## 3 Mechanisms of \"Rock-Support\" Interaction  
To analyze the interaction mechanism of \"surrounding rock-support\" in the broken surrounding rock of a roadway, this paper applies the elasticity theory of thick-walled pipe for analysis and research. As shown in Figure 2, the radius of the roadway is a, the radius of the broken perimeter rock is b, the roadway support resistance is \(p_{i}\), the original rock stress is hydrostatic pressure state, and the pressure of the surrounding rock is \(p_{0}\).  
Figure 2: Mechanical model of “surrounding rock–support” in the broken surrounding rock of a roadway.  
According to the theory of thick-walled tube [30] and [PERSON]'s theorem [31], the stress distribution in the surrounding rock in the elastic zone can be shown as follows:  
\[\begin{array}{c}\sigma_{r}=\frac{a^{2}\left(b^{2}-r^{2}\right)}{r^{2}\left(b^{2} -a^{2}\right)}p_{i}+\frac{b^{2}\left(a^{2}-r^{2}\right)}{r^{2}\left(b^{2}-a^{2 }\right)}p_{0}\\ \sigma_{\theta}=\frac{a^{2}\left(b^{2}+r^{2}\right)}{r^{2}\left(b^{2}-a^{2} \right)}p_{i}-\frac{b^{2}\left(a^{2}+r^{2}\right)}{r^{2}\left(b^{2}-a^{2} \right)}p_{0}\end{array} \tag{15}\]  
From the equality of radial stresses at the junction of the elastic zone and the zone of influence of the crushed enclosing rock, it follows that,  
\[\sigma_{r}^{e}=\sigma_{r}^{p} \tag{16}\]  
Equations (6) and (10) are substituted for Equation (16) to obtain Equation (17); if the radius of the zone of influence of the fractured enclosure is R, Equation (17) can be converted to Equation (18).  
\[p_{0}\left[1-\frac{b^{2}}{r^{2}}+\left(\frac{b}{a}\right)^{\frac{2\sin{\varphi _{0}}}{1-\sin{\varphi_{0}}}}\right]+C_{0}.\frac{b^{2}}{r^{2}}.ctg\varphi_{0} \left[\left(\frac{b}{a}\right)^{\frac{2\sin{\varphi_{0}}}{1-\sin{\varphi_{0} }}}-1\right]=\left(p_{i}+C_{0}.ctg\varphi_{0}\right)\left(\frac{r}{a}\right)^ {\frac{2\sin{\varphi_{0}}}{1-\sin{\varphi_{0}}}}-C_{0}.ctg\varphi_{0} \tag{17}\]  
\[p_{i}=\left(\frac{a}{R}\right)^{\frac{2\sin{\varphi_{0}}}{1-\sin{\varphi_{0} }}}\cdot\left\{p_{0}\left[1-\frac{b^{2}}{R^{2}}+\left(\frac{b}{a}\right)^{ \frac{2\sin{\varphi_{0}}}{1-\sin{\varphi_{0}}}}\right]+C_{0}.\frac{b^{2}}{R^{ 2}}.ctg\varphi_{0}\left[\left(\frac{b}{a}\right)^{\frac{2\sin{\varphi_{0}}}{1- \sin{\varphi_{0}}}}-1\right]\right\}-C_{0}.ctg\varphi_{0} \tag{18}\]  
Using Matlab software (Matlab, 2014a) to analyze Equation (18) yields Figure 3.  
Figure 3: Relationship between support force, surrounding rock pressure, and influence radius of broken surrounding rock.  
From Figure 3, the support force of the broken perimeter rock roadway is inversely proportional to the radius of the broken perimeter rock influence area; with the increase in broken influence area, part of the deformation energy is released, and the pressure of perimeter rock acting on the support becomes smaller. No matter how big the radius of influence of the broken perimeter rock is, the support resistance will not be lower than a certain value; even if the radius of influence of the broken perimeter rock is 0, the support resistance will also be very small. The support force of the broken perimeter rock tunnel is directly proportional to the residual cohesion of the broken rock body, and with the increase in residual cohesion, the greater the support resistance becomes.