# Static Wind Reliability Analysis of Long-Span Highway Cable-Stayed Bridge in Service
## 2 Reliability Theory of Static Wind Instability Analysis for Bridges
### Theory and Method of Reliability Analysis  
The checkpoint method is often used for reliability analysis [24]. When the value of the random variable is at the checkpoint, the structure is in the most unfavorable state. At this moment, the functional function reaches an extreme value, namely, the failure probability of the structure is the largest at this checkpoint. It is worth noting that, before adopting the checkpoint method, if the random variables do not obey the normal distribution, they must be equivalently normalized according to the failure domain area must be equal before and after the equivalent normalization. Note the coordinate value of the checkpoint as \(x_{i}^{*}(i=1,2,\cdots,n)\), thus the expressions of the equivalent normalization condition are:  
\[F_{X_{i}}(x_{i}^{*})=\Phi\left\{\frac{x_{i}^{*}-\mu_{X_{i}}^{\prime}}{\sigma_ {X_{i}}^{\prime}}\right\} \tag{4}\]  
\[f_{X_{i}}(x_{i}^{*})=\frac{\varphi\left\{\frac{x_{i}^{*}-\mu_{X_{i}}^{\prime} }{\sigma_{X_{i}}^{\prime}}\right\}}{\sigma_{X_{i}}^{\prime}} \tag{5}\]  
in which, \(F_{X_{i}}\) stands for the distribution function for non-normal random variables; \(\Phi\) stands for the standard normal distribution function; \(f_{X_{i}}\) stands for the density function for non-normal random variables; \(\varphi\) stands for the density function of \(\Phi\); \(\mu_{X_{i}}^{\prime}\) stands for the average value of the equivalent normal variable \(X_{i}^{\prime}\) and \(\sigma_{X_{i}}^{\prime}\) stands for the variance of the equivalent normal random variable \(X_{i}^{\prime}\). Based on Equations (4) and (5), the average value and variance of the equivalent normal random variable \(X_{i}^{\prime}\) can be obtained as:  
\[\left\{ \begin{array}{l}\mu_{X_{i}}^{\prime}=x_{i}^{*}-\Phi^{-1}\left[F_{X_{ i}}(x_{i}^{*})\right]\sigma_{X_{i}}^{\prime}\\ \sigma_{X_{i}}^{\prime}=\varphi\left(\Phi^{-1}\left[F_{X_{i}}(x_{i}^{*}) \right]\right)/f_{X_{i}}(x_{i}^{\prime})\end{array} \right. \tag{6}\]  
According to the above transformation method, if the random variable \(X\) obeys the log-normal distribution, that is, \(Y=lnX,\ Y\sim N\left(\mu Y,\sigma^{2}Y\right)\), the average value and standard deviation of the normal random variable is calculated as follows:  
\[\left\{ \begin{array}{l}\mu_{Y}=\mu_{lnX}=ln(\frac{\mu_{X}}{\sqrt{1+\phi_{X} ^{2}}})\\ \sigma_{Y}=\sigma_{lnX}=\sqrt{ln(1+\delta_{X}^{2})}\end{array} \right. \tag{7}\]  
where \(\mu_{X}\) is the average value of the random variable \(X\), \(\sigma_{X}\) is the standard deviation of random variable \(X\), and \(\delta_{X}\) is the variation coefficient of random variable \(X\). Let \(Z=g(X_{1},\cdots X_{n})\) as the functional function and \(x_{i}^{*}\) as the design checkpoint, thus the expression of the reliability index \(\beta\) can be written as:  
\[\beta\approx\frac{g_{X}\big{(}x_{1}^{*},x_{2}^{*},\cdots,x_{n}^{*}\big{)}+ \sum_{i=1}^{n}\frac{\partial\beta_{X}}{\partial X_{i}}\Big{|}_{x_{i}^{*}}\big{(} u_{X_{i}}-x_{i}^{*}\big{)}}{\sqrt{\sum_{i=1}^{n}(\frac{\partial\beta_{X}}{ \partial X_{i}}\big{|}_{x_{i}^{*}}\sigma_{X_{i}})^{2}}} \tag{8}\]By introducing the sensitivity coefficient \(a_{i}\), let:  
\[a_{i}=\cos\theta_{i}=\frac{\frac{\partial\chi_{i}}{\partial X_{i}}\Big{|}_{x_{i} ^{*}}\sigma_{X_{i}}}{\sqrt{\sum_{i=1}^{n}(\frac{\partial x_{i}}{\partial X_{i} }\Big{|}_{x_{i}^{*}}\sigma_{X_{i}})^{2}}} \tag{9}\]  
Then there is the following relationship between \(x_{i}^{*}\) and \(\beta\):  
\[x_{i}^{*}=\mu_{x_{i}}-\sigma_{x_{i}}\beta a_{i} \tag{10}\]  
The flow chart of structural reliability calculation by checking point method is shown in Figure 3.  
To ensure the randomness of the parameters affecting the static wind response of the bridge, in this research, the Latin Hypercube Sampling (LHS) method is used to extract the sample of these random parameters, which is a highly efficient and fast sampling method and is a further improvement based on stratified sampling method. The advantage of this method is to greatly reduce the number of samplings while meeting the accuracy requirements.