# Fragility Analysis of Transmission Towers Subjected to Downburst Winds
## 3 Typical Engineering Case Study
### Failure Probability Analysis
#### 3.6.2 Case Study  
There is a 10 km long north-south 220 KV transmission line in southeast Queensland, Australia. Tables 10 and 11 outline the return period \(R_{V}\) and directional probabilities \(Pr(\theta_{v})\) of downburst wind gusts for the region, respectively. Based on the given downburst wind speed and direction data, the transmission line layout conditions of \(22.5^{\circ},67.5^{\circ},112.5^{\circ}\), and \(157.5^{\circ}\) are considered in this paper, as shown in Figure 26.  
According to the above method, the probability of a downburst attacking the transmission line can be calculated. Combined with the fragility curve of 220 KV transmission tower in the collapse state under the downburst, the probability of transmission tower collapse caused by the downburst attacking the transmission line in different directions can be obtained. By adjusting the layout direction of transmission lines, the collapse probability of transmission towers under different layout conditions, different directions, and different wind speeds can be obtained, as shown in Figure 27. The favorable and unfavorable layout conditions for different directions are shown in Table 12.  
Under the condition of different layouts of the transmission line and the same direction angle, the failure probability for a direction angle in the interval 2-6 is the largest and that in the interval 3-7 is the smallest, indicating that the wind directional probability has a great impact on the failure probability.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wind Gust Speed** & **Return Period** & **Wind Gust Speed** & **Return Period** \\ **(m/s)** & **(Years)** & **(m/s)** & **(Years)** \\ \hline
35 & 10.3 & 48 & 100 \\
36 & 11.8 & 49 & 126 \\
37 & 13.5 & 50 & 162 \\
38 & 15.7 & 51 & 212 \\
39 & 18.2 & 52 & 281 \\
40 & 21.3 & 53 & 381 \\
41 & 25.1 & 54 & 526 \\
42 & 29.8 & 55 & 747 \\
43 & 35.7 & 56 & 1090 \\
44 & 43 & 57 & 1650 \\
45 & 52 & 58 & 2600 \\
46 & 64 & 59 & 4300 \\
47 & 79 & 60 & 7700 \\ \hline \hline \end{tabular} Return period = 1/(annual probability of exceedance).  
\end{table}
Table 10: Return periods for downburst wind gusts.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{Pr}(\theta_{v1})\) & \(\mathbf{Pr}(\theta_{v2})\) & \(\mathbf{Pr}(\theta_{v3})\) & \(\mathbf{Pr}(\theta_{v4})\) \\ \(\theta_{v12}\)=22.5\({}^{\circ}\) & \(\theta_{v2}\)=67.5\({}^{\circ}\) & \(\theta_{v3}\)=112.5\({}^{\circ}\) & \(\theta_{v4}\)=157.5\({}^{\circ}\) \\
**Interval 2–6** & **Interval 1–5** & **Interval 4–8** & **Interval 3–7** \\ \hline
0.343 & 0.293 & 0.192 & 0.172 \\ \hline \hline \end{tabular} Note: the downburst gusts in the direction of 22.5\({}^{\circ}\) are added to the downburst gust in the direction of 202.5\({}^{\circ}\), etc.  
\end{table}
Table 11: Directional probabilities for downburst wind gusts.  
Figure 26: Transmission line layout conditions.  
According to the return period of downburst wind gusts, with the increase in wind speed, the return period becomes larger and larger, and the probability of a downburst attacking the transmission line becomes smaller and smaller. Figure 27 reflects the failure probability of the tower under the previous wind speed, and Figure 28 shows the failure probability of the tower under the full wind speed.  
According to the above failure probability results, the average failure probability of four layout conditions is calculated under different wind speeds, and the average fragility curve of the transmission tower under different basic wind speeds is drawn, as shown in Figure 29. The results show that the most favorable arrangement is 157.5\({}^{\circ}\), and the most unfavorable arrangement is 112.5\({}^{\circ}\).  
Figure 27: The probability of transmission tower collapse caused by a downburst attacking the transmission line in different directions.