## 3 Numerical Results
### Strength Requirements of the Backfill Roof with the Basket Weave Drift Layout  
Table 2 presents the iterative search process for assessing the strength requirement of the backfill roof in the case with a drift size of 3.5 m \(\times\) 3.0 m. Initially, the search range for the backfill cohesion was determined as 200-2000 kPa. During the iterative process, this range was progressively narrowed to approximate the critical cohesion of the backfill roof under limit equilibrium. After each simulation was completed, the maximum vertical displacement of the roofs at L2 and L3 was extracted, and the variations of the maximum vertical displacement of the backfill roof with different cohesion were plotted, as shown in Figure 6a.  
Figure 4: Vertical displacement distribution of the backfill roof at (**a**) L3 and (**b**) L2 after the excavation of secondary drifts at L3 (backfill cohesion \(c\) = 425 kPa).  
Figure 5: Vertical displacement distribution of the backfill roof at L3 after the excavation of secondary drifts at L3 (backfill cohesion \(c\) = 200 kPa).  
Figure 6a shows that the vertical displacements of the roofs at both L2 and L3 remain below 10 cm when the backfill cohesion is greater than 350 kPa. Within this range, the vertical displacements show no significant change as the backfill cohesion decreases, indicating that the backfill roof remains in a stable state. However, when the cohesion reaches 350 kPa, a sharp increase in roof displacement occurs, rising from less than 10 cm to over 60 cm for both layers. When the cohesion falls below 350 kPa, the vertical displacements of the roofs at both layers remain consistently above 50 cm, signifying that the roof has failed and become unstable. Consequently, the strength requirement of the backfill roof with a drift size of 3.5 m \(\times\) 3.0 m is determined as \(c_{f}\) = 350 kPa.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Backfill Cohesion** (**kPa**)} & \multirow{2}{*}{**Rood Stability**} & \multirow{2}{*}{**Backfill Cohesion for Next** (**kPa**)} & \multirow{2}{*}{**Note**} \\  & & & & \\ \hline
1 & 200 & unstable & 2000 & confirm the lower bound \\
2 & 2000 & stable & \(1100=(200+2000)\div 2\) & confirm the upper bound \\
3 & 1100 & stable & \(650=(200+1100)\div 2\) & current range [200, 1100] \\
4 & 650 & stable & \(425=(200+650)\div 2\) & current range [200, 650] \\
5 & 425 & stable & \(312.5=(200+425)\div 2\) & current range [200, 425] \\
6 & 312.5 & unstable & \(368.75=(312.5+425)\div 2\) & current range [312.5, 425] \\
7 & 368.75 & stable & \(340.625=(312.5+368.75)\div 2\) & current range [312.5, 368.75] \\
8 & 340.625 & stable & \(\ldots\) & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Iterative search process for evaluation of the strength requirement of the backfill roof with a drift size of 3.5 m \(\times\) 3.0 m.  
Figure 6: Variations in the maximum vertical displacement of backfill roof with different cohesion for cases with drift sizes of (**a**) 3.5 m \(\times\) 3.0 m, (**b**) 4.5 m \(\times\) 4.0 m, and (**c**) 5.5 m \(\times\) 5.0 m.  
Figure 6b,c illustrates the variations in the maximum vertical displacement of the backfill roof with different cohesion in cases with drift sizes of 4.5 m \(\times\) 4.0 m and 5.5 m \(\times\) 5.0 m, respectively. It is evident that a sharp change in displacement occurs at \(c\) = 410 kPa and 620 kPa, and the strength requirements of the backfill roof were determined accordingly. Clearly, the strength requirement increases with the drift size. This aligns with previous reports that the required strength of base-exposed backfill increases with the span [23; 28].