## 4 Application to the Structural Fragility Assessment on a Transmission Line
This joint model is established by combining the Monte Carlo simulation with the two-dimensional non-parametric kernel density estimation [37]:  
Then, we obtain the annual probability of failure of each suspension tower in the normal segment of the transmission line using Equation (25), and the failure probabilities of tension towers and terminate towers are taken to be zero. Clustering the suspension towers based on the line direction and span, we obtain three clusters of the towers, with their centers corresponding to different line directions (Figure 14a). In Figure 14a, \(n\) denotes the number of the towers located in the corresponding cluster. Figure 14b further shows the failure class of each tower in three clusters. The failure class is defined by the annual probability of failure calculated from the Equation (25) and five classes are classified according to the calculation result. In Figure 14a,b, the red cluster accounts for the majority of the towers and their failure probabilities scatter in the five classes. Considering the blue cluster and the black cluster, we find that most of the towers have larger failure probabilities. Defining one strain section of a line as starting from one tension tower and ending with the adjacent tension/terminate tower, there are 14 strain sections in the normal segment of the transmission line. To evaluate each strain section, Figure 15a,b first give the basic information (including the length and the number of the towers) of each strain section, and we have the essential cognition that the longer the strain section is, the more towers are included in the section. Figure 15c,d further show the statistical result of the annual failure probability of the towers in each strain section. Figure 13: Joint probability model of the annual extreme wind speed of typhoons and the wind direction. Comparing the two strain sections with the largest numbers of towers, i.e., section No. 5 and No. 9, we found that although the No. 5 section owns the largest number of towers, its annual failure risk is quite low in terms of both the average and deviation. No. 9 section has the highest failure risk from the perspective of average, while No. 12 and 14 sections have the larger deviations. Furthermore, we considered the failure of stain sections using the bound theory (shown as Equations (26) and (27)):  
\[\text{Lower bound }P_{f,L}\geq\max\Bigl{[}P_{f,T}\bigl{(}T_{j}\bigr{)} \Bigr{]} \tag{26}\]  
Figure 10:0  
where \(P_{f,L}\) is the annual probability of failure of one strain section or line segment, \(P_{f,T}\bigl{(}T_{j}\bigr{)}\) represents the annual probability of failure of the \(j\)-th tower, \(j=1,2,\,\ldots\,\,m\), and \(m\) is the total number of towers in the strain section. Using the Equations (26) and (27), we obtained the failure bounds of each strain section, as shown in Figure 10:1  
Figure 10:2 Clusters of suspension towers and their failure classes. (**a**) clusters of suspension towers; (**b**) failure classes of suspension towers. Figure 10:3 Strain sections and the annual failure probability of towers in each strain section. (**a**) the length of each strain section; (**b**) the number of towers in each strain section; (**c**) the mean of failure probabilities of each strain section; (**d**) the standard deviation of failure probabilities of each strain section. In Figure 16, the No. 9 section has the highest lower bound as well as the highest upper bound among all the strain sections, which indicates that the most dangerous tower is in the No. 9 strain section and most of the towers in this section have a higher risk. Moreover, the lower bounds of No. 10-No. 14 sections are also considerably high, which means that there are also dangerous towers located in these sections. Based on the above assessment, we conclude the following (from the perspective of annual failure risk):  
(i). The primary attention should be drawn on the No. 9 strain section due to its high risk of failure. (ii). The No.