## 4 Effect of Seismic Parameters on Vertical DRFs
### Effect of the Moment Magnitude  
In this section, the vertical response spectra for each earthquake in each moment magnitude bin are calculated at different damping ratios, as seen in Figure 5. Later, the DRF values based on the acceleration, velocity, and displacement spectra are compared and evaluated for different earthquake magnitude bins.  
Figure 6 shows the variation in the mean DRFs at damping ratios (\(\xi\)) equal to 1%, 3%, 10%, 15%, 20%, 30%, and 40% for the three moment magnitude bins (4.5 \(\leq\) M\({}_{\rm w}\)\(\leq\) 5.45, 5.5 \(\leq\) M\({}_{\rm w}\)\(\leq\) 6.46, and 6.5 \(\leq\) M\({}_{\rm w}\)\(\leq\) 7.9). It can be observed from Figure 6a-c that the effect of damping on the DRFa decreases at high magnitudes, especially for longer structural periods. It can be said that the variation in the DRFa values is almost the same for all the magnitude groups up to periods of 0.9 s, 0.7 s, 0.5 s, 0.4 s, and around 0.3 s at damping ratios of 10%, 15%, 20%, 30%, and 40%, respectively. After these periods, the DRFa values diverge, and it can be seen from the figures that the DRFa values for 4.5 \(\leq\) M\({}_{\rm w}\)\(\leq\) 5.45 and 5.5 \(\leq\) M\({}_{\rm w}\)\(\leq\) 6.46 are higher than those of M\({}_{\rm w}\)\(\geq\) 6.5. This situation is more pronounced with increasing damping and structural periods.  
Figure 4: Distribution of (**a**) site classes, (**b**) earthquake magnitudes, and (**c**) rupture distances of the considered vertical ground motion records in the present study.  
Figure 5: (**a–c**) Spectral acceleration spectra of selected earthquakes for different magnitude bins and \(\zeta\) = 0.05, (**d–f**) mean acceleration spectra calculated for different damping ratios.  
Figure 6d-f illustrates that the DRFd values are equal to unity for short periods (T \(<\) 0.02 s) and tend to diverge from unity with an increasing magnitude and longer periods. Figure 6g-i indicate that the DRFv curves are slightly different from the DRFa and DRFd curves, especially for short periods, and the DRFv values approach unity at T \(<\) 0.02 s. Then, the DRFv values diverge from unity with an increasing duration until 0.1 s. After a 1.0 s period, the DRFv values approach unity again, especially for low moment magnitudes M\({}_{\rm w}\)\(\leq\) 6.5.  
In Figure 7, the distributions of the DRFa, DRFv, and DRFd values for the three different magnitude bins are compared against the structural periods. It can be seen from Figure 7 that the DRFd and DRFv curves are almost identical and higher than unity for all the magnitude groups at low damping ratios (i.e., 1.0% and 3.0%). The DRFd and DRFv values are lower than unity at high damping ratios (\(\xi\) \(>\) 5%) and move away from unity with increasing damping ratios.  
Figure 6: The variation in the median DRFs for different damping ratios and moment magnitudes; (a-c) DRFa, (d-f) DRFd, and (g-i) DRFv.  
Figure 7: _Cont._  
Similar observations can be also made for the DRFa values, except for those corresponding the moderate-length to long structural periods. For example, the DRFa values are higher than unity at damping ratios of 1% and 3%, but these values may be below unity at low magnitudes (M\({}_{\rm w}\) < 6.5) and for long structural periods (T > 1.0 s). Also, the DRFa values are lower than unity at high damping ratios (\(\xi\) > 5%), but the DRFa values can be increased by 3 times or more for structural periods (T > 1.0 s). In general, at low damping ratios (i.e., 1.0% and 3.0), a greater M\({}_{\rm w}\) will result in greater DRFs. This situation is reversed at high damping ratios (\(\xi\) > 5%), and this trend has also been observed by various researchers [29, 32, 35, 48, 49].