# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 3 Numerical Study
### Structure Properties  
Buildings \(A\) and \(B\) had the dimensions of 20 m by 15 m and 25 m by 20 m, respectively, with the longer lengths in the longitudinal direction (\(x\)) for each building. The ratio of the foundation mass to the floor mass was 3 for each building. Each story in each building was 2.85 m high. Table 2 provides the basic values describing the structural characteristics that have been used for this study. Based on the reported results [34; 54; 55], we used \(\overline{\beta}=2.75\times 10^{9}\) N/m\({}^{3/2}\) and \(\overline{\zeta}=0.35\) for the bounding force parameters in the nonlinear viscoelastic model, with an established coefficient of friction \(\left(\mu_{f}\right)\) of 0.5. In Equation (13), the translational and torsional stiffness \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Sliding** & **Torsion** & **Rocking** \\ \hline Spring & \(K_{T}=\frac{32(1-\upsilon)G_{\varepsilon}}{7-8\upsilon}\) & \(K_{\theta}=\frac{16G_{\varepsilon}^{3}}{3}\) & \(K_{\phi,\phi}=\frac{8G_{\varepsilon}^{3}}{3(1-\upsilon)}\) \\ Mass Ratio & \(B_{T}=\frac{(7-8\upsilon)M_{\varepsilon}}{32(1-\upsilon)\upsilon\upsilon \upsilon}\) & \(B_{\theta}=\frac{1\upsilon}{\overline{\rho}\overline{\rho}}\) & \(B_{\psi,\phi}=\frac{3(1-\upsilon)I_{\theta\phi}}{8\overline{\rho}\overline{ \rho}}\) \\ Damping & \(D_{T}=\frac{0.288}{\overline{\delta}T}\) & \(D_{\theta}=\frac{0.5}{1+2B_{\theta}}\) & \(D_{\psi,\phi}=\frac{0.15}{(1+B_{\theta,\phi})\sqrt{B_{\theta\phi}}}\) \\ Ratio & \(C_{T}=2D_{T}\sqrt{K_{T}M_{T}}\) & \(C_{\phi}=2D_{\theta}\sqrt{K_{\theta}\phi}\) & \(C_{\phi,\phi}=2D_{\phi,\phi}\sqrt{K_{\phi,\phi}I_{\phi,\phi}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spring and dashpot constants used by [PERSON] et al. [30]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Story No.**} & \multirow{2}{*}{
\begin{tabular}{c} **Story Height** \\ \(\mathbf{h}_{\mu}\) **\(\mathbf{h}_{j}\)** \\ \((\mathbf{m})\) \\ \end{tabular} } & \multicolumn{2}{c}{**Building \(A\)**} & \multicolumn{2}{c}{**Building \(B\)**} \\ \cline{3-6}  & \(\mathbf{h}_{\mu}\) **\(\mathbf{h}_{j}\)** & \(\mathbf{m}_{i}\)\(\times\)10\({}^{\mathbf{6}}\) & \(\mathbf{k}_{i}\)\(\times\)10\({}^{\mathbf{8}}\) & \(\mathbf{m}_{j}\)\(\times\)10\({}^{\mathbf{6}}\) & \(\mathbf{k}_{j}\)\(\times\)10\({}^{\mathbf{8}}\) \\  & \((\mathbf{m})\) & \((\mathbf{kg})\) & \((\mathbf{N}/\mathbf{m})\) & \((\mathbf{m})\) & \((\mathbf{N}/\mathbf{m})\) \\ \hline
1F & 2.85 & 0.30 & 3.46 & 0.4065 & 5.06 \\
2F & 5.7 & 0.30 & 3.46 & 0.4065 & 3.86 \\
3F & 8.55 & 0.30 & 3.46 & 0.4065 & 3.86 \\
4F & 11.4 & 0.30 & 3.46 & 0.4065 & 3.86 \\
5F & 14.25 & 0.30 & 3.46 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Building details [31; 51]. at the center of mass for each story of a building are longitudinally in proportion to the stiffness of the same story [56]. \[\beta_{y}=\frac{k_{yi}}{k_{xi}}=\frac{k_{yj}}{k_{xj}};\ \beta_{t}=\frac{k_{\theta i}}{r_{ \theta}^{2}k_{xi}}=\frac{k_{\theta j}}{r_{\theta}^{2}k_{xj}} \tag{13}\]  
The values of \(\beta_{y}\) and \(\beta_{t}\) were 1.32 and 1.69 for both buildings, respectively. Both buildings were characterized by 2% of the critical damping as a constant of proportionality (\(\alpha\)) in the first vibration mode. Poisson's ratio (\(v\)) was 0.333, and the density of the soil (\(\rho\)) was 1922 kg/m\({}^{3}\). Specifically, four types of soil at shear wave velocities ranging from 65, 130, 200, to 300 m/s as Case I to IV have been examined, respectively. These ranges are described as soft to hard soil, based on [PERSON], et al. [57] and [PERSON] et al. [58]. Soft soil (Case I) indicates the large SSI effect on the building's response, while hard soil (Case IV) indicates the small SSI effect. The original distance, \(D\), between the buildings was 0.04 m. The selection of this value has been based on the previous studies [48; 59; 60].