## 2 Proposed Equations
### Soil Type E  
In this section, the proposed equation for the correlation factor when the colliding buildings are founded on soil type E is presented. Figure 10 presents the plot of the correlation factor when the colliding buildings are founded on soil type D versus \(T_{1}/T_{2}\). It can be seen that the plot is a piecewise function, and it is composed of three different functions. It can also be noticed that the correlation factor follows three different trends depending on whether \(T_{1}\leq 0.2\) s, \(0.2\) s < \(T_{1}\leq 0.4\) s, or \(T_{1}>0.4\) s (see Figure 10). Therefore, the proposed equation for the correlation factor when the colliding buildings are founded on soil type D is composed of three equations depending on whether \(T_{1}\leq 0.2\) s, \(0.2\) s < \(T_{1}\leq 0.4\) s, or \(T_{1}>0.4\) s (see Equation (14)). Figure 11 shows a comparison between the proposed equation (Equation (14)) and the actual values of the correlation factor obtained for different cases. Using Equation (9), the RMS error has been calculated as equal to 7.00% for \(T_{1}\leq 0.2\) s, 2.98% for \(0.2\) s < \(T_{1}\leq 0.4\) s, and 8.30% for \(T_{1}>0.4\) s. It should be noted at the end that, if the calculation based on the proposed correlation factors results in a negative value of the term \(S^{2}={U_{1}}^{2}+{U_{2}}^{2}-2\rho U_{1}U_{2}\), then the absolute value should be taken into account for the calculation of the spacing \(S\). Figure 9: Comparison between the actual trend and the proposed equation for the correlation factor when the colliding buildings are founded on soil type D.  
\[\rho=\left\{ \begin{array}{ccc}\left(\frac{T_{1}}{T_{2}}\right)^{-1.519}&,&T_{1} \leq 0.2\;\mathrm{s}\\ \\ 2531.452\left(\frac{T_{1}}{T_{2}}\right)^{6}-8855.4\left(\frac{T_{1}}{T_{2}} \right)^{5}+12,190\left(\frac{T_{1}}{T_{2}}\right)^{4}-8404.1\left(\frac{T_{1} }{T_{2}}\right)^{3}+3076.1\left(\frac{T_{1}}{T_{2}}\right)^{2}\\ \\ -589.69\left(\frac{T_{1}}{T_{2}}\right)+52.638,&0.2\;\mathrm{s}<T_{1}\leq 0.4 \;\mathrm{s}\\ \\ 78.392\left(\frac{T_{1}}{T_{2}}\right)^{4}-214.39\left(\frac{T_{1}}{T_{2}} \right)^{3}+219.53\left(\frac{T_{1}}{T_{2}}\right)^{2}-99.972\left(\frac{T_{1} }{T_{2}}\right)+17.44,&T_{1}>0.4\;\mathrm{s}\\ \\ \end{array} \right. \tag{14}\]  
Figure 11: _Cont._  
Figure 10: The correlation factor when the colliding buildings are founded on soil type E versus \(T_{1}/T_{2}\). All lines correspond to the same thing (correlation factor). The graph of the correlation factor is not continuous. gap calculated using the SRSS formula and the exact required seismic gap to avoid collisions ranges between 1.18 and 3.53. This means that the SRSS formula significantly overestimates the gap for the considered cases and the proposed equations provide better accuracy than the SRSS formula. Also, Table 7 presents the comparison between the seismic gap calculated using the DDC formula (Equations (3) and (4)) and the exact required seismic gap to avoid collisions based on Equation (8) as well as the ratio between them. It can be seen that the ratio of the gap calculated using the DDC formula and the exact required seismic gap to avoid collisions ranges between 1.18 and 3.53. This means that the DDC formula significantly overestimates the gap for the considered cases and the proposed equations provide better accuracy than the DDC formula. Moreover, Table 8 presents the comparison between the seismic gap calculated using the Australian Code formula (Equation (5)) and the exact required seismic gap to avoid collisions based on Equation (8) as well as the ratio between them. It can be seen that the ratio of the gap calculated using the Australian Code formula and the exact required seismic gap to avoid collisions ranges between 0.74 and 14.81. This means that the Australian Code formula provides both accurate results as well as significantly overestimated results and the proposed equations provide better accuracy than the Australian Code formula.