# Uneven Stiffness Coal Seam: A New Structural Factor Prone to Coal Burst Based on Stiffness Theory
## 3 Uneven Stiffness Coal Seam Structure (USCS)
### Analysis of Stiffness Reduction Function of USCS  
A simplified model of the coal mass-surrounding rock system during the failure process, focusing on the roof deformation while neglecting the influence of gravity and floor deformation, is depicted in Figure 5. When the coal mass reaches its peak pressure, both the internal force (\(F_{\text{b}}\)) of the coal mass and the internal force (\(F_{\text{P}}\)) of roof experience the maximum load, \(F_{\text{max}}\). Post-peak, \(F_{\text{b}}\) decreases due to the strain softening of the coal mass, while \(F_{\text{P}}\) decreases due to roof deformation recovery. During the failure process, the overall amount of roof deformation recovery is denoted by \(\Delta x\). In the case of ESCS, the roof has no bending deformation under uniform overlying loads. At this stage, the amount of recovery is \(\Delta x\) for compression deformation, so the local mine stiffness (\(k_{\text{P}}\)) is equal to the normal stiffness (\(k_{\text{R}}\)) of the roof, which can be computed using Equation (8). \[k_{\text{P}}=k_{\text{R}}=E_{\text{R}}S_{\text{R}}/H_{\text{R}} \tag{8}\]  
In contrast, when USCS sustains uniform overlying loads, both compression deformation and bending deformation of the roof and floors recover. The recovery amount of compression deformation is \(\alpha\Delta x\), where \(\alpha\) is the stiffness reduction coefficient of the USCS and is greater than 0 but less than 1. Consequently, the recovery amount of roof bending deformation is (1-\(\alpha\))\(\Delta x\). The reduction in internal force (\(\Delta F_{\text{P}}\)) of the pressure provider is solely proportional to the recovery amount of compression deformation, with \(\alpha\)\(k_{\text{R}}\)\(\Delta x\) representing the decrease in internal force. So the LMS (\(k_{\text{P}}\)) is determined using Equation (9) and must be less than \(k_{\text{R}}\). \[k_{\text{P}}=\Delta F_{\text{P}}/\Delta x=k_{\text{R}}\alpha\Delta x/\Delta x= \alpha k_{\text{R}}=\alpha E_{\text{R}}S_{\text{R}}/H_{\text{R}} \tag{9}\]  
Equation (9) illustrates that bending of the roof can reduce its internal force reduction under the same amount of deformation recovery after the peak, leading to a decrease in the LMS. Through a stiffness theory analysis, it becomes apparent that a lower LMS yields greater compliance with stiffness criterion, thereby elevating the risk of instability (\(r_{\text{I}}\)). Thus, when compared to the ESCS operating under identical conditions, the USCS exhibits a higher \(r_{\text{I}}\) owing to its stiffness reduction function. Figure 6 illustrates a trisection model of the USCS, where the roof is simplified as a flexible rectangular cross-section beam and the HSZ or LSZ is represented as hard springs  
Figure 5: Simplified model of the coal mass-surrounding rock system with bent roof: (**a**) System at peak: two force equilibrium; (**b**) Post peak system: recovery of bending deformation. (\(k_{\rm H}\)) or soft springs (\(k_{\rm L}\)). Assuming that the discrepancy between the internal force \(F_{\rm L}\) representing the LSZ and the average load is \(\Delta F\), Equation (10) can be deduced. \[\left\{ \begin{array}{l}y_{\rm L}=F_{\rm L}/k_{\rm L}=(F/3-\Delta F)/k_{\rm L} \\ y_{\rm H}=F_{\rm H}/k_{\rm H}=(F/3+2\Delta F)/k_{\rm H}\\ w=y_{\rm L}-y_{\rm H}=\Delta Fl^{3}/24E_{\rm R}I_{\rm R}\end{array} \right. \tag{10}\]  
where, \(F_{\rm H}\) and \(F_{\rm L}\) represent the internal force of the spring in HSZ and LSZs, \(y_{\rm H}\) and \(y_{\rm L}\) represent the compression deformation of the hard and soft springs, \(w\) is the bending deflection of the roof, and \(l\) is the lateral span of the structure. Equation (11) can be inferred from Equation (10). \[\left\{