## 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
\begin{table}
\begin{tabular}{c c c c} \hline
**Rock Mass Description** & **RMR Score Range** & **Q Value Range** & **Corresponding \(k\) Value** \\ \hline Fault influence zone, intensely & 20–40 (Class IV Rock Mass) & 0.1–1.0 & 0.50–0.65 \\ weathered rock zone & & & \\ Crushed limestone with & 40–60 (Class III Rock Mass) & 1.0–4.0 & 0.65–0.75 \\ argillaceous infilling & & & \\ Massive sandstone with closed & 60–80 (Class II Rock Mass) & 4.0–10 & 0.75–0.85 \\ joint development & & & \\ Intact granite with joint spacing \textgreater{} 3 m & 80–100 (Class I Rock Mass) & \textgreater{}10 & 0.85–0.90 \\ \hline \end{tabular}
\end{table} 
Table 4: \(k\) value range. interval in the model before roof breakage is defined as \(L=v_{m}t_{m}\). In practical mining operations, this interval becomes \(L=v_{p}t_{p}\). Since \(v_{m}t_{m}=v_{p}t_{p}\), the model's advance rate can be derived as  
\[v_{m}=\frac{t_{s}}{t_{m}}v_{s} \tag{6}\]  
The ratio of the actual advance time to the modeled advance time is approximately equivalent to the ratio of the actual working face layout length to the width of the rock beam in the model. Let \(\beta=t_{s}/t_{m}\). Given that the rock beam width in the model is defined as a unit width of 1 m, and the working face layout length is \(d\), the ratio of the actual working face layout length to the model rock beam width is defined as \(d\). Thus, \(\beta=d\). Based on the operational parameters of Panel 12308, \(d\) is assigned a value of 300. Substituting the above relational expressions into Equation (3), the maximum tensile stress during rock beam fracture in the initial weighting period is obtained as  
\[\sigma_{\max}=\frac{q(v_{m}t_{m})^{2}}{2h^{2}}=\frac{q(\beta v_{s}t_{m})}{2h^{ 2}} \tag{7}\]  
By solving the partial derivative of \(\sigma_{\max}\) with respect to \(t_{m}\), we derive  
\[\sigma_{\max}{}^{\prime}=\frac{\partial\sigma_{\max}}{\partial t_{m}}=\frac{ \partial\sigma_{\max}}{\partial L}\frac{\partial L}{\partial t_{m}}=\frac{ \beta q Lv_{s}}{h^{2}} \tag{8}\]  
As demonstrated by Equation (8), the rate of change in maximum tensile stress exhibits a positive correlation with the advance rate. Specifically, higher advance rates (\(v_{s}\)) result in accelerated variation rates of \(\sigma^{\prime}_{\max}\), thereby increasing the rock mass failure velocity. Furthermore, experimental studies have demonstrated that the tensile strength of coal-rock masses exhibits an approximately linear relationship with the axial stress rate under uniaxial compression. Based on this finding, the following empirical relationship can be formulated:  
\[R_{T}=a\sigma^{\prime}+b \tag{9}\]  
In the equation, \(\sigma^{\prime}\) denotes the stress rate, MPa/s; \(a\) and \(b\) are constants determined through loading experiments. By combining Equations (8) and (9), the expression for \(R_{T}\) can be derived as  
\[R_{T}=\frac{a\beta q Lv_{s}}{h^{2}}+b \tag{10}\]  
Subsequently, substituting Equation (10) into Equation (5) and solving the quadratic equation yield the computational formula for the initial weighting span of the main roof:  
\[L_{w}=k^{2}a\beta v_{s}+\sqrt{k^{4}a^{2}\beta^{2}v_{s}^{2}+\frac{2k^{2}h^{2} b}{q}} \tag{11}\]  
Equation (11) does not account for the effects of joints and moisture in the rock layers, both of which significantly impact roof stability. Therefore, corrections for joints and moisture must be incorporated. Higher joint density (\(J_{v}\)) degrades rock mass integrity and reduces tensile strength (\(R_{T}\)). A joint influence factor \(k_{j}\) is introduced here, defined as  
\[k_{j}=\sqrt{\frac{1-J_{v}}{I_{v,crit}}} \tag{12}\]  
where\(I_{v}\)--joint density; joints/m\({}^{3}\);  
\(I_{v,crit}\)--critical joint density (threshold at which the rock mass completely loses continuity), joints/m\({}^{3}\).