## 3 Influence of Face Advancing Rate on Roof Caving Interval
### Theoretical Study on the Influence of Advance Rate on the First Weighting Interval of the Main Roof  
During the face advance process, variations in the advance rate significantly influence the weighting interval. Under low and constant advance rates, the overlying strata gradually bend and subside as the face progresses. The slower advance allows sufficient time for strata caving. In contrast, higher advance rates lead to inadequate stress redistribution within surrounding rock masses, resulting in elevated stress concentration coefficients. This inhibits effective pressure relief in the stop periphery, thereby increasing the load borne by the exposed lower strata. As the face advances, the unsupported span progressively extends while roof fracturing remains limited, ultimately enlarging the weighting interval. The kinematic model of overburden strata movement under different advance rates is illustrated in Figure 8.  
As shown in Figure 8, the increase in advance rate leads to an extension of the unsupported span of the main roof (i.e., L1 < L2) due to the hysteresis in load transfer within the main roof. This phenomenon is attributed to the enhanced tensile strength of the strata, which itself results from the increased loading rate. Based on Academician [PERSON]'s \"Voussoir Beam\" theory, the main roof is modeled as a fixed-end beam subjected to uniformly distributed loads at both ends, as schematically illustrated in Figure 9. The critical fracture condition occurs when the maximum tensile stress at the bending moment extremum reaches the rock's tensile strength \(R_{\mathrm{T}}\). Figure 8: Overburden movement models under different advance rates. (**a**) Low-speed mining phase; (**b**) high-speed mining phase. Figure 9: Computational model of fixed-end beam theory. The maximum bending moment occurs at both ends of the fixed-end beam. The formula for the maximum bending moment is  
\[M_{\max}=\frac{1}{12}qL^{2} \tag{1}\]  
The section modulus in bending is  
\[W=\frac{h^{2}}{6} \tag{2}\]  
According to the strength criterion, it can be derived that  
\[\sigma_{\max}=\frac{M_{\max}}{W}=\frac{qL^{2}}{2h^{2}} \tag{3}\]  
When \(\sigma_{\max}=R_{T}\), that is, when the tensile strength of the rock strata is reached at this location, fracture initiates. By combining Equations (1)-(3) through derivation, the critical caving span of the beam at fracture can be obtained as  
\[L_{1}=h\sqrt{\frac{2R_{T}}{q}} \tag{4}\]  
where  
\(q\)--load applied to the main roof (MPa), \(q=\gamma H\);  
\(\gamma\)--unit weight of the strata (kN/m\({}^{3}\));  
\(H\)--overburden depth (m);  
\(h\)--thickness of the main roof (m). Since Equation (4) is derived for an idealized theoretical model, the influence of weak strata necessitates the introduction of a geological weakness coefficient \(k\) based on engineering rock mass classification standards. This leads to the final formula for the initial weighting step distance of the main roof:  
\[L_{w}=kh\sqrt{\frac{2R_{T}}{q}} \tag{5}\]  
The range of \(k\) values is directly correlated with common rock mass classification standards. The \(k\) value increases with improved rock mass integrity, aligning with the scoring trends of classification systems. The specific values are listed in Table 4.  
Next, the advance rate will be incorporated into the model. Assuming the model's working face advance rate is \(v_{m}\) with an advance time \(t_{m}\), the actual working face advance rate is vs. with a corresponding advance time \(t_{s}\), and the working face layout length is \(d\). If the rock beam maintains a constant advance rate prior to fracture, the initial weighting