## 3 Application and Illustration of the Proposed CRITIC-GRA Model
### Stability Index of Open Stoping
#### 3.2.2 Stability Results with FLAC3D Simulations  
According to the preset schemes in Table 1 of different stope dimensions and the mining sequence for typical primary and secondary stopes shown in Figure 1, the stability of the surrounding rockmass in side walls and roof of excavated stope voids was simulated using FLAC3D. An overview of the model and typical divided stopes can be seen in Figure 4, along with two typical cross sections along the stope width and stope length. The Mohr-Coulomb criterion is widely recognized for its simplicity and is commonly employed. Consequently, the backfill and rock was modeled as an elasto-plastic material obeying the Mohr-Coulomb criterion. For the boundary conditions of the numerical model shown in Figure 4c,d, the bottom boundary was fixed in all directions. The four lateral surfaces were applied with the stress boundaries in the normal directions based on the variation of the actual in situ stresses with depth \(h\) from the ground surface in the mine. The investigation and field tests on the in situ stress were previously carried out in the mine from the levels of \(-\)160 m to \(-\)500 m. The obtained maximum horizontal principal stress \(\sigma_{h\ max}\) (MPa), the minimum horizontal principal stress \(\sigma_{h\ min}\) (MPa) and the vertical stress \(\sigma_{v}\) (MPa) of the mine given in Equations (17)-(19), respectively, increased linearly with the mining depth \(h\) from the ground surface. \[\sigma_{h\ max} = 0.0569H-2.1017 \tag{17}\] \[\sigma_{h\ min} = 0.0363H-1.8183\] (18) \[\sigma_{v} = 0.03H-1.1633 \tag{19}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline RQD (\%) & \(J_{n}\) & \(J_{r}\) & \(J_{a}\) & \(J_{w}\) & \(SRF\) & **A** & **Roof B** & **Wall B** & **Roof C** & **Wall C** \\ \hline
90 & 4 & 3 & 1.5 & 1 & 1 & 0.45 & 0.8 & 0.5 & 1 & 8 \\ \hline \end{tabular}
\end{table} 
Table 2: Rockmass properties of illustrated iron mine for Stability Index \(M\). \begin{table}
\begin{tabular}{l l l l l l l l l l l l} \hline \hline
**Preset** & \multicolumn{3}{l}{**Slope Dimensions**} & \multicolumn{3}{l}{**Roof**} & \multicolumn{3}{l}{**Wall**} \\ \cline{2-10}
**Schemes** & **Length** & **Width** & **Height** & **Stability** & **Shape** & **State** & **Stability** & **Shape** & **Stability** & **Shape** & **State** & **Stable** \\ \(L\) & **W** & **H** & **Index** & **M** & **Factor** \(S\) & **State** & **Probability** & **Index** & **M** & **Factor** \(S\) & **State** & **Probability** \\ \hline  
ail & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stability analysis of open stopes with the updated Mathews graph method. For the simulations of horizontal initial stresses in the numerical models, the field-tested maximum principal stress \(\sigma_{h\;max}\) and minimum principal stress \(\sigma_{h\;min}\) were applied to the external lateral boundaries, where the direction of the maximum principal stress \(\sigma_{h\;max}\) was perpendicular to the strike of the ore body and the direction of the minimum principal stress \(\sigma_{h\;min}\) was orthogonal to \(\sigma_{h\;max}\). In the vertical direction, the whole model extended to the ground surface and there was no stress boundary at the top surface boundary which led to an initial vertical stress based on the gravity simulations of the whole model. In addition, as shown in Figure 4c,d, the external expansions from side walls of the stopes to the outer side boundaries were set to be more than 400 m to eliminate the domain effect in numerical simulations. After a series of sensitivity analysis, the mesh for the central stopes was set as a uniform cube grid of 1 m on each side, and the outward mesh was radial to reduce the total number of grids. Based on the lab test results of rock samples which were further calibrated with the geological structures survey and quality evaluation of rockmass, the physical and mechanical parameters of the rockmass used in the numerical models are listed in Table 4, which were consistent with the input parameters for the Mathews graph method.