# Validation and Parametric Study of a Soil-Structure Interaction Nonlinear Simplified Model
## 4 Parametric Study of the Simplified Model
### Equivalent Soil Area  
The mass, stiffness, and damping of the equivalent soil contributed to forming the overall mass, stiffness, and damping matrices of the simplified model. Consequently, selecting different ranges of equivalent soil areas had a certain impact on the overall structural system of the simplified model. In the majority of cases, the footing area of the superstructure was selected as the equivalent soil area. The equivalent soil, treated with group piles and forming an equivalent single pile area, was selected as the reference. The equivalent single pile radius was 0.318 m. Five different equivalent soil areas were selected as five conditions. The specific condition information was shown in Table 22. Under the action of EL-central seismic waves with peak accelerations of 0.1 g, 0.2 g, 0.3 g, and 0.4 g, the frequency changes in the structural system were analyzed. Since the first-order frequencies of the simplified model structural system were all 11.369 Hz under different conditions and different peak seismic waves, only the second-order frequencies of the simplified model structural system were extracted and presented in Table 23.  
From Table 23, it can be observed that when the equivalent soil was not considered, the frequency under different peak acceleration seismic waves were identical. This was because when not considering the equivalent soil, the soil parameters did not contribute to forming the frequency of the structural system. Under different peak acceleration seismic waves, the relative difference between Condition 1 and Condition 2 was significant, with a maximum relative difference of up to 24.69. The difference between Condition 2 and Condition 4 was relatively small, while the difference between Condition 4 and Condition 5 was the smallest, with relative differences all below 1.90. The results of the calculations conducted under the assumption that the equivalent soil was not considered were found to be significantly different from those obtained when the equivalent soil was taken into account. This indicated that the equivalent soil should be included in the simplified model. Upon consideration of the equivalent soil, it became \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Condition**} & \multicolumn{2}{c}{**0.1 g**} & \multicolumn{2}{c}{**0.2 g**} & \multicolumn{2}{c}{**0.3 g**} & \multicolumn{2}{c}{**0.4 g**} \\ \cline{2-9}  & **Frequency** & **Relative** & **Frequency** & **Relative** & **Difference (\%)** & **Relative** & **Frequency** & **Relative** \\  & **(Hz)** & **Difference (\%)** & **(Hz)** & **Difference (\%)** & **(Hz)** & **Difference (\%)** & **(Hz)** & **Difference (\%)** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{21.084} & - & \multirow{2}{*}{21.084} & - & \multirow{2}{*}{21.084} & - & \multirow{2}{*}{21.084} & - \\  & & & \(-\)24.69 & & & & \(-\)23.83 & & \(-\)21.68 \\ \multirow{2}{*}{2} & \multirow{2}{*}{27.997} & \multirow{2}{*}{1.80} & 27.996 & & 27.682 & & \multirow{2}{*}{3.62} & 26.921 & \\  & & & & & & & & \\ \multirow{2}{*}{3} & \multirow{2}{*}{27.503} & \multirow{2}{*}{3.19} & 27.233 & & 26.715 & & 26.127 & \\  & & & & & & & & \\ \multirow{2}{*}{4} & \multirow{2}{*}{26.653} & \multirow{2}{*}{1.37} & 26.281 & & 25.760 & & 25.168 & \\  & & & & & & & & \\ \multirow{2}{*}{5} & \multirow{2}{*}{26.294} & \multirow{2}{*}{1.37} & 25.819 & 1.79 & 25.295 & 1.84 & 24.715 & 1.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 23: The second-order frequencies of structural systems with different equivalent soil areas. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Condition** & **Condition 1** & **Condition 2** & **Condition 3** & **Condition 4** & **Condition 5** \\ \hline Equivalent soil & No consideration & & 10 times the & \\ area & of equivalent soil & & pile area & \\ \hline \hline \end{tabular}
\end{table} 
Table 22: Conditions of equivalent soil area. evident that this had a significant impact on the second-order frequency of the structural system.