## 4 Numerical Simulation of Inclined Shaft Lining
### Results and Analysis of Numerical Simulation  
Two types of models, planar and three-dimensional, were established as shown in Figure 14. The contact model establishes mechanical interaction between the shaft lining \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Elastic Modulus** & \begin{tabular}{c} **Poisson’s Ratio** \\ **of Shaft Lining** \\ \end{tabular} & \begin{tabular}{c} **Elastic Modulus of** \\ **Surrounding Rock** \\ **/GPa** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s Ratio of** \\ **Surrounding Rock** \\ **/GPa** \\ \end{tabular} & \
\begin{tabular}{c} **Allowable Tensile** \\ **Stress** \\ **/MPa** \\ \end{tabular} &
\begin{tabular}{c} **Friction** \\ **Coefficient** \\ \end{tabular} \\ \hline
0.4 & 0.3 & 20 & 18 & 0.2 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters. Figure 12: Test results for elastic modulus and Poisson’s ratio of specimens: (**a**) Elastic modulus test results of concrete specimens; (**b**) Poisson’s ratio test results of concrete specimens; (**c**) Elastic modulus test results of surrounding rock specimens; (**d**) Poisson’s ratio test results of surrounding rock specimens. Figure 13: Finite element model: (**a**) Three-dimensional spatial model; (**b**) Plane model. and surrounding rock, whereas the non-contact model omits such interface behavior. The fundamental distinction lies in whether to consider the rock-lining interaction-neglecting this effect may lead to significant deviations from actual field conditions. As shown in Figure 14a, except for the invert and sidewall areas where the plane model strain exceeds that of the 3D non-contact model, strains in other locations are nearly identical between the 3D and plane models. Due to reduced stress concentration in contact models, the 3D contact model exhibits the smallest strain. Figure 14b shows that at the crown, strains decrease in the following order: 3D contact model, 3D non-contact model, plane non-contact model, and plane contact model. At the invert, the strain trend of the 3D contact model is completely opposite to the other three models. Calculations confirm that the plane contact model aligns with theoretical predictions, making it suitable for analysis. Comparative results for the No. 3 shaft lining are shown in Figures 15 and 16. Figure 14: Finite element model calculation results: (**a**) Circumferential strain at the inner edge of the model; (**b**) Circumferential strain at the outer edge of the model. Figure 15: Comparison between fiber optic measurements and numerical simulations at the inner edge of the NO. 3 shaft lining model: (**a**) X = 0.48 MPa, Y = 0.8 MPa; (**b**) X = 0.96 MPa, Y = 0.1.6 MPa; (**c**) X = 1.44 MPa, Y = 2.4 MPa; (**d**) X = 1.92 MPa, Y = 3.2 MPa; (**e**) X = 2.4 MPa, Y = 4 MPa; (**f**) Specific point numbering of the shaft lining model. As shown in Figure 15, fiber optic measurements at the inner edge of the shaft lining generally align with theoretical predictions. Minor deviations at certain points may result from errors in dimensions, installation, or other factors. The large tensile strain at point A is likely caused by dimensional differences, casting issues, or gaps between the surrounding rock and loading plates. Except for point A, most fiber optic measurements closely match numerical simulation results. This confirms that fiber optics can effectively monitor strain across the entire inner edge of the shaft lining. As shown in Figure 16, fiber optic measurements at the outer edge of the shaft lining generally align with theoretical predictions. Minor deviations at certain points may result from errors in dimensions, installation, or other factors. The large compressive strain at point G, which contradicts numerical simulation results, is likely caused by dimensional differences, casting issues, or gaps between the surrounding rock and loading plates. Except for point G, most fiber optic measurements closely match numerical simulation results.