# Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service
## 2 The Flutter Analysis Framework for Flexible Bridge
### Calculation of the Bridge Flutter Critical Wind Speed  
As mentioned above, in order to realize the time-history analysis of bridge flutter, the flutter derivatives obtained should be time-domain processed, namely, the parameters in Equation (4) should be solved. Rational function parameter fitting was carried out for the flutter derivative of Xiangshan Harbor Bridge. Generally, in the process of rational function fitting, the number of nonlinear terms is two, that is, \(m=4\), which can meet the accuracy requirements. Therefore, each self-excited force expression needed to solve six parameters to be fitted, namely \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\), \(d_{1}\), \(d_{2}\).  
The flutter derivative data of Xiangshan Harbor Bridge calculated by Fluent software was fitted and solved in Matlab, which was divided into the following four steps:  
1. Create the corresponding target function.
2. Read the flutter derivative data and the reduction wind speed data.
3. Determine the initial value of these parameters to be fitted and set the search conditions \(d_{1}d_{2}>0\).
4. Transform the initial value, then start the cycle iteration to find the optimal solution, and take the minimum residual value of the target function as the optimal result.  
Table 4 lists the parameter fitting results of the flutter derivative of Xiangshan Harbor Bridge, which could be substituted into Equation (4) to obtain the calculation formula of the self-excited force on Xiangshan Harbor Bridge for the flutter time-history analysis.  
According to the flutter analysis process and calculation method mentioned above, the flutter time-history analysis of Xiangshan Harbor Bridge was carried out in ANSYS. After removing the unstable part of motion, the time-history curve of the mid-span displacement of Xiangshan Harbor Bridge was obtained, as shown in Figures 10-12 at different wind speeds.  
According to Figure 10, when the wind speed was 50 m/s, the mid-span vertical displacement of the main girder of the cable-stayed bridge and the torsional displacement showed a convergence trend, which shows that no flutter occurs in Xiangshan Harbor Bridge at the wind speed 50 m/s. However, when the wind speed increased to 64 m/s, it was concluded that the cable-stayed bridge reached the critical state of flutter, and its time-history curve of mid-span displacement is shown in Figure 11 where, with the increase of time, the motion state became stable and then the displacement amplitude was a constant. At this time, the cable-stayed bridge is in the critical state of flutter instability, so the flutter wind speed of Xiangshan Harbor Bridge is 64 m/s.  
When further increasing the wind speed, bridge flutter occurs, and the motion state of the cable-stayed bridge after flutter is explored. In order to more obviously display the motion state of the cable-stayed bridge after flutter, the time-history curve of mid-span displacement of the cable-stayed bridge is shown in Figure 12, where the wind speed was 75 m/s. It can be found that the mid-span vertical displacement of the main girder of the cable-stayed bridge and the torsional displacement showed a trend of departure and dispersion, which proves that when the wind speed reaches 75 m/s, the flutter instability of Xiangshan Harbor Bridge has occurred.  
Figure 11: Time-history curve of mid-span displacement of Xiangshan Harbor Bridge when wind speed was 64 m/s. (**a**) Vertical displacement; (**b**) Angular displacement.  
Figure 12: Time-history curve of mid-span displacement of Xiangshan Harbor Bridge when wind speed was 75 m/s. (**a**) Vertical displacement; (**b**) Angular displacement.