## 2 Tower Capacity
### Wind Loading  
Wind loads on structures is generally calculated by:  
\[F_{W}=\frac{1}{2}\rho_{air}V^{2}\cdot C_{D}A \tag{2}\]  
where \(\rho_{air}\) is the density of air, taken to be 1.235 kg/m\({}^{3}\), \(V\) is the wind speed, \(C_{D}\) is the drag coefficient, and \(A\) is the effective area of the wind loading. Here, we set the wind speed \(V\) as the 10-min averaged wind speed \(U\) at 10 m above the ground. The drag coefficient \(C_{D}\) should jointly take into account the vertical variance of wind speed, the combination effect of the wind fluctuation and structural vibration, and the shape factor of structures. Considering the effect of the wind angle of attack \(\varphi\) (i.e., the intersection angle between the wind direction and the direction transverse to the transmission line), the wind load directly acting on transmission towers is given by [20; 21]:  
Figure 1: Classification of tower members.  
\[H_{T}=\frac{1}{2}\rho_{air}U^{2}\cdot\sum_{i}\mu_{z}^{i}\beta_{z}^{i}\left(u_{s,t}^ {i}A_{s,t}^{i}\cos^{2}\phi+u_{s,t}^{i}A_{s,t}^{i}\sin^{2}\phi\right)\cdot\left[1+ 0.2\sin^{2}(2\phi)\right] \tag{3}\]  
where \(i\) is the section number of the tower, \(\mu_{z}\) is a combined wind factor accounting for the altitude and terrain effects, \(\beta_{z}\) is a gust response factor accounting for the wind fluctuation and structural vibration effects, and \(A_{s,t}\) and \(A_{s,t}\) and \(\mu_{s,t}\) and \(\mu_{s,t}\) are the projected areas and shape factors of the transverse and longitudinal faces of the tower, respectively. The term \([1+0.2\sin^{2}(2\phi)]\) is an amplification factor that related to the wind angle of attack. The wind load \(H_{T}\) is commonly decomposed into the transverse and longitudinal directions of the transmission line:  
\[H_{t}=\frac{1}{2}\rho_{air}U^{2}\cdot\sum_{i}u_{z}^{i}\beta_{z}^{i}\left(u_{s, t}^{i}A_{s,t}^{i}\cos^{2}\phi+u_{s,t}^{i}A_{s,t}^{i}\sin^{2}\phi\right)\cdot \left[1+0.2\sin^{2}(2\phi)\right]\cdot\cos\phi \tag{4}\]  
\[H_{t}=\frac{1}{2}\rho_{air}U^{2}\cdot\sum_{i}u_{z}^{i}\beta_{z}^{i}\left(u_{s, t}^{i}A_{s,t}^{i}\cos^{2}\phi+u_{s,t}^{i}A_{s,t}^{i}\sin^{2}\phi\right)\cdot \left[1+0.2\sin^{2}(2\phi)\right]\cdot\sin\phi \tag{5}\]  
where \(H_{t}\) and \(H_{t}\) are the wind load components transverse and longitudinal to the transmission line, respectively. The wind load transferred form transmission wires to the tower structure is given by [20; 21]:  
\[H_{w}=\frac{1}{2}\rho_{air}U^{2}\cdot\sum_{j}\mu_{z}^{i}\beta_{z,w}^{i}u_{s, w}^{j}d^{j}\cdot\sin^{2}(90^{o}-\phi)\cdot\alpha L_{h} \tag{6}\]  
where \(j\) is the number of transmission wires, \(\beta_{z,w}\) is the gust response factor of wires, \(\mu_{s,w}\) is the shape factor of wires, \(d\) is the outer diameter of the wires, \(L_{h}\) is the horizontal span of the transmission line, and \(\alpha\) is a span factor accounting for the span-wise non-uniform distribution of wind speed and determined by the wind speed \(U\). The calculations for the above interaction parameters between the wind and tower, and between the wind and transmission wires, are given in the Table 1 in Appendix A.  
The wind loading to a transmission tower is composed of three parts: \(H_{t}\), \(H_{t}\), and \(H_{w}\). For a given tower and transmission line, with the terrain type known, there are three independent variables that affect the wind loading to the tower structure, i.e., the wind speed \(U\), the wind angle of attack \(\varphi\), and the horizontal span of transmission line \(L_{h}\).