# Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service
## 4 Flutter Reliability Analysis of Xiangshan Harbor Bridge
### Flutter Reliability under Different Conditions  
During the service period of long-span cable-stayed bridges, it is inevitable that there will be various unfavorable situations to affect the wind-resistance stability of bridges, such as wire breakage or even a cable completely broken. When the mean values of these four random variables are changed, the flutter reliability index of the bridge can be also calculated using the same method. Therefore, the statistical eigenvalues of the instability critical wind speed can be obtained by the flutter instability analysis and LHS sampling.  
The flutter uncertainty analysis of the bridge was conducted for 0.5, 1, 1.25 and 1.5 times the initial stiffness of the main girder to obtain the reliability index, and the reliability value was converted into the corresponding failure probability. The reliability value and corresponding failure probability under different stiffnesses of the main girder are shown in Table 11.  
At the same time, the flutter uncertainty analysis of the bridge was conducted for different damping ratios as 0, 0.5% and 1% to obtain the reliability index, and the reliability value was converted into the corresponding failure probability. The reliability value and failure probability under different damping ratios are shown in Table 13.  
Finally, the flutter uncertainty analysis of the bridge was carried out when the cable breakage position is located at the bridge tower, midpoint of the main span and 1/4 of  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Item**} & **Stiffness of Main** & \multirow{2}{*}{**Wire Breaking Rate**} & \multirow{2}{*}{**Damping Ratio**} & **Cable Breakage** \\  & **Girder** & & & **Location** \\ \hline Reliability index & 2.35 & 2.24 & 2.26 & 2.31 \\ \(C_{w}\) & 0.9493 & 0.9525 & 0.9524 & 0.9507 \\ \(U_{s}\) & 60.0317 & 56.827 & 56.9689 & 58.5988 \\ \(G_{G}\) & 1.454 & 1.4503 & 1.4504 & 1.4523 \\ \(U_{b}\) & 39.1955 & 37.3223 & 37.4057 & 38.3604 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Calculation results of flutter reliability.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Stiffness of main & \multirow{2}{*}{0.5} & \multirow{2}{*}{0.75} & \multirow{2}{*}{1.0} & \multirow{2}{*}{1.25} & \multirow{2}{*}{1.5} \\  & girder & & & & \\ \hline Reliability index & 2.28 & 2.35 & 2.48 & 2.64 & 2.73 \\ \hline Failure probability & 0.0144 & 0.0095 & 0.0066 & 0.0042 & 0.0032 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Flutter failure probability corresponding to different stiffnesses of main girder.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{Wire breaking rate} & Initial state of & \multirow{2}{*}{5\%} & \multirow{2}{*}{10\%} & \multirow{2}{*}{15\%} & \multirow{2}{*}{20\%} \\  & bridge (0\%) & & & & \\ \hline Reliability index & 2.48 & 2.47 & 2.45 & 2.38 & 2.24 \\ \hline Failure probability & 0.006 & 0.0066 & 0.0072 & 0.0087 & 0.0125 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Flutter failure probability corresponding to different wire breaking rates.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & **Stiffness of Main** & \multirow{2}{*}{**Wire Breaking Rate**} & \multirow{2}{*}{**Damping Ratio**} & **Cable Breakage** \\  & **Girder** & & & & **Location** \\ \hline Reliability index & 2.35 & 2.24 & 2.26 & 2.31 \\ \(C_{w}\) & 0.9493 & 0.9525 & 0.9524 & 0.9507 \\ \(U_{s}\) & 60.0317 & 56.827 & 56.9689 & 58.5988 \\ \(G_{G}\) & 1.454 & 1.4503 & 1.4504 & 1.4523 \\ \(U_{b}\) & 39.1955 & 37.3223 & 37.4057 & 38.3604 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Calculation results of flutter reliability.  
the main span to obtain the reliability index, and the reliability value was converted into the corresponding failure probability. The reliability value and failure probability under different cable breakage location are shown in Table 14.