## 1 Introduction  
Seismic excitations are often considered as the most dangerous and, at the same time, the most unpredictable loads that can act on civil engineering structures [1, 2, 3, 4]. It is a common situation that collisions between two adjacent structures occur during earthquakes [5, 6, 7]. This phenomenon, called structural bounding, appears when the relative structural displacement exceeds the separation distance between two adjacent buildings or bridge segments [8, 9, 10, 11, 12, 13]. Earthquake-induced structural bounding may lead to serious structural damages and may enlarge the number of casualties [14, 15, 16, 17, 18]. One of the methods that can be used to avoid such situations is filling the in-between gap by using viscoelastic materials (see [19, 20, 21]). Another approach is to increase the gap size so as to avoid structural interactions during earthquakes. A number of scientists have focused their studies on different methods devoted to estimation of the sufficient gap that provides the safety zone [22].  
In order to investigate building pounding, experimental investigations concerning collisions between models of structures, with real and unreal scales, were conducted. Further, extensive numerical analyses were carried out so as to verify different effects related to the phenomenon [9, 10, 23]. [PERSON] et al. [24] estimated impact velocity based on the coefficient of restitution and, subsequently, suggested an equation for the impact damping ratio to be used for modeling purposes. [PERSON] [25; 26] focused deeply on the analysis of the separation distance and suggested some parameters to prevent bounding between structures. [PERSON] [27] suggested a formula to calculate the minimum separation distance. [PERSON] et al. [28] proposed the spectral difference method based on random vibration theory that considers the first mode approximation for displacements of elastic multi-story buildings. [PERSON] et al. [29] improved the equation of the separation distance by adding the effect of the damping ratio. [PERSON] et al. [30] recommended calculating the effective building vibration period to be used in calculations. [PERSON] et al. [31] studied different mitigation measures so as to reduce negative effects of earthquake-induced structural bounding.  
Various researchers studied different methods of evaluating the separation distance for preventing structural collisions during seismic excitations. The sum of the squares of the modal response (SRSS) was often considered for these purposes [32]:  
\[S=\sqrt{\delta_{i}^{2}+\delta_{j}^{2}} \tag{1}\]  
where \(S\) is the separation distance between buildings, and \(\delta_{i}\) and \(\delta_{j}\) denote the peak lateral displacement of buildings \(i\) and \(j\), respectively. [PERSON] et al. [28] introduced a new equation, based on the SRRS formula, that can be presented as  
\[S=\sqrt{\delta_{i}^{2}+\delta_{j}^{2}-2\rho_{op}\delta_{i}\delta_{j}} \tag{2}\]  
where \(\rho_{\text{op}}\) is the gap size index (cross-correlation coefficient). It should be noted that the influence of the seismic excitation is defined in Equation (2) in the form of the peak lateral displacements \(\delta_{i}\) and \(\delta_{j}\) of both buildings, as obtained for a given earthquake. On the other hand, the gap size index, \(\rho_{\text{op}}\), is assumed to depend only on the dynamic properties of adjacent structures.  
The aim of this study is to propose an effective equation that can be used to calculate the gap size index \(\rho_{\text{op}}\) based on the building vibration period and damping ratio so as to provide the optimum separation distance that allows us to avoid structural bounding under different earthquakes.