## 4 Numerical Simulation
### Numerical Modeling
As the underlying hard strata thickness increases (1.5 m), two stress concentration zones appear; strata behavior is distinct in key layer 1; strata behavior seems to appear in the underlying hard strata; the maximum stress is 49.60 MPa, located at 2.5 m from the roadway roof (key layer 1). When the underlying hard strata thickness is 2 m, two distinct stress concentration zones appear above the roadway roof, primarily located near the key layers and the underlying hard strata; the maximum stress concentration zone has moved; strata behavior is distinct in the underlying hard strata, where the stress is as high as 49.68 MPa, located at 2 m from the roadway roof. The plastic zone distribution of the roadway surrounding rock is shown in Figure 8.  
Figure 7: Horizontal stress distribution diagrams of roadway surrounding rock. Note: “\(-\)” for positive compressive stress; “\(+\)” for tensile stress. By comparing Figure 8a-c, when the underlying hard strata thickness is 1 m and 1.5 m, plastic failure occurs above the roadway roof (underlying hard strata, weak interlayer, and key layer 1) and the strata are interconnected. At the thickness of 1 m, the plastic failure in the weak layer gradually extends sideward and its width maximizes; the area of plastic failure in key layer 1 gradually reduces. At the thickness of 1.5 m, the sideward extension width of the plastic failure in the weak interlayer gradually reduces, and the area of plastic failure in key layer 1 gradually reduces. At the thickness of 2 m, plastic failure occurs above the roadway roof (underlying hard strata, weak interlayer), the plastic failure zones are not interconnected due to the influence of the underlying hard strata thickness, but the underlying hard strata thickness can protect key layer 1 from plastic failure. The width of the plastic failure in the weak interlayer further reduces; the roadway is primarily suffering shear failure. The horizontal stress distribution of the roadway surrounding rock is shown in Figure 9.  
Figure 8: Distribution map of plastic zones surrounding rock in roadways of hard rock layers of different thicknesses. Figure 9: Horizontal stress distribution of surrounding rock in roadways of hard rock layers of different thicknesses. From Figure 9, as the distance from the roadway roof increases, the horizontal stress of the roadway surrounding rock first increases then reduces, then increases, then gradually reduces, and finally stabilizes. In the weak interlayer, the horizontal stress gradually reduces because (1) plastic failure in the rock of the weak interlayer leads to a stress reduction, and (2) the rock of the weak interlayer has low strength and loose structure; hence, low physical parameters (cohesion, internal angle of friction, elastic modulus). When the underlying hard strata thickness is 1 m and 1.5 m, strata behavior primarily occurs in key layer 1 because at these thicknesses, plastic failure occurs above the roadway roof (underlying hard strata, weak interlayer, and key layer 1) and the strata are interconnected, causing the stress to move away from the roadway roof. When the underlying hard strata thickness is 2 m, strata behavior primarily occurs in the underlying hard strata because at this thickness, the underlying hard strata do not fail completely, there are some elastic zones. To sum up, it is suggested that:  
1. Underlying hard strata thickness can determine the form of roof fracture. In a mining stress environment, obvious discontinuous failure occurs in the roadway roof containing a weak interlayer. The roof fracture zone shows obvious cross-layer propagation. The existence of more integral hard strata cannot prevent fracture zones from forming in the weak interlayer. Instead, fracture zones will redistribute in the less strong weak interlayer across the more integral hard strata, that is, cross-layer propagation will occur. 2. As the underlying hard strata thickness increases, the maximum horizontal stress of the roadway roof will shift from key layer 1 to the underlying hard strata. 3.