## 4 Discussion  
### Comparison of the Numerical Results with Analytical Solutions  
The strength requirement of the backfill roof with the basket weave drift layout in underhand drift-and-fill mining has been investigated through numerical simulations with FLAC3D. The iterative search method used, which determines the stability state based on sharp changes in roof displacement, can effectively incorporate the influence of specific loading conditions into the estimation of strength requirements for the backfill roof. In addition to numerical simulation methods, analytical methods are also commonly used to assess the strength requirement of the backfill roof. Commonly adopted analytical solutions include the simply supported beam model [29], the fixed-ends beam model [30], the thin plate model [31], and the thick plate model [5; 32]. Using these models, the stress components within the backfill roof under external loads can be calculated, enabling the determination of the maximum tensile stress. Since the tensile strength of cemented backfill is typically about 1/10 of its UCS [20; 21], tensile failure is generally regarded as the dominant factor contributing to roof instability. Therefore, when the maximum tensile stress within the backfill roof exceeds the tensile strength, the roof is considered unstable. Based on this criterion, the strength requirement of the backfill roof can be determined in terms of the tensile strength. (1) The simply supported beam model  
The simply supported beam model is widely used to estimate the strength requirement of the backfill roof, as shown in Equation (3)  
\[\sigma_{t}>\bigg{(}\frac{3W^{2}}{4H^{2}}+\frac{1}{5}\bigg{)}q \tag{3}\]  
where \(\sigma_{t}\) is the tensile strength of the backfill roof (Pa); \(W\) and \(H\) are the span and height of the drift (m); and \(q\) is the uniform load on the top surface of the backfill roof (Pa). (2) The fixed-end beam model  
The fixed-end beam model was proposed by [PERSON] [30] to evaluate the strength requirement of sill mats and has been extended to assess the strength requirement of the backfill roof over the drift void, as shown in Equation (4)  
\[\sigma_{t}>\frac{L^{2}}{2d^{2}}q-\sigma_{c} \tag{4}\]  
where \(L\) and \(d\) are the span and height of the sill mat (m) and \(\sigma_{c}\) is the rock closure stress applied to lateral ends of the sill mat (Pa). (3) The thin plate model  
In the thin plate model, the backfill roof is simplified as a uniformly loaded thin plate on an elastic foundation, and the strength requirement is given by Equation (5)  
(2) The effects of abutment supports on the backfill roof with the basket weave drift layout. As shown in Figure 4, the basket weave drift layout can disperse deformations of the roof exposed in a single void drift. This is achieved through the perpendicular arrangement of drifts between adjacent layers. In Figure 1c, the backfill roof is intermittently supported by the abutments. However, existing analytical solutions neglect the effects of abutment supports by simplifying the boundary conditions as either fixed or simply supported, without considering abutment deformation or abutment width. Previous studies have shown that the stress within the roof can transfer to the abutments after the secondary drifts are excavated [27; 34; 35]. Since the numerical simulation fully incorporates the influence of abutment supports, strength requirements that more closely reflect the actual loading conditions of the backfill roof are yielded. As discussed above, the inherent limitations of traditional analytical solutions can lead to an unreliable evaluation of the strength requirements. In contrast, numerical simulation offers superior capability in modeling the actual loading conditions of backfill roofs, thereby yielding more reliable results. Notably, the iterative search method used in the numerical simulation for the basket weave drift layout can be further extended to determine the strength requirement of the backfill roof across various mining configurations and operational conditions.