## 6 Discussion  
This study proposes a mechanical model for calculating the stability of TSWs under local complex loads. The model computes the stress distribution of TSWs under different operational conditions and determines the yield state of TSWs using the D-P yield criterion. By comparing the results with two existing theoretical models, it is found that when boundary stresses in the y- and z-directions are considered, the deflection and bending moment coefficients of the TSW increase by approximately 5%. Furthermore, because the boundary stresses in the y- and z-directions are different, the bending moment coefficients in each direction are also different. This is because the boundary stresses introduce additional bending deformation, which further increases the stress on the TSW. When the boundary conditions are assumed to be a simple support, a 52% deviation in the bending moment coefficient is observed. This is because the reduced constraint from the simple support leads to more significant bending deformation, resulting in significantly higher calculated stresses. This indicates that the model presented in this study can more effectively and fully account for the influence of in situ stresses and boundary conditions. Based on the analysis of the mechanical model, the influence of various variables, including lateral load position (LP), lateral load width (LW), sidewall thickness (ST), sidewall length (SL), and sidewall height (SH), on the stress distribution and yield zone of TSWs is studied. The results show that as the LP increases, the symmetry of the TSW's stress state and yield zone gradually improves. Moreover, the MPS at characteristic locations of the TSW also becomes more uniform with an increasing LP, as shown in Figures 5-7. As the LP moves closer to the center of the TSW, the stress concentration at the left boundary weakens, and the overall stress distribution becomes more even. Additionally, as the LW increases, both the MPS and yield zone at characteristic locations of the TSW increase, due to the larger lateral load area causing more bending deformation. When the ST and SL increase and the SH decreases, the stress distribution becomes more uniform, with fewer yield zones. This is due to the increased bending stiffness from a larger ST and the enhanced ductility from a larger SL and a smaller SH, which improve the overall stability of the TSW. Calculations of the critical thickness for different operational conditions show that to improve the recovery rate of residual ore, it is necessary to maximize SL and minimize LP, LW, and SH. Finally, using the theoretical model presented in this study, the critical thickness of four TSWs in the residual ore recovery project was calculated. Furthermore, stability analyses were conducted at the V3412 and V3301 mining sites using the theoretical model, numerical simulations, and on-site visual inspections, further verifying the applicability and feasibility of the mechanical model in engineering practice. Furthermore, natural rock masses are heterogeneous and anisotropic, typically containing a series of structural surfaces that control the stability of TSWs. As such, the mechanical model used in this study may provide unrealistic values under certain conditions. However, when a TSW contains a large number of well-developed joints, the model can still offer reasonable solutions. This model is acceptable when the TSW is homogeneous and isotropic. If a TSW contains only a few joints or structural surfaces, the model should not be applied. In future studies, we plan to use theoretical mechanical models, laboratory tests, numerical simulations, and field tests to fully consider the heterogeneity and anisotropy of the rock mass. We aim to develop a more realistic mechanical model, and through the deployment of stress and displacement meters in field experiments, monitor the changes in stress and displacement fields during actual mining operations to better calibrate the theoretical model, improving its accuracy and effectiveness for rock engineering applications.