depth, source mechanism, geological conditions, site conditions, and strongmotion duration. Obviously, an average EPS would be more meaningful for engineering design purposes. In engineering practice, design g-levels are usually specified first. The effects of the magnitude, epicentral distance, and focal depth on the intensity level are taken into account in an implicit manner. The effects of the source mechanism and geological conditions are usually not included since they are not well defined. Therefore, the factors which may influence the EPS as well as the response spectrum generally include the intensity level, site conditions and strongmotion duration. The data base used in this study includes 159 accelerograms recorded from strong-motion seismographs in Taiwan with peak ground acceleration (PGA) larger than 0.05 g (Wu,1989). The data are classified into several types on the basis of site conditions and strong-motion duration. The EPS of any single earthquake record from a particular type is estimated. Choosing an appropriate normalization factor, we build up the shape function of the expected EPS of this type. 775

2 R.Y. Tan The relationship between the normalization factor and the PGA is also investigated. As a result, under a given design g-level, the expected EPS is obtained by multiplying the shape function by the corresponding normalization factor. Finally, based on the expected EPS, the acceleration time histories are simulated to calculate the response spectral values under different structural periods. Comparing these response spectra with the conventional ones, we can gain insight into the strategy of choosing a suitable design response spectrum. II. Multifilter Technique Estimation of the EPS from one single record is by no means a simple task because the associated statistical information is not sufficient. For engineering purposes, several techniques seem to be suitable. The multifilter technique, which estimates the evolutionary process by passing the recorded time history through a filter and making an inference from its output, is a typical one (Kameda, 1975; Scherer et al., 1982). Let a zero-mean random process, a(t), admit a representation of the form a(t)= e iωt f (t,ω)dz(ω), (1) in which f(t,ω) is the amplitude modulating function and dz(ω) the differential of the orthogonal random process Z(ω). The evolutionary power spectrum of a(t), S(t,ω), is given by the following equation: S(t,ω)=E{f 2 (t,ω)dz 2 (ω)}/dω, (2) where E{.} denotes the expected value. The squared envelop of the displacement of a single degree of freedom (SDOF) system subjected to a random base acceleration, a(t), can be represented by R 2 (t)=y 2 (t)+y 2 (t)/ω 0 2, () in which Y(t) = the random relative displacement response and ω 0 = the natural frequency. Accordingly, E[R 2 (t)] = σ Y 2 + σ Y 2 /ω 0 2, (4) in which σ Y and σ Y are the standard deviations of displacement and velocity, respectively. When the damping of the SDOF system, ξ 0, is small, the mean square responses of Y(t) and Y (t) are given approximately by (Scherer et al.,1982) σ Y 2 (t)= πs(t,ω 0) 2ξ 0 ω 0 (1 e 2ξ 0ω 0 t ) (5) σ Y 2 (t)= πs(t,ω 0) 2ξ 0 ω 0 (1 e 2ξ 0ω 0 t ). (6) Substituting Eqs. (5) and (6) into Eq. (4), we have S(t,ω 0 )=ξ 0 ω 0 E[R 2 (t)]/π(1 e 2ξ 0 ω 0 t ). (7) It should be noted that the damping ratio is suggested to be 0.05 in the present study. III. Subprocess Technique In deriving Eqs. (5) and (6),the transient effect is neglected and the parameter of the process is assumed to vary slowly with respect to time and frequency. These assumptions are hardly justified for earthquake records in near field, where short strongly peaked intensities may occur. However, high resolution with respect to both time and frequency is not required for structural engineering purposes. Indeed, for random response analysis of a linear system, it is satisfactory to model an earthquake as a uniformly modulated stationary process, i.e., f(t,ω)=f(t). On the other hand, the response of a nonlinear hysteretic structure is strongly influenced by the low-frequency content of the ground motion. Therefore, it is rather natural to deem the ground motion a(t) as the sum of several subprocesses, a j (t) (Schüeller and Pradlwarter, 1988): a(t)= M a j (t) M=, (8) in which one subprocess represents the low-frequency motion, one the frequency range significant for the linear response analysis and one the high frequency range. The Fourier transform of a(t) can be denoted as A(ω)= M It is clear that and A j (ω). (9) A(ω)= 2π 1 a(t)e iωt dt A j (ω)= 2π 1 a j (t)e iωt dt. (10) Let A j (ω) be obtained by applying filters W j (ω) on A(ω): 776

