## 2 Project Overview
The minimum horizontal tectonic stress \(F_{cr}\), which causes the roof beam to bend and lose stability, is calculated as:  
\[F_{cr}=\frac{\pi^{2}EI}{L^{2}} \tag{5}\]  
If there is no horizontal tectonic stress, then \(q\) acts alone, and Equation (2) can be written as:  
\[\frac{d^{2}w}{dx^{2}}=-\frac{q}{2 EI}(Lx-\frac{qL^{2}}{6}-x^{2}) \tag{6}\]  
The answer to the boundary conditions shows that the deflection is highest at \(x=L/2\), and the value is:  
Figure 2:0  
The buried depth of the coal seam and the actual conditions of the roadway excavation show that the bulk density \(\gamma\) of the overlaying strata is 2500 kN/m\({}^{3}\), the buried depth of the roadway is 800 m, and the horizontal span \(L\) of the roadway roof is 5.5 m. When the roof beam reaches the yield limit under the minimum horizontal tectonic stress, the deflection is \(w=7.72\) cm. This is found by adding each parameter into Equation (7). The deflection limit of 7.72 cm is determined from theoretical analysis as the maximum yield limit of the roof beam under horizontal tectonic stress. Exceeding this threshold indicates the onset of large deformation and potential instability. In engineering applications, the stability control measures proposed in Section 6 have been implemented to constrain roof separation within 77 mm. Field monitoring confirms that, under these measures, the roof deformation remained below the theoretical failure limit, validating the applicability of the criterion. More research shows that after the 1232(3) transportation route has been dug up to the 1242(3) stop line, the horizontal tectonic stress in the rock around the roadway will alter a lot because of the stress transfer from the nearby goat. To find out how the change in the maximum horizontal tectonic stress affects the bending of the road roof, we define the ratio \(R_{w}=w_{max}/w_{0}\) as the buckling deflection variation coefficient and the ratio \(F_{w}=F/F_{\rm cr}\) as the horizontal tectonic stress variation coefficient. More information can be obtained by putting (5), (6), and (7) into it:  
Figure 2:1  
Figure 4 shows that, when \(F_{w}\) is less than 0.6, \(R_{w}\) rises slowly. This is a slow and stable change interval, which means that the change in horizontal tectonic stress does not have much of an effect on the change in roof deflection. The change in the deflection ratio gets bigger when \(F_{w}\) is greater than 0.6. Specifically, when the actual tectonic stress \(F\) exceeds 76% of the critical buckling threshold \(F_{cr}\), which is the minimum horizontal stress required to initiate roof beam buckling, that is, when \(F>0.76F_{cr}\), the roof deformation increases rapidly. This marks the onset of structural yield and indicates a high risk of roof instability or failure. Mechanical Response Characteristics of Roadway Surrounding Rock Under Different Angles Between Tectonic Stress and Roadway  
The third part of the theoretical analysis explores the influence of horizontal tectonic stress on the roof rock beam of the roadway. In practical engineering applications, the orientation of tectonic stress frequently does not align with the configuration of roadway layouts, resulting in altered stress distributions in the surrounding rocks. It is necessary for the theoretical model to be used in engineering practice to find out how the rock around the roadway reacts when the horizontal tectonic stress and the roadway are at different angles.