Seismological Effects on Spectral and Pseudospectral Acceleration Proximities Based on Random Vibration Theory


 Although spectral absolute acceleration, Sa, is a useful tool for estimating the inertial forces of structures in seismic designs, seismic codes typically specify only the pseudospectral acceleration, Spa. Many studies have been performed to clarify the relationship between these two spectra to relate them. A recent study indicated that this relationship could be affected by not only the structural damping ratio and period but also seismological parameters, such as magnitude and distance. However, how seismological parameters affect their relationship is not clearly understood. To clarify this issue, an approach that relates the two spectra, including seismological parameters, is proposed herein based on random vibration theory. The proposed approach is verified by comparison with the results of time-series analysis. Furthermore, the seismological effects are then explored and explained based on the proposed approach. It is found that Spa approaches Sa with increasing moment magnitude and source-to-site distance, particularly at long structural periods, with the main reason being attributable to the increase in the long-period components of earthquake ground motions. Finally, a practical formulation for estimating Sa from Spa considering the seismological effects is constructed and verified using real seismic records.


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Response spectrum is presently the most widely used tool to characterize seismic loads for 28 structural seismic designs. In most seismic codes, particularly those involving force-based 29 design, the response spectrum for the design is typically specified as the 5%-damped 30 pseudospectral acceleration, Spa, along with a damping modification factor for adjustment to 31 other damping levels (Pu et al., 2016;Zhang and Zhao, 2020). Because Spa is defined 32 according to the spectral relative displacement, Sd, as 2 Spa Sd   , with  being the 33 structural circular frequency, the force estimated using Spa is proportional to the relative 34 displacement, thus corresponding to the restoring force of the structure. Therefore, Spa is 35 suitable in cases where the restoring force is of interest in the seismic design, e.g., in the design 36 of structures where the damping is derived from the added energy dissipation devices (Lin and 37 Chang, 2003). Nevertheless, when the inertial force is considered in the seismic design, e.g., to 38 calculate the base shear in the designs of base-isolated structures or in foundation designs, the 39 spectral absolute acceleration, Sa, is more suitable (Sadek et al., 2000;Mentrasti, 2008). 40 Early studies have shown that for small structural damping ratios, Sa can be approximated 41 by Spa; however, when the structural damping ratio is large and the structural period is long, Sa

