## 4 Finite Element Analysis
### Optimization Analysis of Purlin Hangers  
As noted in Section 4.2, one end of the purlin hanger was connected to the purlin, while the other end was connected to the beam, which is critical for the load-bearing capacity of the photovoltaic support structure. Just a slight rotational movement at the purlin hanger could easily compromise the stability of the structure. To enhance the stability of the joint connection of the purlin hanger, an innovative joint connection is proposed to connect the beam and the purlin. Schematic diagrams for the Z-shaped purlin hanger and \(\Pi\)-shaped purlin hanger are shown in Figure 13. Since the failure characteristics of Array A and Array B are similar when other parameters are the same, only Array A was analyzed further. The L-shaped purlin hanger was replaced with Z-shaped and \(\Pi\)-shaped purlin hangers, and the new finite element models were named A3 ZO and A3 JIO, respectively. Hinged connections were modeled when only one bolt was used to connect one side of the purlin hanger to the other side of the purlin. During the modeling process, all other parameters  
Figure 12: Deformation diagram. (**a**) One bolt used at the purlin hanger to connect the purlin and beam. (**b**) Two bolts used at the purlin hanger to connect the purlin and beam. (**c**) End of steel pipe was flattened. were the same as those of model A3 LO. The stress and deformation calculated for the purlin hangers in each model are shown in Figures 14 and 15. According to Figure 14, when the Z-shaped and \(\Pi\)-shaped purlin hangers were used to connect the beams and purlin, the maximum stress in the photovoltaic support structure occurred at the location of the purlin hanger. Under the applied loads, the purlin hanger was prone to deformation. The stress at the location of the purlin hanger and deformation along the purlin (U1) are summarized in Table 4.  
Figure 14: Analysis results for A3 ZO. Figure 13: Innovative purlin hanger connection design. Figure 15: Analysis results for A3 TIO. As shown in Table 4, the maximum stress in A3 ZO is 283.8 MPa, which is approximately 79% of the maximum stress in A2 LO. The maximum stress in A3 TIO is 258.8 MPa, about 72% of that in model A2 LO. Although A3 LO, A3 ZO, and A3 TIO all use three bolts at the location of the purlin hanger, the Z-shaped and \(\Pi\)-shaped purlin hangers effectively reduce the stress concentration at the location of the purlin hanger, with the maximum stress being only about 75% of that in A3 LO. Additionally, the lateral displacement of the purlin hangers in A3 LO, A3 ZO, and A3 TIO are all less than 18% of those in A2 LO. Notably, the lateral displacement of the purlin hanger in A3 TIO is only 9% of that in A2 LO, and the purlin hanger experiences almost no rotation. The maximum stress and U1 in A3 TIO are smaller than those in A3 ZO. The analysis results indicate that the use of three bolts at the location of the purlin hanger is essential to prevent the deformation of the photovoltaic support under load. Moreover, compared to the L-shaped purlin hanger, the Z-shaped and \(\Pi\)-shaped hangers effectively reduce the maximum stress at the location of the purlin hanger. Although the steel usage for the Z-shaped and \(\Pi\)-shaped purlin hangers is slightly higher than that of the L-shaped hanger, this increase is negligible in the overall photovoltaic support structure. A comparative analysis was conducted regarding the cost and practical application of the different purlin hangers. The cost of a steel structure is primarily determined by the amount of steel required. Table 5 presents the steel consumption calculations for the different purlin hangers. To comprehensively evaluate the steel consumption of Array A, the total steel consumption of the photovoltaic array (including purlin hangers) was also quantified.