# Uneven Stiffness Coal Seam: A New Structural Factor Prone to Coal Burst Based on Stiffness Theory
## 4 Simple Numerical Simulation of Uneven Stiffness Coal seam Structure (USCS)  
In previous analysis, it was found that the six key parameters of USCS positively or negatively affect the pressure concentration and stiffness reduction functions of USCS. To validate the previous analytical conclusions and examine the influence of various parameters of USCS on its two specific functions, 6 groups of 14 simplified 2D models using COMSOL 5.2 software was constructed. These models were created based on the controlled variables method, aiming to simulate different responses of the USCS under same working conditions but with varying parameters. To simplify calculations, no failure criteria and floor were established. The bottom of the coal seam restricts vertical displacement, while the top of the roof sustains an evenly distributed pressure of 15 MPa without any constraints on deformation. The the left and right boundaries of the roof and coal seam constrain the lateral displacement. The model's dimensions are 40 m long and 11-15 m high, with a 6 m thick coal seam, a 5-9 m thick roof rock stratum, and a coal rock stratum dip angle of 0\({}^{\circ}\). The mesh division uses extreme fine with triangular elements and unit sizes ranging from 8 \(\times\) 10\({}^{-4}\) m to 0.4 m, resulting in 7082 to 9510 elements. The coal seam of models includes a HSZ and two LSZ with clear boundaries formed by the hard coal in the middle and the soft coal on both sides. The Poisson's ratio for the roof is set to 0.2, while the hard coal and soft coal have Poisson's ratios of 0.22 and 0.3, respectively. Detailed information regarding the model parameters can be found in Table 1. The 14 models were categorized into 6 groups based on different control variables (\(S_{\text{H}}\), \(E_{\text{R}}\), \(E_{\text{H}}\), \(E_{\text{L}}\), \(H_{\text{R}}\), \(S_{\text{R}}\) or \(S_{\text{L}}\)). Model 1 served as a common member for each group, with the other models derived by modifying the values of the control variables in this group. Furthermore, modifying \(E_{\text{H}}\) and \(E_{\text{L}}\) is analogous to altering \(K_{\text{H}}\) and \(K_{\text{L}}\), adjusting \(S_{\text{R}}\) when \(S_{\text{H}}\) remains unchanged would effectively modify \(S_{\text{L}}\). The stress distribution and deformation of the models were computed, as depicted in Figure 7.  
By examining Figure 7, it is evident that:  
1. The normal stress and the Mises stress in HSZ demonstrate notably higher values compared to those in LSZ, providing evidence for the pressure concentration function of USCS. Furthermore, the roof displays bending deformation with its axis of symmetry aligned with the middle axis of the HSZ, substantiating the stiffness reduction function of USCS. 2. At the HSZ-LSZ boundary, significant stress concentrations are present, particularly at the shoulders of the HSZ. These areas are prone to crack development, and the NSC varies as the cracks progress, resulting in a transitional zone where NSC undergoes continuous changes. This aids in reducing stress concentration at the boundary,  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{ \begin{tabular}{c} **Width of Roof** \\ **Sa/m** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Thickness** \\ **of Roof** \\ **\(H_{\text{R}}\)/m** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Young’s** \\ **Modulus of** \\ **Roof** \\ **\(E_{\text{R}}\)/GPa** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Young’s** \\ **of Hard Coal in** \\ **LSZ** \\ **\(E_{\text{H}}\)/GPa** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **Young’s** \\ **Modulus of** \\ **Soft Coal in** \\ **LSZ** \\ **\(E_{\text{L}}\)/GPa** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Width of HSZ** \\ **(One Side)** \\ **\(S_{\text{H}}\)/m** \\ \end{tabular}