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To investigate the seismological effects on the relationship between Sa and Spa, there are 71 two options: using real accelerograms or ground-motion prediction models. To identify 72 consistent patterns within an appreciable margin of variability that is always displayed by 73 strong-motion data, large numbers of records must be employed because one is unlikely to find 74 records that are essentially similar in all but one characteristic. Therefore, the ground-motion 75 prediction models for Spa and Sa are more suitable for use in this study. Although there are 76 many available ground-motion prediction models for the 5%-damped Spa, very few are 77 available for 5%-damped Sa, and even fewer for Sa with various damping ratios. In this study, 78 we adopt a Fourier amplitude spectrum (FAS) ground-motion model and RVT to estimate and 79 relate the two spectra considering the ability of the RVT to relate the FAS to the response where f = frequency (Hz); ρ = mass density of the crust (g/cm 3 ); β = shear-wave velocity of the 91 crust (km/s); R = distance from the source (km); Z(R) = geometric attenuation; κ 0 = site 92 diminution (s); Q(f) = anelastic attenuation; A(f) = crust amplification; M 0 = seismic moment 93 (dyne cm), which is related to the moment magnitude, M, as M 0 = 10 1.5M+10.7 ; f c = corner 94 frequency given as f c = 4.9×10 6 β(Δσ/M 0 ) 1/3 , which represents the frequency below which the 95 FAS decays; Δσ = stress drop. The seismological parameters required in Eq. (1) for central and 96 eastern North America (CENA) are used in this study because of recent work that have updated 97 the seismological parameters for these regions (Boore and Thompson, 2015), as detailed in level. The RVT states that the peak value of a time-series signal is equal to the product of the 103 peak factor and root-mean-square (rms) value, which can be expressed as where a max is the peak value of the signal, pf denotes the peak factor, and the square-root part in 105 Eq. (2) represents the rms value of the signal, which is obtained from the signal duration D and 106 FAS of the signal y(ω), and ω is the circular frequency (ω = 2πf). Since the response spectrum 107 is the peak response value of a single-degree-of-freedom (SDOF) oscillator, according to RVT, 108 the response spectrum should be equal to the product of the peak factor and rms of the 109 oscillator response. Boore (2003) derived an expression for Spa, which is expressed as where, ω and ξ are the SDOF-oscillator circular frequency and damping ratio, respectively; 111 pf ξp is the peak factor of the oscillator response, and the square-root part in Eq. (3) represents 112 the rms value of the oscillator response, which is obtained using the oscillator-response 113 duration, rms D , and oscillator-response FAS, ( , , ) YR ω ω ξ . Here, ( , , ) YR ω ω ξ is equal to the 114 product of the ground-motion FAS Y(ω) and modulus of the oscillator transfer function for Spa, 115 ( , , ) Hpa ω ω ξ , i.e., ( , , ) The oscillator transfer function for Spa, ( , , ) Hpa ω ω ξ , is obtained from the one for Sd, 117 ( , , ) Hd ω ω ξ , by multiplication with 2 ω , i.e., 2 ( , , ) ( , , ) Hpa ω ω ξ ω Hd ω ω ξ  .
( , , ) Hd ω ω ξ 118 is expressed as It should be noted that in RVT analysis, multiplying the spectral relative displacement Sd by 120 2 ω to obtain Spa is equivalent to multiplying the oscillator transfer function ( , , ) Hd ω ω ξ for 121 Sd by 2 ω to obtain Spa. This is attributed to the property of Eq. (3) Then, the expression for the ratio of Sa to Spa can be obtained using the respective spectra: where pf ξ is the oscillator-response peak factor corresponding to Sa. Thus, Eq. (7) theoretically 129 relates to the two spectra. Here, it should be noted that the oscillator-response duration for Sa 130 is assumed to be the same as that for Spa in the derivation of Eq. (7). The expression for Sa/Spa 131 in Eq. (7) can be decomposed into two terms: the first term (i.e., hereafter denoted as R rms ; and the second term (i.e., pf ξ / pf ξp ) is the ratio of the 134 oscillator-response peak factors for Sa and Spa, hereafter denoted as R pf .

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When applying Eq. (7) to estimate the ratio of the two spectra, the value of the peak factor 136 must be determined. Many models have been developed for the estimation of the peak factor 137 (Cartwright and Longuet-Hlggms, 1956;Davenport, 1964;Vanmarcke, 1975). Among these, 138 the model by Vanmarcke (1975) has been found to provide the most reasonable estimations of 139 the response spectra in RVT analysis (Wang and Rathje, 2016). The cumulative distribution 140 function, P, of the peak factor, pf, as provided by Vanmarcke (1975) Here,  is a bandwidth factor, which is defined as a function of the spectral moments: where m 0 , m 1 , and m 2 denote the zeroth-, first-, and second-order moments of the square of the 143 FAS, and the nth-order spectral moment, m n , is defined as In addition, f z denotes the rate of zero crossings, which is also a function of the spectral 145 moments, and is given by The ground-motion duration D gm in Eq. (8) is estimated based on the model of Boore and 147 Thompson (2014Thompson ( , 2015. In RVT analysis, the expect value of the peak factor is always used,

Verification
To investigate the accuracy of the proposed approach, the ratio of the two spectra, Sa/Spa, 152 was calculated using Eq. (7) and compared with that obtained using traditional time-series 153 analysis. The time series for the analysis was generated from ground-motion FAS using the 154 stochastic method SIMulation (Boore, 2005) program via stochastic simulations (Boore, 1983).

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For each FAS, a suite of 100 time-series signals was generated, and the simulated time series  From the figures, the relationship between the two spectra can also be clearly observed. It is 166 observed that the values of Sa/Spa are always greater than unity for all cases, which implies 167 that Sa is always greater than Spa. Figure 1 also indicates that Sa/Spa is nearly equal to unity at 168 small oscillator periods and increases with increasing oscillator periods and damping ratios; 169 these values may be considerably greater than unity for very long oscillator periods and very 170 large damping ratios. These observed properties for the relationship between the two spectra 171 are consistent with those observed by statistical analyses of real seismic records and common 172 knowledge (Jenschke et al., 1964(Jenschke et al., , 1965Veletsos and Newmark, 1964;Newmark and 173 Rosenblueth, 1971;Sadek et al., 2000;Chopra, 2007;Song et al., 2007;Mentrasti, 2008; Figure 3 indicates that the behavior of Sa/Spa with variation of the site-to-source 185 distance R is typically consistent with the variation of the moment magnitude but to a much 186 smaller degree of variation.

