## 2 Structural Model, Wind-Loading Model, and Analysis Procedure
### Wind-Loading Model  
For the development of the wind-loading model and to simplify the parametric study, it is considered that the resultant wind forces, denoted by \(F_{x}(t)\), \(F_{y}(t)\), and \(M(t)\), acting on the structural model can be modeled by using four uncorrelated wind loads (i.e., \(F_{1}(t)\), \(F_{2}(t)\), \(F_{3}(t)\), and \(F_{4}(t)\)) that are acting at different points of the structural model, as shown in Figure 2. The power spectral density (PSD) functions of the uncorrelated wind loads are given by Equation (6):  
\[S_{n}(f)= \begin{cases}Af^{-(1+\alpha)}&f_{L}\leq f\leq f_{u}\\ 0&\text{otherwise}\end{cases} , \tag{6}\]  
where \(f\) is frequency in Hz, \(f_{L}\) and \(f_{u}\) are upper and lower bounds of \(f\), \(\alpha\) is an exponent that depends on the direction of the wind velocity, and \(A\) is a normalization constant such that the integration of the PSD function equals one. This PSD function has been previously used in the studies [13; 14], since boundary layer wind tunnel test results indicate that such a PSD function provides sufficiently accurate characterization of the fluctuating wind force in the inertial subrange, which is of interest in the present study. Based on the uncorrelated wind forces shown in Figure 2, the resultant forces \(F_{x}(t)\) and \(F_{y}(t)\) and the torsional moment with respect to the origin \(M(t)\) are given by  
\[F_{x}(t)=F_{3}(t)+F_{4}(t), \tag{7a}\] \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\] (7b) \[M(t)=F_{2}(t)\cdot L_{x}+F_{4}(t)\cdot L_{y}, \tag{7c}\]  
where \(L_{x}\) and \(L_{y}\) are distances between the lateral forces \(F_{1}\) and \(F_{2}\), and \(F_{3}\) and \(F_{4}\), respectively. Several studies have shown that the resultant wind forces (i.e., \(F_{x}(t)\), \(F_{y}(t)\), and M(\(t))) present a certain degree of correlation. For example, Refs. [15; 16; 17], based on wind tunnel experiments, showed that there is low correlation between the maximum values of the along-wind force and the across-wind force, high correlation between the maximum values of the along-wind force and the torsional moment, and very low correlation between the maximum values of the across-wind force and the torsional moment. If the correlation  
Figure 2: Representation of the uncorrelated wind forces over the structural system. between \(F_{x}(t)\) and \(M(t)\), \(\rho_{xM}\), and between \(F_{y}(t)\) and \(M(t)\), \(\rho_{yM}\), is of interest, Equation (7) can be rewritten as  
\[F_{x}(t)=c_{1}\cdot\sqrt{(1-\rho_{xM}{}^{2})-\left[\rho_{yM}\cdot\rho_{xM}\cdot r _{\sigma}\right]\cdot r_{\epsilon}}\cdot F_{3}(t)+c_{1}\cdot\sqrt{\rho_{xM}{}^ {2}+\left[\rho_{yM}\cdot\rho_{xM}\cdot r_{\sigma}\right]\cdot r_{\epsilon}} \cdot F_{4}(t),\] (8a) \[F_{y}(t)=c_{2}\cdot\sqrt{\left(1-\rho_{yM}{}^{2}\right)-\left[\rho_{yM}\cdot \rho_{xM}\cdot\frac{1}{r_{\sigma}}\right]\cdot\frac{1}{r_{\epsilon}}}\cdot F _{1}(t)+c_{2}\cdot\sqrt{\rho_{yM}{}^{2}+\left[\rho_{yM}\cdot\rho_{xM}\cdot \frac{1}{r_{\sigma}}\right]\cdot\frac{1}{r_{\epsilon}}}\cdot F_{2}(t),\] (8b) \[M(t)=c_{1}\cdot\sqrt{\rho_{xM}{}^{2}+\left[\rho_{yM}\cdot\rho_{xM}\cdot r_{ \sigma}\right]\cdot r_{\epsilon}}\cdot F_{4}(t)\cdot L_{y}+c_{2}\cdot\sqrt{ \rho_{yM}{}^{2}+\left[\rho_{yM}\cdot\rho_{xM}\cdot\frac{1}{r_{\sigma}}\right] \cdot\frac{1}{r_{\epsilon}}}\cdot F_{2}(t)\cdot L_{x},\] (8c) where \[c_{1}\] and \[c_{2}\] are model parameters; \[r_{\sigma}\] is the ratio of the parameters \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\]0 and \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\]1 ; and \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\]2 is the ratio between the lengths \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\]3 and \[F_{y}(t)=F_{1}(t)+F_{2}(t)\,\]4. Details about the development of Equation (8) are given in Appendix A. To evaluate \(F_{x}(t)\), \(F_{y}(t)\), and \(M(t)\), wind forces \(F_{i}(t)\) (\(i=1,\ \ldots,\ 4\)) are first simulated by using the spectral representation method (SRM) [18]. It is noted that more sophisticated simulation methods can also be used [19]; however, for simplicity, the characterization of the wind forces adopted in this study followed a practical approach.