# Fragility Assessment and Reinforcement Strategies for Transmission Towers Under Extreme Wind Loads
## 4 Wind-Induced Response Under Strong Wind  
### Wind Angle Analysis  
The previously calculated wind loads were uniformly applied to the load-bearing nodes of the primary structural components and cross-arm elements. The Newton-Raphson iterative algorithm was adopted to solve the nonlinear equilibrium equations, with the following convergence criteria: residual force tolerance: \(\leq\)1.0 \(\times\) 10\({}^{-4}\); displacement increment threshold: less than 10% of the previous increment, and maximum iterations per load step: 50. The resulting stress and displacement distributions of the transmission tower, subjected to a wind speed of 33 m/s from four distinct wind directions, are illustrated in Figures 9 and 10. The analysis reveals that the maximum stress concentration occurs in the main structural member at the compression zone of the third segment, located near the mid-height of the tower. This region represents a critical structural vulnerability that could compromise overall stability under multidirectional wind loading. Despite the localized stress peaks, the stress levels in all primary members remain below the material yield strength, indicating that the structure operates within the elastic deformation range. As shown in Figure 10, the maximum displacement is observed at the tower top, which aligns with the expected deformation pattern of tall, slender structures subjected to lateral wind forces.  
Figure 8: Illustration of attack wind angles.  
Figure 10: The moment of maximum displacement contour.  
Figure 9: The moment of maximum stress contour.  
As illustrated in Figure 11, the maximum stress and displacement values under different wind directions are summarized to assess the directional sensitivity of the transmission tower's structural response. The analysis reveals that a wind direction of 45\({}^{\circ}\) induces the most severe loading conditions. At this angle, the stress in the primary structural components reaches 259.2 MPa--approximately 75% of the yield strength--while the maximum displacement at the tower top is 458.2 mm, equivalent to 1.14% of the tower's total height. Comparable trends are observed for the 0\({}^{\circ}\) and 60\({}^{\circ}\) wind directions, where stress and displacement values remain elevated but below those observed at 45\({}^{\circ}\). In contrast, the 90\({}^{\circ}\) wind direction results in the lowest stress (168.6 MPa) and minimal top displacement (302.3 mm), indicating a less critical loading scenario. The results exhibit a nonlinear relationship between wind direction and structural response. As the wind direction shifts from 0\({}^{\circ}\) (frontal wind) to 45\({}^{\circ}\), both stress and displacement increase, driven by a corresponding increase in the projected windward area. This area reaches its maximum at 45\({}^{\circ}\), leading to peak aerodynamic loading. Beyond this point, as the wind direction approaches 90\({}^{\circ}\) (lateral wind), the projected windward area decreases, resulting in a reduction in both stress and displacement. These findings highlight the pronounced effect of wind direction on structural performance, identifying 45\({}^{\circ}\) as the most critical wind angle for the tower.