# Study of the Failure Mechanism of Soft Rock Mining Roadways Based on Limit Analysis Theory
## 2 Analysis of the Deformation and Failure of a Soft Rock Mining Roadway
### Velocity Discontinuity Line Height Solution
#### 3.1.2 Results and Discussion  
To verify the reliability of the formula derived in this paper, the soft rock return airway of the 1506 working face in the Anyang Coal Mine is adopted as an engineering example, and the results are compared with the results of the physical similarity simulation test. Therefore, it is necessary to solve objective Function (34) under multivariate constraints, and the maximum value of the objective function is obtained. That is, the maximum value of h (\(\sigma_{\rm s}\)) is the upper bound solution of the height of the velocity discontinuity line. The constraint conditions for solving the height of the velocity discontinuity line are shown in Table 1.  
According to the upper limit objective function (34) of the height of the velocity discontinuity line and the constraint conditions (Table 1) of the height of the velocity discontinuity line, the curve of the failure height of the velocity discontinuity line of the 1506 return airway with the load of the overlying strata can be drawn, as shown in Figure 5.  
According to the upper limit solution of the height of the velocity discontinuity line (Figure 5) and the constraint conditions of the height of the velocity discontinuity line (Table 1), the evolution characteristic diagram of the failure mode of the 1506 return airway can be drawn, as shown in Figure 6. From the diagram, it can be seen that when the overburden load \(\sigma_{\rm s}\) is less than 3.5 MPa, the velocity discontinuity line is mainly distributed on the coal side. When 3.5 MPa \(<\)\(\sigma_{\rm s}\)\(<\) 3.77 MPa, the velocity discontinuity line extends from the coal side to the direct roof. When the overburden load \(\sigma_{\rm s}\) is equal to 3.77 MPa, the velocity discontinuity line penetrates the interface between the direct roof and the basic roof. As \(\sigma_{\rm s}\) continues to increase, the velocity discontinuity line continues to extend upward. When \(\sigma_{\rm s}\) is equal to 9.0 MPa, the two velocity discontinuity lines intersect. At this time, the height of the velocity discontinuity line in the basic roof is 11.32 m.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Density (kg\(\cdot\)m\({}^{-3}\))**} & \multicolumn{3}{c}{**Polar Angle (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Angle of Internal**} & \multicolumn{3}{c}{**Cohesion (MPa)**} & \multicolumn{3}{c}{**Polar Radius (m)**} & \multicolumn{3}{c}{**Pressure**} \\ \hline \(\gamma_{1}\) & \(\gamma_{2}\) & \(\gamma_{3}\) & \(\theta_{\rm A}\) & \(\theta_{\rm B}\) & \(\theta_{\rm C}\) & \(\varphi_{1}\) & \(\varphi_{2}\) & \(\varphi_{3}\) & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(r_{\rm OB}\) & \(r_{\rm OC}\) & \(q\) & \(\sigma\) \\
2450 & 2060 & 1350 & 27 & 15 & 46 & 38 & 34 & 31 & 12.21 & 0.88 & 0.21 & 8.59 & 6.2 & 0.3 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Constraint conditions for solving the height of the velocity discontinuity line.  
Figure 5: Upper bound solution of velocity discontinuity height of the rock surrounding the 1506 return airway in Anyang Coal Mine. (**a**) Immediate roof; (**b**) main roof.