## 3 Homogeneous Site  
### Numerical Model and Input Ground Motion  
The thickness of the homogeneous soil layer is 100 m, the soil density is 1800 kg/m\({}^{3}\), and the shear wave velocity is 220 m/s. In the finite element model, the soil layer is  
When all 40 modes are used in the time-history analysis for modal superposition method, the results are considered the exact solution, denoted as \(r^{*}(t)\). The approximate solution is denoted as \(r(t)\) when \(N\) is less than 40 for the modal superposition method. The error of the approximate solution relative to the exact solution is represented by the cumulative error [27]:  
\[e=\frac{\int\limits_{0}^{\tau}|r^{*}-r|dt}{\int\limits_{0}^{\tau}|r^{*}|dt} \times 100\% \tag{24}\]  
where \(\tau\) is the total duration of the seismic wave.  
Table 3 presents the cumulative errors of the time histories for displacement \(u_{A}\), velocity \(\dot{u}_{\text{A}}\), and acceleration \(\ddot{u}_{\text{A}}\) obtained by the modal superposition method with the first 4-th and first 7-th modes. Figure 4 shows the average errors of displacement, velocity, and acceleration varying with the mode number. Figures 5 and 6 show the time histories and the corresponding Fourier spectra of \(u_{A}\) and \(\ddot{u}_{\text{A}}\). The results of the time history show the following:  
1. When \(N\) = 4, the calculation error of \(u_{A}\) is only 0.42%, indicating that the number of modes selected based on modal participation mass is enough to calculate displacement response.
2. The error of displacement response is the smallest compared with velocity and acceleration, while the error of acceleration response is the largest. The reason is that the higher modes have a greater impact on acceleration than on the displacement. Comparing Figures 5 and 6, it can be observed that the differences in the Fourier spectra of the accelerations are greater than those of the displacement Fourier spectra, which confirms this finding.
3. The error of acceleration, \(\ddot{u}_{A}\), is 13.44% when \(N\) = 4, while the error is less than 5% when \(N\) = 7. Therefore, when modal participation mass and modal contribution factor are used as truncation criteria, the resulting number of modes is suitable for displacement response analysis, but may introduce significant errors in acceleration response. For high-precision acceleration response, the modal contribution factor of acceleration should be adopted as the truncation criterion.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(N\) & **Frequency/Hz** & \(\gamma\) & \(e_{m}\)/\% & \(e_{mA}\)/\% & \(e_{aA}\)/\% \\ \hline
1 & 0.499 & 603.9 & 17.93 & 3.20 & 27.31 \\
2 & 1.499 & \(-\)201.1 & 8.83 & 0.62 & 15.08 \\
3 & 2.496 & \(-\)120.4 & 5.57 & 0.24 & 10.29 \\
4 & 3.489 & \(85.73\) & 3.91 & 0.10 & 7.78 \\
5 & 4.477 & \(66.41\) & 2.92 & 0.02 & 6.22 \\
6 & 5.457 & \(-\)54.05 & 2.26 & 0.01 & 5.17 \\
7 & 6.429 & \(45.44\) & 1.79 & 0 & 4.41 \\
8 & 7.392 & \(39.09\) & 1.46 & 0 & 3.83 \\
9 & 8.343 & \(-\)34.19 & 1.19 & 0 & 3.38 \\
10 & 9.281 & \(-\)30.29 & 0.99 & 0 & 3.01 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Partial frequencies and modal truncation indexes of site.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Seismic Wave**} & \multicolumn{2}{c}{\(u_{A}\)} & \multicolumn{2}{c}{\(\dot{u}_{A}\)} & \multicolumn{2}{c}{\(\ddot{u}_{A}\)} \\ \cline{2-7}  & **4** & **7** & **4** & **7** & **4** & **7** \\ \hline E1 & 0.39 & 0.05 & 3.69 & 0.6 & 3.69 & 0.61 \\ E2 & 0.61 & 0.06 & 4.81 & 0.60 & 23.65 & 4.65 \\ E3 & 0.26 & 0.03 & 2.23 & 0.31 & 12.98 & 2.83 \\ Mean value & 0.42 & 0.047 & 3.58 & 0.51 & 13.44 & 2.69 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Cumulative errors of the soil surface time-history responses (%).  
Figure 4: Mean cumulative errors of modal superposition method of the homogeneous soil.  
Figure 5: Time histories and Fourier spectra of \(u_{A}\).  
Figure 6: Time histories and Fourier spectra of \(\bar{u}_{A}\).