## 2 Methodology
### Analysis and Design of Archetypes
However, the following structure design criteria consider the ACI-318-08 [58] prescriptions in Section 8.8.2, which specify 50% of the gross section properties, which represent the effect of cracking under the effects of a seismic event. The seismic loads corresponded to structures located in seismic zone 3, with a characteristic seismic hazard level of 0.4 g, and soil classification of type C, as specified in the Chilean code. The main results from the modal response spectrum analysis conducted for each archetype and for each analysis direction are summarized as follows. The modal spectral analysis considered a design spectrum prescribed by the NCh433 [55] section SS6.3.5.1, reduced to obtain the inelastic spectral acceleration (\(S_{a}\)), by a response reduction factor \(R^{*}\), which was calculated according to the section of SS6.3.5.3, both dependent on the structure fundamental period and soil properties. This is illustrated in Figure 5. A total of 11 modes were considered to comply with the 90% percentage of participating mass for the seismic action in both directions, as required by NCh433 [55] section SS6.3.3. The first five modes are shown in Table 3. These modes were then combined under CQC to obtain the maximum values in each direction; however, it is important to note that the standard does not prescribe the combination of direction effects in the seismic action. Subsequently, the standard limits for the minimum (\(Q_{min}\)) and maximum (\(Q_{max}\)) values of base shear, as prescribed in section SS6.3.7, were not required to be adjusted for the structure under study. The design base shear (\(Q_{design}\)) considered the response reduction factor and the standard limits. Another parameter of interest was the design roof displacement (\(\delta_{u}\)), which was calculated as 1.3 times the displacement spectra (\(S_{de}\)), as presented in the Supreme Decree DS61 [59; 60], considering the stiffness modifier under the effects of cracking. Finally, the structure design was compliant with the NCh433 prescriptions SS5.9.2 and SS5.9.3, which limit the center-of-mass (CM) interstory drift to a value of 0.2% and the maximum interstory drift to a value of 0.1% (relative to the CM), as shown in Table 4. This is required by the Chilean standard to ensure elevation and plan regularity. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Building** & \multicolumn{2}{c}{\(\mathbf{0}^{\circ}\)} & \multicolumn{2}{c}{\(\mathbf{15}^{\circ}\)} & \multicolumn{2}{c}{\(\mathbf{30}^{\circ}\)} & \multicolumn{2}{c}{\(\mathbf{45}^{\circ}\)} \\
**Topology** & & & & & & & & \\ \hline
**Mode** & **Period [s]** & **Shape** & **Period [s]** & **Shape** & **Period [s]** & **Shape** & **Period [s]** & **Shape** \\ \hline
1 & 0.57 & Translational X & 0.54 & Translational X & 0.48 & Translational X & 0.40 & Translational X \\
2 & 0.36 & Translational Y & 0.34 & Translational Y & 0.30 & Translational Y & 0.25 & Translational Y \\
3 & 0.35 & Translational X & 0.33 & Translational X & 0.29 & Translational X & 0.25 & Translational X \\
4 & 0.14 & Translational X & 0.13 & Translational X & 0.11 & Translational X & 0.10 & Translational X \\
5 & 0.09 & Translational Y & 0.08 & Translational Y & 0.07 & Translational Y & 0.08 & Translational Y \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modal analysis results. It is worth noting that the structural layout was developed with the objective of achieving a regular seismic response, both in elevation and in plan, in accordance with the provisions of the Chilean design code, which establishes limits on interstory drifts. In this context, the code requires that the interstory drift at the center of mass (\(CM\)) and the maximum interstory drift be limited to 0.2% and 0.1%, respectively. Regarding the design of the elements comprising the seismic force-resisting system, the walls were designed as special shear walls, as defined in Chilean Supreme Decrees DS60 and DS61 [59; 60], with the former being an adaptation of ACI 318-08 [58].