## 4 Results and Discussion
### Results of Elastic Analysis  
As previously discussed in Section 2.1, the presence of a slope altered the geometry of the archetype structures. As the slope became steeper, its effect on the structure became increasingly significant. From a dynamic response perspective, this was reflected in a comparison of the fundamental periods of the archetypes founded on sloping ground with those of the archetype on flat terrain. Figure 10 shows the percentage reduction in the fundamental period for both analysis directions of each sloped archetype relative to the flat-ground archetype. It is evident that the slope had a stiffening effect on the structure, which increased with the slope inclination. Moreover, although the geometric effect of the slope was notably different in the direction parallel to the slope plane compared with the orthogonal direction, the variation in the fundamental periods was relatively uniform in both directions. This modification of the fundamental period has a significant impact on the design process, as the estimation of the base shear force used in design depends directly on the fundamental periods. The Chilean seismic code employs response reduction factors (\(R\)) that are period-dependent. Therefore, changes in the fundamental period result in a reduction in the \(R\) factor, as shown in Figure 11.  
Figure 9: Elastic acceleration design spectrum derived from the displacement spectrum, along with the response spectra from the ground motion records used and the resulting average spectrum. In this figure the grey lines represent the response spectra for each acceleration record used. The geometric modification of the structure directly affected its seismic weight, which, in turn, influenced the design base shear force. This relationship is illustrated in Figure 12.  
The structural periods of the archetypes decreased as the slope inclination increased. This reduction was primarily due to two factors. First, the archetypes became progressively stiffer as the structural axes closer to the hillside experienced a decrease in elevation, effectively reducing the total height of the structure along those axes. The most pronounced reduction was observed in the archetype located on a 45\({}^{\circ}\) slope, where axis F exhibited up to two fewer stories than axis A. Second, for the archetypes foun  
Figure 11: The percentage reduction in the response reduction factor relative to the 0\({}^{\circ}\) archetype. Figure 12: The percentage reduction in the seismic weight relative to the 0\({}^{\circ}\) archetype. Figure 10: The percentage reduction in the fundamental period relative to the 0\({}^{\circ}\) archetype. there was a progressive loss of structural elements, including portions of the floor slabs. This effect was particularly evident in the archetypes on slopes of 30\({}^{\circ}\) and 45\({}^{\circ}\). To validate the computed structural periods, expressions from ASCE 7-22 were used to estimate the approximate period of the archetype located on flat ground. It is important to note that these formulas do not include specific considerations for structural systems founded on sloped terrain. The general expression applied was  
\[T_{a}=C_{t}\ h_{n}^{x} \tag{3}\]  
where \(T_{a}\) is the approximate period of the structure, \(C_{t}\) and \(x\) are coefficients depending on the structural system, and \(h_{n}\) is the total height of the structure. Applying this equation yielded \(T_{a}=0.58\) s, which was very close to the fundamental vibration period computed for the archetype on flat ground. A second expression for estimating \(T_{a}\) is  
\[T_{a}=\frac{C_{q}}{\sqrt{C_{w}}}\ h_{n} \tag{4}\]  
where \(C_{q}\) is a coefficient specified according to the unit system used. In imperial units, \(C_{q}=0.0019\ ft\), while in SI units, \(C_{q}=0.00058\) m. Notably, this latter value produced results approximately one order of magnitude lower than expected when used directly with SI units.