# Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service
## 2 The Flutter Analysis Framework for Flexible Bridge
### Overview
\tag{7}\]  
where, when the random variable \(X\) obeys the lognormal distribution; that is, \(Y=lnX,\ Y\sim N\big{(}\mu Y,\sigma^{2}Y\big{)}\); the mean value and standard deviation of \(Y\) can be expressed as:  
\[\left\{ \begin{array}{c}\mu_{Y}=\mu_{H_{X}}=ln\big{(}\frac{\mu_{X}}{\sqrt{1+ \delta^{2}_{X}}}\big{)}\\ \sigma_{Y}=\sigma_{H_{X}}=\sqrt{ln(1+\delta^{2}_{X})}\end{array} \right. \tag{8}\]  
where \(\mu_{X}\) and \(\sigma_{X}\) represent the mean value and standard deviation of random variable \(X\) and \(\delta_{X}\) is the variation-coefficient of the random variable \(X\). Figure 1: Flutter analysis process. Supposing the functional function as \(Z=g(X_{1},\cdots X_{n})\), and \(x_{i}^{*}\) is the design checking point, then the calculation formula of the reliability index \(\beta\) is as follows:  
\[\beta\approx\frac{g_{X}\big{(}x_{1}^{*},x_{2}^{*},\cdots,x_{n}^{*}\big{)}+\sum_{ i=1}^{n}\frac{\partial g_{X}}{\partial X_{i}}\Big{|}\big{(}u_{X_{i}}-x_{i}^{*} \big{)}}{\sqrt{\sum_{i=1}^{n}(\frac{\partial g_{X}}{\partial X_{i}}\big{|}x_{ i}^{*}\sigma_{X_{i}})}} \tag{9}\]  
By introducing sensitivity coefficient \(a_{i}\), let:  
\[a_{i}=cos\theta_{i}=\frac{\frac{\partial g_{X}}{\partial X_{i}}\Big{|}_{x_{i} ^{*}}\sigma_{X_{i}}}{\sqrt{\sum_{i=1}^{n}(\frac{\partial g_{X}}{\partial X_{i} }\big{|}_{x_{i}^{*}}\sigma_{X_{i}})^{2}}} \tag{10}\]  
Then \(x_{i}^{*}\) and \(\beta\) have the following relationship:  
\[x_{i}^{*}=\mu_{x_{i}}-\sigma_{x_{i}}\beta a_{i} \tag{11}\]  
The calculation process of the check point method is shown in Figure 2.  
In order to reflect the randomness of the parameters affecting the flutter response of the bridge, the Latin hypercube sampling method was used to sample the parameters. Latin hypercube sampling (LHS) is an efficient and fast sampling method, which was further improved on the basis of the stratified sampling method. The advantage of this method is that the sampling times can be greatly reduced while the accuracy is maintained. [PERSON] [38] pointed out that when obtaining the same failure probability with the same accuracy, the sample number by the Latin hypercube sampling method is only about \(1/4\) of that by traditional sampling method. This advantage can save a lot of time in the process of numerical simulation. In the subsequent analysis, MATLAB software can be used to conduct Latin hypercube sampling for the parameter uncertainty  
Figure 2: Calculation flow of the checking point method. be effectively combined with the design checking point method to ensure the accuracy of the calculation reliability and greatly improve the calculation efficiency. Determination of Functional Function and Parameters of Wind-Induced Flutter Reliability Analysis 2.3.1. Establishment of Flutter Reliability Functional Function  
The flutter reliability analysis model of the bridge can be described as follows. When the wind speed at the bridge site exceeds the critical wind speed for bridge stability, the bridge structure will have a wind-induced failure risk. Supposing that \(S\) is the wind load effect acting on the bridge and \(R\) is the capacity of the bridge structure itself to resist wind load, then its function is:  
\[Z=R-S \tag{12}\]  
In the reliability analysis of bridge flutter, the limit state function can be expressed by the difference between the critical flutter wind speed \(U_{cr}\) and the design wind speed \(U_{m}\), namely:  
\[Z=U_{cr}-U_{m} \tag{13}\]  
where the designed wind speed can be expressed by the product of the gust coefficient and the 10 min reference wind speed at the bridge site:  
\[U_{m}=G_{v}U_{b} \tag{14}\]  
where \(U_{b}\) is the 10 min time-interval reference wind speed at the height of the main beam at the bridge site; and \(G_{v}\) is the gust coefficient, considering the influence of maximum fluctuating wind speed. The gust coefficient and the reference wind speed at the bridge site need to be fitted by the corresponding probability distribution function.