4.2.1. Time History Analysis of TC-PTS Wind-Induced Vibration Response
The time history analysis of the wind-induced vibration response of TC-PTS was performed by considering the working conditions of the wind speed
and wind direction angle
. The standard deviation (
σ) of the lateral displacement time history results at each node position of a single span conductor in TC-PTS is shown in
Figure 7, and it can be seen that the standard deviation of lateral displacement in the center of the conductor is the largest, which is 2.34 m, indicating that the location in the center of the conductor is more significantly affected by wind load, and the standard deviation values of lateral displacement in the remaining locations are roughly symmetrical along the location in the center of the conductor. The lateral displacement time histories at the center of the conductor are shown in
Figure 8, and the average and maximum values of the lateral displacement time histories in
Figure 8 are listed in
Table 2. The results in
Table 2 and
Figure 8 show that the average value of lateral displacement in the conductor under wind load in this condition is 23.33 m, and its maximum value is 29.70 m. The lateral displacement is distributed from the minimum value of 16 m to the maximum value of 29.70 m, with a large variation.
Similarly, the time history of the midpoint displacement of the conductor computed by ANSYS is plotted in
Figure 8 and
Table 2, whereas the computational efficiency of the proposed model is listed in
Table 3. From
Figure 8 and
Table 2, it can be observed that the relative errors of the mean and maximum values of the lateral displacement time history obtained by the proposed model were 0.13% and 0.83%, respectively, and its accuracy was high. Simultaneously, the lateral displacement time history curves obtained by the proposed model had a consistent trend with the ANSYS results, which had a high degree of agreement. The results in
Table 3 show that the computation time required by ANSYS to analyze the lateral displacement in the conductor was 35.88 min, whereas the computation time of the proposed model was 1.50 min, and its computational cost was only 4.19% of the former. Therefore, the proposed model can be applied to the wind-induced vibration nonlinear finite element analysis for TC-PTS more efficiently.
4.2.2. Effect of Different Wind Direction Angles on TC-PTS Wind-Induced Vibration Response
In this section, the influence of different wind direction angles (
θ taken as 0°, 45°, 60°, and 90°) on the maximum lateral displacement of the transmission line and the maximum tension of the supporting suspension cables of TC-PTS at
are investigated. The variation in the maximum lateral displacement of the transmission line and the maximum tension of the supporting suspension cable with the wind direction angle
θ is shown in
Figure 9. In
Figure 9a, it can be seen that the maximum lateral displacement of the transmission line increases with the increase in
θ, and initially it tends to be faster, and then slower, subsequently. In
Figure 9b, the maximum tension of the supporting suspension cable increases with the increase in
θ. The change in the incremental magnitude of the maximum lateral displacement of the transmission line was also owing to the stress-stiffening effect of the structure. Considering the conductor as an example, the maximum lateral displacements of the conductor were 0.0071, 21.15, 28.12, and 32.96 m for wind direction angles of 0°, 45°, 60°, and 90°, respectively, and the most unfavorable wind direction angle for the maximum lateral displacement of the transmission line was 90°. The maximum tension of the supporting suspension cables increased with the wind direction angle because the downwind projected area of the transmission line was larger than that of the supporting suspension cables. As the wind direction angle increased, the global wind load on the structure increased. Therefore, the wind direction angle had a more significant effect on both the lateral displacement of the transmission line and the tension of the supporting suspension cables, and the most unfavorable wind direction angle for the two-span TC-PTS was 90°.
4.2.3. Effect of Different Wind Speeds on the Wind-Induced Vibration Response of TC-PTS
Based on the conclusions of the above analysis, this section examines the effect of the change in wind speed
V10 on the maximum lateral displacement of the transmission line and the maximum tension of the supporting suspension cables under the most unfavorable wind direction angle of 90° working conditions. The variation in the maximum lateral displacement of the transmission line and the maximum tension of the supporting suspension cable with the wind speed
V10 at
is shown in
Figure 10. In
Figure 10a, it can be seen that the maximum lateral displacement of the transmission line increases with the increase in
V10, and the magnitude of the increase tends to be faster, initially, and becomes slow subsequently. This phenomenon is more obvious on the conductor. In
Figure 10b, the maximum tension of the supporting suspension cables increases with the increase in
V10 all the time. The change in the maximum lateral displacement increment of the transmission line is due to the stress stiffening effect of the structure. According to Equation (16), the element tangent stiffness increases with the increase in displacement, and the lateral stiffness of the structure also increases gradually with the increase in displacement. When the lateral stiffness is larger, the increase in lateral displacement of the transmission line will slow down.
The maximum lateral displacement of a transmission line refers to the greatest horizontal movement experienced by the line due to external forces or factors. In this study, the maximum lateral displacement of the conductor is taken as a case in point. The wind speeds considered are 5, 10, 15, 20, 25, and 30 m/s, while the corresponding maximum lateral displacements of the conductor are measured as 2.83, 11.03, 22.48, 32.96, 40.16, and 44.78 m, respectively. The wind speed is from 5 m/s, and with every increase of 5 m/s, the maximum lateral displacement of the conductor increases by 8.20, 11.45, 10.48, 7.21, and 4.62 m, respectively. Therefore, in the case of smaller wind speed, the increase in wind speed will be higher. The maximum lateral displacement of the conductor increases by 8.20, 11.45, 10.48, 7.21, and 4.62 m. Therefore, the increase in wind speed increases the maximum lateral displacement of the transmission line more significantly when the wind speed is low, whereas the maximum lateral displacement of the transmission line is less affected by the wind speed when the wind speed is high. The change in the maximum tension of the supporting suspension cables is due to the fact that the suspension tension is required to balance the equivalent nodal loads, and the increase in wind speed increases the suspension tension all the time.
For the maximum tension of supporting suspension cables, taking the maximum tension of supporting-conductor suspension cable as an example, the wind speed starts from 5 m/s, and for every increase of 5 m/s, the maximum tension of the supporting-conductor suspension cable increases by 6.59, 11.05, 15.62, 20.38, and 25.47 kN, respectively, and the increment of the maximum tension is approximately linearly related to the wind speed, and the maximum tension and the quadratic of the wind speed are approximately linearly correlated. When the wind speed is small, the maximum tension of the supporting suspension cable is less affected by wind speed, while when the wind speed is larger, the maximum tension of the supporting suspension cable is more significantly affected by wind speed.
Therefore, at lower wind speeds V10, the lateral displacement of the transmission line was more affected by the wind speed than the tension of the supporting suspension cable, and at higher wind speeds V10, the tension of the supporting suspension cable was more notably affected by wind speeds than the lateral displacement of the transmission line.