# Stability Analysis of Isolated Roof in Overlapping Goal Based on Strength Reduction
## 1 Introduction  
The goat formed by ore mining is a major potential hazard, which seriously threatens the safety of mine production [1, 2, 3]. During the mining process, two or more layers of goat are formed at different spatial heights, which is called overlapping goat. The support system of overlapping goat is composed of upper pillars, isolated roof, and lower pillars. Among them, the isolated roof is located between the upper and lower pillars, which is the weak part of the whole support system and the key to maintaining the stability of overlapping goat [4]. The stability of the roof is closely related to the relative position of the upper and lower pillars. From the point of view of the force relationship, the relative positions of the upper and lower pillars can be divided into three cases: full alignment, partial alignment,and misalignment (Figure 1). In the first case, the upper and lower pillars are perfectly aligned, and the pressure from the overlying strata is directly transmitted to the lower pillars through the upper pillars. In this case, the roof of the partition is almost unloaded, and naturally, the role of the roof of the partition in the support system is almost negligible. In the second case, in the downward transmission process of the overlying pressure, part of the pressure is transferred directly from the upper pillar to the lower pillar, and the other acts on the roof of the isolated roof, causing the roof to bear part of the load. In other words, the isolated roof shares part of the pressure in the support system. In the third case, the upper and lower pillars are completely misaligned, and the pressure of the overlying strata can only be transferred to the lower pillars through the isolated roof. The roof bears the largest load comparatively, and therefore, instability and collapse are most likely to occur. Regarding the stability of overlapping goat, scholars have carried out relevant research on the instability prediction of isolated roof with pillar overlap rate as variable [5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. These studies mainly focus on the changes in the stability state of the isolated roof during the process, as shown in Figure 1a-c, when the overlap rate of the top and bottom pillars decreases from 100% to 0% [15; 16; 17]. However, it can be seen from Figure 1 that the overlap rate cannot truly reflect the relative position relationship between the upper and lower pillars. In fact, when the pillar overlap rate is 0%, it is not only the state shown in Figure 1c. Theoretically, as long as the lower pillar is located at any position below the upper goat, the overlap rate of the pillar is 0% (Figure 2). Obviously, the different relative positions of pillars will inevitably lead to different stress for the isolated roof, and the stability of the roof will inevitably change as well. In Figure 3, \(c\) is the length between the center lines of the upper and lower pillars, \(a\) is the span of the goat, and \(b\) is the width of the pillars, \(h\) is the thickness of the isolated roof. When \(c\) is 0, the lower pillar is completely aligned without dislocation. with the increase of \(c\), the lower pillar moves to the middle of the goat. When \(c=(a+b)/2\), the lower pillar is located in the middle of the upper goat. If \(c\) is further increased, the lower pillar moves to the right side of the upper goat. Therefore, as long as \(c\) meets \(b\leq c\leq a\), the overlap rate of the pillars is 0%. However, in the process of \(c\) changing from \(b\) to \(a\), the stress of the roof must also change because the relative positions of the pillars in the upper and lower layers have changed. Obviously, when the upper and lower pillars are not aligned, the stability state of the roof is not invariable. Focusing on the influence of misaligned support and pillar size on the stability of the isolated roof, this study established a stability analysis method for the roof based on strength reduction.