Cracked pillars have limited capabilities for the bearing of dynamic and static loads, consequently threatening the stability of entire goafs. In the current study, we establish a structural mechanical model for cracked strip pillars and determine a mathematical formula for the total energy based on an underground bauxite mine in Shanxi, China. Furthermore, the instability criterion is employed to derive an analytical method for the estimation of the critical blasting load q_cr for cracked strip bauxite pillars. The relationship between q_cr and pillar height (H), pillar width (b), pillar length (L), and the crack loosen zone width (d) is quantitatively evaluated. Results demonstrate that when L and d are constant, q_cr decreases as H increases. Moreover, q_cr is observed to increase with b for constant H and d, with a continuous reduction in curvature fluctuations. Although q_cr increases with L for fixed H and b, the increase amplitude is small, indicating the limited influence of L. For constant L and b, q_cr decreases as d increases, and when the value of d exceeds that of the inflection point, the rate of change is steeper. The proposed analytical method is validated using numerical results and a case study. Our work provides theoretical support for the safe production of bauxite mining.

破裂的支柱承受动态和静态载荷的能力有限，因此威胁到整个采空区的稳定性。在当前的研究中，我们建立了裂纹带状柱的结构力学模型，并基于山西地下铝土矿开采确定了总能量的数学公式。此外，采用不稳定性判据推导了一种分析方法，用于估算裂纹带状铝土矿立柱的临界爆破荷载q _cr。之间的关系q _CR和支柱高度（ħ），柱宽度（b），柱长度（大号（），和裂纹松开区宽度d）进行定量评估。结果表明，当L和d恒定时，q _cr随着H的增加而减小。此外，对于恒定的H和d，观察到q _cr随着b的增加而增加，并且曲率波动不断减小。尽管对于固定的H和b，q _cr随L增大，但增大幅度很小，表明L的影响有限。对于常数L和b，q _crd随d的增加而减小，并且当d的值超过拐点的值时，变化率更陡。数值结果和案例研究验证了所提出的分析方法。我们的工作为铝土矿安全生产提供理论支持。