# Dynamic Response of Slender Vertical Structures Subjected to Thunderstorm Outflows
## 2 Analytical Formulation
### Maximum Dynamic Response: Thunderstorm Gust Response Factor Technique  
In this section, the derivation of the maximum dynamic response is outlined by employing the TGRF technique [13; 38], benefiting from the closed-form solution proposed by [PERSON] and [PERSON] [25]. The TGRF constitutes a generalization of the gust response factor from Davenport [22] for thunderstorms, accounting for the transient dynamic effects due to the non-stationarity of the loading and the presence of a background wind [13; 38]. It is defined as follows:  
\[G_{x,T}=1+2g_{x}(\widetilde{v}_{x}\widetilde{T}_{eq})I_{v}(z_{eq})\mathcal{C }\sqrt{B^{2}+R^{2}} \tag{24}\]  
where \(g_{x}\) is the Davenport peak factor [39], \(B\) and \(R\) are the background and resonance factors, respectively, and \(\widetilde{v}_{x}\) is the normalized expected frequency. They read as follows [40; 41; 42]:  
\[B^{2}=\frac{1}{1+0.9\Big{[}\frac{b+H}{L_{v}(z_{eq})}\Big{]}^{0.63}} \tag{25}\]  
\[R^{2}=\frac{\pi}{4\xi}S_{\overline{x}}(z_{eq},n_{1})\chi_{H}(\eta_{H})\chi_{b }(\eta_{b}) \tag{26}\]  
\[\eta_{\varepsilon}=\frac{4n_{1}\varepsilon}{\overline{v}_{\max}(z_{eq})} \qquad\varepsilon=H,b \tag{27}\]  
\[\widetilde{v}_{x}=\sqrt{\frac{R^{2}}{B^{2}+R^{2}}} \tag{28}\]  
where \(\chi_{\varepsilon}\) (\(\varepsilon=H,b\) ) accounts for the partial correlation of the turbulence along \(H\) and \(b\) in virtue of the equivalent wind spectrum (EWS) technique [40; 42], which operates solely on the reduced turbulent fluctuations \(z_{eq}=0.6H\), and \(L_{v}\) is the integral length scale of the reduced turbulence. In Equation (24), \(\mathcal{C}\) and \(\widetilde{T}_{eq}\) are equivalent parameters that can be estimated by employing the closed-form solution proposed by [25]:  
\[\mathcal{C}^{2}=\frac{\gamma^{*20}\Big{(}\frac{\widetilde{T}_{\max}}{T_{tot} }-1\Big{)}+\Lambda^{5}-\frac{5}{2}\Lambda^{4}\Phi+\frac{10}{3}\Lambda^{3}\Phi^ {2}-\frac{5}{2}\Lambda^{2}\Phi^{3}+\Lambda\Phi^{4}-\frac{1}{6}\Phi^{5}}{\gamma^ {*16}\Big{(}\frac{\widetilde{T}_{\max}}{T_{tot}}-1\Big{)}+\Lambda^{4}-2\Lambda ^{3}\Phi+2\Lambda^{2}\Phi^{2}-\Lambda\Phi^{3}+\frac{1}{5}\Phi^{4}} \tag{29}\]  
\[\widetilde{T}_{eq}=\frac{\gamma^{*16}\Big{(}\widetilde{T}_{\max}-\widetilde{T }_{tot}\Big{)}+\widetilde{T}_{tot}\Big{(}\Lambda^{4}-2\Lambda^{3}\Phi+2\Lambda ^{2}\Phi^{2}-\Lambda\Phi^{3}+\frac{1}{5}\Phi^{4}\Big{)}}{\left[\mathcal{C}^{2 }\right]^{4}} \tag{30}\]  
where \(\gamma^{*}\) is defined as the ratio between the background mean wind speed and the maximum mean wind velocity of the thunderstorm outflow [20; 38], and the parameters \(\widetilde{T}_{\max}\), \(\widetilde{T}_{tot}\), \(\Lambda\), and \(\Phi\), which are expressed as follows:  
\[\widetilde{T}_{\max}=T_{\max}n_{1} \tag{31}\]  
\[\widetilde{T}_{tot}=\widetilde{T}_{1}+\widetilde{T}_{N} \tag{32}\]\[\Lambda=\gamma^{*4}+\Phi \tag{33}\]  
\[B^{2}=\frac{1}{1+0.9\Big{[}\frac{b+H}{L_{v}(z_{eq})}\Big{]}^{0.63}} \tag{25}\]0  
where \(T_{\max}=600\) s, and:  
\[B^{2}=\frac{1}{1+0.9\Big{[}\frac{b+H}{L_{v}(z_{eq})}\Big{]}^{0.63}} \tag{25}\]1  
\[\widetilde{T}_{N}= \begin{cases}\widetilde{T}_{2},&\widetilde{T}_{2}\leq \widetilde{T}_{\max}/2\\ \widetilde{T}_{\max}/2,&\text{otherwise}\end{cases}  \tag{36}\]  
\[\widetilde{T}_{2}=\widetilde{T}_{1}\bigg{(}1+\frac{1}{2\widetilde{\xi} \widetilde{T}}\bigg{)} \tag{37}\]  
\[\beta=\frac{1}{1+\frac{1}{4\widetilde{\xi}\widetilde{T}}} \tag{38}\]  
\[\widetilde{T}=Tn_{1} \tag{39}\]  
where \(T\) represents the duration of the intense phase of the outflow [20; 38]. Once the TGRF is calculated, the maximum response is estimated as follows:  
\[x_{\max}=\overline{x}_{\max}G_{x,T} \tag{40}\]  
where \(\overline{x}_{\max}\) is the maximum value of the mean part of the response evaluated at \(z=H\) (\(\psi_{1}(H)=1\)), that reads (Equations (11) and (20)):  
\[\overline{x}_{\max}=\frac{1}{2m_{1}(2\pi n_{1})^{2}}\rho\overline{v}_{\max} ^{2}(h)a_{1} \tag{41}\]  
Finally, it is worth observing that for the stationary case, \(\gamma^{*}\to 1\); thus, \(\widetilde{T}_{eq}\rightarrow\widetilde{T}_{\max}\) and \(\mathcal{C}\to 1\).