# Stability Analysis of Isolated Roof in Overlapping Goal Based on Strength Reduction
## 3 Stability Analysis of Isolated Roof Under the Staggered Support  
### Relationship Between the Safety Factor and the Relative Position of the Pillars  
The relative positions of the upper and lower pillars in the overlapping goat are random rather than fixed, but the distance between the center lines of the upper and lower pillars will not exceed the width of goaf no matter how it changes. To illustrate the problem, the concept of dislocated width is introduced, and it is defined as the distance between the center lines of the upper and lower pillars, which is \(c\) in Figure 3. It is not difficult to determine from the figure that when the upper and lower pillars are perfectly aligned, the dislocated width is the smallest, and \(c_{min}=0\). When the lower pillar moves to the position  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **E/GPa** & \(\sigma_{t}\)/MPa & **Poisson’s Ratio** & **Cohesion /MPa** & **Friction/\({}^{\circ}\)** \\ \hline Ore & 12.3 & 0.7 & 0.25 & 3.8 & 27.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical parameters of materials.  
Figure 5: Numerical model of overlapping goat.  
directly below the adjacent upper pillar, the dislocated width is the largest, and \(c_{max}=a+b\); in this case, the upper and lower pillars are also completely overlapped.  
The calculation results of each scheme and the safety factor curve are shown in Figure 6. The safety factor of the isolated roof is closely related to the dislocated width, and the curve shows a clear trend from falling to rising, which looks like a parabola with an upward opening and has remarkable symmetry. Analysis of the curve shows that with the increase of \(c\), the curve can be roughly divided into three stages with \(c=12\) and \(c=28\) as the separatrix:  
1. The first stage (L1) is the rapid decline of the safety factor. The safety factor of the isolated roof decreases rapidly with the increase of the dislocated width, and the safety factor decreases almost in a steep straight line during this stage. The upper and lower pillars are gradually misaligned from complete alignment, which causes the pressure from the overlying strata to transfer from the upper pillar to the isolated roof and then to the lower pillar. The roof begins to bear part of the load, and its safety reserve drops sharply from large to small.
2. The second stage is the L2 where the safety factor changes from down to up. When the dislocated width continues to increase, the safety factor no longer shows the rapid decline in the L1 stage but instead enters the interval from decline to rise. In this interval, the curve first slowly drops and reaches the bottom when the dislocated width is 20 m and then begins to enter the rising period. During this stage, although the variation range of the safety factor is not significant, its value is less than 1.5. Obviously, the continuous low movement of the safety factor may threaten the stability of the roof and even easily lead to roof collapse.
3. The third stage (L3) is the rapid increase of the safety factor. Due to the symmetry of the upper and lower pillars, if the dislocated width continues to increase, the spacing between the upper and lower pillars will be reduced, which is equivalent to the pillars moving towards each other. The load transfer is basically the same as that in the first stage, mainly between the upper and lower pillars, the stress on the roof is greatly reduced, and the safety factor is rapidly increased.