## 4 Numerical Simulation and Results
### Analysis of the Effect of Slate Crushing on the Force Deformation of the Roof Slab
To analyze the influence of different degrees of damage on the stability of the roof plate of Yuanjue Cave, the mechanical parameters such as the modulus of elasticity of the cracked damaged slates were reduced to 1/2, 1/5, 1/10, 1/50, and 1 \(\times\) 10\({}^{-6}\) of the original ones (where 1 \(\times\) 10\({}^{-6}\) can be equated to complete damage). The overall equivalent stiffness of each group of slates was calculated, and the results are shown in Table 4 (only the slates with changed equivalent stiffness are listed). Figure 17 shows the vertical displacement of the roof slab of the cave from left to right when the damage factor was 0.5, 0.8, 0.9, and complete damage. With the increase of the damage of the cracked slate, the uneven settlement of the roof plate of the cave occurred, and the maximum settlement position was shifted from the central area to the left front of the cave, which had a significant effect on the force deformation and stability of the roof plate of the cave. According to the numerical simulation results, the vertical displacement curve of each measurement point with the damage factor was made in the horizontal coordinate of the cracked stone damage factor (Figure 18a). 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. The change curve of the width of the L6 crack above and below the damage factor was obtained (Figure 18b). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Damage** & & & & & \\
**Factor** & & & & & \\ \hline Grouping & K(N/m) & K(N/m) & K(N/m) & K(N/m) & K(N/m) \\ TS01 & \(1.71\times 10^{10}\) & \(6.84\times 10^{9}\) & \(3.42\times 10^{9}\) & \(6.84\times 10^{8}\) & \(\approx\)0 \\ TS10 & \(7.77\times 10^{9}\) & \(3.11\times 10^{9}\) & \(1.55\times 10^{9}\) & \(3.11\times 10^{8}\) & \(\approx\)0 \\ TS12 & \(1.78\times 10^{10}\) & \(7.12\times 10^{9}\) & \(3.56\times 10^{9}\) & \(7.12\times 10^{8}\) & \(\approx\)0 \\ TS14 & \(4.71\times 10^{9}\) & \(2.87\times 10^{9}\) & \(1.73\times 10^{9}\) & \(4.17\times 10^{8}\) & \(\approx\)0 \\ TS16 & \(8.91\times 10^{9}\) & \(6.74\times 10^{9}\) & \(4.79\times 10^{9}\) & \(1.45\times 10^{9}\) & \(\approx\)0 \\ TS29 & \(1.89\times 10^{10}\) & \(1.09\times 10^{10}\) & \(6.66\times 10^{9}\) & \(1.64\times 10^{9}\) & \(\approx\)0 \\ TS30 & \(5.23\times 10^{9}\) & \(3.30\times 10^{9}\) & \(2.11\times 10^{9}\) & \(5.52\times 10^{8}\) & \(\approx\)0 \\ TS34 & \(4.00\times 10^{9}\) & \(2.50\times 10^{9}\) & \(1.54\times 10^{9}\) & \(3.78\times 10^{8}\) & \(\approx\)0 \\ TS49 & \(1.04\times 10^{10}\) & \(9.55\times 10^{9}\) & \(9.13\times 10^{9}\) & \(8.73\times 10^{9}\) & \(\approx\)0 \\ TS51 & \(1.21\times 10^{10}\) & \(6.05\times 10^{9}\) & \(3.30\times 10^{9}\) & \(7.13\times 10^{8}\) & \(\approx\)0 \\ TS52 & \(1.28\times 10^{10}\) & \(6.43\times 10^{9}\) & \(3.66\times 10^{9}\) & \(8.48\times 10^{8}\) & \(\approx\)0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Equivalent bearing stiffness of the cracked boulders. Figure 16: Distribution of the location of the current cracked and damaged boulders. The settlement of Y6, Y8, and Y9 gradually increased and the settlement of Y10 decreased with the increase of damage factor, and the change rate increased with the increase of the damage factor of L6. Y9 settlement started to increase sharply and entered the stage of rapid change. At the same time, the L6 cracks still showed the typical tensile fracture characteristics of \"wide at the top and narrow at the bottom\", and the width of the cracks kept increasing as the damage of the cracked rocks intensified. When the damage factor of the cracked slate exceeded 0.9, the L6 crack width started to increase sharply and entered the stage of rapid change.