## 7 Numerical Examples
### 3D Analysis of an Exacavation in Layered Rock  
This example investigates the impact of out-of-plane layers on the deformation and stability of an excavation. It highlights the effectiveness and precision of the Cos-FEA method in predicting the behavior of layered structures. Figure 6 illustrates the geometry, boundary conditions, loading configuration and definition of layer orientation for this problem. Due to the symmetry of the problem, we study only a quarter of it, as shown in Figure 7. The basic parameters of the interstratified rock mass are listed in Table 3. The length of the chamber is set to 30 m, and a uniform pressure of 2.0 MPa is applied to the top surface. Using its symmetry, half of the model is simulated with 1800 eight-node equal-parameter brick elements. Element type: eight-node brick element;  
Number of elements: 1800;  
Number of elements in out-of-plane direction: 20;  
Number of FEM nodes: 2394;  
Number of degrees of freedom in 3D space: 11,357. \begin{table}
\begin{tabular}{c c c} \hline
**Timoshenko Solution** & **Cosserat FEM of Solution** & **Traditional FEM of** \\
**(mm)** & **(mm)** & **Solution (mm)** \\ \hline
2.854 & 2.794 & 2.524 \\ \hline \end{tabular}
\end{table} 
Table 2: The deflection of the tip of the cantilever layered plate. \begin{table}
\begin{tabular}{c c c} \hline
**ParametersName** & **Sandstone** & **Sandy Mudstone** \\ \hline Volume concentration (\(\mathbf{n}\))/\(\%\) & 25.0 & 75.0 \\ Volume density (\(\mathbf{k}\))/kN\(\cdot\)m\({}^{-3}\) & 24.9 & 25.8 \\ Elastic modulus (\(\mathbf{E}\))/GPa & 21.78 & 5.15 \\ Internal friction angle (\(\mathbf{\gamma}\))/\({}^{\circ}\) & 26.0 & 34.0 \\ Cohesive strength (\(\mathbf{\sigma_{c}}\))/MPa & 1.40 & 0.75 \\ [PERSON]’s ratio (\(\mathbf{\mu}\)) & 0.23 & 0.31 \\ \hline \end{tabular}
\end{table} 
Table 3: Basic parameters of the layered rock mass. The Cosserat FEM (CFEM) mesh discretization of the structure is shown in Figure 8. The numerical deformed configuration diagram is exhibited in Figure 9. Moreover, the contour maps of vertical displacement, the maximum principal stress and the maximum shear strain at failure were obtained by using the M-C criterion-based CFEM (CFEM M-C) and the D-P criterion-based CFEM (CFEM D-P), as shown in Figures 10-13. The mesh grid of structural deformation is enlarged by 200 times, where red stars represent the yielded Gaussian points, as shown in Figure 13. As can be seen from Figures 10-13, the results calculated by the D-P criterion are larger, and the damage area of the cavern is predicted to be wider. Due to the stress concentration, the maximum compressive stress occurs in the area of abrupt shape, that is, the four corners of the cavern. The maximum tensile stress is concentrated on the side wall, top and bottom of the cavern, which is consistent with the location of the failure phenomena, such as rock bending, that occur when the cavern is excavated in the layered rock mass. In addition, due to the equivalence of the Cosserat extended model, the contour maps calculated using the two intensity criteria appear to be relatively smooth. The load factors are accessed on the first iteration of each load increment. In this program, they were specified as 0.8 and 0.2, so the total load acting on the structure during the second increment is 1.0 times the calculated loads. Figure 14 indicates that when  
Figure 6: The geometric dimensions, boundary conditions and loads of the model. Figure 7: Calculating schematic of a square tunnel in an interlayered rock mass. computation convergence is attained, the M-C criterion-based Cosserat FEM (CFEM M-C) need to finish 102 iterations, and the D-P criterion-based Cosserat FEM (CFEM D-P) only needs 18 iterations. The former was approximately six times more than the latter. Figure 8: Cos-FEA mesh discretization of structure. Figure 10: Contour map of vertical displacement (\(u_{3}\)) using D-P method. Figure 9: Numerical deformed configuration diagram. The red frame represents the dimensions of the structure that are not deformed.