## 1 Introduction
A large amount of research focuses on elucidating the rock burst mechanism triggered by fault activation during mining near fault protection coal pillars. For instance, [PERSON] et al. [27] and [PERSON] et al. [28] posit that variations in displacement and stress within the overlying strata above the working face are the primary factors contributing to the activation of adjacent faults. [PERSON] et al. [29] revealed that the loose space formed by the falling of overlying strata on the working face is necessary for fault activation and stress release. The unloading of horizontal and vertical principal stresses in the mining area serves as the primary factor influencing fault sliding. As the unloading of horizontal principal stresses intensifies, the possibility of fault sliding increases, exhibiting a trend that can be characterized as exponential [6; 16; 23; 30; 31]. [PERSON] et al. [32] simplified the fault as a discontinuous boundary and studied the activation law of faults near the working face in response to decreasing stress levels within the activity of key strata. [PERSON] et al. [17] verified this activation law through the variation of microseismic b-values on the fault plane. [PERSON] et al. [33] and [PERSON] et al. [34] revealed, through numerical simulations, that the stress evolution of surrounding rock during tunnel excavation through a penetrating fault comprises three distinct stages: reduction, rising, and stability. The research presented above indicates that mining disturbances outside the fault protection coal pillar can elicit the large-scale activation of a penetrating fault. Conversely, when tunneling through the penetrating fault, the stress evolution within the surrounding rock in response to small-scale fault movements appears relatively simple. Unfortunately, this simplified stress evolution does not provide sufficient guidance for implementing effective rock burst prevention measures during the mining of the 501 working face in TB Coal Mine, as it mines through the penetrating fault. Due to the irregular cutting of the working face [35] and the penetrating fault [36; 37], triangular coal pillars are locally formed within the coal seam. Numerous scholars have delved into the influence of coal pillar width, height, and shape on its stress distribution characteristics. For instance, as the width of the coal pillar progressively increases, the length of the hanging roof above the working face gradually increases, subsequently stabilizing at a particular length. The stress evolution within the coal pillar exhibits three distinct stages: an initial low-stress phase, followed by a high-stress phase, and ultimately culminating in a stable stress state [38; 39; 40; 41; 42; 43; 44]. [PERSON] et al. [45], through their experiments, revealed that the rock burst risk of coal pillars initially intensifies and then gradually diminishes as the width-to-height ratio of the coal pillar increases. Notably, when the width-to-height ratio exceeds 3:1, the risk of coal pillar failure is minimized. [PERSON] et al. [45] posit that there exists a notable disparity in the overlying strata structure on either side of an irregular coal pillar, resulting in an asymmetric stress distribution within the coal pillar. Furthermore, they observed a phenomenon of stress mutation where the shape of the coal pillar undergoes a transition [44]. The fault plane parameters exert additional influences on the stress within the coal pillar, particularly when the fault and triangular coal pillar combine to form a 'fault-coal pillar' structure [46], which can induce rock burst. For instance, [PERSON] et al. [47] experimentally demonstrated that the activation of fault planes with high and low roughness is stick-slip and stable slip, respectively, implying that the corresponding dynamic stress required for fault activation must differ. [PERSON] et al. [48] investigated the influence of fault dip angle and azimuth angle on coal pillar stress through numerical simulation.