## 5 Stability Analysis of TSWs
### Engineering Application of the Model and Its Reliability Verification  
While idealized mechanical models can accurately describe TSW deflection and bending in general, actual mining conditions involve complex secondary stress fields, requiring numerical simulations for comprehensive analysis [39; 40]. To validate the theoretical model's accuracy and applicability, four residual ore recovery projects located at an elevation of 330 m (average burial depth: 547 m) were examined. The projects were designed in consideration of ore body strike and inclination. Figure 12 shows the 3D models, depicting stope-goaf spatial relationships. The initial TSW stresses were calculated using the tributary area method. The input parameters from Table 5 were applied to the theoretical model to compute MPS values, stress locations, critical thicknesses, and yield zones. The outcomes are summarized in Table 6.  
According to Table 6, the reserved thicknesses in the V3412 and V3429 stopes are too large, while the reserved thickness for the V3301 stope is too small. The reserved thickness of the V32010 slope is close to the critical thickness. Recently, V3412 and V3301 stopes have been fully mined. To assess the validity of the theoretical model, numerical simulations and theoretical calculations of the plastic zone range for the V3412 and V3301 TSWs were conducted using Flac3D6.0 numerical simulation software. These results were then compared with on-site visual inspection to verify the accuracy of the theoretical model. The spatial configurations of the stope and goaf used in the numerical simulation model are illustrated in the 3D model diagrams in Figure 12a,d. The numerical simulation model has dimensions of 600 m \(\times\) 400 m \(\times\) 340 m and utilizes tetrahedral cells with a grid  
Figure 11: Contour image of the critical thickness of the TSW as the lateral load parameters. size of 2 m. The Mohr-Coulomb constitutive model is employed for this analysis. The rock mechanical parameters for the ore body and surrounding rocks are those listed in Table 2. Clamped boundary conditions are used for all boundaries except the top boundary. The initial ground stress conditions applied are denoted as \(\sigma_{\mathrm{z}}=15\)MPa, \(\sigma_{\mathrm{x}}=\sigma_{y}=12\) MPa. The stability analysis results for the TSWs of the V3412 and V3301 slopes are presented in Figures 13 and 14, respectively. A comparison of the maximum principal stresses (MPS) on TSWs at the V3412 and V3301 mining sites using both the mechanical model and numerical simulation is shown in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stope Number** & **V3412** & **V3429** & **V32010** & **V3301** & **Units** \\ \hline MPS & 1.46 & 2.05 & 0.49 & 6.94 & MPa \\ extremal point & (20.7, 0) & (22, 0) & (72, 0) & (0, 24) & m \\ critical thickness & 4.6 & 4.2 & 2.6 & 11.9 & m \\ yield or not & Not & Not & Not & Yes & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 6: TSW stress state, stability analysis and critical thickness. Figure 12: Three-dimensional design model diagram of residual ore recovery projects. (**a**) V3412 Slope; (**b**) V3429 Slope; (**c**) V32010 Slope; (**d**) V3301 Slope. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stope Number** & **V3412** & **V3429** & **V32010** & **V3301** & **Units** \\ \hline SL & 61 & 71 & 118 & 56 & m \\ Stope span & 2.3 & 2.6 & 3.0 & 1.9 & m \\ SH & 43 & 42 & 45 & 48 & m \\ Stope Inclination angle & 83 & 85 & 84 & 85 & \({}^{\circ}\) \\ LP & (20.7, 21.5) & (22, 21) & (72, 22.5) & (10.3, 24) & m \\ LW & 3.3 & 3.6 & 2.9 & 11 & m \\ ST & 9 & 6 & 10.7 & 6.4 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Characteristic parameters of the TSWs. The error between the mechanical model and numerical simulation results is no greater than 4%. The stability analysis results for the TSW at the V3412 mining site are shown in Figure 13. Both the numerical simulation, theoretical calculations, and on-site visual