# 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
### Energy Consumption Analysis of Roadway-Adjacent Rock
#### 2.3.1 Calculation of the Gravity Power of the Slider in the Velocity Discontinuity Line  
The gravity power of coal and rock sliders in the discontinuous line is calculated. First, the model is discretized, as shown in Figure 2. The gravity power \(P_{\gamma}\) can be obtained by adding the gravity power \(P_{\rm ABC}\), \(P_{\rm CDGH}\), and \(P_{\rm DEFG}\), as shown in Formula (4). \[P_{\gamma}=2P_{\rm ABC}+P_{\rm CDGH}+P_{\rm DEFG} \tag{4}\]  
In this formula, \(P_{\gamma}\) is the total gravity power of the coal and rock mass in the velocity discontinuity line; \(P_{\rm ABC}\) is the gravity power of the coal in the ABC of the roadway slider block; \(P_{\rm CDGH}\) is the gravity power of the direct top slider CDGH; and \(P_{\rm DEFG}\) is the gravity power of the basic top slider DEFG. (1) ABC gravity power \(P_{\rm ABC}\) of the spiral slider on the roadway side  
OA and AC divide the block OBC into three parts: OAC, OAB, and ABC. Therefore, the gravity power \(P_{\rm ABC}\) of the spiral slider ABC can be obtained by \(P_{\rm OBC}\), \(P_{\rm OAC}\), and \(P_{\rm OAB}\), which can be expressed as:  
\[P_{\rm ABC}=P_{\rm OBC}-P_{\rm OAC}-P_{\rm OAB} \tag{5}\]  
In this formula, \(P_{\rm OBC}\) is the gravity power of coal in sliding block OBC; \(P_{\rm OAC}\) is the gravity power of coal in slider OAC; and \(P_{\rm OAB}\) is the gravity power of coal in slider OAB. Figure 2: Gravity power calculation diagram. (a) Roadway side; (b) immediate roof; (c) main roof. The power \(P_{\rm OBC}\) of the gravity of coal in the OBC slider is  
\[P_{\rm OBC}=\frac{1}{3}\gamma_{3}\omega r_{\rm OB}{}^{3}\int_{\theta_{\rm B}}^{ \theta_{\rm C}}\left[\mathrm{e}^{(\theta-\theta_{\rm B})\tan\varphi_{3}}\right] ^{3}\sin\theta\mathrm{d}\theta=\frac{1}{3}\gamma_{3}\omega r_{\rm OB}{}^{3}f_{1} \tag{6}\]  
The powers \(P_{\rm OAC}\) and \(P_{\rm OAB}\) of coal gravity in the OAC and OAB sliders are solved similarly. The specific expressions can be expressed as  
\[P_{\rm OAC}=\frac{1}{3}\gamma_{3}\omega h_{2}{}^{3}\int_{\theta_{\rm A}}^{ \theta_{\rm C}}\frac{\sin\theta}{\cos^{3}\theta}\mathrm{d}\theta=\frac{1}{3} \gamma_{3}\omega h_{2}{}^{3}f_{2} \tag{7}\]  
\[P_{\rm OAB}=\frac{1}{3}\gamma_{3}\omega h_{0}{}^{3}\int_{\theta_{\rm B}}^{ \theta_{A}}\frac{1}{8\mathrm{sin}^{2}\theta}\mathrm{d}\theta=\frac{1}{3} \gamma_{3}\omega h_{0}{}^{3}f_{3} \tag{8}\]  
In this formula, \(\gamma_{3}\) is the bulk density of coal, \(\omega\) is the angular velocity, and \(f_{1},f_{2}\), and \(f_{3}\) are the coefficients. \[f_{1}=\frac{1}{1+9\mathrm{tan}^{2}\varphi_{3}}\Big{[}(\cos\theta_{\rm B}{-}3 \sin\theta_{\rm B}\tan\varphi_{3})-\mathrm{e}^{3(\theta_{\rm C}{-}\theta_{\rm B })\tan\varphi_{3}}(\cos\theta_{\rm C}{-}3\sin\theta_{\rm C}\tan\varphi_{3}) \Big{]} \tag{9}\]  
\[f_{2}=\frac{\mathrm{tan}^{2}\theta_{\rm C}{-}\mathrm{tan}^{2}\theta_{\rm A}}{2} \tag{10}\]  
\[f_{3}=\frac{\cot\theta_{\rm B}{-}\cot\theta_{\rm A}}{8} \tag{11}\]  
where \(\theta_{\rm C}\) can be obtained according to Formula (12). \[ \begin{cases}r_{\rm OC}=r_{\rm OB}\mathrm{e}^{(\theta_{\rm C}{-}\theta_{\rm B })\tan\varphi_{3}}\\ r_{\rm OC}=r_{\rm OA}\cos\theta_{\rm A}/\cos\theta_{\rm C}\end{cases}  \tag{12}\]  
(2) Gravity power \(P_{\rm CDGH}\) of the direct roof slider CDGH of the roadway  
The direct roof slider CDGH is divided into three parts: OCD, OAC, and AOJ. Therefore, the gravity power \(P_{\rm CDGH}\) of the direct roof slider CDGH can be obtained by calculating \(P_{\rm OCD}\), \(P_{\rm OAC}\), and \(P_{\rm AOJ}\), which can be expressed as  
\[P_{\rm CDGH}{=2(P_{\rm OCD}+P_{\rm OAC})+P_{\rm AOJ}} \tag{13}\]  
In this formula, \(P_{\rm CDGH}\) is the power of gravity of the direct roof slider CDGH; \(P_{\rm OCD}\) is the power of gravity of the direct roof slider OCD; \(P_{\rm OAC}\) is the power of gravity of the direct roof slider OAC; and \(P_{\rm AOJ}\) is the power of gravity of the direct roof slider AOJ.