# Validation and Parametric Study of a Soil-Structure Interaction Nonlinear Simplified Model
## 4 Parametric Study of the Simplified Model
### Free Field System
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Peak Velocities in (m/s)**} & \multicolumn{4}{c}{**Relative Difference between**} \\
**Soil Layer** & \multicolumn{4}{c}{**for Condition**} & \multicolumn{4}{c}{**Adjacent Conditions (\%)**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **1-2** & **2-3** & **3-4** \\ \hline
1 & 0.008613 & 0.005641 & 0.004810 & 0.004715 & 52.70 & 17.27 & 2.01 \\
2 & 0.007393 & 0.004967 & 0.004260 & 0.004177 & 48.83 & 16.60 & 1.99 \\
3 & 0.006062 & 0.004169 & 0.003580 & 0.003508 & 45.40 & 16.46 & 2.05 \\
4 & 0.004828 & 0.003421 & 0.002944 & 0.002885 & 41.12 & 16.21 & 2.07 \\
5 & 0.003661 & 0.002699 & 0.002340 & 0.002294 & 35.67 & 15.33 & 2.02 \\
6 & 0.002620 & 0.002054 & 0.001817 & 0.001785 & 27.54 & 13.06 & 1.79 \\
7 & 0.001540 & 0.001302 & 0.001177 & 0.001159 & 18.27 & 10.65 & 1.55 \\
8 & 0.000738 & 0.000676 & 0.000624 & 0.000616 & 9.23 & 8.29 & 1.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Peak displacements of each layer in the free field system (for input 0.2 g). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Peak Velocities in (m/s)**} & \multicolumn{4}{c}{**Relative Difference between**} \\
**Soil Layer** & \multicolumn{4}{c}{**for Condition**} & \multicolumn{4}{c}{**Adjacent Conditions (\%)**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **1-2** & **2-3** & **3-4** \\ \hline
1 & 0.095767 & 0.080551 & 0.079388 & 0.078713 & 18.89 & 1.47 & \(-\)0.86 \\
2 & 0.081936 & 0.070831 & 0.069967 & 0.069460 & 15.68 & 1.24 & \(-\)0.73 \\
3 & 0.066947 & 0.059182 & 0.058586 & 0.058629 & 13.12 & 1.02 & 0.07 \\
4 & 0.053203 & 0.048217 & 0.047762 & 0.048140 & 10.34 & 0.95 & 0.78 \\
5 & 0.040326 & 0.037993 & 0.037521 & 0.037755 & 6.14 & 1.26 & 0.62 \\
6 & 0.028869 & 0.028602 & 0.028397 & 0.028581 & 0.93 & 0.72 & 0.64 \\
7 & 0.017027 & 0.017974 & 0.017867 & 0.017902 & \(-\)5.27 & 0.60 & 0.19 \\
8 & 0.008271 & 0.009192 & 0.009249 & 0.009300 & \(-\)10.02 & \(-\)0.62 & 0.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Peak velocities of each layer in the free field system (for input 0.2 g). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Peak Velocities in (m/s)**} & \multicolumn{4}{c}{**Relative Difference between**} \\
**Soil Layer** & \multicolumn{4}{c}{**for Condition**} & \multicolumn{4}{c}{**Adjacent Conditions (\%)**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **1-2** & **2-3** & **3-4** \\ \hline
1 & 0.012988 & 0.008544 & 0.007284 & 0.007148 & 52.01 & 17.31 & 1.89 \\
2 & 0.011159 & 0.007511 & 0.006461 & 0.006342 & 48.58 & 16.24 & 1.88 \\
3 & 0.009165 & 0.006308 & 0.005443 & 0.005340 & 45.28 & 15.89 & 1.94 \\
4 & 0.007315 & 0.005189 & 0.004491 & 0.004405 & 40.95 & 15.5 Based on Tables 5-12, for peak displacements under seismic effects at various peak accelerations, the relative difference between Condition 1 and Condition 2 exceeded 20%, except for the bottom layer, with a maximum of 52.94%. The relative difference between Condition 2 and Condition 3 ranged from 8% to 20%, with a maximum of 17.49%. The relative difference between Condition 3 and Condition 4 was within 2.20%, with a maximum of only 2.15%. The maximum relative difference between Condition 1 and Condition 2 for peak velocities under seismic effects at various peak accelerations was 23.36%. The maximum relative difference between Condition 2 and Condition 3 was 4.72%. The maximum relative difference between Condition 3 and Condition 4 was 1.51%. In summary, the relative difference between Condition 1 and Condition 2 was substantial, indicating that when the side length of the free field system increased from 20 m to 50 m, the resulting differences were significant. The maximum relative differences in displacement and velocity between Condition 2 and Condition 3 were 17.49% and 4.72%,