## 6 Stress and Energy Evolution of Triangular Coal Pillars
### Numerical Modelling of Triangular Coal Pillars  
FLAC3D (V5.0), as a continuous element numerical simulation software [72], can focus on exploring the stress evolution of triangular coal pillars as their width varies. Based on the geological conditions of the TB coal mine and the characteristics of the 501 working face, a numerical simulation model of a triangular coal pillar was established, as shown in Figure 9. The numerical model has dimensions of 1000 \(\times\) 600 \(\times\) 230 m in length, width, and height, respectively. It is internally divided into the 412 gob and the 501 working face, both of which have a strike and inclination length of 800 m and 200 m, respectively. The width of the coal pillar situated in the section between the two is 40 m.  
In the numerical model, the thickness of the overlying strata above the M4-5 coal seam is approximately 200 m. Considering the depth of the 501 working face is 500 m, a vertical stress of 7.5 MPa was applied to the top of the model. According to in-situ stress tests at the TB coal mine, the horizontal stress is 1.5 times the vertical stress. Therefore, a horizontal stress of 11.25 MPa (calculated as 7.5 MPa \(\times\) 1.5) should be applied to the model. The coal and rock masses within the 501 working face and adjacent goaf in the numerical model are modeled using the Mohr-Coulomb constitutive model [73; 74]. The numerical model initially excavates the adjacent goaf areas, resulting in natural subsidence and damage to the overlying strata. This process aims to simulate and obtain the conventional lateral abutment stress characteristics of the adjacent goaf areas on the 501 working face. The dip angles of the T6 and T5 penetrating faults in the numerical model are 75\({}^{\circ}\) and 85\({}^{\circ}\), respectively, with drops of 2 m and 4 m, respectively. The distance between the faults is 50 m. Coal seam stress monitoring lines 1\(\#\), 2\(\#\), and 3\(\#\) are arranged in the 501 working face and the triangular coal pillar along the transport roadway, the central part, and the air-return roadway. Additionally, a numerical model without faults was also established. Figure 8: Rock burst mechanism of triangular coal pillar. (**a**) stress distribution of triangular coal pillar. (**b**) rock burst mechanism. Both the numerical models, with and without faults, initially excavate the 412 working face. After stress equilibrium of models were reached, the 501 working face was excavated from the boundary in 10 m steps until it completely passed through T5 and T6 penetrating faults. At the same time, the evolution of stress and energy within the coal seam was monitored. The numerical simulation model employed the Mohr-Coulomb strength criterion. There was a notable difference in the mechanical parameters between the rock and rock mass. Based on the physical and mechanical parameters of the rock obtained from the 42-6 borehole, by utilizing the \(RQD\) index to convert the elastic modulus of the rock and the rock mass [75], the compressive strength of the rock mass could be calculated, as shown in Equation (7) and Table 1. The physical and mechanical parameters of the model are presented in Table 2.  
\[\left\{ \begin{array}{l}E_{m}/E_{r}=10^{0.0186 RQD-1.91}\\ \left(\frac{\sigma_{cm}}{\sigma_{c}}\right)=\left(\frac{E_{m}}{E_{r}}\right)^{ n}\end{array} \right. \tag{7}\]  
where \(E_{m}\) and \(E_{r}\) are elastic modulus of the rock and rock mass, respectively, GPa; \(\sigma_{cm}\) and \(\sigma_{c}\) are the uniaxial compressive strength of the rock and rock mass, respectively, MPa; and \(n\) is the coefficient of weakening of the rock, which takes the value of 0.63; \(RQD\) index is determined by the 42-6 borehole.