## 3 Validations and Applications
### Numerical Results of Wind-Induced Vibration Responses
Furthermore, when the incoming wind speed is 45 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the girder have an obvious flutter divergence phenomenon. Therefore, it can be concluded that the flutter critical wind speed of the main girder of GBEB is about 40 m/s. Figure 8: _Cont._  
The structural dynamic parameters of the main girder of the GBEB in Table 3 are applied to the FSI program, based on the Volterra model and Newmark-\(\beta\) method. It should be noted that the first-order kernel function of the Volterra series of the aerodynamic force of the main deck section model is identified for the wind speed of 5 m/s. Here, assuming that the first-order kernel of the Volterra series for the wind speed of 5 m/s is  
Figure 8: Comparisons of results of force coefficients (**a**) in Case 1; (**b**) in Case 2; (**c**) in Case 3; (**d**) in Case 4; (**e**) in Case 5; and (**f**) in Case 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Unit**} & \multicolumn{2}{c}{**Prototype Main Deck**} & **Scale** & **Main Deck** \\  & & **Section** & **Ratio** & **Section MODEL** \\ \hline Width, \(B\) & m & 31.0 & 1:80 & 0.3875 \\ Height, \(H\) & m & 4.4 & 1:80 & 0.0550 \\ Mass per unit length, \(m\) & kg/m & \(17.8\times 10^{3}\) & 1:80\({}^{2}\) & 2.781 \\ Torsion mass moment of inertia per unit & kg\(\cdot\)m\({}^{2}\)/m & \(2.173\times 10^{6}\) & 1:80\({}^{4}\) & 0.053 \\ Vertical bending frequency, \(f_{h}\) & Hz & 0.099 & 10:1 & 0.99 \\ Torsional frequency, \(f_{\alpha}\) & Hz & 0.186 & 10:1 & 1.86 \\ Vertical bending–damping & \% & 0.5 & / & 0.5 \\ ratio, \(\zeta_{h}\) & \% & 0.5 & / & 0.5 \\ Torsional damping ratio, \(\zeta_{\alpha}\) & \% & 0.5 & / & 0.5 \\ Wind velocity, \(U\) & m/s & / & 1:8 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main parameters of the main deck section of GBEB. equal to that for the wind speeds of 4.375 m/s, and 5.625 m/s, respectively. Figure 10 shows the numerical results of wind-induced vibration responses of the main deck section by ROM method for incoming wind speeds of 35 m/s, 40 m/s, and 45 m/s for the prototype main deck section, respectively. As shown in Figure 10, when the incoming wind speed is 35 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the prototype girder have no obvious flutter divergence phenomenon. However, when the incoming wind speed is 40 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the girder show a trend of slow divergence. Furthermore, when the incoming wind speed is 45 m/s for the prototype main deck section, the torsional displacement and vertical displacement of the girder have an obvious flutter divergence phenomenon. The calculation result of the flutter critical wind speed of the main girder section of GBEB obtained by the ROM method is about 40 m/s. Meanwhile, Figures 9 and 10 have consistency for the flutter analysis results. Figure 9: Vertical and torsional displacements of main deck section of GBEB, based on the FSI method for incoming wind speeds of (**a**) 35 m/s, (**b**) 40 m/s, and (**c**) 45 m/s for the prototype bridge, respectively. Table 4 summarizes the flutter critical wind speed and flutter frequency results of the main girder of the GBEB calculated by FSI numerical simulation based on CFD and the aeroelastic analysis method in this study and other examples of research. Table 4 represents that the flutter critical wind speed of the girder of GBEB obtained by different methods are similar, and the flutter frequency is also in good agreement. Among the different methods, the numerical simulation method has a long calculation period. Although the wind tunnel test method has a short test period, there is complicated preparation work in the early stage of the test, which requires more workforce and material resources.