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The effects of the site conditions on the relationship between the two spectra were also presented. It is evident from the figure that Sa/Spa varies with the soil parameters, i.e., 197 impedance ratio, soil damping ratio, and site fundamental period, but to a much smaller degree 198 than with the moment magnitude (Fig. 2). In addition, the effects of the site conditions on 199 Sa/Spa are complex, and no simple and regular pattern is readily observed. The values of 200 Sa/Spa for the soil sites may be larger or smaller than those for the rock sites, depending on the parameters of the soil sites. It is noted that Sa/Spa for the soil site in Fig. 4(b) is nearly the same 202 as that for the rock site. When Ip is larger ( Fig. 4(a)) or the site fundamental period T s is longer 203 (Fig. 4(c)), Sa/Spa for the soil site decreases compared with that for the rock site. When the soil 204 damping ratio h is smaller ( Fig. 4(d)), Sa/Spa for the soil site increases compared with that for 205 the rock site. In addition, from comparison of the results in Fig. 4, it is observed that the effect The observed phenomena can be theoretically explained based on the proposed approach.

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The representative results of the two terms in Eq. (7), i.e., R rms and R pf , for the cases in adequately by R rms . In addition, Fig. 6 indicates that the value of R pf is close to unity compared 216 to R rms . These observations indicate that Sa/Spa is dominated by R rms , which facilitates 217 explanation of the phenomenon based on R rms .

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To facilitate further analysis, the integral terms in the numerator and denominator of R rms , i.e., , corresponds to Spa. Thus, the proximity between the two spectra or 223 the value of Sa/Spa is determined by the proximity between these two areas. The closer the two 224 areas are, the more similar are the two spectra, and the closer is the value of Sa/Spa to unity. It 225 can be further observed from the first term R rms in Eq. (7) that the difference between the two 226 areas is caused by the difference between the oscillator transfer functions of the two spectra, 227 i.e., ( , , ) Ha ω ω ξ and ( , , ) Hpa ω ω ξ , because the ground-motion FAS Y(ω), which affects the 228 oscillator-response FAS, is the same for the two spectra.

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To investigate the differences between the oscillator transfer functions for Sa and Spa, their 230 values for two oscillator damping ratios, namely 5% and 30%, are compared in Fig. 7. The with increasing oscillator damping ratio, the difference between the two integral areas, i.e., , will increase. This causes an 244 increase in the difference between the two spectra with increases in the oscillator damping 245 ratios (Fig. 1). Further, because the value of the oscillator transfer function for Sa is always 246 greater than that for Spa, value of Sa is always larger than that of Spa ( Fig. 1-3).

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To explain the Sa/Spa trend with variation in oscillator period, the values of the two oscillator transfer functions for different oscillator periods (0.1 and 1 s) are compared in Fig. 8.

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The oscillator damping ratio is set as 30% for comparison. It is observed that owing to the 251 increase in the oscillator period, the region of the oscillator transfer functions with periods less 252 than the oscillator period T 0 increases. Because the values of these two transfer functions are 253 very different in this region ( ( , , ) Ha ω ω ξ > ( , , ) Hpa ω ω ξ ), as shown in Fig. 8, the differences 254 between the two areas given by the numerator term. This conclusion can be easily verified by multiplying the FAS by a constant value; since 266 a constant value exists in both the numerator and denominator of R rms , they will cancel each 267 other and will not affect R rms . Therefore, the key factor affecting the seismological effects is the 268 distribution of ground-motion FAS with frequency, i.e., the frequency content of the ground 269 motions.

