## 2 Project Overview
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Monitoring** \\ **Station** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Depth of** \\ **Burial/m** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Vertical** \\ **Stress/MPa** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maximum** \\ **Horizontal** \\ **Principal** \\ **Stress/MPa** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Minimum** \\ **Horizontal** \\ **Principal Stress/MPa** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Direction of** \\ **Maximum** \\ **Principal** \\ **Stress/MPa** \\ \end{tabular} } \\ \hline No. 1 station & \(-801\) & 20.03 & 29.37 & 8.83 & N9\({}^{\circ}\)E \\ No. 2 station & \(-794\) & 19.85 & 20.46 & 3.69 & N46\({}^{\circ}\)E \\ No. 3 station & \(-806\) & 20.15 & 33.25 & 9.36 & N25\({}^{\circ}\)E \\ \hline \hline \end{tabular}
\end{table} 
Table 1: In situ stress test results. Figure 2: Comprehensive histogram of coal seam ceiling and base. strata. In the project area, the roof strata consist of multiple thin rock layers with differing mechanical and deformation properties. Due to the geological complexity and the difficulty in accurately characterizing the mechanical interactions and potential slip surfaces between layers, the layered structure was simplified into a single equivalent beam with averaged mechanical properties. Additionally, to facilitate the development of the mechanical model and reduce computational complexity, the roof material was assumed to follow linear elastic behavior. This simplification allows for tractable analysis of stress transfer and deformation mechanisms. However, it does not fully capture the nonlinear, anisotropic, or time-dependent characteristics of deep soft rocks under high in situ stress, particularly in tectonically active zones. The process of creating a mechanical model of the roadway roof strata is illustrated in Figure 3.  
The reaction moment \(M_{A}\)\(M_{B}\) and the support reaction force \(N_{A}\)\(N_{B}\) come from the two ends of the roadway support. This happens because of the self-weight \(q\) and the horizontal tectonic stress \(F\). The equation for how the roof beam bends is:  
\[\frac{d^{2}w}{dx^{2}}=-\frac{Mx}{EI} \tag{1}\]  
We can find \(M_{x}\) in Equation (1) when \(M_{x}=N_{A}x-M_{A}-\frac{1}{2}qx^{2}+Fw\):  
\[\frac{d^{2}w}{dx^{2}}=-\frac{1}{EI}(\frac{1}{2}qlx-\frac{qL^{2}}{12}-\frac{1} {2}qx^{2}+Fw) \tag{2}\]  
\(E\) is the roof's elastic modulus. \(I\) is the section's moment of inertia. \(F\) is the tectonic stress acting on both ends of the roof. \(L\) is the horizontal span of the roadway roof. Set \(K^{2}=\frac{F}{EI}\). Equation (2) can be simplified as:  
\[\frac{d^{2}w}{dx^{2}}+K^{2}w=-\frac{qK^{2}}{2F}(lx-x^{2}-\frac{L^{2}}{6}) \tag{3}\]  
The general solution of the homogeneous equation \(\frac{d^{2}w}{dx^{2}}+K^{2}w=0\) is \(w_{1}=A\cos Kx+B\sin Kx\), where A and B are two whole numbers. The non-homogeneous special solution to Equation (3) is \(w_{2}=\frac{q}{2F}(x^{2}-Lx+\frac{L^{2}}{6}-\frac{2}{K^{2}})\). We can find constants A and B by looking at the boundary conditions. By putting A and B into Equation (3), we can find the bending deformation of roof strata caused by both \(q\) and \(F\). It is clear that the deflection is biggest when \(x=L/2\), and its value is:  
\[w_{max}=\frac{q}{FK^{2}}\bigg{[}\bigg{(}1-\frac{L^{2}K^{2}}{12}\bigg{)}\sec( \frac{KL}{2})-\frac{L^{2}K^{2}}{24}-1\bigg{]} \tag{4}\]  
Figure 3: A mechanical model of the roof layers. Based on this linear elastic assumption and the beam-like geometry of the roof strata, the Euler-Bernoulli beam theory is adopted to analyze the buckling behavior of the roof under axial tectonic stress.