# Fragility Analysis of Transmission Towers Subjected to Downburst Winds
## 2 Fragility Analysis Framework
### Downburst Wind Load Model
Assume that the nonuniform modulated evolutionary power spectral density function (EPSD) \(S(z_{j};\omega,t)\) of the point at the height \(z_{j}\) can be prescribed by  
\[S(z_{j},\omega,t)=\left|A(z_{j},\omega,t)\right|^{2}S(z_{j},\omega) \tag{6}\]  
where \(A(z_{j},\omega,t)\) and \(S(z_{j},\omega)\) are, respectively, the modulation function and the spectrum of the corresponding stationary process, which can be expressed as [38]. \[A(z_{j},\omega,t)=\sqrt{\frac{\overline{U}(z_{j},t)}{\overline{U}(z_{j})} \left[\left(1+50\frac{\omega z_{j}}{2\pi\overline{U}(z_{j})}\right)/\left(1+ 50\frac{\omega z_{j}}{2\pi\overline{U}(z_{j},t)}\right)\right]^{5/3}} \tag{7}\]  
\[S(z_{j},\omega)=\frac{1}{2}\frac{200}{2\pi}u_{*}^{2}\frac{z_{j}}{\overline{ U}(z_{j})}\frac{1}{\left(1+50\frac{\omega z_{j}}{2\pi\overline{U}(z_{j})}\right)^{5/3}} \tag{8}\]  
where \(\omega\) is the circular frequency; \(\overline{U}(z_{j})=\frac{1}{T}\int_{0}^{T}\overline{U}(z_{j},t)dt\) is the temporally average wind speed of the time-varying mean wind speed; \(u_{*}=k\cdot\overline{U}(z_{j})\cdot\left[\ln(z/z_{0})\right]^{-1}\) is the friction velocity; \(z_{0}\) is the roughness of the ground; \(k\) is the Von Karman constant. Apart from the auto EPSD, the cross EPSD between the points at the height of \(z_{j}\) and \(z_{k}\) can be given by  
\[S(z_{j},z_{k},\omega,t)=A(z_{j},\omega,t)A(z_{k},\omega,t)\sqrt{S(z_{j},\omega )}\sqrt{S(z_{k},\omega)}\Gamma(\omega) \tag{9}\]where \(S(z_{j},z_{k},\,\omega,t)\) is the cross EPSD; \(\Gamma(\omega)\) is the Davenport time-invariant coherence function model, which can be expressed as  
\[\Gamma(\omega)=\exp\left\{-\frac{\omega}{2\pi}\frac{\sqrt{C_{y}^{2}(y_{j}-y_{k} )^{2}+C_{z}^{2}(z_{j}-z_{k})^{2}}}{\frac{1}{2}\big{[}\overline{U}(z_{j})+ \overline{U}(z_{k})\big{]}}\right\} \tag{10}\]  
where \(y_{j}\) and \(y_{k}\), respectively, denote the coordinate of the spatial point along the transmission line; \(C_{y}\) and \(C_{z}\) are, respectively, the decay factor corresponding to the \(y\) and \(z\) direction. Given the spectral matrix of downburst winds, the spectral representation method can be used to generate the wind speed time history [39; 40]. Once the wind speed time history has been determined, the wind load can be obtained according to quasi-steady assumptions and the Chinese specifications (DL/T 5154-2012) [41]. When the wind attack angle is zero degrees, the wind load of the transmission tower can be expressed as  
\[W_{sb}(z,t)=1/2C_{Db}\rho U^{2}(z,t)A_{b} \tag{11}\]  
where \(\rho=1.225\,\mathrm{kg/m^{3}}\) is the air density; \(C_{Db}=2.5\) is the drag coefficient corresponding to a zero-degree wind attack angle; \(A_{b}\) is the structural projected area of windward corresponding to a zero-degree wind attack angle. Similarly, when the wind attack angle is 90 degrees, the wind load of the transmission tower can be expressed as  
\[W_{sa}(z,t)=1/2C_{Da}\rho U^{2}(z,t)A_{a} \tag{12}\]  
where \(C_{Da}\) is the drag coefficient corresponding to a 90-degree wind attack angle; \(A_{a}\) is the structural projected area of windward corresponding to a 90-degree wind attack angle. For the results of other wind attack angles, the reader can be referred to as Table 1, which is used in this study. The wind load of transmission line is given below. \[W_{x}=\alpha\mu_{z}\mu_{sc}w_{0}\beta_{c}dl_{p}sin^{2}\theta \tag{13}\]  
where \(W_{x}\) represents the standard value of wind load perpendicular to the transmission line; \(\alpha\) is the wind pressure nonuniformity coefficient; \(\mu_{z}\) is the height variation coefficient of wind pressure; \(\mu_{sc}\) is the shape coefficient on wind load of the transmission line; \(w_{0}\) is basic wind pressure; \(\beta_{c}\) is the wind vibration coefficient of the transmission line; \(d\) is the outer diameter of the transmission line; \(l_{p}\) is the horizontal span of the tower; \(\theta\) is the included angle between the direction of transmission line and the wind direction.