## 2 Proposed Equations
Then, the curve defined by the proposed equation has been fitted into the data set of actual values using the method of least squares. The difference between the actual results and the results based on the proposed equation has been assessed by calculating the normalized root mean square (RMS) error presented in Equation (9) (see [82]):  
\[\text{RMS}=\frac{\sqrt{\sum\limits_{i=1}^{NV}\left(H_{i}-\overline{H}_{i} \right)^{2}}}{\sqrt{\sum\limits_{i=1}^{NV}H_{i}^{2}}}\times 100\% \tag{9}\]  
where \(H_{i}\), \(\overline{H}_{i}\) are the actual value and the value obtained by using the proposed equation, respectively, and \(NV\) denotes the number of values in the data set. Several techniques have been followed for fitting the curves, including equations with different types: polynomial, power, linear, logarithmic, and exponential. The chosen equation is the one that leads to the lowest percentage of error that is reported in the paper. The whole procedure has been performed for five soil types, A, B, C, D, and E, defined in the ASCE 7-10 code [69] (see Table 3). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Earthquake** & **Magnitude** & **PGA (g)** & **Station** & **Year** \\ \hline Kobe & 6.9 & 0.27577 & Kobe University & 1995 \\ \hline Parkfield & 6.19 & 0.01175 & San Luis Obispo & 1966 \\ \hline San Fernando & 6.61 & 0.02576 & 2516 Via Tejon PV & 1971 \\ \hline Loma Prieta & 6.93 & 0.07871 & APEEL 3E Hayward CSUH & 1989 \\ \hline Imperial Valley & 6.53 & 0.28726 & Agrarias & 1979 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Earthquake records used in the study. \begin{table}
\begin{tabular}{c c} \hline \hline
**Site Class Description** & **Site Class Definition** \\ \hline A & Hard rock \\ \hline B & Rock \\ \hline C & Very dense soil and soft rock \\ \hline D & Stiff soil \\ \hline E & Soft clay soil \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of the site classes. The soil type/site class has been considered by defining the response spectrum in ETABS software, and then by matching the earthquake records (defined in Table 2) with the target response spectrum. In the definition of the response spectrum, several parameters are required to be defined, which are the site class and the site properties. In this article, the value of \(S_{s}\) (mapped risk-targeted maximum considered earthquake spectral response acceleration parameter at short period) has been considered to be equal to 1.25, the value of \(S_{1}\) (mapped risk-targeted maximum considered earthquake spectral response acceleration parameter at 1 s period) has been considered to be equal to 0.5, and the value of \(T_{L}\) (long transition long period) has been taken as equal to 8 s (see [66; 67] for details). These values have been taken into account based on the studies conducted by [PERSON] and [PERSON] [66; 67], involving extensive analysis on several scaling parameters, with the conclusion that these values lead to the highest and most significant responses. After defining these parameters, the response spectrum has been defined and the five ground motions have been scaled to the target response spectrum. The structural response has been obtained by applying the fast nonlinear analysis (FNA) method developed by [PERSON] and [PERSON] [83]. [PERSON] and [PERSON] [84] conducted a pilot test using the direct integration, by [PERSON] (1959) and FNA methods. The test was conducted for the top floor relative displacement time histories for the no-pounding case. The results revealed a very good agreement between the two methods. Because of that, and considering that the FNA method consumes much less time unlike the Newmark method (which requires long period of time), the FNA method has been used in this study. In this method, the nonlinearity is considered for the gap and support elements while the linearity is considered for other elements. The dynamic equilibrium equation of the vibrating structure based on this method is shown in Equation (10).