## 4 Conclusions  
In this paper, new equations for the evaluation of the optimum seismic gap preventing earthquake-induced bounding between adjacent buildings founded on different soil conditions have been proposed. The DDC formula has been taken into consideration and the modification of the correlation factor has been introduced. In the study, 1260 cases of bounding between different concrete buildings with various dynamic properties have been considered under five ground motions. Five soil types have been taken into account, as defined in the ASCE 7-10 code [69], i.e., hard rock, rock, very dense soil and soft rock, stiff soil, and soft clay soil. The normalized RMS errors have been calculated for the proposed equations. The results of the study indicate that the error ranges between 2% and 14%, confirming the accuracy of the approach. Therefore, the proposed equations can be effectively used for the determination of the optimum seismic gap preventing earthquake-induced bounding between buildings founded on different soil types. The current equations existing in the literature consider only one equation for the separation distance regardless of the soil type, which means that the same equation is supposed to be valid no matter the soil type. However, the proposed equations are multiple equations where every equation corresponds to a certain soil type, which means that the selection of the equation to be used is dependent on the soil type that the buildings are founded on. By designing the buildings and providing separation between them based on the proposed equations, no collisions will occur between them, which means that providing spacing based on the proposed equations will provide more safety to the vibrating buildings during earthquakes. Since only concrete buildings are considered in this study, it can be said that the accuracy of the formulas is verified for concrete buildings and for concrete-to-concrete bounding. The accuracy of these formulas has to be investigated for other kinds of bounding, such as steel-to-steel bounding and timber-to- timber bounding, and compared with the formulas existing in the literature. The current formulas existing in the literature can be less or more accurate than the proposed equations in other kinds of bounding, such as steel-to-steel bounding and timber-to- timber bounding. Since the equations are based on natural periods, they should also be valid for steel and timber structures. However, further checking and verification  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Building 1** & **Building 2** & **Soil Type** & **U1 (mm)** & **U2 (mm)** & **Naderpour** & **Required** & **Ratio** \\
**Number** & **Number** & **Soil Type** & **U1 (mm)** & **U2 (mm)** & **Naderpour** & **Required** & **Required** & **Ratio** \\ \hline
21 & 22 & A & 2.6 & 11.4 & 11.4 & 4.1 & 2.81 \\ \hline
21 & 23 & B & 7.8 & 54.2 & 60.  
are necessary. Also, checking the accuracy of the proposed equations compared with the current formulas existing in the literature in the case of direct integration nonlinear analyses rather than fast nonlinear analyses (FNAs) is required. Finally, an experimental verification of the proposed equations is necessary. The experimental verification can be completed using the shaking table by simulating two models placed with a separation distance based on the proposed equations and checking if bounding occurs or not.  
M.M.: Conceptualization, Methodology, Formal analysis, Investigation, Writing--Original Draft. R.J.: Supervision, Writing--Review & Editing. All authors have read and agreed to the published version of the manuscript.  
The first author ([PERSON]) gratefully acknowledges the financial support of this research from the \"Doctoral Scholarship\" awarded from Gdansk University of Technology.  
Not applicable.  
All the data are included in the article.  
The authors declare no conflict of interest.