## 3 Numerical Simulation Results
### Realization of Strain-Softening  
In the current numerical simulation studies, the post-peak strain softening process of supporting materials, based on the Mohr-Coulomb yield criterion, was rarely considered. Therefore, some discrepancies may exist between numerical modeling and field monitoring. The linear strain-softening equation of the supporting materials was herein deduced based on previous studies [31; 32; 33]. The following assumption was made before the equation was derived: the supporting materials were in unidirectional stress during the supporting process. The linear strain-softening equation was constructed in the form of Figure 6.  
Figure 5: Model setup and supporting structures of the roadway. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Lithology** & \begin{tabular}{c} **Elastic Modulus** \\ **(GPa)** \\ \end{tabular} & **Poisson Ratio** & \begin{tabular}{c} **Unit Weight** \\ **(kN-m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Cohesion** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Friction** \\ **Angle (\({}^{\circ}\))** \\ \end{tabular} & **Tension (MPa)** \\ \hline Silstone & 13.0 & 0.25 & 26.9 & 9.8 & 37 & 6.4 \\ Sandy & 4.50 & 0.30 & 26.1 & 1.9 & 30 & 2.45 \\ mudstone & 1.12 & 0.22 & 22.9 & 0.58 & 32 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of the surrounding rocks. A general equation can be developed from Figure 6 and shown as follows in Equation (1). \[\sigma=\left\{ \begin{array}{c}\frac{\sigma_{\min}-\sigma_{\max}}{d_{1}-d_{0}}d+ \sigma_{\max}-\frac{\sigma_{\min}-\sigma_{\max}}{d_{1}-d_{0}}d_{0},\ d_{0}\leq d <d_{1}\\ \sigma_{\min},\ d\geq d_{1}\end{array} \right. \tag{1}\]  
where \(\sigma_{\max}\) and \(\sigma_{\min}\) are the yield stress and the residual stress, respectively; \(d_{0}\) and \(d_{1}\) are displacements corresponding to the yield stress and the residual stress, respectively, and \(d\) is the compressive displacement. Based on the results of the above compression test, the optimum content of basalt fibers in BFRGC was 0.4%; therefore, A2, B2, C2, and D2 were selected to further study the optimal mixing amounts of rubber particles. The \(\sigma_{\max}\) and \(\sigma_{\min}\) in Equation (1) are determined by fitting the results of the above triaxial test and the uniaxial test obtained by [PERSON] et al. [27]. The relationships between confining pressure and yield stress and residual stress are shown in Figure 7, and the corresponding fitting equations are listed in Equation (2). \[\left\{ \begin{array}{c}\sigma_{y\_A2}=34.55+1.47\sigma_{3},\sigma_{r\_A2}=20.40+2.54\sigma_{3}\\ \sigma_{y\_B2}=21.64+2.81\sigma_{3},\sigma_{r\_B2}=9.11+3.94\sigma_{3}\\ \sigma_{y\_C2}=14.58+1.98\sigma_{3},\sigma_{r\_C2}=4.86+2.61\sigma_{3}\\ \sigma_{y\_D2}=10.19+2.07\sigma_{3},\sigma_{r\_D2}=3.98+2.42\sigma_{3}\end{array} \right. \tag{2}\]  
where \(\sigma_{y}\) is the yield stress, MPa; \(\sigma_{3}\) is the confining pressure, MPa; \(\sigma_{r}\) is the residual stress, MPa. Figure 6: The linear strain-softening equation. Figure 7: The fitting results. (**a**) The relationships between confining pressure and yield stress; (**b**) the relationships between confining pressure and residual stress. It could be determined from Equation (2) that the yield stresses of A2, B2, C2, and D2 under the uniaxial stress were about 34.55 MPa, 21.64 MPa, 14.58 MPa, and 10.19 MPa, and the residual strengths were about 20.40 MPa, 9.11 MPa, 4.86 MPa, and 3.98 MPa, respectively. Additionally, based on the uniaxial test results [27], the \(d_{0}\) of A2, B2, C2, and D2 under the uniaxial stress was about 0.23 mm, 0.53 mm, 0.64 mm, and 0.82 mm, and the \(d_{1}\) was about 0.46 mm, 1.05 mm, 1.27 mm, and 1.63 mm. The derived equations of A2, B2, C2, and D2 are thus given in Equations (3)-(6), respectively. \[\sigma_{A2}=\left\{ \begin{array}{rl}48.70-61.52d,&0.23\leq d<0.46\\ 20.40,&d\geq 0.46\end{array} \right. \tag{3}\]  
\[\sigma_{B2}=\left\{ \begin{array}{rl}34.41-24.10d,&0.53\leq d<1.05\\ 9.11,&d\geq 1.05\end{array} \right.