## 3 Mechanical Analysis and Numerical Simulation of Roadway Wall Rock
### Wall Rock Support System and Its Mechanical Analysis
According to the theoretical calculation model, when the side wall of the roadway is unstable, the maximum rise \(b_{1}\) of the natural balance arch is as follows  
\[b_{1}=\frac{a_{1}}{f_{p}} \tag{1}\]  
where \(a_{1}\) is the half-span of the natural balanced arch and \(f_{p}\) is the rock mass firmness coefficient. The maximum span of natural balanced arch \(2a_{1}\) is:  
\[2a_{1}=2\Big{[}a+h\tan\Big{(}45^{\circ}-\frac{\varphi_{c}}{2}\Big{)}\Big{]} \tag{2}\]  
where \(a\) is the half-width of the roadway; \(h\) is the roadway height; and \(\varphi_{c}\) is the angle of internal friction of the rock mass. The maximum wall rock pressure \(Q\) is:  
\[Q=\gamma b_{1} \tag{3}\]  
where \(\gamma\) is the average weight of the rock mass. Solution of wall rock pressure based on [PERSON]'s theory  
Based on the principle of stress transmission, [PERSON]'s theory [39] believes that the wall rock pressure acting on temporary support equipment comes from the stress transmission of the self-weight of the overlying soil, and the size of the wall rock pressure depends on such factors as the self-weight of the soil in the loose area and the internal friction angle of the rock mass [40]. In this theory, the differential element body is taken in the sliding block for the stress analysis, and the maximum wall rock pressure of the roadway is calculated. The calculation model is shown in Figure 3.  
Figure 2: Theoretical calculation model of equilibrium arch. In the [PERSON]'s calculation model, when the overlying strata subside, the two sides of the roadway form a staggered surface. The included angle between \(OA\) and \(OB\) and the vertical plane in Figure 3 is \(45^{\circ}-\varphi_{c}/2\). The staggered surface starts at point \(O\) of the roadway floor, reaches points \(A\) and \(B\) along curve \(OA\) and \(OB\), and then reaches points \(E\) and \(I\) along curve \(AE\) and \(BI\). The trajectory of curve \(AE\) and \(BI\) is difficult to calculate. Generally, approximate vertical curves \(AD\) and \(BC\) are used to replace curves \(AE\) and \(BI\), respectively. The area \(ABCD\) between the roadway top and the ground is the sliding block, and its width \(2b\) is as follows  
\[2b=2\Big{[}a+h\tan\Big{(}45^{\circ}-\frac{\varphi_{c}}{2}\Big{)}\Big{]} \tag{4}\]  
where \(b\) is the half-width of the sliding block. The differential element body with a thickness of \(dz\) and a buried depth of \(z\) is cut in the sliding block. The stress of the element body is shown in Figure 3. The overburden rock pressure \(P_{\rm s}\) applied vertically downward on the upper surface of the unit is as follows  
Figure 2.0  
where \(q_{r}\) is the vertical initial geostress. The dead weight \(G\) of the unit is:  
Figure 2.1  
where \(dz\) is the unit thickness. The vertical uplift force \(T\) applied on the lower surface of the unit is as follows  
Figure 2.2  
Figure 2.3 [PERSON]’s theoretical model. The vertical sidewall rock friction \(F_{c}\) applied on the side of the unit is as follows  
Figure 2.4  
where \(c\) is the cohesion of the rock mass and \(k_{0}\) is the coefficient of side pressure. The balance condition of the unit in the vertical direction is as follows  
Figure 2.5  
By introducing Formulas (7)-(10) into Formula (11), we get the following  
Figure 2.6  
where \(z\) is the buried depth of the unit. According to the characteristics of [PERSON]'s theoretical model, when \(z=0\), \(q_{r}=P_{0}\), and \(P_{0}\) is the additional load on the surface. \(P_{0}\) is calculated as follows  
Figure 2.7  
Substituting Formula (13) into (12), we get the following  
Figure 2.8  
Taking \(z=L\) into Formula (14), the maximum wall rock pressure \(Q\) can be obtained as follows  
Figure 2.9  
where \(L\) is the roadway burial depth.