## 4 Numerical Study  
Numerical analysis was conducted for five different earthquake records, i.e., for the El Centro (1940), Parkfield (1966), San Fernando (1971), Kobe (1995) and Kocaeli (1999) earthquakes--see Table 2. It was assumed in the analysis that the direction of ground vibrations during each earthquake (see components in Table 2) is convergent with the longitudinal direction defined for adjacent buildings. It should be added that the influence of the direction of ground vibrations can be taken into account by considering two horizontal components of each earthquake (usually NS and EW) and defining different layout angles for adjacent structures in relation to these horizontal directions. In the first stage of the investigation, all five structural models were analyzed individually. As the result of this analysis, the top story maximum lateral displacement for each model was determined under various ground motions (see Table 3). It can be seen from Table 3 that the largest maximum lateral displacement (equal to 36.18 cm) was observed for model B\({}_{1}\) under the Kocaeli earthquake record, while the lowest value (equal to 2.95 cm) was determined in the case of model B\({}_{4}\) under the San Fernando earthquake. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Name**} & **Stiffness of Stories** & **Mass of Stories** & **Building Vibration Period** \\  & **(N/m)** & **(kg)** & **(s)** \\ \hline B\({}_{1}\) & \(4.5\times 10^{6}\) & 65,000 & 1.69 \\ B\({}_{2}\) & \(3.46\times 10^{6}\) & 25,000 & 1.20 \\ B\({}_{3}\) & \(1.1\times 10^{8}\) & 315,000 & 0.75 \\ B\({}_{4}\) & \(5.2\times 10^{7}\) & 130,000 & 0.70 \\ B\({}_{5}\) & \(8.5\times 10^{7}\) & 114,000 & 0.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of five different models. Figure 2: Schematic models of two adjacent three-story buildings. Then, Equation (1) was used to calculate the required separation distance between two models. The results of these calculations are presented in Table 4. As it can be seen from the table, the largest required separation distance (equal to 51.17 cm) was obtained for two adjacent buildings B1 exposed to the Kocaeli earthquake, while the lowest value (equal to 4.17 cm) was calculated for two adjacent buildings B4 under the San Fernando earthquake. It is quite obvious, however, that the optimal separation distance between two identical structures should be equal to zero, since they will always vibrate in-phase during the same ground motion. Therefore, in the next stage of the investigation, all combinations between two adjacent B1 and Bj models were analyzed under different earthquakes using the CRVK program, and the separation distance between them was slowly decreased by 0.01 cm. When the first impact was observed, the previous separation distance was automatically selected as the lowest possible value of the distance between buildings so as to avoid bounding during a particular ground motion. The separation distance values estimated in this way are summarized in Table 5. Moreover, for better interpretation, the results of the analysis are also presented in Figure 3. The results clearly indicate that Equation (1) may substantially overestimate the in-between gap size in some cases, since it does not take into account the relation between vibration periods of adjacent buildings. Therefore, Equation (2), with an appropriately defined gap size index, \(\rho_{\text{op}}\), should be used instead.