3 Construction of Design Response Spectrum estimated as follows: E{a(t)a * (t)} = E{ a j (t) a * k (t)} = f j (t) f k (t)e{dz j (ω)dz k * (ω)} k =1 Fig. 1. Selection of filter. A j (ω)=a(ω)w j (ω) and M W j (ω) =1. (11) The filter W j (ω) should be selected such that the correlation between two distinct amplitudes A(ω) is essentially preserved in the subprocesses. An example of filter selection is given in Fig. 1. When the subprocess a j (t) is a uniformly modulated random process, it can be expressed as a j (t)=f j (t)z j (t) z j (t)= e iωt dz j (ω), (12) where dz j (ω) are orthogonal increments and z j (t) is a stationary random process with spectral density S j (ω). Such a spectrum is usually evaluated from the raw or sample spectrum: S j,t (ω)= A j (ω) 2 /2πT, (1) in which T is the duration of the process a j (t). A more reliable estimate for S j (ω) can be obtained if the raw spectrum is convoluted with a spectral window W S : S j (ω)= 1 2π S j,t (ω α)w s (α)dα. (14) Furthermore, application of the random process theorem yields f j 2 (t)=e{a j 2 (t)}/ S j (ω)dω. (15) The orthogonal processes Z j (ω) correlate fully with each other under the same frequency ω, and the following equation holds : E{dZ j (ω)dz k * (ω)} = S j (ω)s k (ω) dω. (16) In addition, dz(ω)= dz j (ω). (17) Thus, the evolutionary power spectrum S(t,ω) can be = [ f j S j (ω) ] 2 dω. (18) In accordance with this, we have S(t,ω)=[ f j S j (ω) ] 2. (19) IV. Classification of Data The data base used in this study includes 159 accelerograms recorded from strong-motion seismographs in Taiwan with horizontal peak values greater than 0.05 g. These data are classified into two categories based on the site conditions at the recording stations. The alluviam site is deemed to be a soft site; all the others are considered to be hard sites. The effective strong-motion duration of an accelerogram can be used as a basis for further classification. This duration refers to the time interval based on the 5% to 95% energy fraction (Trifunac and Brady, 1975). The data in the soft site can then be grouped into seven types and designated as types S5, S10, S15, S20, S25, S0, and S5. The Arabic numerals in the above notations refer to the effective duration in seconds. For instance, type S10 represents the group with duration ranging from 7.5 sec. to 12.5 sec., type S15 represents that with duration from 12.5 sec. to 17.5 sec. etc. In addition, the duration associated with type S5 is over 2.5 sec. The number of accelerograms collected for each type is, 12, 10, 5, 5, 6, and 10, respectively. Similarly, for the hard site, there are seven types, namely, H5, H10, H15, H20, H25, H0, and H5. The number for each type is 17, 27, 24, 10, 10, 2, and 18, respectively. A typical recorded accelerogram from type S25 is shown in Fig. 2. Its EPS is estimated using Eq. (7) and shown in Fig. (Tan and Chao,1992). This figure clearly indicates the time variation of the mean square intensity for each specific frequency. A similar trend is observed when the EPS is estimated using Eq. (19). V. Expected Power Spectra 1. Method 1 Once the EPS of any single earthquake record of 777

4 R.Y. Tan 4. Such a relationship is also observed for other frequencies and, therefore, is not shown herein. If the design PGA is 100 gal, the corresponding peak factor is estimated to be 5 cm 2 /s at ω=7π rad/sec. When similar estimations for other frequencies are made, the design peak factor D(ω) is obtained and is delineated in Fig. 5. Finally, the expected EPS, S (t,ω) is obtained using S (t,ω)=g(t,ω)d(ω). (21) Fig. 2. Typical accelerogram from S25. Figure 6 presents the expected EPS of type S10 when the design PGA is 0.1 g. It should be noted, however, that the present method may be invalid if the design PGA is over 0.8 g, which is the largest ground motion acceleration in the available data set. 2. Method 2 The EPS of any accelerogram for a certain type can also be obtained using Eq. (19). The shape function of the expected EPS is, then, defined as G(t,ω)= N i =1 S i (t,ω)/d * i (ω)n, (22) where D i * (ω) is the energy factor obtained from Fig.. Evolutionary power spectrum using multifilter technique. a particular type is obtained using Eq. (7), the expected EPS of this type can be constructed. First, the shape function of the expected EPS is defined by G(t,ω)= N i =1 S i (t,ω)/d i (ω)n, (20) where N is the number of records for this type, S i (t,ω) is the EPS of the i-th accelerogram with a PGA of P i, and D i (ω) is the peak factor. The factor is defined as the maximum value of the EPS at a specific frequency. The relation between the peak factor D(ω) and PGA can be obtained by performing a regression of D i (ω) on P i. As an example, at ω=7π rad/sec (i.e..5 Hz) and for type S10, this relationship is shown in Fig. Fig. 4. Relationship between the peak factor and PGA for S10 at.5 Hz. Fig. 5. Design peak factor for S10 when the design PGA is 0.1 g. 778

