# Study on the Evolution Law of Overlying Rock Collapse Induced by Mining Based on BOTDR
## 4 Analysis of Horizontal Optical Fiber Strain Monitoring in Slope  
### Analysis of Overburden Collapse and Fiber Optic Mechanism  
As the working face advances, the overlying strata undergo progressive collapse. When fiber optics are effectively embedded within the rock mass, they experience axial tensile strain as the strata collapse, thereby reflecting the internal strain variations within the overburden. By analyzing the spatiotemporal evolution of overburden strain in response to mining operations, the movement and failure mechanisms of the disturbed overburden can be revealed, providing insights into the failure modes under different excavation conditions. Since fiber optics directly capture strain evolution but do not explicitly measure displacement variations caused by rock collapse, a mechanical model is established based on the principles of material mechanics. In the model, the red line represents the initial placement of the fiber optic cable, while the green line represents its deformed shape after overburden collapse. For analytical convenience, it is assumed that the overburden load is uniformly distributed, the fiber optic material is isotropic, and the fiber remains intact and continuously deployed within the rock mass. When the overlying strata collapse is shown in Figure 4, the following model is established. Assuming that the length between AB is 1\({}_{1}\), the length between AB' is 1\({}_{1}\), BB' is the displacement h\({}_{1}\), and the strain of fiber AB' is \(\varepsilon_{1}\), it can be obtained from the definition of strain \(\varepsilon_{1}\):  
\[\varepsilon_{1}=\frac{l_{1}-l_{1}}{l_{1}} \tag{2}\]  
Figure 3: Test model test layout. Then, the length \(1^{\prime}_{1}\) between the collapse backward AB\({}^{\prime}\) is:  
\[\textit{I}\textit{r}_{1}=\textit{I}_{1}(1+\varepsilon_{1}) \tag{3}\]  
According to the Pythagorean theorem, the displacement h\({}_{1}\) is:  
\[\textit{h}_{1}{}^{2}=\textit{l}_{1}{}^{2}-\textit{l}_{1}{}^{2}=\textit{l}_{1}{ }^{2}(1+\varepsilon_{1})^{2}-\textit{l}_{1}{}^{2} \tag{4}\]  
The displacement h\({}_{1}\) is:  
\[h_{1}=\pm l_{1}\sqrt{\varepsilon_{1}{}^{2}+2\varepsilon_{1}} \tag{5}\]  
As shown in Figure 5, the slope of \(\Delta\)ABB\({}^{\prime}\) is k \(<\) 0 (the collapse slope of the rock stratum is k \(<\) 0), and the collapse displacement of the rock stratum increases, so the positive sign is taken; otherwise, the negative sign is taken. From this, it can be seen that the displacement h\({}_{1}\) is:  
\[h_{1}=l_{1}\sqrt{\varepsilon_{1}{}^{2}+2\varepsilon_{1}} \tag{6}\]  
As shown in Figure 5, a single rock layer collapses, and the optical fiber in the rock does not undergo tensile changes, so h\({}_{2}\) = h\({}_{1}\). When the overlying strata fall, as shown in Figure 5, the two ends are hinged with each other, and the following model is established. The length between AB is l\({}_{1}\), BB\({}^{\prime}\) is the displacement h\({}_{1}\), and the strain of optical fiber AB\({}^{\prime}\) is \(\varepsilon_{1}\). The backward displacement h\({}_{1}\) can be obtained from the above formula:  
\[h_{1}=l_{1}\sqrt{\varepsilon_{1}{}^{2}+2\varepsilon_{1}} \tag{7}\]  
Similarly, the length of BC is l\({}_{2}\), CC\({}^{\prime}\) is the displacement h\({}_{2}\), and the strain of B\({}^{\prime}\)C\({}^{\prime}\) is \(\varepsilon_{2}\). At this time, the total displacement of the fiber is h\({}_{C}\) = h\({}_{1}\) + h\({}_{2}\), and the displacement variation h\({}_{2}\) can be obtained from the above formula:  
\[h_{2}=l_{2}\sqrt{\varepsilon_{2}{}^{2}+2\varepsilon_{2}} \tag{8}\]  
The displacement of model D is h\({}_{D}\) = h\({}_{1}\) + h\({}_{2}\) + h\({}_{3}\). The length between the CD is l\({}_{3}\), DD\({}^{\prime}\) is the displacement h\({}_{3}\), and the strain of the optical fiber C\({}^{\prime}\)D is \(\varepsilon_{3}\). It can be seen from the diagram that the slope of the C\({}^{\prime}\)D of the overburden collapse is k \(>\) 0, the displacement is reduced, and h\({}_{3}\) is negative.