## 3 Vertical Ground Motion Database  
In this study, numerous real earthquake records, namely 775 vertical ground motion records, were selected from the PEER [47] strong motion database. The locations of the selected earthquakes are shown in Figure 2. A total of 171 earthquake events were considered to represent a wide range of site distances, average shear wave velocity (V\({}_{\rm S30}\)) intervals, and earthquake moment magnitudes (M\({}_{\rm w}\)). The moment magnitudes of the selected records range between 4.5 and 7.9, and their source-to-site distances (i.e., rupture distances R\({}_{\rm rup}\)) span up to 161.23 km. The site conditions of all the accelerograms are identified according to V\({}_{\rm S30}\) (the average shear wave velocity of the upper 30 m of the soil layer). The V\({}_{\rm S30}\) interval of the database is between 108.21 and 2016.13 m/s. Based on the \(V_{\rm S30}\) intervals, selected ground motions were classified as A, B, C, and D regarding the site classification method given in Eurocode 8. Soil type A is rock or very stiff (V\({}_{\rm S30}\)\(\geq\) 800 m/s), soil type B is very dense and or gravel or stiff clay (360 m/s \(\leq\) V\({}_{\rm S30}\) < 800 m/s), soil type C is dense or medium dense sand or gravel (180 \(\leq\) V\({}_{\rm S30}\) < 360 m/s), and soil type D is loose to medium cohesionless soil (V\({}_{\rm S30}\) < 180 m/s). The percentages of normal, reverse, and strike-slip events correspond to 21.05%, 30.4%, and 48.55% of the database, respectively.  
The distribution of certain seismic features of selected earthquakes is presented in Figure 3. The number of events corresponding to different M\({}_{\rm w}\), V\({}_{\rm S30}\), and R\({}_{\rm rup}\) bins is given in Figure 3a-c. The distributions of M\({}_{\rm w}\) versus R\({}_{\rm rup}\) and M\({}_{\rm w}\) versus PGA are plotted in Figures 3d and 3e, respectively. It can be observed that the considered earthquakes are well distributed with respect to M\({}_{\rm w}\), varying between 4.5 and 7.9. The PGA values of the selected strong motion records vary between 0.09 g and 0.35 g, which are also consistent with the minimum and maximum effective peak ground acceleration (EPGA) values, i.e., 0.05 g (corresponding to the design basis for earthquake hazards in areas of low seismicity) and 0.36 g (corresponding to the maximum considered earthquake hazard in high-seismicity areas), respectively, as recommended by several seismic design codes.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Equation** & **Code Source** & **Reference** \\ \hline \multirow{2}{*}{(1)} & \multirow{2}{*}{DRF = \(\sqrt{10/(5+\xi)}\)} & Eurocode 8 (2004) & [13] \\  & & Italian Code (2018) & [43] \\ \multirow{2}{*}{(2)} & DRF = \(150/(100+10\xi)\) & Japan JPN (2001) & [44] \\  & DRF = \(1+(5-\xi)/(6+1.4\xi)\) & China (2010) & [14] \\ \multirow{2}{*}{(4)} & DRF = \((5.6-\ln(\xi))/4\) & ASCE/SEI (2014) & [12] \\  & DRF = \((5/\xi)^{0.3}\) & AASHTO (2010) & [45] \\ \multirow{2}{*}{(6)} & DRF = \(\sqrt{7/(2+\xi)}\) & Algerian RPA99 (2003) & [46] \\ \hline \hline \end{tabular}  
**Note:**\(\xi\) should be taken as percentage, e.g., \(\xi\) = 10 for a damping ratio of 0.10.  
\end{table}
Table 1: Expressions for period-independent DRFs based on displacement response spectra defined in various design codes.  
Figure 1: Damping reduction factors (DRFs) according to formulas proposed in seismic design codes [12; 13; 14; 44; 45; 46].