# Numerical Analysis of Seismic Pounding between Adjacent Buildings Accounting for SSI
## 2 Theoretical Model Framework
### Approximate Normal Modes of Adjacent Buildings
\begin{bmatrix}M^{a}_{n}&0\\ 0&M^{b}_{n}\end{bmatrix}\begin{bmatrix}\ddot{D}^{a}_{n}(t)\\ \ddot{D}^{b}_{n}(t)\end{bmatrix} + \begin{bmatrix}C^{a}_{n}&0\\ 0&C^{b}_{n}\end{bmatrix}\begin{bmatrix}\dot{D}^{a}_{n}(t)\\ \dot{D}^{b}_{n}(t)\end{bmatrix} + \begin{bmatrix}K^{a}_{n}&0\\ 0&K^{b}_{n}\end{bmatrix}\begin{bmatrix}D^{a}_{n}(t)\\ D^{b}_{n}(t)\end{bmatrix} + \begin{bmatrix}F^{ap}_{n}(t)\\ -F^{bp}_{n}(t)\end{bmatrix}  \tag{7}\] \[=- \begin{cases}M^{a}_{n}t\Big{(}\Gamma^{a}_{xn}\ddot{x}_{S}+ \Gamma^{a}_{yn}\ddot{y}_{g}\Big{)}\\ M^{b}_{n}t\Big{(}\Gamma^{b}_{xn}\ddot{x}_{S}+\Gamma^{b}_{yn}\ddot{y}_{g}\Big{)} \end{cases} \]  
where \(M^{a}_{n}=T^{aT}_{n}M^{a}T^{a}_{n}\), \(C^{a}_{n}=T^{aT}_{n}\Gamma^{a}_{Tn}\), \(K^{a}_{n}=T^{aT}_{n}K^{a}T^{a}_{n}\), \(M^{b}_{n}=T^{bT}_{n}M^{b}T^{b}_{n}\), \(C^{b}_{n}=T^{bT}_{n}\Gamma^{b}_{xn}\) and \(K^{b}_{n}=T^{bT}_{n}K^{b}T^{b}_{n}\) are sized \(8\times 8\); \(i\) a vector sized as an \(8\times 1\) column with all elements the same as unity; the \(3N+5\) and \(3S+5\) multi-DOF modal equations of motion, as given in Equation (7) for both buildings, comprise a nonproportionally damped system. \[ \begin{split} C^{a}&=T^{aT}_{n}(\alpha M^{a}+ \beta K^{a})T^{a}_{n}=\alpha M^{a}_{n}+\beta K^{a}_{n}\\ C^{b}&=T^{bT}_{n}\Big{(}\alpha M^{b}+\beta K^{b} \Big{)}T^{b}_{n}=\alpha M^{b}_{n}+\beta K^{b}_{n}\end{split}  \tag{8}\]  
[PERSON] and [PERSON] [52] found that the Rayleigh damping model underestimates the damping of higher modes that contribute to the seismic response, resulting in an overestimation of the seismic response. This manuscript does not address the validity of the mass-proportional and stiffness-proportional assumptions in Rayleigh damping. The approximation approach and rigorous methods are contrasted in the first three modes of this study. If \(C^{a}\) and \(C^{b}\) are proportionally damped, i.e., \(C^{a}=(\alpha M^{a}+\beta K^{a})\) and \(C^{b}=\Big{(}\alpha M^{b}+\beta K^{b}\Big{)}\), the \(n^{th}\) modal damping matrices for each building can be described as Equation (8). In here, \(\alpha\) and \(\beta\) are coefficients found by the damping ratios of the two specific modes, since the original SSI system is not proportionally damped, i.e., \(C^{a}\
eq(\alpha M^{a}+\beta K^{a})\), \(C^{b}\
eq\Big{(}\alpha M^{b}+\beta K^{b}\Big{)}\). In approximate method, Equation (8) converts \(C^{a}\
eq\alpha M^{a}_{n}+\beta K^{a}_{n}\) and \(C^{b}\
eq\alpha M^{b}_{n}+\beta K^{b}_{n}\) based on [35]. The nonlinear viscoelastic model is only modified for the approximate method used in this study, as given in Equation (9). Notably, the elements of \(D^{a}_{n}\) and \(D^{b}_{n}\) are not the same, even in an elastic state. \[F^{p}_{xijn}(t)=0\text{ for }\delta_{ijxn}(t)\leq 0; \tag{9}\] \[F^{p}_{xijn}(t)=\overline{\beta}\big{(}\delta_{ijxn}(t)\big{)}^{3/2 }+\bar{c}_{ijn}(t)\dot{\delta}_{ijxn}(t)\text{ for }\delta_{ijxn}(t)>0\text{ and }\dot{\delta}_{ijxn}(t)>0;\] \[F^{p}_{xijn}(t)=\overline{\beta}\big{(}\delta_{ijxn}(t)\big{)}^{3/2 }\text{ for }\delta_{ijxn}(t)>0\text{ and }\dot{\delta}_{ijxn}(t)\leq 0;\] \[\delta_{ijxn}(t)=\delta_{ijyn}(t)=D^{a}_{xn}\varphi^{a}_{ixn}(t)- D^{b}_{xn}\varphi^{b}_{jxn}(t)-D;\;\delta_{ij\theta n}(t)=D^{b}_{\theta n} \varphi^{a}_{i\theta n}(t)f^{a}-\] \[D^{b}_{\theta n}\varphi^{b}_{j\theta n}(t)f^{b}-D;\] \[\dot{\delta}_{ijxn}(t)=\dot{D}^{a}_{xn}\varphi^{a}_{ixn}(t)- \dot{D}^{b}_{xn}\varphi^{b}_{jxn}(t);\;\dot{\delta}_{ijxn}(t)=D^{a}_{\theta n} \varphi^{a}_{ijxn}(t)-\dot{D}^{b}_{yn}\varphi^{b}_{jyn}(t);\] \[\dot{\delta}_{ij\theta n}(t)=\dot{D}^{a}_{\theta n}\varphi^{a}_{ i\theta n}(t)f^{a}-\dot{D}^{b}_{\theta n}\varphi^{b}_{j\theta n}(t)f^{b};\] \[F^{ap}_{ijn}=\Big{[}F^{ap}_{xijn}\ F^{ap}_{yjn}\ F^{ap}_{\theta ijn }0\ 0\ 0\ 0\Big{]}^{T}_{(3N+5)\times 1}\]  
Here, \(\delta_{ij}(t)\) is the total displacement between the buildings with respect to the foundation; \(\dot{\delta}_{ij}(t)\) is their velocity; \(\overline{\beta}\) is the damping; and \(\overline{c}_{ij}(t)\) is the stiffness of the impact element.