# Fragility Analysis of Transmission Towers Subjected to Downburst Winds
## 2 Fragility Analysis Framework
### Downburst Wind Load Model  
In this study, only the static downburst is considered. The location of the center of the downburst relative to the center of the tower is defined using the radial distance \(R\) and the angle \(\theta\). The wind speed of the downburst changes with the change in the radial distance, which is one of the main factors affecting wind speed. Due to the influence of the climate, terrain, and other factors, the direction of the incoming flow at each transmission tower location is random and unpredictable. It is essential to consider the effect of wind attack angle. Assume that the downburst touchdown occurs at a point whose horizontal radial distance and wind attack angle with the transmission tower are \(R\) and \(\theta\), respectively, as shown in Figure 2. In this case, the downburst wind field acting on the transmission tower can then be modeled as  
\[U(z,t)=\overline{U}(z,t)+u(z,t) \tag{1}\]  
where \(U(z,t)\) is the wind speed at the height \(z\) of the transmission tower; \(\overline{U}(z,t)\) is the time-varying mean wind speed; \(u(z,t)\) is the corresponding fluctuating wind. For the time-varying mean wind speed, it can be expressed as the product of a time-invariant vertical wind profile model and a time modulation function [19; 34], i.e.,  
\[\overline{U}(z,t)=\overline{U}(z)f(t) \tag{2}\]  
where \(\overline{U}(z)\) is the vertical wind profile; \(f(t)\) is a modulation function whose maximum is one. This way of modeling the mean wind speed has an underlying assumption that the mean wind speeds reach their maximum values simultaneously. Figure 2: Relative position of transmission tower and downburst. Currently, there are many empirical models that have been used to describe the time-invariant mean wind profile of downburst winds [35; 36]. In this study, the Vicroy wind profile model is chosen, which can be expressed as [37]  
\[\overline{U}(z)=1.22\times\left[e^{-0.15z/z_{\max}}-e^{-3.2175z/z_{\max}}\right] \times V_{\max}(R) \tag{3}\]  
where \(z_{\max}\) denotes the height corresponding to the maximum horizontal wind speed; \(V_{\max}(R)\) represents the maximum horizontal wind speed under the case of radial distance \(R\), which can be given by  
\[V_{\max}(R)=\left\{ \begin{array}{ll}V_{R,\max}\times(R/R_{\max}),&0\leq R<R_ {\max}\\ V_{R,\max}\times\exp\Bigl{\{}-\bigl{[}(R-R_{\max})/R_{r}\bigr{]}^{2}\Bigr{\}}~{},&R\geq R_{\max}\end{array} \right. \tag{4}\]  
where \(V_{R,\max}\) is the maximum wind speed at a certain height in the wind speed field; \(R_{\max}\) is the horizontal distance between the maximum wind speed point and the downburst collision point; \(R_{r}\) is the radial length scale factor. For the modulation function, the following model is employed  
\[f_{(t)}=\left\{ \begin{array}{ll}t/300,&0\leq t\leq 300\\ \exp\left[-(t-300)/1800\right],&t>300\end{array} \right. \tag{5}\]  
For stationary downbursts, the time modulation function does not change with changes in the radial distance. For mobile downbursts, the time modulation function changes with the radial distance. Once the radial distance is prescribed, the time-varying mean wind profile of the downburst can be obtained. When it comes to the fluctuating wind, it is generally modeled as a nonstationary random process, which is described by the time-varying spectrum. Currently, there are many different ways to define the time-varying spectrum of nonstationary random processes. Among them, the evolutionary spectrum is widely utilized due to its clear physical significance, which is also employed in this study. For simplicity, the extended Kaimal spectrum is used to describe the target evolutionary spectrum of fluctuating wind acting on the transmission tower.