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To investigate the variation of the frequency content of ground motions with the moment 271 magnitude and site-to-source distance, the values of the ground-motion FAS for two moment 272 magnitudes and two site-to-source distances are compared in Fig. 9. It is noted that although increasing site-to-source distances, the decreases at short periods are more significant. This 278 means that the long-period components of the earthquake ground motions increase relative to 279 the short-period components with increasing site-to-source distances. In addition, it is observed 280 that the degree of variation of the frequency content of the ground motions with the 281 site-to-source distances is smaller than that of the moment magnitude.

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Based on the above properties of variation of the ground-motion frequency content with 283 moment magnitude and site-to-source distance, the Sa/Spa variation trends can be explained. It 284 is worth emphasizing again that the spectral ratio Sa/Spa can be understood by investigating 285 the differences between the two areas of the square of the oscillator-response FAS in Eq. (7).

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The difference between these two areas is attributed to the difference between their oscillator 287 transfer functions over short periods. In addition, the two transfer functions are very similar at 288 periods around and longer than the oscillator period T 0 . It can be seen from Fig. 9 that when the 289 components are concentrated more over a long period, in which the two oscillator transfer 290 functions are similar, the two spectra will be more similar. Therefore, when the long-period 291 components of the ground motions increase with increasing moment magnitudes and 292 site-to-source distances, similar portions of the two areas will increase relatively, as shown in 293 Fig. 9. This explains why Spa approaches Sa with increasing moment magnitudes and 294 site-to-source distances (Fig. 2).

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To explain the effects of site conditions on the relationship between the two spectra, the FAS 296 values for the soil sites in Fig. 4 are compared with those for the rock sites, which is shown in 297 Fig. 10. It is noted that the effects of the soil conditions on the ground-motion frequency content are not as simple as those of the moment magnitude and site-to-source distance. The 299 site effect on the frequency content is strongly dependent on the period T. The components 300 around the first few site resonance periods are amplified; however, those between these site 301 resonance periods or at very short periods are de-amplified, as reported in previous studies  (c)).

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Based on the properties of the effects of the site conditions on the frequency content, the 309 Sa/Spa trends with variation of site conditions can be explained. When the impedance ratio and 310 soil damping are large ( Fig. 10 (a)) or the site fundamental period is long (Fig. 10 (c)), the 311 decreases in the components in the short period are more significant than the increases in the 312 components around the site resonance periods. This means that the short-period components of 313 the ground motion decrease. Because the two oscillator transfer functions are different at short 314 periods and are very similar at periods around and greater than the oscillator period T 0 , when 315 the short-period components of the ground motion decrease, the different parts of these two 316 areas will decrease. Therefore, the difference between the two areas in R rms decreases, and 317 Sa/Spa approaches unity, as shown in Figs. 4 (a) and (c). When the soil damping is very small 318 ( Fig. 10 (d)), the degree of increase in the components around the site resonance periods is 319 more significant than the degree of decrease of the components over short periods. This implies 320 that the short-period components of the ground motions increase. Therefore, the difference 321 between the two areas in the first term increases, and Sa/Spa increases compared to that for the 322 hard rock site, as shown in Fig. 4 (d). where ζ is a parameter reflecting the seismological effects on Sa/Spa, which is detailed below.

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This equation satisfies the boundary condition that, when the oscillator period T 0 and 333 damping ratio ξ decrease to zero, Sa/Spa equals unity. Equation (12)  increase. Therefore, ζ typically increases with the long-period components and can simply 363 reflect the frequency content of the ground motion. In fact, the use of the Spa values at 364 periods 0 and 6 s was determined by trialing numerous values at different periods; it is found 365 that these two values result in Eq. (12) generating the best precision.

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The Sa/Spa results calculated by Eq. (12) were compared with those in Section 3; some 367 representative comparisons are shown in Fig. 11. It is noted that Eq. (12) performs very well 368 in the prediction of Sa/Spa, particularly in the period range, T 0 < 6 s, which is generally of 369 interest in practical seismic design.

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To investigate the effects of seismological parameters on the relationship between spectral 392 and pseudospectral accelerations, i.e., Sa and Spa, an approach relating the two spectra 393 including seismological parameters is proposed based on random vibration theory. Using the 394 proposed approach, the effects of the seismological parameters on the relationship between the 395 two spectra are systematically explored and theoretically explained. Finally, a practical formulation for estimating Sa from Spa considering the seismological effects is constructed.

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The main conclusions of this study are summarized as follows: