## 5 Results
### Influence of Wind Fluctuating Effect
#### 5.2.2 Failure Probability Analysis  
By substituting the fragility functions derived from the static and dynamic analyses into Equations (2) and (3), respectively, the corresponding failure probability densities are calculated and presented in Figure 16. Within the loading range selected in this paper, the shapes of the failure probability density surfaces obtained from the static and dynamic analyses exhibit significant differences. Specifically, due to the fluctuating effect of the wind, the failure probability peaks at an ice thickness of 10.125 mm in the dynamic analysis, whereas in the static analysis the peak failure probability density aligns with an ice thickness of 25 mm. The corresponding wind speeds for these maximum failure probability densities are 10.08 m/s and 9.96 m/s, respectively. It can be observed that in the dynamic analysis, the failure probability density corresponding to an ice thickness of 25 mm is lower than that of 10.125 mm. The reason for this phenomenon is that when ice thickness increases from 10.125 mm to 25 mm, although the exceedance probability increases from 0.0217 to 0.1156, the joint probability density of ice and wind decreases from 4.193 \(\times\) 10\({}^{-5}\) to 5.774 \(\times\) 10\({}^{-6}\). In this case, the joint probability distribution of ice and wind exerts a more significant influence than the fragility, resulting in a downward trend in the failure probability density. This phenomenon indicates that, after considering the fluctuating effect of wind, even smaller ice loads can potentially lead to the failure of the transmission tower-line system.  
Table 8 presents the cumulative failure probabilities of the tower-line system obtained from both static and dynamic analyses. After considering the fluctuating effect of wind, the cumulative failure probability of the tower-line system increases by 4.7 times. Furthermore, for the scenario examined in this paper, accounting for the joint probability distribution of wind speed and direction results in a notable reduction in the failure probability of the transmission line under combined ice and wind loads. This phenomenon underscores the significance of considering joint distribution, as neglecting it can yield inaccurate failure probability estimates for transmission tower-line systems.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ice** & \multicolumn{2}{c}{**Wind Speed 10 m/s**} & \multicolumn{2}{c}{**Wind Speed 15 m/s**} & \multicolumn{2}{c}{**Wind Speed 24 m/s**} \\
**Thickness** & **Dynamic** & **Static** & **Dynamic** & **Static** & **Dynamic** & **Static** \\ \hline
5 mm & 0.77\% & 0.07\% & 9.51\% & 2.12\% & 49.26\% & 24.43\% \\
10 mm & 2.02\% & 0.33\% & 17.46\% & 5.91\% & 63.88\% & 41.12\% \\
15 mm & 3.95\% & 0.91\% & 26.03\% & 11.37\% & 74.19\% & 55.2\% \\
20 mm & 6.86\% & 2.06\% & 35.5\% & 18.72\% & 82.11\% & 67.35\% \\
25 mm & 11.14\% & 4.15\% & 45.81\% & 28.11\% & 88.23\% & 77.58\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: The comparison of exceedance probabilities.  
Figure 16: The failure probability density surfaces: (**a**) static analysis; (**b**) dynamic analysis.