## 4 Effect of Supporting Force on the Plastic Zone of the Surrounding Rock  
An implicit equation for the plastic zone boundary of the rock surrounding the roadway in a nonuniform stress field under dynamic pressure and subject to the action of supporting force can be deduced after following the steps outlined in Section 3. First, a roadway under a dynamic pressure mechanical model considering supporting force is established, as shown in Figure 7a, and then decomposed into two situations, as shown in Figure 7b,c. Through calculation, the binary implicit function of \(r\) and \(\theta\) can be obtained as shown in Formula (15). \[ \begin{array}{ll}f(\theta,r)=&[\frac{P_{1}r_{0}^{2}}{r^{2}}-\frac{qr_{0}^{2} }{r^{2}}+P_{2}(1-\frac{2r_{0}^{2}}{r^{2}}+\frac{3r_{0}^{4}}{r^{4}})\cos 2 \theta]^{2}+{P_{2}}^{2}(1+\frac{2r_{0}^{2}}{r^{2}}-\frac{3r_{0}^{4}}{r^{4}}) ^{2}\sin^{2}2\theta\\ &-(P_{1}+\frac{2P_{2}r_{0}^{2}}{r^{2}}\cos 2\theta)^{2}\sin^{2}\varphi-(2P_{1} +\frac{4P_{2}r_{0}^{2}}{r^{2}}\cos 2\theta)c\sin\varphi\cos\varphi\\ &-c^{2}\cos^{2}\varphi\end{array}  \tag{15}\]  
In the formula, \(P_{1}=(K_{1}\ P_{0}+K_{2}\lambda P_{0})/\ 2\), and \(P_{2}=(K_{1}\ P_{0}-K_{2}\lambda P_{0})/\ 2\). When \(f(\theta,r)=0\), an implicit equation for the boundary of the plastic zone of the rock surrounding the roadway under dynamic pressure and in a nonuniform stress field subject to supporting force can be obtained. When using the existing technology and equipment, the supporting force is not of the same order of magnitude as the resistance needed to completely control the deformation of the surrounding rock. The supporting force applies to active types of support that can exert preload force, including anchor bolts and anchor cables, and it is approximately a few percent of the in situ stress. For example, the supporting force range of anchor bolts is 0.05-0.8 MPa, and the maximum support resistance that hydraulic supports can provide is only 0.6 MPa [28]. At the same time, in the current situation, the maximum support resistance provided by support equipment materials generally does not exceed 1 MPa. Figure 7: Mechanical model of the roadway under dynamic pressure when considering the supporting force. (**a**) Mechanical model of the roadway with supporting force under dynamic pressure (**b**) Part one of the mechanical model in Figure 7a. (**c**) Part two of the mechanical model in Figure 7a. Therefore, the distribution range and change rules of the plastic zone boundary of the rock surrounding the roadway under dynamic pressure were studied when the supporting forces were 0, 0.3 MPa, 0.5 MPa, 0.8 MPa, and 1.0 MPa, respectively. When \(\lambda\) is 0.8, the relationship between the supporting force and the boundary of the plastic zone of the surrounding rock is shown in Figure 8. When the roadway is not affected by mining, the shape of the plastic zone is elliptical, with larger plastic zones at both sides of the roadway and smaller plastic zones at the roof and floor. As the supporting force increases, there is a certain reduction in the size of the plastic zone, but the decrease is very small, as shown in Figure 8a. When the roadway is only affected by horizontal mining, the shape of the plastic zone is shown in Figure 8b,c. As the horizontal mining influence coefficient increases, the shape of the plastic zone changes from elliptical to butterfly, and the plastic zone gradually expands to the roof and floor corners of the roadway. As the supporting force increases, the plastic zone decreases slightly, but its shape does not change. When the roadway is only affected by vertical mining, the shape of the plastic zone is shown in Figure 8d. It is butterfly shaped, and the two butterfly wings are symmetrical. As the supporting force increases, the plastic zone gradually decreases, but the decrease is very small.