## 3 Numerical Simulation Analysis
### Numerical Simulation Results and Analysis
#### 3.3.2 Evolution Characteristics of the Plastic Zone under a \(\sigma_{Z}\)-Dominant Stress Field  
Figure 9 shows the distribution pattern of the plastic zone under the \(\sigma_{Z}\)-dominant stress field when the angle \(\alpha\) between the roadway axial direction and the maximum horizontal principal stress direction changes between 0\({}^{\circ}\) and 90\({}^{\circ}\). When \(\alpha\) is 0\({}^{\circ}\), the plastic zone in rock surrounding the circular roadway is an irregular vertical butterfly shape. With an increase in the axial angle of the roadway \(\alpha\), the wing angle of the roadway begins to decrease. When \(\alpha\) is 50\({}^{\circ}\), the butterfly shape of the roadway plastic zone is not obvious. The area of the two sides of the roadway is significantly larger than the area of the roof and the floor, which is similar to the flat ellipse. When \(\alpha\) is 60-90\({}^{\circ}\), the area of the plastic zone of the roof and the floor of the roadway surrounding rock increases gradually, and the plastic zone is close to a ring in shape. Figure 8: Variation in the depth of the plastic zone with axial angle \(\alpha\). Figure 10 shows the variation curve of the plastic zone size of the roadway with the axial angle \(\alpha\) of the roadway. The following can be seen from Figure 11:  
1. When \(\alpha=0^{\circ}\), the maximum size of the plastic zone of roadway surrounding rock occurs at the wing angle and the plastic zone size is 10.71 m, i.e., 3.57 times that of the roadway radius (3.57 a). With the increase in the axial angle \(\alpha\) of the roadway (10-40\({}^{\circ}\)), the failure depth at the wing angle decreases sharply, and the failure radii are 9.12, 6.90, 4.39, and 3.00 m (i.e., decreases by 1.18, 1.55, 2.44, and 3.57 times, respectively). When \(\alpha\) is 40-70\({}^{\circ}\), the shoulder size is in a relatively stable stage, only decreasing by 0.4 m. When \(\alpha\) is 80-90\({}^{\circ}\), the plastic zone size of the roadway slowly increases by 0.86 m.
2. With the increase in the roadway axial \(\alpha\) (0-40\({}^{\circ}\)), the plastic zone size of the roadway roof decreases from 1.28 to 0.42 m, a decrease by 3.04 times. When \(\alpha\) is 40-90\({}^{\circ}\), the plastic zone depth slowly increases, from 0.42 to 2.78 m, increasing by 2.36 m.
3. When the roadway axial angle \(\alpha\) increases from 0\({}^{\circ}\) to 70\({}^{\circ}\), the size of the plastic zone in the middle of the roadway sidewall does not change, and the size of the plastic zone is roughly 3 m. When \(\alpha\) increases from 70\({}^{\circ}\) to 90\({}^{\circ}\), the size of the plastic zone in the sidewall increases from 3 to 3.8 m, an increase of 0.8 m.
4. In the process of axial rotation of the roadway, when \(\alpha\) is between 0\({}^{\circ}\) and 40\({}^{\circ}\), the maximum plastic zone depth of the roadway occurs at the roadway wing angle, and from 50\({}^{\circ}\), the maximum failure depth of the roadway occurs at the roadway side. 5. In the three positions of the roadway roof, sidewall, and four wings, as the roadway axial angle \(\alpha\) increases, the curve slope of the wing angle changes the most, followed by the change in the roof, and the slope of the sidewall changes the least. Therefore, in the \(\sigma_{Z}\)-dominant stress field, the sensitivity of different positions of the roadway to angle \(\alpha\) is wing angle \(>\) roof \(>\) side. Figure 10: Variation in the depth of the plastic zone with axial angle \(\alpha\). Figure 9: Distribution pattern of the roadway plastic zone under different values of angle \(\alpha\) under the dominant stress field of \(\sigma_{Z}\). 3.3.3 Evolution Characteristics of the Plastic Zone under a \(\sigma_{\rm H}\)-Dominant Stress Field  
Figure 11 shows the distribution pattern of the plastic zone under the \(\sigma_{\rm H}\)-dominated stress field when the angle \(\alpha\) between the roadway axial direction and the maximum horizontal principal stress direction changes between 0\({}^{\circ}\) and 90\({}^{\circ}\).