# Fragility Analysis of Transmission Towers Subjected to Downburst Winds
## 3 Typical Engineering Case Study
### Failure Probability Analysis  
#### 3.6.1 Evaluation Framework Based on Fragility Analysis  
The downburst footprint is approximately a rectangle with length \(l_{V}\) and width \(w_{V}\) and area \(a_{V}\)[50]. The direction of the long axis of the footprint can be described by \(\theta_{V}\), with a size between \(\pi\) and \(2\pi\), as shown in Figure 25. The impact of the downburst on a transmission line is mainly concentrated in the area where the wind speed exceeds \(V\). It is assumed that the annual average occurrence rate of downburst footprint with specific parameter \((VJ_{V},w_{V},\theta_{v})\) in area \(A\) is \(\tau(V\theta_{v})\). The rate of intersection between a point in the area and downburst footprint in one year is:  
\[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]  
Figure 23: Fragility curve of collapse state of 35 KV transmission tower under ABL wind. Figure 24: Fragility curve of collapse state under downburst. Suppose a transmission line with a length of \(L\) is arranged in area A, and the line with orientation \(\psi\) lies between 0 and \(\pi\). The directional probability \(Pr(\theta_{v})\) of downburst wind is considered. The rate of intersection per year between the transmission line and downburst footprints in area \(A\) is given as follows:  
1. Normal to the transmission line \[n_{l,w,v,\theta_{v}}=\frac{\tau(V\theta_{v})l_{V}LPr(\theta_{v})}{A}=\frac{v_{ l,w,v,\theta_{v}}LPr(\theta_{v})}{w_{V}}\] (18)
2. Oblique to the transmission line \[n_{l,w,v,\theta_{v}}=\frac{\tau(V\theta_{v})l_{V}L|\sin(\theta_{v}-\psi)|Pr( \theta_{v})}{A}=\frac{v_{l,w,v,\theta_{v}}L|\sin(\theta_{v}-\psi)|Pr(\theta_{ v})}{w_{V}}\] (19)
3. Parallel to the transmission line \[n_{l,w,v,\theta_{v}}=\frac{\tau(V\theta_{v})(l_{V}+L)w_{v}Pr(\theta_{v})}{A}= (1+\frac{L}{l_{V}})v_{l,w,v,\theta_{v}}Pr(\theta_{v})\] (20)  
It is assumed that the occurrences of intersections with wind speeds exceeding \(V\) obey the Poisson distribution, and the probability of no intersections in any one year is equivalent to the cumulative distribution of the annual maximum wind speed in such events. It can be expressed as:  
\[1-P_{w\theta_{v}}(v>V)=\exp(-v_{w,v,\theta_{v}})\cong 1-v_{w,v,\theta_{v}} \tag{21}\]  
Hence  
\[P_{w\theta_{v}}(v>V)=v_{w,v,\theta_{v}} \tag{22}\]  
The same method applies to the intersections with a transmission line:  
\[1-P_{w\theta_{v}}(a_{V}\cap L)=\exp(-n_{w,v,\theta_{v}})\cong 1-n_{w,v,\theta_{ v}} \tag{23}\]  
\[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]0 Oblique intersection of transmission line with downburst footprint. Hence  
\[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]1  
where \(P_{w\theta_{v}}(v>V)\) represents the probability that the annual maximum wind speed from downburst footprints with characteristics \(w_{V}\) and \(\theta_{V}\), and having any \(l_{V}\), intersecting a point in area A exceeds \(V\). Correspondingly, \(P_{w\theta_{v}}(a_{V}\cap L)\) represents the probability that the annual maximum wind speed from downburst footprints intersecting the transmission line exceeds \(V\). Accordingly, the intersection rate between the downburst footprint and the transmission line under each condition is:  
1. Normal to the transmission line \[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]2 (25)
2. Oblique to the transmission line \[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]3 (26)
3. Parallel to the transmission line \[v_{l,w,\theta_{v}}=\frac{l_{V}w_{V}\tau(V\theta_{v})}{A} \tag{17}\]4 (27)  
As for the width of the downburst footprint, Equations (25)-(27) show that a decrease in \(w_{V}\) is associated with an increase in the rate of intersection \(n_{l,w,v,\theta_{v}}\). If the width is less than the span, the action range of the downburst may only on one tower or none at all. Therefore, it can be assumed that the width of the downburst footprint is the transmission line span.