5 Construction of Design Response Spectrum S (t,ω)=g(t,ω)d * (ω). (24) Figure 9 presents the expected EPS of type S10 when the design PGA is 0.1 g. As expected, this figure and Fig. 6 differ in local appearance. VI. Response Spectra Based on the expected EPS, the simulated accelerogram can be generated by the following equation (Shinozuka and Jan, 1972) : n a(t)= 2 S (t,j ω) ω cos (j ωt + Φ j ), (25) where ω= the interval of the frequency chosen to be 0.08π rad/sec, i.e., 0.04 Hz, n = the number of super- Fig. 6. Expected EPS for S10 using Method 1 when the design PGA is 0.1 g. Fig. 8. Design energy factor for S10 when the design PGA is 0.1 g. Fig. 7. Relationship between the energy factor and PGA for S10 at 1.1 Hz. D i * = 0 T S i (t,ω)dt/t. (2) The relation between the energy factor and PGA is obtained through regression analysis. As an example, at ω=2.2π rad/sec (i.e. 1.1 Hz) and for S10, this relationship is shown in Fig. 7. If the design PGA is 100 gal, the corresponding peak factor is estimated to be cm 2 /s at ω=2.2π rad/ sec. When similar estimations for other frequencies are made, the design energy factor D * (ω) is determined and is shown in Fig. 8. Finally, the expected EPS, S (t,ω), is obtained using Fig. 9. Expected EPS for S10 using Method 2 when the design PGA is 0.1 g. 779

6 R.Y. Tan obtained by averaging all the spectra based on different data sets. Taking the number of records in each type into account, we can construct the response spectrum of the soft site when the design PGA is 0.1 g, which is shown by the solid curve in Fig. 12. While in the short period, spectral acceleration obtained using the old method (dashed curve) is conservative, the spectral value obtained using the present method is significantly larger when T>0.70 sec. Figure 1 delineates the response spectra of the Fig. 10. Response spectra for S10 using Method 1 when the design PGA is 0.1 g and the damping ratio is posed harmonic components, and Φ j = the independent random angle distributed uniformly between 0 and 2π. 1. Method 1 According to Eqs. (21) and (25), N simulated accelerograms are produced to construct the response spectrum. As a demonstration, the acceleration-response spectrum, Sa (T), for type S10 with a design PGA of 0.1 g, was calculated on the basis of these generated accelerograms (the solid curve in Fig. 10) and contrasted with its estimate obtained using the conventional or old method (the dashed curve in Fig. 10). The latter was obtained by correcting the average response spectrum, which was normalized for a maximum ground acceleration of 1.0 g, by a factor of 0.1. The unit of the spectral values was cm/sec 2. This figure shows that when the period T is greater than 0.65 sec., the spectral value obtained using the EPS method is larger. The damping ratio is 0.05 here. The response spectra of other types can be obtained in the same way. For instance, Fig. 11 shows the spectral acceleration of type S5. It is seen that the duration of the earthquake affects not only the spectral values, but also the discrepancy between two spectra estimated using different methods. With a limited number of records (note that the number of records for type S10 and type S5 is 12 and 10, respectively), however, we are not in a position to deduce how the duration affects this discrepancy. To circumvent the scarcity of data and to reduce the associated uncertainty, a response spectrum can be Fig. 11. Response spectra for S5 using Method 1 when the design PGA is 0.1 g and the damping ratio is Fig. 12. Response spectra for the soft site using Method 1 when the design PGA is 0.1 g and the damping ratio is

