# Static Wind Reliability Analysis of Long-Span Highway Cable-Stayed Bridge in Service
## 2 Reliability Theory of Static Wind Instability Analysis for Bridges
### Aerostatic Instability Analysis Theory for Bridges
#### 2.1.2 Calculation of Aerostatic Instability wind Speed of Bridges  
So far, the accepted solution method for the aerostatic instability wind speed of bridges is the UL incremental method [23], that is, a method combining the incremental method with the internal and external two-iterative procedure is used to solve the nonlinear incremental equation to obtain the displacement response of the bridge. When increasing the static wind speed based on a predetermined step size, the iteration in the inner layer finishes the geometric nonlinear calculation of the bridge structure, and the iteration in the outer layer finds the equilibrium position of the bridge structure under this wind speed. The specific process is as follows [8]:  
1. Assume the initial static wind speed \(V_{0}\), and the initial wind attack angle \(\alpha\);
2. Calculate the static wind force acting on each bridge component including the main girder, stay cables, piers, and pylons at current wind speed and wind attack angle;
3. Establish the finite element model of the bridge, then apply the static wind load calculated in step (2) to each member of the bridge, and the Newton-Rapson method is adopted to solve the static response of the bridge such as the lateral displacement (\(U_{x}\)), vertical displacement (\(U_{y}\)) and torsion angle (\(Rot_{z}\)) of each node of the main girder at current wind speed and wind attack angle;
4. Extract the maximum torsion angle of the main girder from the calculation result in step (3), and recalculate the static wind load on the bridge considering the variation of the wind attack angle;
5. Judge whether the Euclidean norm of the three-component force coefficient is no more than the allowable value: \[\left\{\frac{\sum_{j=1}^{N}\left[C_{k}(\alpha_{j})-C_{k}(\alpha_{j-1}) \right]^{2}}{\sum_{j=1}^{N}\left[C_{k}(\alpha_{j-1})\right]^{2}}\right\}^{1/ 2}\leq\varepsilon_{k}(k=H,V,M)\] (3) where \(C_{k}(\alpha_{j})\) is the three-component force coefficient corresponding to the effective wind attack angle \(\alpha_{j}\) of each node of the main girder after applying the \(j\)th static wind load; \(C_{k}(\alpha_{j-1})\) is the result after applying the \((j-1)\)th static wind load; \(N\) is the total number of nodes on the main girder in the finite element model of the bridge; \(\varepsilon_{k}\) is the allowable value of the Euclidean norm of the three-component force coefficient which is taken as 5%.
6. If all three-component force coefficients are less than the allowable value \(\varepsilon_{k}\), improve the wind speed in accordance with the predetermined step size (\(\Delta v=20\) m/s in this research), and repeat steps (2)-(5), otherwise repeat steps (3)-(5);
7. If the number of iterations is greater than a predetermined time at a certain wind speed, the convergence requirement is still not satisfied, restore the previous wind speed, shorten the step size of the wind speed and recalculate, until the difference between the two adjacent wind speeds is no more than the predetermined value.