# 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
#### 2.3.3 Energy Dissipation Rate on the Velocity Discontinuity Line  
The velocity discontinuity lines of the basic roof, immediate roof, and roadway side are divided into three parts: \(D_{1}\), \(D_{2}\), and \(D_{3}\). Therefore, the energy dissipation rate \(D\) of the  
Figure 3: Support pressure and overlying strata load power calculation diagram. (**a**) Roadway side support pressure \(\sigma\); (**b**) immediate roof support pressure \(q\); (**c**) overburden rock load \(\sigma_{\rm s}\).  
whole velocity discontinuity line can be obtained by adding \(D_{1}\), \(D_{2}\), and \(D_{3}\), which can be expressed as  
\[D=D_{1}+D_{2}+D_{3} \tag{24}\]  
In this formula, \(D\) is the total energy dissipation rate on the velocity discontinuity line; \(D_{1}\) is the energy dissipation rate on the basic vertex velocity discontinuity line; \(D_{2}\) is the energy dissipation rate on the direct roof velocity discontinuity line; and \(D_{3}\) is the total energy dissipation rate on the velocity discontinuity line of the roadway side.  
1. Internal energy dissipation rate of the velocity discontinuity line in the roadway sidewall The internal energy dissipation rate of discontinuity line BC is \[D_{\rm BC}=\int_{\theta_{\rm B}}^{\theta_{\rm C}}c_{3}\omega r_{\rm OB}{}^{2} {\rm e}^{2[(\theta-\theta_{\rm B})\tan\varphi_{3}]}{\rm d}\theta=\frac{c_{3} \omega r_{\rm OB}{}^{2}}{2\tan\varphi_{3}}g_{1}\] (25) And the internal energy dissipation rate of the discontinuous line AC is obtained similarly. \[D_{\rm AC}=c_{3}\omega\int_{\theta_{\rm A}}^{\theta_{\rm C}}\left(\frac{h_{2} }{\cos\theta}\right)^{2}{\rm d}\theta=c_{3}\omega h_{2}{}^{2}(\tan\theta_{ \rm C}-\tan\theta_{\rm A}\right)\] (26) where the expression of \(g_{1}\) is \[g_{1}={\rm e}^{2(\theta_{\rm C}-\theta_{\rm B})\tan\varphi_{3}}-1\] (27) The total energy dissipation rate within the velocity discontinuity line of the roadway side is \[D_{3}=2\bigg{(}c_{3}\omega h_{2}{}^{2}(\tan\theta_{\rm C}-\tan\theta_{\rm A}) +\frac{c_{3}\omega r_{\rm OB}{}^{2}}{2\tan\varphi_{3}}\left({\rm e}^{2(\theta_ {\rm C}-\theta_{\rm B})\tan\varphi_{3}}-1\right)\bigg{)}\] (28) 2. Internal energy dissipation rate of the direct roof velocity discontinuity line The internal energy dissipation rate generated on the logarithmic spiral CD is equal to the integral value of its tangential velocity and cohesion to the entire logarithmic spiral discontinuity. The internal energy dissipation rate on CD can be obtained as \[D_{\rm CD}=\int_{\theta_{\rm C}}^{\theta_{\rm D}}c_{2}\omega r_{\rm OC}{}^{2} {\rm e}^{2[(\theta-\theta_{\rm C})\tan\varphi_{2}]}{\rm d}\theta=\frac{c_{2} \omega r_{\rm OC}{}^{2}}{2\tan\varphi_{2}}g_{2}\] (29) In this formula, \(g_{2}\) is expressed as \[g_{2}={\rm e}^{2(\theta_{\rm D}-\theta_{\rm C})\tan\varphi_{2}}-1\] (30) The energy dissipation rate of discontinuous line OA is \[D_{\rm OA}=c_{2}v_{0}\sqrt{h_{2}{}^{2}+h_{0}{}^{2}/4}\] (31) Then, the total energy dissipation rate of the direct roof velocity discontinuity is \[D_{2}=2\bigg{(}\frac{c_{2}\omega r_{\rm OC}{}^{2}}{2\tan\varphi_{2}}g_{2}+c_{ 2}v_{0}\sqrt{h_{2}{}^{2}+h_{0}{}^{2}/4}\bigg{)}\] (32) 3. Internal energy dissipation rate of the main roof velocity discontinuity line \[D_{1}=c_{1}v_{0}\bigg{(}\frac{h_{1}}{\cos\varphi_{1}}+4r_{\rm OC}{}^{(\theta _{\rm D}-\theta_{\rm C})\tan\varphi_{2}}-h_{1}\tan\varphi_{1}\bigg{)}\] (33)