# Study on the Seismic Effect of the Pebble Soil Site in the Zhongwei Basin
## 3 Research Method and Input Seismic Load
The target response spectrum was converted to the corresponding power spectrum by Equation (1): \[S(\omega)=\frac{\varepsilon}{\pi\omega}S_{a}^{2}(\omega)/\left\{-Ln\Big{[}- \frac{\pi}{\omega T}Ln(1-r)\Big{]}\right\}\] (1) where \(S(\omega)\) is the power spectrum; \(\varepsilon\) is the damping ratio; \({S_{a}}^{2}(\omega)\) is the target response spectrum; \(T\) is the duration; and \(r\) is the exceeding probability of the target reaction spectrum. 2. Stable Gaussian processes with zero mean values were generated by the trigonometric series superposition method, using Equation (2): \[X(t)=\sum_{i=0}^{n}A_{i}\cos(\omega_{i}t+\phi_{i})\] (2) where \(A_{i}=[4S(\omega_{i})\Delta\omega]^{0.5}\), and the initial phase \(\phi_{i}\) is a uniformly distributed random variable during \([0-2\pi]\). 3. The stationary time history was multiplied by the nonstationary strength envelope to obtain the nonstationary acceleration time history, using Equation (3): \[a(t)=\Psi(t)\cdot X(t)\] (3) \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Soil Type** & \multirow{2}{*}{**Soil Name**} & \multirow{2}{*}{**Parameter**} & \multicolumn{8}{c}{\(\gamma_{4}\) (10\({}^{-4}\))} \\ \cline{3-10}
**Number** & & & **0.05** & **0.1** & **0.5** & **1** & **5** & **10** & **50** & **100** \\ \hline
1 & Silty clay & G\({}_{\rm d}\)/G\({}_{\rm dmax}\) & 0.9949 & 0.9899 & 0.9514 & 0.9072 & 0.6617 & 0.4944 & 0.1636 & 0.0891 \\  & & \(\lambda\) & 0.0042 & 0.0068 & 0.0213 & 0.0339 & 0.0865 & 0.1156 & 0.1664 & 0.177 \\
2 & Slightly & G\({}_{\rm d}\)/G\({}_{\rm dmax}\) & 0.9931 & 0.9866 & 0.9459 & 0.9017 & 0.7101 & 0.5633 & 0.2405 & 0.1299 \\  & dense pebbled & \(\lambda\) & 0.0118 & 0.0163 & 0.0339 & 0.0478 & 0.0861 & 0.109 & 0.1552 & 0.1697 \\  & Medium & G\({}_{\rm d}\)/G\({}_{\rm dmax}\) & 0.9952 & 0.9889 & 0.9624 & 0.9326 & 0.8191 & 0.7041 & 0.3987 & 0.2456 \\
3 & dense pebbled & \(\lambda\) & 0.0206 & 0.0252 & 0.0403 & 0.0504 & 0.0831 & 0.1006 & 0.1346 & 0.1478 \\
4 & Tightly & G\({}_{\rm d}\)/G\({}_{\rm dmax}\) & 0.9988 & 0.9924 & 0.9751 & 0.951 & 0.8662 & 0.7871 & 0.5127 & 0.3461 \\  & dense pebbled & \(\lambda\) & 0.0115 & 0.0142 & 0.0254 & 0.0336 & 0.0621 & 0.0805 & 0.117 & 0.1328 \\
5 & Sticky sand & G\({}_{\rm d}\)/G\({}_{\rm dmax}\) & 0.9919 & 0.9839 & 0.9246 & 0.8597 & 0.5508 & 0.38 & 0.1092 & 0.0578 \\  & & \(\lambda\) & 0.0089 & 0.0126 & 0.0275 & 0.0376 & 0.0679 & 0.0799 & 0.096 & 0.0987 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic nonlinear parameters for different soil types. The nonstationary envelope function was determined to be of the following form, using Equation (4):  
\[\Psi(t)= \begin{cases}\left(\begin{array}{c}\frac{t}{t_{1}}\\ \frac{1}{t_{1}}\end{array}\right)^{2}&0<t\leq t_{1}\\ 1&t_{1}<t\leq t_{2}\\ e^{-c(t-t_{2})}&t_{2}<t\leq T\end{cases}  \tag{4}\]  
where \(t_{1}\) is the rising part of the peak, \(t_{1}\)-\(t_{2}\) is the stationary segment of the peak, \(T\) is the duration, and \(c\) is the attenuation coefficient of the peak. Eighty-five control points of the target response spectrum were selected within 0.04\(\sim\)6 s to ensure the precision of the target response spectrum fitting during the synthetic ground motion. During synthesis, the time history of the synthesized acceleration accurately met the peak acceleration of the target and approximated the response spectrum of the target acceleration by approximating the target spectrum step by step. To maximize the reflection of the actual structure subjected to seismic action, site-specific ground motions are necessary. In this paper, artificial ground motions were generated by gradually approaching the target design spectrum in this area. Four ground motions with different amplitudes were generated based on the return period of this area, as shown in Figure 3. They corresponded to exceeding probabilities of 63% (frequently occurring earthquakes), 10% (normally occurring earthquakes), and 2% (rarely occurring earthquakes) in 50 years, and 1% in 100 years (extremely rarely occurring earthquakes).