# Dynamic Response of Slender Vertical Structures Subjected to Thunderstorm Outflows
## 3 Numerical Application
### Maximum Dynamic Response to Thunderstorms: Comparison among Different Approaches  
In this section, the maximum dynamic response estimated with the TGRF technique (Section 2.3) is compared with the one obtained using TDA and TRS [16]. The maximum response is estimated via TDA, starting from the set of wind speed records outlined in Section 3.2. To simplify the calculations, the EWS technique by [PERSON] [42] is employed, which consists of replacing the actual reduced turbulent fluctuation--the random function of time and space--with an equivalent turbulent fluctuation perfectly coherent in space. Assuming that the coherence function of the reduced turbulent fluctuation is expressed by Equation (6), the equivalent turbulent fluctuation is obtained following the steps reported by [PERSON] [18] and [PERSON] [16]. Once the equivalent wind speed is determined, the wind loading and the first modal force are derived by Equations (8)-(10) and (13), and the dynamic response is calculated by integrating the equation of motion into Equation (12). Finally, the maximum dynamic response is estimated by taking the maximum value of Equation (11), considering \(z=H\). Considering the TRS technique, the diagrams from [PERSON] and [PERSON] [16] are adopted. Since [16] does not furnish values of the TRS for \(\xi=0.005\), for the telecommunications tower, it is estimated through a linear interpolation between the values \(\xi=0.002\) and \(\xi=0.01\). Moreover, in order to provide a consistent comparison with the TRS, the turbulence intensity is assumed to be constant with the height--i.e., \(\theta(z)=1\)--and equal to the average value extracted from the data (Section 3.2). For the TDA, the maximum dynamic response is reported in terms of average value together with a confidence interval of one standard deviation. The comparison in terms of maximum response is reported in Figure 8 on varying the height of the nose \(z_{m}\).  
It can be observed that the three approaches are in good agreement with each other, especially for the telecommunications tower (Figure 8b). More specifically, for the lighting pole (Figure 8a), TDA and TRS furnish very close trends, almost constant with \(z_{m}\), while the TGRF furnishes slightly more conservative results which, however, lie within the confidence interval of one standard deviation obtained using TDA. Instead, the results obtained for the telecommunications tower (Figure 8b) show an almost perfect agreement. In this case, the maximum dynamic response tends to increase upon increasing \(z_{m}\). The different behavior of the trends with \(z_{m}\) observed for the two structures is explained by  
Figure 8: Maximum response evaluated with the three approaches (TDA in black lines and its value plus/minus one standard deviation in the grey area; TRS—blue lines; and TGRF—red lines): (**a**) lighting pole; (**b**) telecommunications tower.  
the proximity of the tip of the nose-shaped profile of the mean wind speed with their height, which produces a greater wind loading. For the lighting pole--which can be regarded as a low-rise structure--the more \(z_{m}\) is reduced, the more it approaches its height, and thus provides greater wind loading on the structure (Figure 7a). Conversely, for the telecommunications tower, the more \(z_{m}\) is increased, the more the nose tip reaches its top, providing greater wind speeds, and thus results in a greater wind loading on the structure (Figure 5b). This aspect reflects the well-known characteristic of thunderstorm outflows, i.e., the threat they provide for mid-low structures.  
Overall, from the results shown in Figure 8, it can be concluded that the three methods adopted for the estimate of the maximum response are in good agreement with each other. In particular, TRS furnishes a good estimate, but its derivation may not be so straightforward due to it being based on diagrams; the TGRF may provide some overestimation for low-damped structures; however, it constitutes a reliable and handy tool for engineers in virtue of the closed-form solution provided.