# 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)
\begin{array}{l}f_{\rm scH}=\overline{\sigma}_{\rm yH}/P\\ \overline{K}_{\rm pH}=\overline{\pi}K_{\rm R}=\overline{\frac{\eta_{\rm yRH}}{ U_{\rm yRH}}}\cdot\frac{E_{\rm R}}{T_{\rm R}}=\frac{\overline{\pi}_{\rm yRH}}{ \overline{\eta}_{\rm yRH}+\overline{\pi}_{\rm RH}}\cdot\frac{E_{\rm R}}{T_{\rm R}} \end{array} \right. \tag{13}\]  
where, \(\overline{\sigma}_{\rm yH}\) is the average normal stress value in HSZ; \(P\) is the overlying pressure, set to 15 MPa in the simulation; And \(\overline{\pi}_{\rm yRH}\) and \(\overline{\pi}_{\rm RH}\) are the average value of compression deformation and bending deflection of the roof in HSZ, respectively. Figure 7: Simulation results of different USCS models (stress unit: Pa): (**a**) Normal stress cloud map of model 1 (common member); (**b**) Deformation and the Mises stress cloud map of model 1; (**c**) Normal stress cloud map of model 3 (\(S_{\rm H}\) = 20 m); (**d**) Deformation and the Mises stress cloud map of model 3; (**e**) Normal stress cloud map of model 8 (\(E_{\rm L}\) = 1 GPa); (**f**) Deformation and the Mises stress cloud map of model 8; (**g**) Normal stress cloud map of model 12 (\(H_{\rm R}\) = 9 m); (**h**) Deformation and the Mises stress cloud map of model 12; (**i**) Normal stress cloud map of model 14 (\(S_{\rm R}\) = 12 m); (**j**) Deformation and the Mises stress cloud map of model 14. Analyzing the information provided in Table 2 reveals the following findings:  
1. Across all models of USCS, it is evident that the stress concentration factor of the HSZ (\(f_{\text{scH}}\)) consistently exceeds 1. Additionally, the LMS in HSZ (\(K_{\text{pH}}\)) remains lower than the stiffness of the roof (\(K_{\text{R}}\)). These findings indicate that USCS exhibits the dual characteristics of pressure concentration and stiffness reduction functions. 2. The \(\overline{\sigma}_{\text{yH}}\) (average normal stress in HSZ) positively correlates with \(E_{\text{R}}\), \(H_{\text{R}}\), \(K_{\text{H}}\), and \(S_{\text{L}}\), but negatively correlates with \(K_{\text{L}}\) and \(S_{\text{H}}\); The \(\overline{K}_{\text{pH}}\) (average LMS per unit area in HSZ) negatively correlates with \(H_{\text{R}}\), \(K_{\text{H}}\), \(S_{\text{L}}\), and \(S_{\text{H}}\), but positively correlates with \(E_{\text{R}}\) and \(K_{\text{L}}\). 3. An increase in the \(E_{\text{R}}\) leads to a relatively smaller growth in \(K_{\text{pH}}\). When switch Model 1 to Model 5, where \(E_{\text{R}}\) increases from 10 GPa to 20 GPa, the stiffness of the roof (\(K_{\text{R}}\)) shows a 100% increase. In contrast, the relative increase in \(K_{\text{pH}}\) is only 41.7%. The reduction in the value of \(\alpha\) helps partially offset the influence of roof hardening on the stiffness of the surrounding rock. Figure 8: \(Cont.\)  
In the mechanism analysis and simplified simulation, six key parameters of USCS are involved, namely, thickness (\(H_{\text{R}}\)) of roof, elastic modulus (\(E_{\text{R}}\)) of roof, NSC of HSZ (\(K_{\text{H}}\)), NSC of LSZ (\(K_{\text{L}}\)), area (\(S_{\text{H}}\)) of HSZ, and area (\(S_{\text{L}}\)) of LSZ. These parameters can have either positive or negative effects on the pressure concentration function and stiffness reduction function of the USCS, thereby influencing the risk of failure, instability, and coal burst in HSZ. By considering mining, fracturing, blasting, drilling, and slot cutting as a reduction in stiffness of coal seam, and considering support and backfilling as an increase in stiffness of coal seam, and treating goal or roadway as an LSZ of the artificial USCS, it is possible to reconsider past engineering experience and summarize the impact of these six key parameters on indicators related to impact risk. Table 3 presents the correlations between the six key parameters and risk indicators of coal burst, which are derived from the analysis results, simulation results, and engineering experience. From Table 3, it can be seen that:  
1.