## 3 Parametric Analysis and Results  
### Structural Characteristics and Peak Responses  
The analyses were carried out with six structural models, which represent actual structures approximated by 3 DOF systems whose characteristics are defined by the generalized properties of actual tall buildings. The dynamic and geometric characteristics of the structural systems are summarized in Table 1. The parameters \(L_{x}\) and \(L_{y}\) employed in the wind-loading model are also included in Table 1.  
For all the numerical analyses, the model parameters \(c_{1}\) and \(c_{2}\) were selected such that the mean peak acceleration was within acceleration thresholds of perception [3,7]. For the parametric analysis, the natural frequency in torsion (\(\omega_{\theta}\)) varied for each model as a function of the uncoupled torsional ratio (\(\Omega_{\theta}=\omega_{\theta}/\omega_{x}\)), which varied from 0.8 to 2.0 with increments equal to 0.1. The damping ratios for the sway modes (\(\zeta_{\mathrm{x}}\) and \(\zeta_{\mathrm{y}}\)) and torsional mode (\(\zeta_{\theta}\)) were set equal to 1%. To assess the influence of the correlation between \(F_{x}(t)\) and \(M(t)\), and \(F_{y}(t)\) and \(M(t)\) (i.e., \(\rho_{fxM}\) and \(\rho_{FyM}\)) on the mean peak responses, for the moment consider that the structural model 1 of Table 1 is subjected to wind loads calculated with Equation (8) for five different correlation pairs [i.e., (0, 1), (0, 0.25), (0.25, 0), (0.5, 0.5), and (0.75, 0.5)] and the structural response is calculated by solving Equation (5). To account for the uncertainty in the wind-induced response, the analyses were repeated 30 times to evaluate the mean peak responses. To show the impact of \(\rho_{fxM}\) and \(\rho_{FyM}\) on the mean peak responses, Figure 3 presents the variation of the mean peak displacement, rotation, and acceleration of the structural model 1 without the A-\(\Delta\) effect for \(\Omega_{\theta}=1\). It can be observed in Figure 3 that the mean peak responses shown are affected by \(\rho_{fxM}\) and \(\rho_{FyM}\). The analysis carried out for the structural model 1 was repeated, but considering the rest of the structural models shown in Table 1. Figure 4 presents a comparison of the mean peak displacement, rotation, and acceleration for all the models presented in Table 1 for \(\Omega_{\theta}=1\). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Mass** & \(\omega_{\mathrm{x}}\) & \(\omega_{\mathrm{y}}\) & \multirow{2}{*}{\(H\) (m)} & \multirow{2}{*}{\(L\) (m)} & \multirow{2}{*}{\(W\) (m)} & \multirow{2}{*}{\(H\)/\(L\)} & \multirow{2}{*}{\(H\)/\(W\)} & \multirow{2}{*}{\(L_{x}\) (m)} & \multirow{2}{*}{\(L_{y}\) (m)} & \multirow{2}{*}{\(r_{e}\)=\(L_{x}\)/\(L_{y}\)} \\  & **(kg)** & **(rad/s)** & **(rad/s)** & **(rad/s)** & & & & & & & \\ \hline
1 & 2,281,509 & 0.925 & 0.945 & 170 & 40 & 33 & 4.3 & 5.2 & 12 & 11 & 1.09 \\ \hline
2 & 3,516,347 & 1.259 & 0.788 & 262 & 55 & 25 & 4.8 & 10.5 & 35 & 30 & 1.17 \\ \hline
3 & 2,180,727 & 1.179 & 1.033 & 140 & 38 & 34 & 3.7 & 4.1 & 15 & 13 & 1.15 \\ \hline
4 & 827,663 & 0.980 & 1.045 & 116 & 36 & 19 & 3.2 & 6.1 & 28 & 25 & 1.12 \\ \hline
5 & 479,315 & 1.098 & 1.559 & 101 & 29 & 14 & 3.5 & 7.2 & 20 & 18 & 1.11 \\ \hline
6 & 2,519,208 & 0.753 & 0.753 & 200 & 30 & 30 & 6.7 & 6.7 & 20 & 20 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the structural and wind model. It can be observed in Figure 4 that the consideration of \(\rho_{FxM}\) and \(\rho_{FyM}\) affects the mean peak response. It is worth noting that the dimensions of the buildings modelled as 3 DOF systems with generalized properties have important implications in the wind-induced response. As shown in Table 1, the heights of the buildings range from 101 to 262 m, and the aspect ratios (i.e., \(H/L\) and \(H/W\)) are within 3.2 to 10.5, which indicate that the structural models are very sensitive to the dynamic effects of the wind loading.