## 6 Investigation on the Accuracy of Proposed Formula  
In order to investigate the accuracy of the proposed formula for the gap size index, \(\rho_{\text{op}}\), defined in Equation (10), four different configurations were considered. In the case of the first one, two specific models of three-story buildings were analyzed: one with a lumped mass, stiffness and damping ratio of each story equal to 7500 kg, 4.65\(\cdot\)10\({}^{6}\) N/m and 0.05, respectively; the second one with a lumped mass, stiffness and damping ratio of each story equal to 15,700 kg, 1.2\(\cdot\)10\({}^{8}\) N/m and 0.05, respectively. Vibration periods of buildings were calculated as equal to 0.56 and 0.16 s, for the first and second models, respectively. Both structures have been exposed to different ground motions; however, the most representative results obtained for the El Centro earthquake excitation are shown herein. The maximum lateral displacement under this earthquake was determined as equal to 7.96 cm for the first model and 0.6 cm for the second model. The separation distance was calculated using Equations (2) and (10) as equal to 7.98 cm. The top story displacement time histories, separated by this gap size, are shown in Figure 8. It can be seen from the figure that the structures do not collide during the whole time of the seismic excitation. Then, a five-story building was considered to be located next to another two-story structure. Both models were analyzed for two different scenarios so as to verify the accuracy of the proposed Equation (10). Firstly, it was assumed that the properties of each story of both buildings are the same. The lumped mass, stiffness and damping ratio of all stories for both models were defined as equal to 25,000 kg, 3.46 \(\times\) 10\({}^{6}\) N/m and 0.05, respectively. The building vibration period was determined as equal to 0.86 s for the two-story model and 1.87 s for the five-story structure. Both structures have been exposed to different ground motions; however, the most representative results obtained for the Parkfield earthquake excitation are shown herein. The maximum lateral displacement under this earthquake was determined as equal to 9.17 cm for the two-story building and 23.45 cm for the five-story structure. The required separation distance was calculated using Equations (2) and (10) as equal to 36.14 cm. The story displacement time histories at the level of the second story, separated by this gap size, are shown in Figure 9. As it can be seen from the figure, an acceptable safety separation distance is provided between the two buildings analyzed. Figure 8: The lateral displacement time histories of buildings separated by the seismic gap of 7.98 cm. The third stage of the investigation was focused on adjacent two-story and five-story buildings with substantially different values of the story mass and stiffness. The following structural properties were considered:  
\(m_{1}^{2s}=m_{2}^{2s}=25,000\) kg \(m_{1}^{5s}=m_{2}^{5s}=m_{3}^{5s}=m_{4}^{5s}=m_{5}^{5s}=1,000,000\) kg \(k_{1}^{2s}=k_{2}^{2s}=3.46\times 10^{6}\) N/ m \(k_{1}^{5s}=k_{2}^{5s}=k_{3}^{5s}=k_{4}^{5s}=2.215\times 10^{9}\) N/m \(\zeta_{1}^{2s}=\zeta_{2}^{2s}=0.05\) \(\zeta_{1}^{5s}=\zeta_{2}^{5s}=\zeta_{3}^{5s}=\zeta_{4}^{5s}=5s=0.05\)  
\(T^{2s}=0.86\) s \(T^{5s}=0.46\) s  
where \(m_{i}^{js}\), \(k_{i}^{js}\) and \(\zeta_{i}^{js}\) are the lumped mass, stiffness and damping ratio of the \(i\)-th story of a \(j\)-story building, respectively, whereas \(T^{js}\) denotes the natural vibration period of a \(j\)-story structure. The maximum lateral displacement under the Parkfield earthquake was determined as equal to 9.17 cm for the two-story building and 7.08 cm for the five-story structure. The required separation distance was calculated using Equations (2) and (10) as equal to 11.47 cm. The story displacement time histories at the level of the second story, separated by this gap size, are shown in Figure 10. It can be seen from the figure that the structures do not collide during the whole time of the seismic excitation.