## 1 Introduction
They determined that when the fault dip angle and azimuth angle at a Hengda coal mine was \(75^{\circ}\) and \(50^{\circ}\), respectively, the risk level associated with the rock burst of the triangular coal pillar reached its peak. A fault plane represents a discontinuous interface formed by loose rock masses [32]. The existence of faults exhibits a barrier effect on stress transmission [49]; currently, the study of the stress evolution in triangular coal pillars with varying widths, subject to the barrier effect of faults, remains unclear. This lack of understanding hinders the development of a theoretical foundation for targeted prevention measures aimed at mitigating rock burst in triangular coal pillars containing faults. The stress concentration within the triangular coal pillar is primarily influenced by roof abutment stress [38]. Therefore, the prevention and control of rock burst in triangular coal pillars primarily focus on two aspects: (1) Reducing the length of the hanging roof and the excess stress on the triangular coal pillar by implementing roof cutting measures. The primary methods for this include roof blasting [50; 51] and hydraulic fracturing [52]. (2) However, the implementation of roof cutting measures can compromise the support of the roadway. Furthermore, rock burst roadways require higher support strength to resist dynamic stress. Consequently, rock mass injections [36; 53] are employed to control roof fragmentation in the triangular coal pillar area of the roadway, while additional cables [54; 55; 56] are used to reinforce the support and enhance the roadway's resistance to dynamic stress. The current research focuses on elucidating the rock burst mechanism in the working face located outside the fault protection coal pillar, whereas the rock burst within the working face mining through the fault remains largely unexplored. While research examining the influence of coal pillar size, fault dip angle, and azimuth angle on the stress within triangular coal pillars adjacent to faults is relatively abundant, there is a notable lack of relevant research on the stress evolution of triangular coal pillars with varying widths under the barrier effect of faults. Rock burst cases [7; 8] indicate that, although the microseismic triggering of the rock burst was located on the fault plane, the damage area was primarily concentrated on the triangular coal pillars. The triangular coal pillar, being a rock burst risk area, is influenced by the superposition of tectonic stress, abutment stress, and fault barrier effects. As the working face mines through the penetrating fault, the width of the coal pillar varies, and under the barrier effect of the fault, the stress evolution within the triangular coal pillar exhibits increasingly complex characteristics. This study focuses on the behavior of triangular coal pillars in the 501 working face of the TB mine during mining through the T6 penetrating fault, taking into account the fault barrier effect. Considering that surrounding rock stress contributes to the occurrence of rock bursts [57], the concept of fault barrier is introduced. Through theoretical analysis, the stress evolution under such conditions is investigated. Additionally, it reveals the specific rock burst mechanism in fault triangular coal pillars and enables the determination of rock burst risk levels for various pillar widths. Energy accumulates within the coal and rock mass, which serves as an indicator of the degree of damage caused by rock bursts [58; 59]. Through numerical simulation, the accumulation pattern of energy within the fault triangular coal pillar is investigated, thereby identifying critical regions susceptible to rock burst damage. Based on these findings, targeted prevention and control measures are proposed to mitigate the risk of rock burst. This study aims to provide a useful reference for preventing rock bursts in similar fault triangular coal pillars.