# Design Method of Primary Structures of a Cost-Effective Cable-Supported Photovoltaic System
## 3 Bearing Capacity and Failure Modes
### Failure of Triangle Brackets  
Triangle brackets are analyzed in this subsection, and lateral connectors are not considered because they only enhance lateral stability. In Case 180\({}^{\circ}\), the axial force of the lateral connectors and the triangular brackets are close to 0 when the wind load is greater than 0.238 kN/m\({}^{2}\) because cable 3 no longer works; in Case 0\({}^{\circ}\), the axial force of the lateral connectors is much smaller than that of the triangular brackets. Normally, the lateral connectors are set the same as those of the triangular brackets. Case 0\({}^{\circ}\) was the controlling case for the triangle brackets, and the axial forces of all triangle brackets in Case 0\({}^{\circ}\) were investigated in detail. Each triangular bracket is composed of three slender steel pipes that bear only axial forces. The failure modes of the triangular brackets were either yielding or buckling. When the internal force of the steel pipes is larger than their yielding strength or buckling strength, the steel pipes fail and the CSPS breaks. The internal forces of the triangular brackets at the 1/5 and 2/5 spans, as shown in Figure 7, were calculated when the wind load \(w_{\rm k}\) varied from 0 to 2.0 kN/m\({}^{2}\). The results for Case 0\({}^{\circ}\) are shown in Figure 8. Rods 1, 3, 4, and 6 are in compression, whereas rods 2 and 5 are in tension. The maximum compressive force appears in rod 3, whereas the maximum tensile force appears in rod 2; the compression force is much higher than the tensile force. The internal forces of the rods increased with the wind load. Figure 6: Required cross-sectional area. The internal forces in rods 1-6 are considered in relation to the load capacity of the rods to ensure the structural safety of the proposed system and are referred to as \(P_{cr}^{1}\) through \(P_{cr}^{6}\), respectively. The rods should have sufficient cross-sectional area to withstand the required internal forces. The load capacity of the rod, \(P_{cr1}\), was calculated using Equation (3) while considering the yield strength of the rod. In addition, according to [PERSON]'s formula, the load capacity of a slender compression rod, \(P_{cr2}\), which is hinged at both ends, is calculated using Equation (4) while considering the buckling strength. The required load capacity of the rods is the minimum of \(P_{cr1}\) and \(P_{cr2}\):  
\[P_{cr1}=E\pi\Big{(}D^{2}-d^{2}\Big{)}\sigma \tag{3}\]  
\[P_{cr2}=\frac{\pi^{2}EI}{l^{2}}=\frac{\pi^{3}\big{(}D^{4}-d^{4}\big{)}E}{64l^{2}} \tag{4}\]  
where \(E\) is the elastic modulus of the rod, \(I\) is the minimum inertia moment of the cross section, \(l\) is the calculation length, \(\sigma\) is the yielding strength, and \(D\) and \(d\) are the outer and inner diameters, respectively. In this study, \(d\) is set equal to 0.01 m, and the required load capacity \(P_{cr}\) and outer diameter \([D]\) are calculated as follows:  
\[P_{\sigma}= \begin{cases}P_{cr1}&(\text{Tension rod})\\ \min(P_{cr1},P_{cr2})&(\text{Compression rod})\end{cases}  \tag{5}\]  
\[[D]= \begin{cases}\sqrt{\frac{P_{cr}}{E\pi\sigma}+d^{2}}&(\text{Tension rod}) \\ \max\biggl{(}\sqrt{\frac{P_{cr}}{\pi\sigma}+d^{2}},\sqrt[4]{\frac{64P_{cr}l^{2 }}{\pi^{3}E}+d^{4}}\biggr{)}&(\text{Compression rod})\end{cases}  \tag{6}\]  
Using Equation (6), the internal forces in Figure 8 were transformed into the required outer diameters. Figure 9 shows the required outer diameters of rods 1-6 (\(D_{1}\) to \(D_{6}\)) as the wind load increases. Rod 6 had the maximum required outer diameter, and rods 2 and 5 had the minimum required outer diameter. Wind strongly affects the required outer  
Figure 8: Internal force of the six slender rods in Case 0\({}^{\circ}\). Figure 7: Triangle brackets. diameters of the compression rods but has little effect on the tensile rods. The required outer diameters of the compression rods increased nonlinearly as the wind load increased. Triangle brackets exhibit two different types of failure: yielding and buckling.