# Influence of Directionality on the Seismic Response of Typical RC Buildings
### [PERSON] and [PERSON]  
The authors propose addressing the problem of directionality through modal response spectrum analysis, solving five load cases [21]. The proposed method consists of the following:  
* Solving the five spectral modal load cases to obtain the response parameters. * Using Equation (3) to obtain the critical angle of incidence. Using Equation (2) to obtain the maximum responses for each critical angle of incidence. It should be noted that each parameter can have a different critical angle, where \(R\) is the peak response of the chosen parameter, \(C\) is the participation coefficient, and \(\theta_{crit}\) is the critical incidence angle, calculated using the following equations:  
\[ \begin{gathered} R(\theta)=\Big{\{}\Big{[}\big{(}R^{1x}\big{)}^{2 }+\big{(}R^{2y}\big{)}^{2}\Big{]}cos^{2}(\theta)+\Big{[}\big{(}R^{1y}\big{)}^{2 }+\big{(}R^{2x}\big{)}^{2}\Big{]}sin^{2}(\theta)+2 sin\ (\theta)\\ cos(\theta)\bigg{[}\sum\limits_{i}\sum\limits_{j}C_{ij}R_{i}^{1x}R_{ j}^{1y}-\sum\limits_{i}\sum\limits_{j}C_{ij}R_{i}^{2y}R_{j}^{2x}\bigg{]}+ \big{(}R^{3}\big{)}^{2}\Bigg{\}}^{1/2}\\ \theta_{crit}=\frac{1}{2}tan^{-1}\left(\frac{2\sum_{i}\sum_{j}C_{ ij}\Big{[}R_{i}^{2y}R_{j}^{2x}-R_{i}^{1x}R_{j}^{1y}\Big{]}}{\big{(}R^{1y}\big{)}^{2 }+\big{(}R^{2y}\big{)}^{2}-\big{(}R^{1x}\big{)}^{2}-\big{(}R^{2y}\big{)}^{2}} \right)\end{gathered}  \tag{2}\]  
In Equations (2) and (3), \(R_{j}^{2x}\) and \(R_{i}^{2y}\) are the peak modal responses obtained along the structural reference axes \(x\) and \(y\), respectively. Finally, \(R^{3}\) is the peak modal response in the vertical axis \(z\). ### [PERSON] and [PERSON]  
The authors propose the use of the CQC3 combination as an alternative to the multi-component combination (instead of the use of SRSS, CQC, and rules of 30% or 40%) [27]. Additionally, a formula is proposed for the calculation of the critical angle, where \(k\) and \(l\) represent the axes and the seismic components considered, \(\alpha\) is the coefficient of participation related between the modes and the indicated direction, \(\rho\) corresponds to the correlation coefficient between modes, and \(\gamma\) is the ratio between the design spectra used. Finally, \(S_{ki}\) correspond to the kth principal axis of the ground motion. The equations are presented below:  
\[R_{k}^{2}=\sum\limits_{i}\sum\limits_{j}\rho_{ij}\alpha_{kj}\alpha_{ki}S_{ki}S_ {kj} \tag{4}\]  
\[R_{kl}^{2}=\sum\limits_{i}\sum\limits_{j}\rho_{ij}\alpha_{lj}\alpha_{ki}S_{ki} S_{lj} \tag{5}\]  
\[\theta_{\sigma}=\frac{1}{2}tan^{-1}[\frac{\frac{2}{7}R_{12}}{R_{1}^{2}-\frac{1 }{\gamma^{2}}R_{2}^{2}}] \tag{6}\]  
### [PERSON] and [PERSON]  
This method is based on previously published considerations presented by [PERSON] and [PERSON] [24], which are used by means of the response spectra when considering the existence of a major axis in the direction of the epicenter, intermediate transversal, and minor vertical, this being valid only when the effects of proximity to the fault are absent. The proposed seismic design is based on determining the critical orientation that provides the greatest response, on calculating maximum and minimum response values and on the application of the extreme stress or extreme force method to determine the most unfavorable combination of the resulting stresses acting on specific structural elements [22]. To determine the critical direction, the reference systems indicated in Figure 1 and the following equations should be used:  
\[\theta_{cr}=\left\{ \begin{array}{c}\theta_{0}=\frac{1}{2}tan^{-1}\Big{(}2 \frac{R_{xy}}{R_{x}^{2}-R_{y}^{2}}\Big{)}\\ \theta_{0}+\pi/2\end{array} \right.