## 4 Finite Element Analysis
### Damage Analysis  
The calculation results for models A2 LO and A3 LO are shown in Figures 9 and 10, respectively. The results indicate that the finite element analysis results calculated by ABAQUS were generally consistent with the results obtained from SAP2000. The maximum stress occurred at the purlin hanger joint connections, while the maximum deformation appeared in the mid-span region of the purlins. However, the finite element analysis results were slightly different from those from SAP2000. This discrepancy was because finite element analysis could not account for detailed structural measures of the joints or accurately capture minor deformations at the location of the purlin hanger. The deformation at the connections between the purlins and the beams, as well as at the ends of the braces, is illustrated in Figure 11.  
Figure 8: FEA model established using ABAQUS. Figure 10: Analysis results for A3 LO. Figure 9: Analysis results for A2 LO. As shown in Figure 11, the deformation occurred at the purlin hanger and the end of the braces, which is consistent with the damage observed in Figure 5. The finite element analysis effectively validated the relationship between the deformation of photovoltaic supports and their connection configurations. When the purlin hanger was connected using two bolts, significant lateral displacement along the purlin and rotation around the bolt were observed in A2 LO and B2 LO. Additionally, the lateral displacement at the end of the braces in A2 LO and B2 LO was significantly greater compared to models with three bolts at the purlin hanger. The lateral displacement and rotation angles of the purlin hanger in A2 LO and B2 LO were substantially higher than those in A3 LO and B3 LO. This indicated that the connection with the two bolts at the purlin hanger failed to provide effective constraints between the beams and purlins, resulting in the considerable rotation of the purlin hanger and beam around the bolt. The maximum rotation angles of the purlin hanger in A2 LO and B2 LO were 6.21\({}^{\circ}\) and 5.78\({}^{\circ}\), respectively, which were more than 11 times greater than those in A3 LO and B3 LO. Additionally, when the purlin hanger rotated, the brace connected to the beam experienced out-of-plane deformation. The deformation that occurred at the end of the brace was related to the deformation of the purlin hanger. The lateral displacement of the braces  
Figure 11: Deformation at the joints of the support. in A2 LO and B2 LO was significantly greater than in A3 LO and B3 LO. The maximum lateral displacements of the braces in Array A and Array B were 5.03 mm and 4.73 mm, respectively. The lateral displacements of the end of the braces in A3 LO and B3 LO were less than 38% of those in A2 LO and B2 LO. This demonstrates that using three bolts to connect the beam and purlin at the purlin hanger effectively limited the deformation of the purlin hangers. The on-site damage observations and finite element analysis results are consistent. The deformations of photovoltaic supports were primarily due to inadequate joint connections. As depicted in Figure 12a, when the purlin hanger was connected to the purlin using a single bolt, the connection lacked sufficient stability, causing the purlin hanger to rotate under construction deviations and applied loads. Conversely, using two bolts for the joint connection, as shown in Figure 12b, effectively restricted the rotation of the purlin hanger, ensuring greater stability. Additionally, as illustrated in Figure 12c, flattening the ends of the round steel pipe braces reduced their out-of-plane stiffness, making them susceptible to out-of-plane instability and excessive deformation when the purlin hanger rotated. This, in turn, could lead to damage to the photovoltaic components and fluctuations in the panel surface, compromising the overall reliability of the photovoltaic supports.