## 4 Application to the Structural Fragility Assessment on a Transmission Line  
The above study constructed a capacity function for the limit state of transmission towers subjected to winds, and solved the capacity function by the proposed kriging-based surrogate modeling. Considering the capacity surface \(\theta_{L}\), the limit state function is written as:  
\[\text{g}=U_{\text{lim}}-U=\psi(\phi,\ L_{h};\ \theta_{S})-U \tag{24}\]  
where \(\varphi\) and \(L_{h}\) are the normal variables; \(\theta_{S}\) is a structural vector containing stochastic structural parameters. Based on the above limit state function, the fragility curves can be simulated through the workflow shown in Figure 10 (\(\Theta_{S}\) is the sample assembly of the stochastic structural vector \(\theta_{S}\); \(N_{s}\) is the scale of Monte Carlo sampling; \(N(\text{g}<0)\) means that the number of elements that less than zero in the limit state vector \(\text{g}\)). Figure 10: The flow for generating the structural fragility of towers under winds. Figure 9: The surrogate result for capacity surfaces. Depending on the established kriging surrogate model of capacity surface \(\theta_{L}\), we can generate multiple groups of fragility curves for the transmission tower with its structural uncertainties considered and the external variables (i.e., the wind environment and line span) involved. Figure 11 further gives the fragility curves of tower ZY achieved by the above flow, in which different wind angles of attack and three horizontal spans are of concern. Taking the practical transmission line (Table 1) as the example and considering the normal segment of the line (i.e., the blue section in Figure 12), we conducted the structural fragility assessment on the towers and the line segment under typhoons by implementing the workflow of the fragility calculation, shown in Figure 10.  
In the normal segment of the line, there were 86 transmission towers in total, including 71 suspension tangent towers, 14 tension towers, and 1 terminate tower. Suspension tangent towers, as the wind-sensitive structure, are concerned in the fragility assessment. Tension towers and terminate towers generally have a higher safety level under winds and are not considered here. Taking the wind and the line information into account simultaneously, the  
Figure 11: Structural fragility curves of tower ZY. (**a**) \(L_{h}\) = 150 m; (**b**) \(L_{h}\) = 275 m; (**c**) \(L_{h}\) = 400 m.  
Figure 12: A practical transmission line. * The background image is captured from Google Earth Pro [33]. annual probability of failure of each tower can be estimated by convolving the structural fragility with the joint model of the annual extreme wind speed of typhoons (\(v\)) and the corresponding wind direction (\(\beta\)) as:  
\[P_{f,T}=\int_{0}^{360}\int_{0}^{v_{\max}}F_{T}(v,\phi|L_{h},\alpha)f_{v,\beta}(v,\beta)dvd\beta \tag{25}\]  
in which \(P_{f,T}\) is the annual probability of failure of the tower structure and \(F_{T}\) represents the cumulative probability of failure of the tower with the horizontal span and line direction taken as \(L_{h}\) and \(\alpha\), respectively. This cumulative probability is a function of the annual extreme wind speed of typhoons \(v\) and the corresponding wind angle of attack \(\varphi\) (\(\varphi\) is calculated according to the line direction \(\alpha\) and the direction of annual extreme wind \(\beta\)), which can be calculated using the workflow shown in Figure 10. \(f_{v,\theta}\) is the joint probability density of the annual extreme wind speed (\(v\)) and its wind direction (\(\beta\)), in which the wind speed \(v\) and direction \(\beta\) range from 0-\(v_{\max}\) and 0\({}^{\circ}\)-360\({}^{\circ}\), respectively. Here, we adopt the joint probability model of \(v\) and \(\beta\) that are generated based on the 60-year typhoon on-site data (Figure 13) [34; 35; 36].