# 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 & 0.142888 & 0.116661 & 0.115396 & 0.116920 & 22.48 & 1.10 & 1.30 \\
2 & 0.122457 & 0.103125 & 0.101950 & 0.102169 & 18.75 & 1.15 & 0.21 \\
3 & 0.100316 & 0.086415 & 0.085703 & 0.086548 & 16.09 & 0.83 & 0.98 \\
4 & 0.079948 & 0.071252 & 0.070221 & 0.070757 & 12.21 & 1.47 & 0.76 \\
5 & 0.060770 & 0.056297 & 0.055373 & 0.055733 & 7.95 & 1.67 & 0.65 \\
6 & 0.043550 & 0.042440 & 0.041911 & 0.042179 & 2.62 & 1.26 & 0.63 \\
7 & 0.025707 & 0.026608 & 0.026439 & 0.026585 & \(-\)3.39 & 0.64 & 0.55 \\
8 & 0.012435 & 0.013686 & 0.013722 & 0.013792 & \(-\)9.14 & \(-\)0.26 & 0.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Peak velocities of each layer in the free field system (for input 0.3 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 & 0.189622 & 0.153720 & 0.149134 & 0.150906 & 23.36 & 3.08 & 1.17 \\
2 & 0.162823 & 0.134586 & 0.132064 & 0.133476 & 20.98 & 1.91 & 1.06 \\
3 & 0.133791 & 0.113253 & 0.111477 & 0.112493 & 18.13 & 1.59 & 0.90 \\
4 & 0.106995 & 0.093698 & 0.091829 & 0.092491 & 14.19 & 2.04 & 0.72 \\
5 & 0.081611 & 0.074298 & 0.072705 & 0.073152 & 9.84 & 2.19 & 0.61 \\
6 & 0.058564 & 0.056117 & 0.055160 & 0.055459 & 4.36 & 1.74 & 0.54 \\
7 & 0.034603 & 0.035260 & 0.034927 & 0.035095 & \(-\)1.86 & 0.95 & 0.48 \\
8 & 0.016761 & 0.018174 & 0.018183 & 0.018267 & \(-\)7.78 & \(-\)0.05 & 0.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Peak velocities of each layer in the free field system (for input 0.4 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 & 0.189622 & 0.153720 & 0.149134 & 0.150906 & 23.36 & 3.08 & 1.17 \\
2 & 0.162823 & 0.134586 & 0.132064 & 0.133476 & 20.98 & 1.91 & 1.06 \\
3 & 0.133791 & 0.113253 & 0.111477 & 0.112493 & 18.13 & 1.59 & 0.90 \\
4 & 0.106995 & 0.093698 & 0.091829 & 0.092491 & 14.19 & 2.04 & 0.72 \\
5 & 0.081611 & 0.074298 & 0.072705 & 0.073152 & 9.84 & 2.19 & 0.61 \\
6 & 0.058564 & 0.056117 & 0.055160 & 0.055459 & 4.36 & 1.74 & 0.54 \\
7 & 0.034603 & 0.035260 & 0.034927 & 0.035095 & \(-\)1.86 & 0.95 & 0.48 \\
8 & 0.016761 & 0.018174 & 0.018183 & 0.018267 & \(-\)7.78 & \(-\)0.05 & 0.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Peak velocities of each layer in the free field system (for input 0.3 g). respectively, indicating that there was still a noticeable difference when the soil body side length was 50 m compared to 100 m. The maximum relative differences in displacement and velocity between Condition 3 and Condition 4 were 2.15% and 1.51%, respectively, showing that the results were very similar when the soil body side length was 100 m compared to 200 m. Therefore, it can be concluded that the results obtained with a 100 m side length free field system were essentially consistent with those obtained with a 200 m side length system, and a free field side length of 100 m meets the requirements without needing to increase the size further. The superstructure's span and depth were both 2 m, and the 100 m \(\times\) 100 m free field system's side length was 50 times the layout size of the superstructure.