## 3 Rock Burst Mechanism of Triangle Coal Pillar with Penetrating Fault
### Abutment Stress Characteristics of Triangle Coal Pillar with Penetrating Fault  
During the excavation of the 501 working face near the T6 penetrating fault, the key stratum sank and fractured under gravity. The hanging roof behind the workface was supported by the gob gangue, the triangular coal pillar, and the T6 fault plane [64]. Figure 3: 42-6 Borehole. Consequently, the mechanical model of triangle coal pillar abutment stress was established, as shown in Figure 4.  
The stress transfer characteristics of the \"vousoir beam\" indicate that half of the hanging roof load was supported by the gangue, and the remaining half was supported by the triangular coal pillar and the fault plane [65]. The rock mass along the fault plane exhibits characteristics of weakness and fragmentation. This weak rock mass has the ability to impede the propagation of mining-induced stresses, leading to asymmetry in stress distribution on either side of the fault. When a fault transfers high stress from one plate to another, the fault plane will attenuate the transfer of mining stress to the other plate. The fault plane exhibits a barrier effect on stress transfer due to the inclusion of weakened and fractured rock [49; 66]. Assuming a stress transfer coefficient \(K\) for the fault plane, where the stress on the hanging wall of the fault is \(\sigma_{1}\), and the stress transmitted to the footwall through the barrier effect of the fault plane is \(\sigma_{2}\), the load transfer coefficient can be expressed as \(K=\sigma_{2}/\sigma_{1}\). Here, \(\sigma_{1}\) is a constant value and \(\sigma_{2}\) ranges from 0 to \(\sigma_{1}\), meaning that \(K\) values range from 0 to 1. When \(K=1\), it indicates that the fault has no barrier effect, and \(\sigma_{2}=\sigma_{1}\). When \(K=0\), it indicates that the fault has a complete barrier effect, and at this time \(\sigma_{2}=0\). The load of roof In the gob area and the overlying triangular coal pillar transmitted across the fault to the hanging wall of the T6 fault can be expressed as:  
\[Q=K\bigg{(}qL_{1}+\frac{Q_{c}}{2}\bigg{)}=K\bigg{(}qL_{1}+\frac{qL_{1}}{2} \bigg{)}=\frac{3 KqL_{1}}{2} \tag{1}\]  
In the formula, \(Q\) represents the load transmitted from the sub key stratum to the hanging wall of T6 fault, kN/m; \(Q_{c}\) denotes the load transmitted from the hanging roof to the working face, kN/m; \(K\) stands for the load transfer coefficient of the fault plane, which is assigned a value of 0.8 based on the fragmentation observed on the T6 fault plane; \(q\) is the load of the sub key stratum and is calculated as \(q=\gamma h\), where \(\gamma\) is the average bulk density of rock, 25 kN/m\({}^{3}\); \(h\) represents the thickness of the rock stratum controlled by the sub key stratum, 75 m; and \(L_{1}\) is the periodic weighting step of the 501 working face, taken as 20 m.  
The load carried by the triangular coal pillar Iles the load that was not transmitted to the hanging wall of the T6 fault, and the gravity of the underlying rock of sub key stratum, as shown in Equation (2):  
\[Q_{B}=\frac{3(1-k)qL_{1}}{2}+\gamma h_{z}\bigg{(}\frac{h_{z}\tan\theta}{2}+B \bigg{)} \tag{2}\]  
In the formula, \(Q_{B}\) is the load of the triangular coal pillar, kN/m; \(H_{z}\) is the thickness of the underlying rock, which is 28 m; \(\theta\) is the dip angle of the fault; and \(B\) is the width of the triangular coal pillar, m.  
Figure 4: Mechanical model of triangle coal pillar abutment stress. The depth \(h\) of the 501 working face is 500 m, then the abutment stress of the triangular coal pillar \(q_{B}\) is determined by Equation (3):  
\[q_{B}=\frac{\frac{3(1-k)qL_{1}}{2}+\gamma_{z}h_{z}\Big{(}\frac{h_{z}\tan\theta}{ 2}+B\Big{)}}{B}+\gamma h \tag{3}\]  
The characteristics of abutment stress influenced by the width of the triangular coal pillar and the load transfer coefficient are analyzed in combination with Equation (3), as shown in Figure 5: (1) The abutment stress exhibits a negative correlation with the width of the coal pillar.