# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 2 Theoretical Model Framework
### Equation of Motion  
Figure 1 shows a structural model of two adjacent buildings on a half-space with elastic properties. Subscripts \(i\) and \(j\) in Figure 1 indicate the numbers of stories in the buildings. These are \(1,\ 2,\ \dots,\ N\) for Building \(A\) and \(1,\ 2,\ \dots,\ S\) for Building \(B\). In this study, the following values have been determined: the mass, stiffness, damping coefficients, and moment of inertia of the floor with respect to the axes parallel to the \(x\) and \(y\) axes of Building \(A\) and in relation to the center of mass. In addition, Buildings \(A\) and \(B\) are represented either by the subscript or superscript \(a\) and \(b\), respectively. A building with \(N\) stories has \(3N+5\) DOF; hence, Buildings \(A\) and \(B\) have \(3N+5\) and \(3S+5\) equations, respectively. Here, only the equations for Building \(A\) are highlighted. The equation of motion noted from previous studies [30; 35; 42; 43; 44] is briefly presented herein for completeness. The three equations of motion of each floor of the building may be written in a matrix form as \((3\,N)\):  
\[ \begin{split}&[M_{a}]\left\{\ddot{x}_{ic}^{t}\right\}+[C_{ax}] \left\{\dot{x}_{i}\right\}+[K_{ax}]\left\{x_{i}\right\}+\left[F_{xij}^{p}(t) \right]=\left\{0\right\}\\ &[M_{a}]\left\{\ddot{y}_{ic}^{t}\right\}+\left[C_{ay}\right] \left\{\dot{y}_{i}\right\}+\left[K_{ay}\right]\left\{y_{i}\right\}+\left[F_{ yij}^{p}(t)\right]=\left\{0\right\}\\ & r_{a}^{2}[M_{a}]\left\{\ddot{\theta}_{i}^{t}\right\}+f_{a}[C_{ ax}]\left\{\dot{x}_{i}\right\}-e_{a}[C_{ay}]\left\{\dot{y}_{i}\right\}+ \left[C_{\theta R}^{a}\right]\left\{\dot{\theta}_{ic}\right\}+f_{a}[K_{ax}] \left\{x_{i}\right\}\\ &\qquad\qquad\qquad\qquad\qquad-e_{a}[K_{ay}]\left\{y_{i}\right\} +\left[K_{\theta R}^{a}\right]\left\{\theta_{ic}\right\}+\left[F_{\theta ij}^{p}(t) \right]=\left\{0\right\}\end{split}  \tag{1}\]  
The mass, damping, and stiffness of Building \(A\) in the submatrices are denoted as \(M_{a}\), \(C_{ax}\), \(C_{ay}\), \(K_{ax}\), and \(K_{ay}\), in relation to both directions. These matrices are sized with \(N\times N\) dimensions. As described by Equation (2), the parameters \(x_{ic}^{t}\), \(y_{ic}^{t}\), and \(\theta_{ic}^{t}\) are the displacements of the center of mass of the floors along the longitudinal and transverse axes, and their twists about the upward axis (\(z\)) in Building A, respectively. The forces used for bounding in the longitudinal direction \(F_{xij}^{p}(t)\) are obtained from the nonlinear viscoelastic model developed by [PERSON] [45]. \(F_{yij}^{p}(t)\) and \(F_{\theta ij}^{p}(t)\) are derived by the Coulomb friction model used by [PERSON] [46]. The impact model (nonlinear viscoelastic model) is used in  
Figure 1: Modeling of asymmetric adjacent buildings. this study for the rigorous model as it is. On the other hand, the impact model for the approximate method is deeply examined. \(x_{i},y_{i}\), and \(\theta_{ic}\) are the displacement vectors of the center of resistance (CR) in the \(x\) and \(y\) directions and the twist of each floor with respect to the base, respectively. Additionally, two-way asymmetric buildings have been considered such that the CRs and center of masses (CMs) are not symmetrical along two axes of the horizontal plane as shown in Figure 2. The center of rigidity is the centroid of stiffness in a floor-diaphrag layout. The floor diaphragm experiences translational displacement in two directions and rotation when the center of stiffness is subjected to lateral loading. The static eccentricity of the center of rigidity from the center of mass (\(e\) and \(f\)) for each story level does not differ, although the CR varies from floor to floor. Hence, the CR lies at the co-ordinates \(e_{av},\ f_{a}\) of Building \(A\) and \(e_{b},\ f_{b}\) of Building \(B\). The radius of gyration of the floor mass is taken from the center of mass. The torsion stiffness matrix defined about the CR is denoted as \(K_{\theta R}^{a}\), whereas \(K_{\theta M}^{a}\) in Equation (1) is defined in relation to CM.