# Influence of Directionality on the Seismic Response of Typical RC Buildings
## 3 Definition of Case Studies
Table 3 shows the periods of the first six vibration modes (obtained as a result of the dynamic analysis), specifying whether each mode is translational (along the \(x\) or \(y\) axis) or rotational (along the \(z\) axis). Figure 9 shows the design inelastic spectrum reduced by a regulatory response reduction factor of 11, and presents the vibration periods of the first modes in the \(x\) and \(y\) direction. Some of the most noteworthy findings are the special difference among the periods of Archetype 3 and the practically equal periods in both directions for Archetype 4. It should also be noted that some archetypes did not reach the minimum seismic coefficient in some directions (in which cases, the response reduction factors were adjusted to ensure compliance). The stiffness and torsional properties of the archetypes were calculated and are summarized in Table 4. The results show that there is a significant difference between the stiffness of Archetypes 1 and 3. Archetype 2 presents a less notable difference in stiffness than the previous one, while Archetype 4 presents practically the same stiffness in both directions. The imbalance in the rigidity will prove to be significant in determining the influence of the angle of incidence on the response of the archetypes. The calculated value of torsional stiffness, as expected, is significantly higher in Archetype 2 (whose structure is based on shear walls), and to a lesser extent in Archetype 3 (which has shear walls only on the right side of the plan). This irregular distribution is what produces the eccentricity of 8.7 m in this archetype. It is important to point out that Archetype 1 presents insufficient torsional rigidity, when verifying that \(r_{yc}<l_{s}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & \\ \hline \multicolumn{8}{c}{**Archetype**} \\ \hline  & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**3**} & \multicolumn{2}{c}{**4**} \\ \hline
**Mode** & **T(s)** & **Mode Type** & **T(s)** & **Mode Type** & **T(s)** & **Mode Type** & **T(s)** & **Mode Type** \\ \hline  & 1 & 0.239 & y & 0.103 & y & 0.17 & y & 0.348 & y \\  & 2 & 0.205 & z & 0.077 & x & 0.093 & x & 0.346 & x \\  & 3 & 0.116 & x & 0.057 & z & 0.074 & y & 0.283 & z \\  & 4 & 0.082 & y & 0.037 & y & 0.06 & y & 0.12 & y \\  & 5 & 0.07 & z & 0.032 & z & 0.042 & z & 0.119 & x \\  & 6 & 0.05 & y & 0.03 & x & 0.037 & z & 0.097 & z \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Periods of the first six modes of vibration of the archetypes. Figure 9: Inelastic design spectrum with the predominant structural periods in the directions of analysis. The nonlinear behavior of the archetypes was determined using fiber finite elements, with a structural damping equal to 5% of the critical damping. The structural members were incorporated into the model including the characteristics of the reinforcement, both longitudinal and transverse, previously designed according to current regulations in Chile for a high level of seismic hazard (0.4 g). In both the static and dynamic analyses, the P-\(\Delta\) effect was included. As a first approximation to the seismic response, the conventional pushover analysis was carried out in both directions for each archetype, whose capacity curves are summarized in Figure 10. The pushover analysis was carried out considering a distribution of lateral forces corresponding to the first vibration mode. This procedure, however, is not the most appropriate to determine the torsional behavior of the archetypes studied. The results of the pushover analysis show capacity curves in which it is possible to appreciate the difference in lateral stiffness of archetypes 1, 2, and 3 in the two directions of analysis. Note that the capacity curves were represented in a normalized way with respect to height and seismic weight, so that the response of the different archetypes can be compared.