# 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 Pressure Concentration Function of USCS  
Many scholars attribute the occurrence of coal burst to stress concentration and link the cause of stress concentration to tectonic stress without delving into the topic extensively. However, the natural USCS represents a range of tectonic structures. Under conditions of uniform distribution of overlying loads, stress tends to concentrate from LSZ to HSZ, particularly at the boundary between HSZ and LSZ (HSZ-LSZ boundary). Figure 4 illustrates the simplified analysis conducted to study the stress distribution in an uneven stiffness coal seam under a uniformly distributed load. The approach involves fixing the floor and considering only the deformation of the coal seam and roof. The focus is on examining the normal deformation and normal stress distribution in the coal seam under three different scenarios. Figure 4: Stress distribution and deformation of uneven stiffness coal seam under three kinds of roof: (**a**) discontinuous roof and coal seam. (**b**) continuous elastic roof and coal seam. (**c**) rigid roof and continuous elastic coal seam. Figure 3: Three kinds of typical USCS in coal measure strata: (**a**) Coal seam thinning zone (a certain coal burst accident in Tianchi Coal Mine, Sichuan) [3]. (**b**) Coal seam bifurcation area. (**c**) Mignatic intrusion area (2015 \"8.5\" coal burst accident in a coal mine in Inner Mongolia) [34]. In the first scenario, if the roof and coal seam exhibit extreme flexibility (with a bending stiffness of 0) or if there are discontinuities at the HSZ-LSZ boundaries, the normal stress in each area of the coal seam will be uniformly distributed and denoted as P. However, a macro deformation gap will be present between the HSZ and LSZ, as depicted in Figure 4a. Moving to the second scenario, when the roof is continuous and possesses a finite bending stiffness greater than 0, it will bend to eliminate the macroscopic deformation gap and maintain a continuous coordination of deformation. This bending action leads to increased compression deformation in HSZ, particularly at its boundary, and reduced compression deformation in LSZ, especially at its boundary. As a result, the normal stress in the coal seam will be greater than \(P\) in HSZ and less than \(P\) in LSZ, as shown in Figure 4b. Considering the premise of maintaining the bending stiffness of the roof, the greater the difference in normal stiffness between the HSZ and LSZ, the more significant the bending deflection of the roof, leading to a higher concentration of normal stress in HSZ. The third scenario involves a continuous roof with infinite bending stiffness, thereby preventing any bending from occurring. Consequently, the deformation in both the HSZ and LSZ will be forced to be consistent, as depicted in Figure 4c. In this case, the normal stress \(\sigma_{\rm{yH}^{\prime}}\) in HSZ and \(\sigma_{\rm{yL}^{\prime}}\) in LSZ will satisfy Equation (6). \[\left\{ \begin{array}{l}\sigma_{\rm{yH}^{\prime}}S_{\rm{H}}+\sigma_{\rm{yL}^ {\prime}}S_{\rm{L}}=P(S_{\rm{H}}+S_{\rm{L}})\\ \sigma_{\rm{yH}^{\prime}}S_{\rm{H}}/k_{\rm{H}}=\sigma_{\rm{yL}^{\prime}}S_{ \rm{L}}/k_{\rm{L}}=\sigma_{\rm{yH}^{\prime}}/K_{\rm{H}}=\sigma_{\rm{yL}^{ \prime}}/K_{\rm{L}}\end{array} \right. \tag{6}\]  
where, \(K_{\rm{H}}\) and \(S_{\rm{H}}\) is the NSC and area of the HSZ, respectively; and \(K_{\rm{L}}\) and \(S_{\rm{L}}\) is the NSC and area of LSZ, respectively. The solution of Equation (6) can be writed as follows:  
\[\left\{ \begin{array}{l}\sigma_{\rm{yH}^{\prime}}=\frac{P(S_{\rm{H}}/S_{\rm{L }}+1)}{(S_{\rm{H}}/S_{\rm{L}}+K_{\rm{L}}/K_{\rm{H}})}=P+\frac{P(1-K_{\rm{L}}/ K_{\rm{H}})}{(S_{\rm{H}}/S_{\rm{L}}+K_{\rm{L}}/K_{\rm{H}})}\\ \sigma_{\rm{yL}^{\prime}}=\frac{P(S_{\rm{L}}/S_{\rm{H}}+1)}{(S_{\rm{L}}/S_{\rm {H}}+K_{\rm{H}}/K_{\rm{L}})}=P+\frac{P(1-K_{\rm{H}}/K_{\rm{L}})}{(S_{\rm{L}}/S_ {\rm{H}}+K_{\rm{H}}/K_{\rm{L}})}\end{array} \right.