# Fragility Analysis of Transmission Towers Subjected to Downburst Winds
## 2 Fragility Analysis Framework
### Capacity Curve of Structural Model  
The purpose of buckling analysis is to obtain the critical load of the structure from a stable equilibrium state to an unstable equilibrium state. The two commonly used analysis methods are eigenvalue buckling analysis of ideal structures and nonlinear buckling analysis of defective structures. Eigenvalue buckling analysis can be used to predict the theoretical yield strength of an ideal linear structure. The advantage of eigenvalue buckling analysis is to obtain the critical load and buckling shape of the structure without complex nonlinear analysis. It can provide reference load values for nonlinear buckling analysis. Here, the first-order eigenvalue of the structure is solved by the block Lanczos method, and its governing equation is:  
\[([K_{\beta}]+\lambda[K_{\sigma}])\{\psi\}=0 \tag{15}\]  
where \(\lambda\) is the characteristic value, that is, the load factor in common sense; \(\left[K_{\beta}\right]\) is the elastic stiffness matrix of the structure; \(\left[K_{\beta}\right]\) is the reference initial stress matrix. \(\{\psi\}\) is the characteristic displacement vector. In order to consider the effect of initial geometric defects on the theoretical buckling strength of the structure, the nonlinear buckling analysis of the structure based on the large deflection finite element theory must be carried out. The cylindrical equal arc length method is widely used to track the nonlinear equilibrium route. In the iterative process, the load convergence criterion is selected as the criterion of convergence, i.e.,  
\[\sqrt{\{g\}^{T}\{g\}}\leq\beta\sqrt{\{q\}^{T}\{q\}} \tag{16}\]  
where, \(\{g\}\) is the unbalance node force vector; \(\{q\}\) is the reference load vector; \(\beta\) is a parameter, which can be taken as \(10^{-5}\). If the number of iterations has exceeded a predetermined maximum or the displacement vector is becoming larger and larger, it is regarded as a divergence. Transmission towers are often damaged due to instability. The accident investigation shows that many failures of transmission towers are caused by the buckling of compressed leg members or support members. This can be unveiled by analyzing its force-deformation curve or capacity curve. Nonlinear buckling analysis uses the static analysis method with gradually increasing load to determine the critical load that makes the structure unstable. This analysis method can consider the effects of geometric nonlinearity, material nonlinearity, and initial defects at the same time. By analyzing the relationship between the component failure and overall failure, this paper reveals the whole process of transmission tower failure under the action of a downburst so as to establish a reasonable failure criterion to classify the failure level of transmission towers. Nonlinear buckling analysis is performed on the transmission tower to obtain the top displacement base shear bearing capacity curve. At the same time, the maximum Mises stress of all main and diagonal components of the transmission tower can be extracted at each step of the loading process to obtain the top displacement main material maximum Mises stress curve and the top displacement diagonal material maximum Mises stress curve. The three curves are drawn and analyzed simultaneously. For different tower types and due to structural reasons, there are two situations: the first is that the main material yields before the diagonal material, as shown in Figure 3a; the second is that the diagonal material yields before the main material, as shown in Figure 3b. When the main material yields before the diagonal material, it can be observed that as the wind load continues to increase, when the top displacement of the tower reaches XI, the maximum stress of the main material reaches the yield strength before the diagonal material. As the main load-bearing component of the transmission tower is the main material, the overall structure of the tower begins to enter nonlinearity.