## 5 AAF Numerical Application
### Buffeting Response Obtained Using Multiple AAF Types  
The buffeting responses of the case cable-stayed bridge were calculated using a nonlinear transient FE approach. Three types of AAFs were taken into consideration in the calculation, i.e., AAF = 1.0, the Sears function and the identified AAFs previously mentioned. AAF = 1.0 corresponds to completely  
Figure 16: Sample of turbulent wind velocity obtained by the described procedure (U = 10 m/s). Figure 17: PSDs of different turbulent components in the simulated wind field. neglecting the correction of the unsteady effect on the quasi-steady buffeting forces in the AAFs. In the calculation, the wind speed at the girder is set at the design speed of 49.8 m/s, and the oncoming wind is lateral and normal to the bridge span. The calculated root-mean square (RMS) values of the lateral, vertical and torsional displacements under the service and construction stages are plotted in Figures 18 and 19, respectively. The PSDs of the buffeting displacements at the mid-span centre point are plotted in Figure 20.  
Figure 19: RMS of the truss girder displacements at the construction stage under different AAFs. Figure 18: RMS of the truss girder displacements at the service stage under different AAFs. According to Figures 18 and 19, the calculated vertical, lateral and torsional buffeting responses based on the identified AAFs deviate from those based on the Sears function and 1.0 AAFs. The RMS reflects the vibration amplitude of the DDTG in turbulence. The RMS of the Sears function is very close to the RMS obtained when neglecting the correction of the unsteady force. However, the RMS by these two AAF types always have disparity with identified AAFs. Furthermore, because the Sears function is greater than the identified AAF in terms of \(|\chi_{D}|\) for both the service and construction stages mentioned above, the lateral response calculated based on the identified AAFs is much lower in each stage. Similarly, the Sears function is less than the identified AAF in terms of \(|\chi_{L}|\), so the vertical RMS of the identified AAFs is 1.75 times larger than that of the Sears function in the service stage and 2.0 times in the construction stage (see Table 8). Hence, if the wind resistant design for the DDTG using the Sears function or AAF = 1.0, an unreasonable and unsafe stiffness design for the girder is produced with severe low vertical stiffness and an excessive safety factor for lateral stiffness. Due to the high torsional rigidity of the DDTG, the torsion RMS is relatively small, and the difference among each AAF of \(|\chi_{M}|\) is almost negligible. Figure 20: Buffeting response PSDs at the mid-span centre point under different AAFs. For long-span bridges, like the case one, the vibration of the girder is most controlled by the modal in the low frequency region (\(f\leq 2.0\)Hz for the case bridge). In Figure 20, some obvious differences are observed between the buffeting response PSDs obtained using the three kinds of AAFs. When ignoring the correction of the unsteady effect on the quasi-steady buffeting forces using AAF = 1.0, the buffeting response PSDs of the DDTG are the largest for all of the structural stages. It follows that the energy transfer from the wind spectrum to the buffeting force spectrum is incomplete, and the energy loss cannot be ignored. Usually, the Sears function is set as the AAF for a single deck section and derived from the wind lift pressure/force on a single deck. Although the decks of the DDTG are wide and flat, the wind flow in the space between the double decks also has the wind pressure/force on the roadway and railway decks. Combined with Table 7, lateral vibration easily occurs in the long-span cable-stayed bridge with the DDTG.