7 Construction of Design Response Spectrum In summary, within the range where the fundamental periods of most civil structures lie, the spectra of the hard site obtained using the two proposed methods are consistently greater than that obtained using the old method. In regard to the spectrum of the soft site, the conventional method should be used tentatively as the spectral values obtained using the two proposed methods lack consistency. VII. Conclusions On the basis of recorded accelerograms, this study Fig. 1. Response spectra for the hard site using Method 1 when the design PGA is 0.1 g and the damping ratio is hard site when the design PGA is 100 gal. The results obtained using the two methods are fairly close when T<0.6 sec. Again, over a critical period of 0.6 sec., the spectral values from Method 1 dominate. In other words, the major characteristics of the spectrum are similar to those in the soft site spectrum. 2. Method 2 Based on the corresponding expected power spectra obtained using Method 2 and simulated accelerograms, the response spectra for all types can be established. An average of these spectra yields a design response spectrum. The solid curve in Fig. 14 presents the proposed spectrum of the soft site when the design PGA is 0.1 g. The related conventional spectrum is also illustrated by the dashed curve for comparison. In the range of short period, i.e., less than 0.7 sec., the spectral values obtained using the old method are conservative. This result is identical to that shown in Fig. 12. However, when T>0.7 sec., the old method also dominates, which is inconsistent with the outcome shown in Fig. 12. As for the spectrum of the hard site, Fig. 15 shows the two spectra obtained using the two methods. When T>0.45 sec., the spectral values obtained using Method 2 are generally higher. This coincides with the previous result obtained using Method 1, shown in Fig. 1. On the other hand, in the range of short period, the spectral values obtained using the proposed method are also higher. This is contrary to the results shown in Fig. 1, where the spectra obtained using the two methods are close. Fig. 14. Response spectra for the soft site using Method 2 when the design PGA is 0.1 g and the damping ratio is Fig. 15. Response spectra for the hard site using Method 2 when the design PGA is 0.1 g and the damping ratio is

8 R.Y. Tan has utilized evolutionary power spectra and frequencydependent normalization factors to construct design response spectra. Two techniques, namely, a multifilter technique and subprocess technique have been employed to evaluate the evolutionary power spectra. The essential ingredients presented in this paper are the introduction and application of the peak factor and the energy factor. This enables the two proposed methods to deal with variation of motion amplification among different excitation components in a rational and systematic manner. Both of the proposed methods provide a better physical interpretation than does the conventional method, in which each individual response spectrum is normalized for a PGA of 1.0 g before statistical analysis is performed, and in which the motion amplification for all components is implicitly a constant. The numerical results show that the discrepancy between the spectra estimated using the two proposed methods and that obtained using the old method depends on the site conditions. While the spectral values of the soft site obtained using the two proposed methods are far from consistence, within the range of periods of interest, the spectral responses of the hard site estimated using the two EPS methods are generally greater than those obtained using the traditional method. Since most important civil structures are built on hard sites, from the viewpoint of safety design, the spectrum obtained using the old method is not conservative. In conclusion, the spectra obtained using the two proposed EPS methods are preferred in the design of tall buildings and some important or critical structures. Acknowledgment The numerical calculation is due to many graduate students, Mr. U.T. Shiou, Mr. Y.T. Chao and et al. Their effort is gratefully acknowledged. References Kameda, H. (1975) Evolutionary spectra of seismogram by multifilter. J. of Engineering Mech., ASCE, 101, Newmark, N. M., J. A. Blume, and K. K. Kapur (197) Seismic design spectra for nuclear power plant. J. of Power Div., ASCE, 99, Priestley, M. B. (1965) Evolutionary spectra and nonstationary process. J. of the Royal Statistical Soc. Series B., 27, Scherer, R. J., R. I. Schüeller, and J. D. Reida (1982) Estimation of the time-dependent frequency content of earthquake accelerations. Nuclear Engineering and Design, 71, Schüeller, G. I. and H. J. Pradlwarter (1988) Estimation of the evolutionary process of strongly nonstationary earthquake records. Proc. of the 9th WCEE, 2, pp Tokyo-Kyoto, Japan. Shinozuka, M. and C. M. Jan (1972) Digital simulation of random process and its applications. J. of Sound and Vibration, 25, Tan, R. Y. and Y. T. Chao (1992) Construction of a design response spectrum from evolutionary spectra of seismogram. Proc. of the 10th WCEE, pp Madrid, Spain. Trifunac, M. D. and A. G. Brady (1975) On the correlation of seismic intensity scales with the peaks of recorded strong ground motion. Bull. Seism. Soc. Amer., 65, Wu, C. L. (1989) A Study on Seismic Response Spectra for Taiwan. M.S. Thesis. Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. 782

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