## 2 Proposed Equations
For each bounding scenario, the displacement time histories for the level of possible contact (the level of the top storey of the shorter building) have been firstly obtained for both buildings vibrating independently under the specified ground motion. Then, the spacing required to avoid collisions has been calculated using Equation (7). In the next step, the peak displacements \(U_{1}\) and \(U_{2}\) have also been obtained for each building vibrating separately from the time history analyses. In this study, the DDC formula has been used (Equation (3)). The value of the correlation factor \(\rho\) has been calculated based on Equation (8) (obtained from re-arranging of Equation (3)). This procedure has been performed for 1260 bounding cases for buildings exposed to 5 different earthquakes and founded on 5 different soil types. \[S=max\,|\,U_{2}{}^{*}(t)-U_{1}(t)\,| \tag{7}\]  
\[\rho=\frac{{U_{1}}^{2}+{U_{2}}^{2}-S^{2}}{2U_{1}U_{2}} \tag{8}\]  
where \(U_{1}(t)\) is the displacement time history of the shorter building at the top storey and \({U_{2}}^{*}(t)\) is the displacement time history of the taller building at the storey corresponding to the top storey of the shorter building; e.g., if bounding occurs between 4- and 6-storey buildings, \(U_{1}(t)\) and \(U_{2}{}^{*}(t)\) concern the 4 th storey of these two buildings, respectively. Figure 1: Plan views of the considered models. For instance, in the case of bounding between the 10-storey and 12-storey buildings (10-12 pounding scenario), the structures have been firstly studied when they vibrate independently. The displacement time histories of both buildings at the possible contact level (the level of the top storey of the shorter building, i.e., the level of 10 th storey) have been obtained. In 10-12 pounding scenario, the \(U_{1}(t)\) and \(U_{2}{}^{*}(t)\) correspond to the displacement time histories at the 10 th storey of the 10-storey and 12-storey building, respectively. In the next step, using Equation (7), the spacing required to avoid collisions between them has been calculated. Then, \(U_{1}\) and \(U_{2}\), corresponding to the peak displacement of the 10-storey and the 12-storey buildings, respectively, have been determined. After calculating the values of \(U_{1}\) and \(U_{2}\), as well as the spacing \(S\), the calculation of the correlation factor has been conducted using Equation (8). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Building** & **Number of** & **Period (s)** & **Frequency** & **Building** & **Number of** & **Period (s)** & **Frequency** \\
**Number** & **Storeys** & **Period (Hz)** & **Number** & **Storeys** & **Period (Hz)** \\ \hline
1 & 1 & 0.20 & 4.902 & 31 & 11 & 1.979 & 0.505 \\ \hline
2 &  
Most seismic codes require a number of 2 to 4 independent ground motion simulations so as to obtain the average responses (see [78] for details). Therefore, five ground motions have been taken into account in this study (see Table 2) downloaded from the PEER website [79]. These ground motions are ground surface records. The authors have intentionally adopted different ground motions with different PGAs and frequency content to obtain insight into the issue of how different PGAs and frequency content may contribute to the dynamic response of colliding buildings. Also, different ground motions with significantly different PGAs have been considered to ensure that the proposed equations will be valid for large range of PGAs and not limited to a specific range of PGAs. The correlation factor has been calculated for each of the 1260 cases under these five ground motions and then the average value has been determined. The analysis has been performed using ETABS software v.18 [80]. Then, the correlation factor has been plotted as a function of the ratio of the natural periods of both buildings \(T_{1}/T_{2}\) (\(T_{1}<T_{2}\)). The ratio of the natural periods has been taken into account since it is the primary factor affecting the earthquake-induced structural bounding [81].