## 5 Proposed Formula  
In order to suggest an effective equation to calculate the gap size index, \(\rho_{\text{op}}\), let us firstly consider the dynamic response of the \(n\)-degrees of freedom system in mode \(i\)[27]:  
\[R(t)=\sum_{i}\Psi_{i}S_{i}(t) \tag{3}\]  
Figure 3: Estimated separation distance using the CRVK program for different models under various ground motions. where \(\Psi_{i}\) is the effective participation factor for mode \(i\) and \(S_{i}(t)\) denotes the \(i\)-th normal coordinate. Then, the power spectral density of \(R(t)\) can be expressed by the formula [27]  
\[G_{\rm{R}}(w)=\sum_{i}\sum_{j}\Psi_{i}\Psi_{j}G_{\rm{F}}(w)H_{i}(w)H_{j}^{*}(w) \tag{4}\]  
where \(G_{\rm{F}}(w)\), \(H_{i}(w)\) and \(H_{j}^{*}(w)\) are the power spectral density of a stationary input \(F(t)\), the complex frequency response function of the displacement response of mode \(i\) and the complex conjugate of \(H_{i}(w)\), respectively. According to Equation (4), moments of the response power spectral density about the frequency origin are obtained as [27]  
\[\lambda_{m}=\int\limits_{0}^{\infty}w^{m}G_{R}(w)dw=\sum_{i}\sum_{j}\Psi_{i} \Psi_{j}\lambda_{m,ij} \tag{5}\]  
where \(\lambda_{m,ij}\) is defined as [27]  
\[\lambda_{m,ij}=\rho_{m,ij}\sqrt{\lambda_{m,ii}\lambda_{m,ij}} \tag{6}\]  
where \(\rho_{m,ij}\) is parametrically illustrated as the gap size index. Substituting Equation (6) into Equation (5) gives  
\[\lambda_{m}=\sum_{i}\sum_{j}\Psi_{i}\Psi_{j}\rho_{m,ij}\sqrt{\lambda_{m,ii} \lambda_{m,ij}} \tag{7}\]  
According to [PERSON] [34], the power spectral density can be written as  
\[G_{\rm{F}}(w)=\frac{w_{g}^{4}+4\zeta_{g}^{2}w_{g}^{2}w^{2}}{\left(w_{g}^{2}-w ^{2}\right)^{2}+4\zeta_{g}^{2}w_{g}^{2}w^{2}}C_{0} \tag{8}\]  
where \(G_{0}\), \(w_{g}\) and \(\zeta_{g}\) are described to be a scale factor, circular frequency and damping ratio, respectively. Therefore, a normal equation to determine the gap size index is suggested as [34]  
\[\rho_{op}=\frac{2\sqrt{\zeta_{i}\zeta_{j}}\Big{(}\big{(}w_{i}+w_{j}\big{)}^{2 }(\zeta_{i}+\zeta_{j})+\left(w_{i}^{2}-w_{j}^{2}\right)(\zeta_{i}-\zeta_{j}) \Big{)}}{4\big{(}w_{i}-w_{j}\big{)}^{2}+\left(w_{i}+w_{j}\right)^{2}(\zeta_{i }+\zeta_{j})^{2}} \tag{9}\]  
It should be underlined that the gap size index, \(\rho_{\rm{op}}\), expressed by Equation (9) depends only on the dynamic properties of adjacent structures (vibration periods of buildings and structural damping ratios) and it is independent from the seismic excitation used in the analysis. In the second stage, by using the CRVK program, all the required separation distances between two adjacent buildings were listed and the trend to predict the optimum separation distance was created. The lowest values of the distance, which prevent collisions of different models under various earthquakes (see Table 5), were used as inputs. Based on all inputs and the estimated trend of the solution, the effective equation to calculate the gap size index was determined using the process described in detail in Figure 4. It should be added that \(\alpha\), \(\beta\) and \(\sigma\) shown in the figure are the parameters of the iterative process which are considered by the CRVK program during the analysis. The obtained formula for the gap size index, \(\rho_{\text{op}}\), has the following form:  
\[\rho_{op}=\frac{(\zeta_{i}+\zeta_{j}\mu^{0.118})}{\mu^{1.47}+\zeta_{i}+\mu^{0.097 }(\zeta_{j}\mu^{0.383}-1)} \tag{10}\]  
where \(\mu\) is a relation between vibration periods of adjacent buildings, as defined by the formula \(\mu=\frac{T_{j}}{T_{i}}\rightarrow\left(T_{j}\geq T_{i}\right)\), and \(\zeta_{i}\) and \(\zeta_{j}\) denote the damping ratios of structures. It should be underlined that the gap size index, \(\rho_{\text{op}}\), expressed by Equation (10) depends only on the dynamic properties of adjacent structures and it is independent from the seismic excitation used in the analysis. In addition, Equation (10), as proposed by the authors in the present paper, shows the originality of the treatment described herein.