## 2 Proposed Equations  
In the process of modification of the correlation factor, 1260 bounding cases have been taken into account. In the study, 60 three-dimensional numerical models of concrete buildings have been considered. Table 1 presents the number of storeys, natural period, and frequency for each building. It should be noted that the natural period has been evaluated in the direction of possible bounding. Among these 60 numerical models, 1260 bounding cases have been considered. Table A1 in Appendix A presents a detailed description of these 1260 bounding cases, including the colliding buildings of every case as well as the period ratio between them. To generalize the proposed equations, the authors intended to consider multiple cases with varied situations and scenarios. The authors considered low-rise, mid-rise, and high-rise buildings (from 1-storey buildings up to 20-storey buildings). The torsional bounding was taken into account as well. All the cases have been studied considering five earthquake excitations. Considering the combination of all these factors has led to the generalizability of the proposed equations. These cases have concerned collisions between concrete buildings with different dynamic properties (see details in [66; 67; 72]), including different number of storeys (ranging from 1 storey up to 20 storeys). Among these buildings, 20 buildings have identical inertia in both directions x and y (ranging from 1 storey up to 20 storeys), 20 buildings have higher inertia in the x-direction (ranging from 1 storey up to 20 storeys), and 20 buildings have higher inertia in the y-direction (ranging from 1 storey up to 20 storeys). Figure 1 presents the plan views of the considered models. These buildings are reinforced concrete structures with a storey height of 3 m and with different lengths and widths. The shortest buildings were 3 m (one storey) and the highest buildings were 60 m (20 storeys). All the storey cases in between have been considered (buildings with 2 storeys, 3 storeys, 4 storeys, 5 storeys, up to 20 storey buildings). The properties of the material used in the models in this study are as follows: concrete with the compressive strength of 35 Mpa and the modulus of elasticity of 27.8 Gpa, steel (grade 60)with the yield strength of 420 Mpa and the modulus of elasticity of 200 Gpa. Indeed, the live load was taken to be equal to 4 kN/m\({}^{2}\) and the superimposed dead load was taken to be equal to 2 kN/m\({}^{2}\). These values were taken considering the frequent use of such materials in construction sites. However, it should be underlined that they will not affect the accuracy of the proposed equations since the authors considered a wide range of natural structural periods, which is considered to be the main factor influencing the earthquake-induced structural bounding. Moreover, the buildings were designed to satisfy the minimum reinforcement requirements based on the ACI code (American Concrete Institute). The ACI code (in Section 10.9.1) states that the reinforcement ratio should be between 1% and 8% of the concrete area. In this study, the reinforcement ratio in the columns has been taken as equal to 1% to ensure both optimum and economic design. Also, the columns and beams were defined as frame elements, while slabs were modelled as shell elements. In this study, the frame element uses a general three-dimensional beam-column formulation that involves the effects of biaxial bending, torsion, axial deformation, and biaxial shear deformations (see [77] for details). Several damping ratios have been considered in this study so that the proposed equations will be valid for all ranges of damping ratios. All the buildings considered in this study have been modelled and designed solely for this study. The criteria of modelling have been verified using the results of shaking table experimental study [71].