Design of fault water-resisting coal pillars based on deep-beam analysis: A comparison of two analytical methods

The rational design and mechanical assessment of fault water-resisting coal pillars are essential for effective disaster prevention and mitigation. In the mechanical analysis of water-resisting coal (rock) pillars, deep-beam effects can significantly influence stress distribution, yet the applicability of existing analytical methods under deep-beam conditions has not been systematically compared or clearly defined. Classical elastic beam theory is widely used to evaluate pillar stresses, but it can yield substantial errors at non-slender geometries. Its limitations become pronounced when interlayer shear transfer and vertical compression cannot be neglected, which typically occurs when h/L > 0.2 (equivalently, h/L < 5).The aim of this paper is to compare the applicability and accuracy of analytical methods for deep-beam problems. We develop a layered deep-beam decomposition method, where the coal pillar-floor system is idealized as a simply supported rock beam under a uniformly distributed hydraulic load and discretized through the thickness into interacting shallow-beam layers to account for interlayer shear transfer and vertical compression. Based on a deep-beam model of the faulted floor, analytical solutions are obtained using both the classical elastic stress-function method and the proposed layered deep-beam decomposition method, and are validated against FLAC3D numerical simulations. Representative comparisons show that, for h/L = 0.3-1.0, the proposed approach (using 10, 16, and 20 layers) predicts mid-span normal stresses with relative errors of 6.0%-9.9%. In contrast, the classical elasticity solution deteriorates rapidly as h/L increases: the relative error can exceed 100% at larger h/L, and the solution fails to capture the downward migration of the neutral axis. Application to an engineering case from a deep coal mine in northern Anhui Province further indicates that, after incorporating a safety factor, the predicted pillar width is consistent with empirical design guidelines, supporting the method's engineering applicability. Overall, this study focuses on the comparison of applicability and computational accuracy between the two analytical methods, which helps to clarify the applicable conditions and advantages of each method for deep-beam models in water-resisting coal pillar analysis.