## 4 Numerical Method of Support Design  
### Basic Assumptions and Methodology for Numerical Modeling  
Numerical modeling is a widely adopted design tool in rock engineering, enabling precise calculations of stress and displacement states around underground or surface \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Symbol** & **Unit** & **A** & **B** \\ \hline \multicolumn{5}{c}{**1. Input parameters**} \\ \hline Contour radius of the excavation opening & \(r_{w}\) & m & 4.58 & 3.87 \\ Characteristic compressive strength of shotcrete & \(f_{ck}\) & MPa & 35.00 & 40.00 \\ Characteristic tensile strength of shotcrete & \(f_{ctd}\) & MPa & 2.20 & 2.50 \\ Partial safety factor for fiber-reinforced shotcrete & \(\gamma_{c}\) & - & 1.40 & 1.40 \\ Design compressive strength of shotcrete & \(f_{cd}\) & MPa & 25.00 & 28.57 \\ Desing tensile strength of shotcrete & \(f_{ctd}\) & MPa & 1.57 & 1.79 \\ Shotcrete thickness & \(g\) & m & 0.05 & 0.05 \\ Characteristic value of the ultimate strength of steel & \(f_{yk}\) & MPa & 650 & 650 \\ Partial safety factor for steel & \(\gamma_{s}\) & - & 1.25 & 1.25 \\ Design value of the ultimate strength of steel & \(f_{d}\) & MPa & 520 & 520 \\ Cross-sectional area of the steel set profile & \(A\) & m\({}^{2}\) & 0.00363 & 0.00452 \\ Steel set spacing & \(e\) & m & 0.600 & 0.500 \\ \hline \multicolumn{5}{c}{**2. Calculation results**} \\ \hline Support capacity due to compression failure & \(p_{sup1}\) & MPa & 0.960 & 1.584 \\ Support capacity due to shear failure & \(p_{sup2}\) & MPa & 0.481 & 0.821 \\ \hline \end{tabular}
\end{table} 
Table 8: Input data and calculation results of the composite support capacity for the hoisting machine chamber (excavation A) and the technological shaft inset (excavation B). excavations. This methodology is particularly useful in evaluating the stability of rock masses under complex conditions, such as those encountered in advanced tunneling and mining operations. In such cases, traditional empirical methods or analytical closed-form solutions often fall short due to the intricate nature of geological formations and loading conditions. By integrating numerical models, engineers gain a deeper understanding of the behavior of rock masses, allowing for more reliable predictions and informed decision-making in design and construction. In the present study, a continuum analysis method based on the finite element method (FEM) was implemented to model the behavior of the rock mass. Continuum analysis assumes the domain to be a homogeneous and isotropic medium, thereby neglecting the explicit representation of individual discontinuities. This simplification enables the efficient simulation of large-scale problems but requires appropriate adjustments to account for the inherent heterogeneity and discontinuities present in natural rock masses. To address this limitation, the mechanical properties of the rock mass were adjusted from laboratory-derived properties obtained from intact rock samples, as described in Sections 2.3 and 2.4. While discontinuity methods, such as the Distinct Element Method (DEM), can provide a more detailed representation of the rock mass's behavior, capturing the effects of rock mass anisotropy, they are often less practical in scenarios where input data are scarce or difficult to obtain. This limitation, combined with the increased computational complexity, often makes such methods less suitable for large-scale simulations typically required in coal mining projects. In contrast, continuum methods, although less detailed in their representation of fractures and joints, offer a more feasible solution in cases where data availability and computational resources are limited. Although the stress and displacement state in the vicinity of an excavated tunnel, roadway, or chamber is inherently a three-dimensional problem, a 2D plain-strain analysis can, under certain conditions, be effectively applied for the practical assessment of excavation stability and support behavior [46].