## 4 Numerical Simulation and Results
### Analysis of the Effect of Slate Overhang on the Force Deformation of the Roof Slab  
The study on the effect of overhanging slates showed an increase in the number of overhanging slates sequentially from the drainage of J10 behind the cavern of Yuanjue Cave to the direction of the light hole under the current condition. The empty model in FLAC 3D numerical analysis software was used to simulate the state of the overhanging slates without bearing pressure, and the change in the number of overhanging stones was simulated by changing the intrinsic structure of the slates ([PERSON] et al. [15]).  
Figure 10 shows, from left to right, the vertical displacement clouds of the top slab of Yuanjue Cave in the case of 1, 4, 10, and 15 overhanging slates. It can be seen that with the increase in the number of overhanging boulders, the top slab of Yuanjue Cave underwent uneven settlement, and the maximum settlement position was shifted from the central area to the left rear of Yuanjue Cave (boulder overhanging side).  
According to the numerical simulation results, the variation curve of vertical displacement of each measurement point with the number of overhanging slate blocks was made  
Figure 10: Cloud diagram of vertical displacement of the top slab when the number of overhanging slates is different.  
Figure 9: Three-dimensional geological model. (**a**) Schematic diagram of the development of overhanging slates. (**b**) Location layout of the monitoring points.  
with the number of overhanging slate blocks as the horizontal coordinate (Figure 11a). The horizontal displacement in the x and y directions of the two pairs of measurement points above and below the L6 crack was decomposed and differentiated along the vertical direction of the L6 crack to obtain the variation curve of the width above and below the L6 crack with the number of overhanging slate blocks (Figure 11b).  
With the increase of the number of overhanging slates, the settlement of Y6 and Y10 measuring points on the roof of the Yuanjue Cave decreased, the settlement of Y8 and Y9 measuring points increased, the width of L6 cracks kept increasing, and the change rate increased with the increase of the number of overhanging boulder blocks. The settlement of the Y8 measuring point was the largest and the most significant change (Figure 11a), and the settlement of Y6, Y9, and Y10 were smaller because the Y8 measuring point was close to the development area of overhanging slates. The settlement of Y8 and Y9 began to increase sharply when the overhanging boulders exceeded 12 (TS30) and entered the stage of rapid change.  
The L6 crack pattern was \"wide at the top and narrow at the bottom\", which is a typical tensile crack, and the L6 crack width increased with the increase of overhanging slates, and when the overhanging slates exceeded 12 pieces, the L6 crack width started to increase sharply and entered the rapid change stage (Figure 11b). The development process of the settlement of the roof slab and the L6 crack width of Yuanjue Cave was similar, and both can be divided into three stages, i.e., the initial uniform change stage (the number of overhanging sized slates is less than 3), the middle stable change stage (the number of overhanging sized slates is between 3 and 12), and the later rapid change stage (the number of overhanging sized slates exceeds 12). When the number of overhanging slates exceeds 12, the vertical displacement and L6 crack width of each measurement point enter the late rapid change stage. Based on the principle of preventive protection, the number of overhanging slates reaching 12 can be taken as a sign that the roof of Yuanjue Cave enters the early warning state of instability. The critical number of overhanging stones was 12 (TS30 location in Figure 12).  
Figure 11: (**a**) The variation curve of vertical displacement of each measurement point with the number of overhanging slate blocks; (**b**) the variation curve of the increase of L6 crack width with the number of overhanging slate blocks.