# Stability Analysis of Isolated Roof in Overlapping Goal Based on Strength Reduction
## 4 Effect of Width Ratio and Dislocated Width on the Stability of the Isolated Roof  
### Characterization of the Variation in the Safety Factor for Different Width Ratios  
In order to deeply explore the change characteristics of the safety factor of the isolated roof in different sizes of excavation space, the concept of the width ratio is proposed. The ratio of the goat width to the pillar width is defined as the width ratio, which is used to analyze the stability of the roof. \[\lambda=\frac{a}{b} \tag{4}\]  
In the equation, \(\lambda\) is the width ratio, which is a parameter indicating the size of the goat; \(a\) and \(b\) are the width of the goat and the width of the pillar, respectively. The numerical calculation results are extracted to draw the curve diagram of the relationship between the safety factor and the width ratio, as shown in Figure 8. The safety factor \(F\) of the isolated roof has a significant stage change characteristic, with \(\lambda\) = 2 as the dividing line, the safety factor curve can be divided into the safety reserve decreasing stage A and the safety reserve stabilizing stage B. When \(\lambda\) < 2, \(F\) is larger in general, and the roof has sufficient safety reserve. However, \(F\) decreases sharply with the increase of \(\lambda\), as shown in the following: the tensile safety factor \(F_{t}\) decreases from 13.28 to 2.72, and the shear safety factor \(F_{s}\) decreases from 7.78 to 1.82. On the contrary, when \(\lambda\) > 2, \(F\) is smaller in general, and the curve becomes very smooth, and within this stage, the increase of \(\lambda\)  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(c\)/m & \(F_{t}\) & \(F_{s}\) & **Status** & **Depth/m** & \(c\)/m** & \(F_{t}\) & \(F_{s}\) & **Status** & **Depth/m** \\ \hline
10 & 3.96 & 3.98 & stability & & 10 & 0.95 & 0.94 & stability & \\
12 & 3.32 & 3.13 & stability & & 12 & 0.70 & 0.55 & stability & \\
14 & 2.67 & 2.29 & stability & & 14 & 0.53 & 0.37 & stability & \\
16 & 2.18 & 1.83 & stability & & 1  
only makes \(F\) decrease slowly and tend to 0. Therefore, it is very clear that the width ratio \(\lambda\) is an important factor affecting the stability of the roof in the overlapping goal. According to Equation (3), \(\lambda\) is jointly determined by the width of the goaf and the width of the pillar, and its essence is a characteristic parameter indicating the size of the overlapping goaf. The greater \(\lambda\) means that the goaf is relatively larger, or the pillar is relatively smaller. But in either case, the isolated roof is bound to bear greater loads. On the other hand, the strength of the roof does not change due to \(\lambda\). Therefore, the increased load naturally causes a decrease in the safety factor \(F\), which further leads to the evolution of the roof from stability to instability. This is the basic law of the influence of \(\lambda\) on the stability of the roof. Obviously, a reasonable width ratio is crucial to guarantee the stability of the roof. The smaller \(\lambda\) is, the larger \(F\) is, and the better the stability of the roof is. From Figure 7, it is not difficult to determine that when \(\lambda<2\), \(F\) is greater than 1.5, and when \(\lambda>2\), \(F\) tends to 0. Accordingly, in order to make the roof be in a stable state, the width ratio should not be greater than 2. Characteristics of the Destabilization Evolution of the Isolated Roof Under Different Dislocated Widths  
The plastic zone is the most direct reflection of rock mass failure. Extracting numerical simulation calculation results and analyzing the changes in the plastic zone of the isolated roof (Figures 9 and 10) with dislocated width, the following characteristics were found:  
1. The distribution of the plastic zone of the isolated roof is closely related to the dislocated width, and it has extremely significant symmetry. The smaller the dislocated width, the more obvious the bearing characteristics of the pillar, and the more sparse the failure units in the roof of the interlayer.