# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 2 Theoretical Model Framework
### Approximate Normal Modes of Adjacent Buildings  
Equation (4) can be expressed in the following form, as proposed by [PERSON] and [PERSON] [51]:  
\[P^{a}(t)=\sum_{n=1}^{3N+5}s_{n}^{a}\Big{(}\Gamma_{xn}^{a}\ddot{x}_{g}+\Gamma_{ yn}^{a}\ddot{y}_{g}\Big{)};P^{b}(t)=\sum_{n=1}^{3S+5}s_{n}^{b}\Big{(}\Gamma_{xn}^{b} \ddot{x}_{g}+\Gamma_{yn}^{b}\ddot{y}_{g}\Big{)} \tag{4}\]  
The \(n^{th}\)-mode modal inertia force distribution can be calculated as \(M^{a}\varphi_{n}^{a}\) for vector \(s_{n}^{a}\) and \(M^{b}\varphi_{n}^{b}\) for vector \(s_{n}^{b}\). The \(\varphi_{n}^{a}\) calculated from \(K^{a}\) and \(M^{a}\) is the \(n^{th}\)-mode shape without damping. \(\varphi_{n}^{b}\) is also obtained in the same way as \(\varphi_{n}^{a}\). \(\Gamma_{xn}^{a}\), \(\Gamma_{yn}^{a}\), \(\Gamma_{xn}^{b}\), and \(\Gamma_{yn}^{b}\) in the longitudinal and perpendicular directions for the \(n^{th}\) modal contribution values for both buildings. The output values of \(U(t)\) contain translations in the \(x\) and \(y\) directions, rotations for each floor of the superstructure and the foundation, and the rocking angles for the foundation only in both directions. The \(n^{th}\) mode of the contribution factor in the modal analysis is calculated as  
\[ \begin{split}\Gamma_{xn}^{a}=\frac{\varphi_{n}^{a\,T}\times M^{a }\times\left|\mathbf{1}^{T}\mathbf{0}^{T}\mathbf{0}^{T}\mathbf{1}\mathbf{0} \mathbf{0}\mathbf{0}\right|^{T}}{\varphi_{n}^{a\,T}\times M^{a}\times\varphi_ {n}^{a}};\\ \Gamma_{yn}^{a}=\frac{\varphi_{n}^{a\,T}\times M^{a}\times\left| \mathbf{0}^{T}\mathbf{1}^{T}\mathbf{0}^{T}\mathbf{0}\mathbf{1}\mathbf{0} \mathbf{0}\right|^{T}}{\varphi_{n}^{a\,T}\times M^{a}\times\varphi_{n}^{a}} \end{split}  \tag{5}\]  
where \(\mathbf{1}\) and \(\mathbf{0}\) are the unit and zero vectors, respectively, sized as \(N\times 1\). Equation (5) proves that the \(n^{th}\) modal participation factors rely on the path of the horizontal ground motion excitations. The 1940 El Centro (117 El Centro Array-9 station) and the 1995 Kobe (KJMA station) ground motions have been used for the analysis of both buildings, which is shown in Figure 3. The top accelerations of the related excitations were scaled to 0.3 g and 0.2 g for the 1940 El Centro NS-EW and 0.8 g and 0.6 g for the 1995 Kobe NS-EW, respectively. \(U_{n}^{a}\) and \(U_{n}^{b}\) are defined in Equation (4) by the time variations in \(\ddot{x}_{g}(t)\) and \(\ddot{y}_{g}(t)\). In the analysis, the vertical motion of the ground was not in use. The force distribution can be expressed with the sum of the modal inertia forces, as given in Equation (4). \(T_{n}^{a}\) and  
Figure 3: (**a**) 1940 El Centro and (**b**) 1995 Kobe earthquakes with maximum ground acceleration scaled to 0.3 g and 0.8 g, respectively. are \((3N+5)\times 8\) and \((3S+5)\times 8\) diagonal matrices for both buildings, respectively. \(\varphi^{a}_{xn}\), \(\varphi^{a}_{yn}\), and \(\varphi^{a}_{bn}\) in \(T^{a}_{n}\) are subvectors of the \(n^{th}\) natural vibration mode in the superstructure, sized as \(N\times 1\). They are related to the displacement in both directions and the rotating DOF. \(\varphi^{a}_{x_{n}a}\), \(\varphi^{a}_{y_{n}a}\), \(\varphi^{a}_{b_{n}a}\), \(\varphi^{a}_{b_{n}a}\), and \(\varphi^{a}_{b_{0}n}\) in \(T^{a}_{n}\) are five subvectors of the mode shapes of the SSI system for Building \(A\). With generalized modal co-ordinates for both buildings, the \(n^{th}\) undamped modal displacement responses, \(U^{a}_{n}\) and \(U^{b}_{n}\), can be redefined as  
\[ \begin{cases}U^{a}_{n}(t)\\ U^{b}_{n}(t)\end{cases} = \begin{bmatrix}T^{a}_{n}&0\\ 0&T^{b}_{n}\end{bmatrix}\begin{bmatrix}D^{a}_{n}(t)\\ D^{b}_{n}(t)\end{bmatrix}  \tag{6}\]  
where \(D^{a}_{n}\) and \(D^{b}_{n}\) are the \(n^{th}\) generalized modal co-ordinate of both buildings. By substituting Equation (6) and rearranging each side of the equation of motion by \( \begin{bmatrix}T^{a}_{n}&0\\ 0&T^{b}_{n}\end{bmatrix} ^{T}\), we obtain  
\[