## 4 Results and Discussion
### Results of Nonlinear Static (Pushover) Analysis  
From the results of the nonlinear pushover analysis, two key parameters can be highlighted: the displacement ductility and the overstrength, both calculated from the capacity curve of each archetype in the two orthogonal directions. The displacement ductility was objectively determined using the area equivalence method proposed by [PERSON] [73], applied to the segment of the capacity curve up to the point of maximum base shear. Additionally, the capacity curves were normalized with respect to the seismic weight and the height of each archetype in order to enable a direct comparison between them (see Figure 13) [74]. From the capacity curves, it was possible to determine several relevant coefficients that provided insight into the seismic performance of the structure. These included the displacement ductility \(\mu\) and the overstrength factor \(\Omega_{0}\), which were calculated according to the following equations:  
\[\mu=\frac{\delta_{u}}{\delta_{y}} \tag{6}\]  
\[\Omega_{0}=\frac{V_{max}}{V_{design}} \tag{7}\]  
The values obtained from the nonlinear pushover analysis for the displacement ductility and overstrength are summarized in Tables 8 and 9, for the x- and y-directions, respectively. The displacement ductility values revealed a marked difference between the x-direction (longitudinal) and the y-direction (transverse) responses of the building. This difference was largely attributed to the geometric adaptation of the structural configuration to the slope, which more significantly affected the transverse direction. Nonetheless, the overall displacement ductility behavior exhibited trends consistent with those reported in previous nonlinear analyses of buildings with similar structural typologies, particularly shear wall systems. Comparable ductility patterns have been identified in studies that applied performance-based methodologies and nonlinear time-history analyses to reinforced concrete buildings with complex wall configurations [63; 75; 76]. Using the calculated overstrength values for each performance group (x- and y-directions), the average values were determined to be 6.5 and 6.2 for the x- and y-directions, respectively. To proceed with the methodology, it was essential to calculate the period-based ductility. This calculation was carried out by defining the global yield displacement of the structural system, based on the following equation:  
\[\delta_{y,\;eff}=C_{0}\frac{V_{max}}{W}\Big{[}\frac{g}{4\pi^{2}}\Big{]}\left( max(T,T_{1})\right)^{2} \tag{8}\]  
where \(C_{0}\) is a coefficient that correlates the roof displacement of a single-degree-of-freedom (SDOF) system, \(\frac{V_{max}}{W}\) is the ratio of the maximum base shear to the seismic weight, \(g\) is the acceleration due to gravity, \(T\) is the fundamental period calculated using the prescribed expression, and \(T_{1}\) is the first-mode period obtained from an eigenvalue analysis. Figure 14 presents the period-based ductility values for each archetype, calculated in both analysis directions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Archetype** & \(\delta_{y}\) & \(\delta_{u}\) & \(\mu\) & \(V_{design}\) & \(V_{max}\) & \(\Omega_{0}\) \\ \hline
0\({}^{\circ}\) & 0.516 & 0.919 & 1.780 & 0.087 & 0.424 & 4.893 \\
15\({}^{\circ}\) & 0.356 & 0.787 & 2.214 & 0.088 & 0.626 & 7.110 \\
30\({}^{\circ}\) & 0.310 & 0.700 & 2.261 & 0.100 & 0.707 & 7.102 \\
45\({}^{\circ}\) & 0.206 & 0.540 & 2.627 & 0.109 & 0.741 & 6.788 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Overstrength and displacement ductility values obtained from the pushover analysis in the x-direction.