## 3 Mechanical Analysis and Numerical Simulation of Roadway Wall Rock
### Numerical Simulation of Roadway Wall Rock Based on FLAC 3D  
In view of the complex geological conditions of the coal mine roadway, it was difficult for the theoretical calculation to accurately reflect the real stress, deformation, and displacement of the wall rock support system in the roadway. A numerical simulation method of roadway wall rock support system based on FLAC 3D was constructed to analyze the stress, displacement, and plastic failure characteristics of the roadway roof before and after temporary support. A numerical simulation method can fully reflect the stress, displacement, plastic failure, and other characteristics of the wall rock within the range of the geotechnical model and can modify the model parameters according to the actual situation of the rock mass to simulate the different working conditions and geological conditions of the wall rock. Common numerical simulation methods include the finite element method, boundary element method, finite difference method, etc. Among them, the finite difference method uses a difference instead of the differential, uses a Taylor series method to replace the differential quotient in the original equation and the conditions of a definite solution, and uses an integral sum to replace the integral, so as to establish difference equations instead of differential equations, which is more suitable for geotechnical research [41]. Rock has the properties of a high compressive strength but low tensile and shear strength, and the stress-strain relationship of the rock mass has complex nonlinear characteristics. The rock mass bearing complex stress will have different forms of damage, plastic failure, weak plane shear failure, etc. Based on the generalized concept of plasticity theory, the above failure forms of the rock mass can be considered as \"plastic failure\", so the corresponding constitutive relationship can be established by using the plasticity theory. Before reaching the yield limit, the medium can be approximately regarded as elastic, and after reaching the yield limit, it can be regarded as plastic. When the material reaches the strength limit, it is destroyed, and plastic flow occurs. The Mohr-Coulomb plastic failure criterion model can better represent the stress failure characteristics of a soil mass and is generally used for underground excavation and slope stability analysis. Its theory is that materials change from elastic deformation to plastic deformation [42], and yield failure occurs when the stress reaches the yield limit. Its yield function is as follows  
\[\left|\tau_{q}\right|=c+\sigma_{n}\tan\varphi \tag{14}\]  
where \(\tau_{q}\) is the shear force of the yield surface and \(\sigma_{n}\) is the yield surface normal stress. The parameters of the Mohr-Coulomb plastic failure criterion model mainly include bulk modulus (\(K\)), shear modulus (\(G\)), [PERSON]'s modulus \(\mathbf{\epsilon}\), and [PERSON]'s ratio (\(\mu\)). The relationship between these parameters is as follows  
\[K=\frac{E}{3(1-2\mu)} \tag{15}\]  
\[G=\frac{E}{2(1+\mu)} \tag{16}\]  
In the numerical simulation of roadway wall rock, the commonly used geotechnical analysis software includes FLAC 3D, UDEC, 3 DEC, etc. FLAC 3D was selected as the software foundation to build our numerical simulation method of a roadway wall rock support system. It is based on the Lagrangian finite difference method, which can accurately simulate the yield and plastic flow of materials and can effectively deal with elastoplastic materials, large deformation, and other problems, which is suitable for the research content of this paper. The size of the elastic-plastic area and stress concentration area during roadway excavation is affected by factors such as the roadway section size, burial depth, and mechanical properties of coal and rock. Combined with the geological data of the 7900 mining area of a mine, FLAC 3D was used to build the roadway wall rock support system model, as shown in Figure 4, wherein (a) is the initial model of wall rock and (b) is the model of roadway after excavation.