## 5 A New Prediction Model for Vertical DRFs
### Comparison of the Models  
Several DRF prediction models have been suggested in the literature, as discussed in the introduction. However, the accuracy of these models is dependent on multiple factors, such as the database considered, the limitations of the analysis models, and the use of different vertical spectra. To illustrate the accuracy of the proposed model, some of the published models are used for comparison. Since the proposed vertical DRF models considered in this study are derived for the acceleration, displacement, and velocity spectra, existing predictive models given in Table 5 used for distinct vertical response spectra are considered. Figure 14 shows a comparison of the mean DRFa values of the proposed model with those of existing models from the literature at damping ratios of 2%, 3%, 10%, 20%, 30%, and 40%. It can be noted from the figure that the existing DRFa models may have different predictions at various damping ratios. According to the database of the Lin and Chang model, the effect of the damping ratio was also not obvious since all the DRF values were almost identical at a damping ratio lower than 5%, and the model was developed for T \(\geq\) 0.1 s. For this reason, [PERSON] and Chang's model completely differs from the other models when the damping ratio is less than 10%. Except for the [PERSON] and Chang model, the proposed model and the existing models yield similar results with relatively small differences at \(\xi\leq\) 10%. The predicted DRFa curves have a similar pattern when the damping ratio is higher than 10%. On the other hand, the predictions of the models separated and thus the differences between the models increased when the damping ratio was higher than 10%. The proposed model has the highest accuracy at all damping ratios and for all period ranges, compared to the existing models. In Figure 15, DRFd predictive models from various studies [40, 54, 55, 56] are compared with the developed model at damping ratios of 1%, 3%, 10%, 20%, 30%, and 40%. In general, all the models have similar trends for distinct period ranges. Due to its limited  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{1}{c}{**Equation**} & **Ref. ** & **Limit Condition** \\ \hline \multirow{5}{*}{\ \begin{tabular}{c} (1) \\ (2) \\ (3) \\ (4) \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} DRFa = \\ (3) \
umber of ground motion records and its consideration of the Algerian code in terms of the soil classification and the response spectrum [46], the Benahmed model differs from all the other models. The differences between the existing models change depending on the damping ratios and the period range. There is no apparent harmony between the existing models and the proposed model, especially at 1% damping. However, the proposed model has very good agreement with the mean DRFd ratios determined from the analysis. Except for at 1% damping, the Saez model has good compatibility with the proposed model for the DRFd. It is also obvious from the figures that the Hatzigeorgiou [40] model does not fulfill the theoretical boundary restriction that the DRF should approach one as the natural vibration period approaches zero or infinity, and this may cause disparity [33]. In addition, this model was improved to account for the influence of the soil conditions and ground motion type (i.e., near- or far-fault earthquakes) as well as the effect of the damping ratio. It is also  
Figure 14: Comparison of proposed DRFa with equations given in Table 5[32; 50; 53; 54] for different damping values. Figure 15: Comparison of proposed DRFd values with equations given in Table 5[40; 54; 55; 56] for different damping values. noteworthy that this model has relatively large errors when the natural vibration period is less than 0.5 s at all damping ratios.