# 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
### Construction of the Surrounding Rock Failure Mode of the Roadway  
Using the upper bound limit analysis method, it is necessary to establish an accurate failure mode and the corresponding velocity field. The failure mode is composed of a series of rigid sliders. The absolute velocity and relative velocity of the rigid slider are obtained by using the velocity compatibility condition and the plastic flow equation. The failure mode reflects the essential characteristics of the surrounding rock when the roadway is under load, and its specific geometric shape is determined by a series of geometric parameters.  
When the support strength of the roadway side is insufficient or the strength of the surrounding rock is small, a shear failure zone appears beside the roadway. The shear failure zone gradually extends upward with increasing confining pressure until the shear zone forms a stable pressure arch effect above the roadway. According to the failure characteristics of the rock surrounding the roadway, a calculation model of the failure mode of the rock surrounding the soft rock mining roadway is proposed, as shown in Figure 1. For the immediate roof and two sides with poor mechanical properties, the logarithmic spiral line is used to simulate the shear failure zone. For the basic roof with good mechanical properties, a straight line is used to simulate the shear failure zone. The two sides are composed of logarithmic spiral rigid sliders, and the velocity discontinuity line of the sliders on the roadway side is composed of two logarithmic spiral lines BC and HI. The velocity discontinuity line of the immediate top slider consists of two logarithmic spiral lines (CD and GH) and two straight lines (OA and OJ). Under the limit condition, the spiral slider rotates rigidly with angular velocity \(\omega\) around point O, and relative motion occurs with the surrounding rock mass. The velocity vector of the slider velocity discontinuity line is not the same. The velocity discontinuity line of the main roof is composed of DE, EF, and FG. The velocity is \(v_{0}\) and the direction is vertical downward.  
In the figure, \(h_{3}\) is the height of the roadway, \(h_{2}\) is the height of the direct roof of the roadway, and \(h_{1}\) is the height of the main roof failure. \(\sigma\) is the uniform support strength acting on the roadway side and \(\sigma_{\text{s}}\) is the load of the overlying strata. \(q\) is the support strength of the roadway roof. The shear strength parameters of the immediate roof and roadway side are \(c_{2}\), \(\varphi_{2}\) and \(c_{3}\), \(\varphi_{3}\). It follows that the logarithmic spiral CD expression \(r_{2}(\theta)\) and the BC expression \(r_{3}(\theta)\) are [38; 39]:  
\[\begin{cases}r_{2}(\theta)=&r_{\text{OCE}}\text{e}^{(\theta-\theta_{\text{C}} )\tan\varphi_{2}}\\ r_{3}(\theta)=&r_{\text{OBE}}\text{e}^{(\theta-\theta_{\text{B}})\tan\varphi_{ 3}}\end{cases} \tag{2}\]  
In this formula, \(r_{\text{OC}}\) is the length of polar OC, m; \(r_{\text{OB}}\) is the length of the polar radius OB, m; \(\theta_{\text{C}}\) is the angle between OC and the vertical direction, \({}^{\circ}\); \(\theta_{\text{B}}\) is the angle between OB and the vertical direction, \({}^{\circ}\); \(\theta\) is the variable polar angle, \({}^{\circ}\); \(\varphi_{2}\) is the internal friction angle of the immediate roof, \({}^{\circ}\); and \(\varphi_{3}\) is the internal friction angle of the coal side, \({}^{\circ}\).  
Figure 1: Calculation model of surrounding rock failure in a soft rock mining roadway.