# Dynamic Response of Slender Vertical Structures Subjected to Thunderstorm Outflows
## 2 Analytical Formulation
### Dynamic Response  
Let us consider a slender vertical structure schematized as a linear elastic continuous system. The alongwind force per unit length due to the wind speed in Equation (1) reads:  
\[f(z,t)=\frac{1}{2}\rho v^{2}(z,t)b(z)c_{D}(z) \tag{7}\]  
where \(b(z)\) is the width of the face exposed to the wind. Substituting Equation (1) into Equation (7), the wind loading can be expressed as the sum of a mean and fluctuating part as follows:  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]  
Invoking the hypothesis of small turbulence, the mean, and the fluctuating parts of the loading read, respectively:  
\[\overline{f}(z,t)=\frac{1}{2}\rho\overline{v}_{\max}^{2}(h)\gamma^{2}(t) \alpha^{2}(z)b(z)c_{D}(z) \tag{9}\]  
\[f^{\prime}(z,t)=\rho\overline{v}_{\max}^{2}(h)\overline{I}_{\
u}(h)\gamma^{2} (t)\alpha^{2}(z)\theta(z)b(z)c_{D}(z)\bar{\
u}^{\prime}(z,t) \tag{10}\]  
It can be observed in Equation (10) that the fluctuating part of the loading is a uniformly modulated process. Let us assume that the dynamic response to thunderstorm outflows is dominated by the first mode of vibration; therefore, the alongwind displacement reads:  
\[x(z,t)=\psi_{1}(z)p_{1}(t) \tag{11}\]where \(\psi_{1}(z)\) is the first modal shape and \(p_{1}(t)\) is the first principal coordinate, the solution of the equation of motion:  
\[\ddot{p}_{1}(t)+2\omega_{1}\xi\dot{p}_{1}(t)+\omega_{1}^{2}p_{1}(t)=\frac{1}{m_{ 1}}f_{1}(t) \tag{12}\]  
where \(\omega_{1}=2\pi n_{1}\) is the first circular modal frequency, \(m_{1}\) is the first modal mass, and \(f_{1}(t)\) is the first modal force defined as:  
\[f_{1}(t)=\int\limits_{0}^{H}f(z,t)\psi_{1}(z)dz \tag{13}\]  
where \(H\) is the height of the structure. Substituting Equations (8)-(10) in Equation (13), the modal force can also be written as the sum of a mean and fluctuating part:  
\[f_{1}(t)=\overline{f_{1}}(t)+f_{1}^{\prime}(t) \tag{14}\]  
where \(\overline{f_{1}}(t)\) and \(f_{1}^{\prime}(t)\) read:  
\[\overline{f_{1}}(t)=\frac{1}{2}\rho\overline{v}_{\max}^{2}(h)\gamma^{2}(t)a_{1} \tag{15}\]  
\[f_{1}^{\prime}(t)=\rho\overline{v}_{\max}^{2}(h)\overline{I}_{v}(h)a_{1} \gamma^{2}(t)\tilde{a}_{1}^{\prime}(t) \tag{16}\]  
respectively, where \(a_{1}\) has the dimension of an area and can be interpreted as the area perceived by the wind on the first mode, while \(\tilde{a}_{1}^{\prime}(t)\) is a non-dimensional function of time. The two quantities are defined as follows:  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]0  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]1  
Substituting Equation (14) in Equation (12), the first principal coordinate can be decomposed as follows:  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]2  
where the mean part \(\overline{p}_{1}(t)\) is assumed as the static response to the mean part of the loading \(\overline{f}_{1}(t)\) (Equation (15)):  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]3  
The fluctuating part \(p_{1}^{\prime}(t)\) is characterized by its EPSD:  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]4  
where \(Z_{1}(n,t)\) is the evolutionary frequency response function (EFRF) related to the first mode, and \(S_{\tilde{a}_{1}^{\prime}}(n)\) is the PSD of \(\tilde{a}_{1}^{\prime}(t)\):  
\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]5\[f(z,t)=\overline{f}(z,t)+f^{\prime}(z,t) \tag{8}\]6  
where \(h_{1}(t)\) is the impulse response function associated with the first mode of vibration. It is worth pointing out that the generalization to MDOF of the EFRF method is pursued here following a mode-by-mode approach (Equation (21)), which offers the advantage of overcoming potential limitations of the EFRF when applied to MDOF systems [37].