# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 3 Numerical Study
### Response Analysis  
The results have been compared with those reported by [PERSON] et al. [42], [PERSON] and [PERSON] [23], and [PERSON] et al. [35], who did not address the effect of bounding. As the reference building (RB) in this study, another four-story building without considering SSI has been chosen. The first six natural frequencies of the eight-story building-foundation system used by [PERSON] et al. [42] are shown in Table 3. As can be seen from the table, the \(x\)-direction displacement component is dominant in the first and fourth modes, while the \(y\)-direction displacement component is dominant in the second and fifth modes. The major component in the third and sixth modes corresponds to the rotation around the vertical axis. Building B has a modal response that is less than that of Building A, but with the same trend; therefore, the results of Building B are not provided here. Figure 4 shows the mode shapes of Building A placed on rigid ground as well as the mode shapes of each case using the approximate method. The thin lines in Figure 4 represent the translations and rotations of the base, as shown by the offsets and slopes. In Figure 4, the first to third mode shapes of the dominant motions are the \(x\)- and \(y\)-translation and rotation around the vertical axis for both the RB without the effects of SSI and for all cases. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \begin{tabular}{c} **Cases** \\ **Frequency** \\ \end{tabular} & **Case I** & **Case I *** & **Case IV** & **Case IV *** & **RB** & **RB *** & \
\begin{tabular}{c} **Case I *** \\ **RB *** \\ \end{tabular} &
\begin{tabular}{c} **Case IV *** \\ **RB *** \\ \end{tabular} \\ \hline W\({}_{1}\) (Hz) & 0.698 & 0.685 & 0.788 & 0.787 & 0.794 & 0.792 & 0.865 & 0.994 \\ \hline W\({}_{2}\) (Hz) & 0.783 & 0.787 & 0.932 & 0.931 & 0.941 & 0.940 & 0.837 & 0.990 \\ \hline W\({}_{3}\) (Hz) & 1.120 & 1.107 & 1.224 & 1.222 & 1.233 & 1.231 & 0.899 & 0.993 \\ \hline W\({}_{4}\) (Hz) & 1.896 & 1.874 & 1.942 & 1.937 & 1.943 & 1.941 & 0.965 & 0.998 \\ \hline W\({}_{5}\) (Hz) & 2.227 & 2.175 & 2.301 & 2.297 & 2.306 & 2.303 & 0.944 & 0.997 \\ \hline W\({}_{6}\) (Hz) & 2.905 & 2.795 & 3.013 & 3.007 & 3.019 & 3.015 & 0.927 & 0.997 \\ \hline \hline \end{tabular}  
* Obtained by current studies; RB: reference building.  
\end{table} 
Table 3: First six natural frequencies of an eight-story building-foundation system considered in [38] compared with those in the current study. However, the third mode shape in Case I is characterized by a rocking movement in the \(y\)-direction. In Cases I to IV, the first to third modes of shapes are consistent with those of the reference building. Figure 5 shows that the modal dislocation-time interactions for each of the eight DOFs for Case I are different. In Case IV, the modal responses for the eight DOFs display a similar pattern to those of the second and third modes. Case I shows that the modal responses for the first three modes are mainly affected by SSI effects, and they are out of phase. Figure 6 shows the response of the first mode in Case IV without any pounding. Figure 4: Building \(A\) in the first to third modes without considering SSI effects and for all cases. Figure 5: First to third modes during soft and hard soil earthquakes used in this study. As shown in Figure 6, when the necessary distance is present between the buildings to avoid rounding, the modal responses of the eight DOFs for each vibration mode are similar to those in Case IV. Figure 7 illustrates the response histories of Cases I and IV obtained with the proposed method under the 1995 Kobe earthquake. The dashed line in Figure 7 denotes the approximate method (App.) given by Equation (7), whereas the solid line denotes the rigorous method (Rig.) obtained by Equation (3). Figure 6: First-mode responses for El Centro earthquake in 1940 and Kobe earthquake in 1995 without rounding. Figure 7: Results for both approximate and rigorous solutions on the fourth floor for the 1995 Kobe earthquake.