# Stability Analysis of Isolated Roof in Overlapping Goal Based on Strength Reduction
## 5 Discussion  
### Criteria for Judging the Critical Equilibrium State of the Isolated Roof  
The key to analyzing the stability of the isolated roof using the strength reduction method lies in how to determine the critical equilibrium state. From related research of geotechnical mechanics, the criteria for determining the instability of the rock mass can be divided into three categories: plastic zone penetration criteria, critical point displacement mutation criteria, and calculation non-convergence criteria. These three standards have different judgments on the critical equilibrium state of the rock mass. The plastic zone penetration criterion considers that when the plastic zone penetrates through the rock mass, it can be judged that the rock mass has entered the critical equilibrium state [25; 26; 27; 28; 29]. The critical point displacement criterion is to determine the point of displacement mutation by monitoring the displacement change at the critical point, which is regarded as the critical equilibrium point of the rock mass stability state. The judgment method of the non-convergence criterion is more direct. By continuously reducing the strength of the rock mass until the numerical calculation does not converge, the non-convergence state is determined as the critical equilibrium state. Each of the above three criteria has its own focus, and the plastic zone penetration criterion is the most clear explanation for rock mass instability, which is the criterion used in this study. However, it is not clear whether the critical point displacement criterion and the non-convergence criterion are applicable to the strength reduction analysis of  
Figure 10: Evolution of failure units in the roof with increasing dislocated width by reducing shear strength. the isolated roof. For this reason, it is necessary to discuss the applicability of different criteria separately. During the numerical calculation process, six monitoring lines A, B, C, D, E, and F were set up to monitor the displacement changes in key areas of the isolated roof, as shown in Figure 11a. The calculation results indicate that the displacement change of the roof is stable, without significant fluctuations, and there are no sudden changes. Taking monitoring line F as an example, regardless of how the strength is reduced, the displacement remains around 61.5 mm. Therefore, the displacement curve is basically a horizontal straight line, and the displacement cloud diagrams with different reduction factors are also roughly the same (Figure 11b). It should be noted that the lack of significant change in displacement does not mean that the rock mass is in a stable state [30; 31; 32]. Due to the continuous reduction of strength, the roof continues to weaken its ability to resist damage, leading to its stability deterioration, which is evidenced by the distribution of the plastic zone (Figure 11c); the range of the plastic zone continues to expand in the process of the increase of the strength reduction factor. According to the plastic zone penetration criterion, when the reduction factor increases to 4.0, the roof has already been damaged, and it has experienced the critical equilibrium state before that. However, since the displacement has never undergone a sudden change, the critical point displacement criterion considers that the roof has still not reached critical equilibrium. Obviously, the critical point displacement criterion is not applicable to the analysis of the stability strength reduction method of the roof. In numerical simulation analysis, the phenomenon of non-convergence of the calculation process is often regarded as a distinctive feature of the rock mass in critical equilibrium [33]. In this study, it was found that no matter what the dislocation of the upper and lower pillars is, non-convergence occurs when the destruction rate of the units reaches 80% to 85%. Such a high destruction rate makes most of the units become plastic units, and the roof has been penetrated by the plastic zone.