# Design Method of Primary Structures of a Cost-Effective Cable-Supported Photovoltaic System
## 3 Bearing Capacity and Failure Modes
### Bearing Capacity  
The internal forces of cables 1, 2, and 3 are calculated when the wind load \(w_{\rm k}\) varies from 0 to 2.0 kN/m\({}^{2}\) in the two control cases. The cross-sectional areas of the cables changed  
Figure 4: Overview of the new prototype CSPS and its FEM model.  
as the wind load increased. The internal forces of cables 1, 2, and 3 are also considered in relation to the required breaking forces of cables 1, 2, and 3 to ensure structural safety and are referred to as \(F_{1,2}\) and \(F_{3}\), respectively.  
The results of previous studies indicate that the \(F_{1,2}\) and \(F_{3}\) have different trends when the wind load is in the range of 0.15-2.0 kN/m\({}^{2}\). The static wind load is supported by three cables in Case 0\({}^{\circ}\), whereas only cables 1 and 2 bear the static wind load when the wind load is greater than 0.238 kN/m\({}^{2}\) in Case 180\({}^{\circ}\). In addition, the required breaking force of cables 1 and 2 in Case 0\({}^{\circ}\) (\(F_{1,2}^{0.1}\)) is always smaller than that in Case 180\({}^{\circ}\) (\(F_{1,2}^{180}\)). In contrast, the required breaking force of cable 3 in Case 0\({}^{\circ}\) (\(F_{3}^{0}\)) is larger than that in Case 180\({}^{\circ}\) (\(F_{3}^{180}\)), which means that Case 180\({}^{\circ}\) is the controlling condition of cables 1 and 2, and Case 0\({}^{\circ}\) is the controlling condition of cable 3 when the wind \(w_{\rm k}\) is in the range of 0 to 2.0 kN/m\({}^{2}\)[31].  
To ensure the structural safety of a given CSPS, the required breaking force should be lower than the cable resistance force. The load-bearing capacity was determined by the resistance forces of cables 1, 2, and 3. In Case 180\({}^{\circ}\), the bearing capacity of the CSPS (\(B_{1,2}\)) is calculated by setting the resistance force of cables 1 and 2 equal to \(F_{1,2}^{180}\), and setting the resistance force of cable 3 to be higher than \(F_{3}^{180}\). In Case 0\({}^{\circ}\), the load-bearing capacity of the CSPS (\(B_{3}\)) was calculated by setting the resistance force of cable 3 equal to \(F_{3}^{0}\), and setting the resistance force of cables 1 and 2 to be higher than \(F_{1,2}^{0}\). The load capacity of the CSPS was the minimum between \(B_{1,2}\) and \(B_{3}\).  
Figure 5 shows the vertical deformation at the midspan of the structure as the wind load increased in the two cases. The vertical deformation in Case 0\({}^{\circ}\) was downwards, and that in Case 180\({}^{\circ}\) was upward. In both cases, the midspan vertical deformation increased nonlinearly with increasing wind load, and the deformation increment decreased when the wind load increased. These results indicate that the vertical stiffness of the PV system increased with increasing deformation. When the wind load increases from 0.3 to 2.0 kN/m\({}^{2}\), the midspan vertical deformation increased by 226.5% and 190%, respectively. The midspan vertical deformation in Case 180\({}^{\circ}\) was always greater than that in Case 0\({}^{\circ}\) when the wind load was higher than 0.238 kN/m\({}^{2}\).