## 4 Results
### Developmental Characteristics of the Plastic Zone  
According to the Kastner Formula (1), the radius of the plastic zone in the roadway surrounding rock is 1.91 m. According to Equation (2), the plastic zone depth of the surrounding rock is approximately 1.62 m, which has a weak relationship with the polar coordinates of the surrounding rock. The plastic zone of the surrounding rock, calculated using the formula, is large. Owing to the different simplification degrees of the stress  
Figure 8: Relationship between distribution of principal stress and position of working face: (**a**) primary mining influence; (**b**) secondary mining influence. fields in the roadway, the calculation results differ significantly. Therefore, further research is necessary. The results of the broken rock zone detection are shown in Figure 9. As shown in Figure 9a, the acoustic propagation velocities at the four test holes on the coal wall side remain essentially unchanged in the plastic zone with a low hole depth; furthermore, the wave velocity is low. As the distance from the hole opening exceeds 1.8, 1.6, 1.2, and 1.0 m, the wave velocity increases sharply, indicating that the coal wall enters the elastic zone. Therefore, it is concluded that the broken rock zone ranges for the four measuring holes on the coal wall are 1.8, 1.6, 1.2, and 1.0 m. Similarly, as shown in Figure 9b, the broken rock zone ranges for the three measuring holes on the coal pillar side are 1.4, 1.2, and 0.8 m.  
The statistics of the broken rock zone detection results are presented in Table 2. Based on the test results, it is evident that, the shorter the distance is from the working face, the greater is the range of the broken rock zone. From 45 m to 15 m in front of the working face, the broken rock zone of the coal pillar side increases from 0.8 m to 1.4 m, while the broken rock zone of the coal wall side increases from 1.0 m to 1.8 m. The range of the broken rock zone of the coal wall is larger than that of the coal pillar. Figure 9: Acoustic propagation velocity of surrounding rock: (a) coal wall side hole; (b) coal pillar side hole. The plastic zone range, calculated by the formula, is 1.91 m and 1.62 m, and the maximum plastic zone range, obtained via field observations, is 1.8 m; these values are highly similar. It can be considered that the range obtained using the calculation formula is the plastic zone range of the surrounding rock with the most significant damage. Thus, it is safe to design the support according to the maximum plastic zone of the surrounding rock; this, however, is not necessary. This will waste materials and equipment, resulting in high support costs. Therefore, it is essential to obtain the range of the plastic zone and also further study its distribution characteristics. To better understand the distribution characteristics of the plastic zone, two small FLAC\({}^{\text{3D}}\) numerical models of the 21309 roadway under the two mining influences were established. The dimensions of the numerical model were 70 m along the \(X\)-axis, 3 m along the \(Y\)-axis, and 40 m along the \(Z\)-axis. The Mohr-Coulomb constitutive model was adopted as the failure criterion. The displacement boundary conditions were used in the \(Y\)-axis direction, whereas the stress boundary conditions were used in the other directions. The stress determined from the large numerical model was applied to the small model. The required stress value was selected from the location where the ratio of the maximum principal stress to the minimum principal stress was the largest; a distance of 70 m behind the working face was selected for primary mining. The initial stresses applied to the small numerical model were \(\sigma_{\text{x}}=-5.32\) MPa, \(\sigma_{\text{y}}=-4.32\) MPa, \(\sigma_{\text{z}}=-5.25\) MPa, and \(\tau_{\text{xx}}=-9.95\) MPa.