## 4 Results and Discussion
### Maximum Story Shear Forces  
Figure 8(a)-c shows the maximum story shear forces endured by prototypes N3, N6 and N9, respectively, together with the prediction of shear forces calculated using Equations (20) and (21). The results indicate that Equation (21) provides satisfactory (upper bound) maximum shear forces in the upper stories. Figure 8: Residual drift ratios for prototypes N3 (**a**), N6 (**b**) and N9 (**c**). Proposal of \(n_{eq}\) for RC Frames with a Soft First Story and Dampers under Near-Fault Earthquakes  
The maximum drift of the first story predicted with Equation (19) using the values of \(n_{eq}\) proposed by [PERSON] in Equation (12) are shown with vertical dashed lines in Figure 6. In general, they are seen to be underestimated. More precisely, the prediction approaches the mean value obtained with the time history analyses for near-fault earthquakes but substantially underestimates the mean plus one SD. The reason is that Equation (12) was proposed for conventional structures subjected to general ground motions, that is, not specifically near-fault earthquakes; therefore, it is not appropriate for impulsive earthquakes. Near-fault ground motions typically concentrate the energy demand in one or a few pulses [10]. In terms of the number of equivalent cycles, this means that the structure has to dissipate the same amount of energy input with a smaller number of cycles of greater amplitude when compared to far-field earthquakes. It is worth emphasising that the reason behind the discrepancy between the maximum first-story drifts obtained with the time history analyses and the prediction given by Equation (19) with \(n_{eq}\) calculated using Equation (12) lies in Equation (12) itself, not the formulae given by Equation (19). To justify this assertion, first, the actual \(n_{eq}\) obtained for each prototype and for each ground motion was calculated, being referred to as \(n_{eq,NTH}\). Next, the maximum first-story drift was recalculated using Equation (19) using this \(n_{eq,NTH}\) and is referred to as \(\delta_{max1,prediction}\) hereafter. In Figure 10, \(\delta_{max1,prediction}\) is plotted against the actual maximum first-story drift obtained for each ground motion with the NTH analyses, \(\delta_{max1,NTH}\). As can be seen in Figure 10, all points lie very close to the 45\({}^{\circ}\) line that corresponds to \(\delta_{max1,prediction}=\delta_{max1,NTH}\). In other words, Figure 10 shows that if the  
Figure 9: Maximum shear forces for prototypes N3 (**a**), N6 (**b**) and N9 (**c**). actual value of \(n_{eq}\) (=\(n_{eq,NTH}\)) corresponding to each prototype and to each ground motion is used in Equation (19), then the maximum displacement predicted with Equation (19) \(\delta_{max1,prediction}\) (vertical axis of Figure 10) is very close to the maximum displacement obtained from the time history analyss \(\delta_{max1,NTH}\) (horizontal axis of Figure 10). Here, the actual value of \(n_{eq}\) (=\(n_{eq,NTH}\)) means the ratio between the plastic strain energy dissipated by the damper \(W_{p}\) and \({}_{s}\delta_{yls}\delta_{y1}\), as defined in Section 2.4 by Equation (11). Therefore, the maximum first-story drift in the event of near-fault ground motions can be well predicted with Equation (19) if an appropriate value is adopted for \(n_{Equation}\). For RC frames with a first soft story retrofitted with hysteretic dampers, and based on the results of this study, the following new expression is proposed for \(n_{eq}\):  
\[n_{eq}=\left\{ \begin{array}{c}2+2r_{q1}\;\forall\;r_{q1}<1\\ 4\;\forall\;r_{q1}>1\end{array} \right. \tag{22}\]  
The above expression was obtained by multiplying Equation (12) by a factor so that the maximum displacement \(\delta_{max1}\) predicted with Equation (19) with the \(n_{eq}\) given by the new Equation (22) surpasses the 90 th percentile of the responses obtained from the time history analyses. This percentile is above the 84 th percentile that is commonly considered to provide the seismic design with an appropriate level of confidence [34].