# Stability Analysis of Isolated Roof in Overlapping Goal Based on Strength Reduction
## 6 Conclusions  
1. It is proposed to adopt the dislocated width to express the relative position relationship between the upper and lower pillars. The study shows that the increase of the dislocated width will lead to the continuous change of the safety factor of the isolated roof. The curve of the safety factor can be divided into three stages: the sharp decline stage, the decline-to-increase stage and the rapid increase stage. In the first stage, the safety factor is inversely correlated with the dislocated width, and the safety reserve drops sharply; In the second stage, the safety factor changes only slightly, and the curve firstly decreases slowly to the inflection point, and then turns to rise, and the stability of the roof is the worst; In the third stage, the safety factor is positively correlated with the dislocated width, and the increase of the dislocated width makes the safety factor rise rapidly, which leads to the obvious enhancement of the stability of the roof. 2. The overlap rate of pillars determines the type of failure of the roof but not the decisive factor of the safety factor. When the overlap rate is \(\eta>0\), the tensile safety factor is less than the shear safety factor, which leads to the tensile failure of the roof more easily. The greater the value, the greater the safety factor of the roof. However, when the overlap rate is \(\eta=0\), the shear safety factor is less than the tensile safety factor, resulting in the roof being more prone to shear failure. During this stage, the safety factor changes with the change in the dislocated width but is not related to the overlap rate. Figure 14: Safety factor of the roof supported by hourglass and funnel pillars. (**a**) Hourglass-shaped pillar; (**b**) Funnel-shaped pillar; (**c**) Safety factor of the roof supported by an hourglass-shaped pillar; (**d**) Safety factor of the roof supported by a funnel-shaped pillar. 3. The width ratio is the key factor affecting the stability of the roof. It is found that the width ratio \(\lambda\) = 2 can be the critical value of the safety reserve. When \(\lambda\) < 2, the safety factor \(F\) of the roof is large, and the safety reserve is sufficient, but \(F\) decreases sharply with the increase of \(\lambda\) in this stage. On the contrary, when \(\lambda\) > 2, with the increase of the width ratio \(\lambda\), \(F\) slowly decreases and gradually tends to 0. Accordingly, reducing the width ratio helps to improve the stability of the roof, and in order to prevent it from destabilizing, the width ratio should not be larger than 2. 4. The study reveals the evolution of the plastic zone of the roof. The failure units are concentrated and symmetrically distributed on both sides of the center line of the upper and lower pillars. The plastic zone is born around the pillar at the beginning, where the tensile failure zone is like a fold line, and the shear failure zone is an oblique strip. With the increase of the dislocated width, resulting in the accumulation of failure units, which promotes the acceleration of the expansion of the plastic zone. Gradually, the tensile failure zone changes into a wavy line, and the width of the shear failure zone becomes a square strip. Writing--original draft preparation, C.L. ; writing--review and editing, K.Z. ; methodology, P.Z. ; investigation, C.G. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Key Research and Development Program of China (No. 2017 YFC0804601) and the National Natural Science Foundation of China (No. 51664018). Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. The authors are very grateful for the financial contributions and convey their appreciation to the organizations for supporting this basic research. Author [PERSON] was employed by the company Changsha Institute of Mining Research Co., Ltd.