# Numerical Simulation-Based Study on the Arching Effect in Subsequent Backfill
## 3 Numerical Simulation
### Analysis of Stress Distribution Patterns with Respect to Backfill Internal Friction Angle  
As shown in Figure 6, whether it is the vertical stress calculated by the weight method or the stress distribution under different internal friction angles (\(25^{\circ}\), \(30^{\circ}\), \(35^{\circ}\), \(40^{\circ}\), \(45^{\circ}\)), the phi in the chart represents the friction angle inside the filling body; their values increase significantly with the increase of the depth of the filling body. This variation law is consistent with the mechanical response characteristics of the filling body under the combined action of its own weight and the surrounding rock in actual engineering. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Definition** \\ \hline Internal Friction Angle & \(\varphi\) & \(\circ\) & The angle representing the shear resistance of the backfill material due to interparticle friction. \\ \hline Cohesion & c & MPa & The inherent shear strength of the backfill when no normal stress is applied. \\ \hline Elastic Modulus & E & GPa & Measures the stiffness of the backfill material under elastic deformation. \\ \hline Poisson’s Ratio & \(\mu\) & & Ratio of transverse strain to axial strain when a material is subjected to uniaxial tension or compression. \\ \hline Interface Friction Angle & \(\delta\) & \(\circ\) & The friction angle at the contact between backfill and surrounding rock. \\ \hline Interface Cohesion & c\({}_{\text{k}}\) & MPa & The bonding strength at the backfill–rock interface. \\ \hline Normal Stiffness & k\({}_{\text{n}}\) & GPa/m & Resistance of the interface to compression/separation. \\ \hline Shear Stiffness & k\({}_{\text{s}}\) & GPa/m & Resistance of the interface to tangential sliding. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Glossary of key parameters in FLAC3D numerical simulation of backfill structures. Figure 6: Comparative analysis of the influence of internal friction angle on backfill stress distribution. The vertical stress and horizontal stress values obtained through FLAC3D numerical simulation are both lower than the calculation results of the traditional overweight method. This difference mainly stems from the fact that the overweight method only considers the self-weight effect of the filling body, while ignoring the significant influence of the internal friction angle on the stress distribution. The numerical simulation results show that as the internal friction angle decreases, the stress distribution of the filling body gradually approaches the calculation result of the overweight method. This phenomenon reveals the widespread arching effect in the mining plate area slope, which leads to the redistribution of internal stress in the filling body, thereby forming a characteristic stress transfer phenomenon. As the friction angle in the filling body gradually increases from 25\({}^{\circ}\) to 45\({}^{\circ}\), both the vertical stress and the horizontal stress show a decreasing trend. This phenomenon stems from the fact that an increase in the internal friction angle significantly enhances the friction between the contact surfaces of the filling body, thereby effectively reducing the stress level. It is notable that the sensitivity of stress to the change of the internal friction angle shows a decreasing characteristic: when the internal friction angle increases from 25\({}^{\circ}\) to 30\({}^{\circ}\), the vertical stress change amplitude reaches 40.2 kPa; when the internal friction angle increases from 40\({}^{\circ}\) to 45\({}^{\circ}\), the variation range decreases to 25.6 kPa. This indicates that within the range of the lower internal friction angle, the stress value is more sensitive to the change of the internal friction angle. The research results show that the internal friction angle is a key parameter controlling the stress distribution of the filling body.