Analyzing Pillar Strength and Behavior using Wolfram Mathematica code: Effects of Cracks, Size and Discretization

The stability of fractured rock masses and the modeling of underground mining pillars necessitate a comprehensive understanding of behavioral models and mechanical properties. This study employs the Wolfram Mathematica code to investigate mining pillar reliability, specifically focusing on elucidating the influence of scale and shape on pillar strength. Drawing inspiration from methodologies in the existing literature, our approach is based on the Mohr-Coulomb theory and Griffiths's random field of rock strength. This study highlights the significance of shape, where pillar strength exhibits exponential growth with increasing width-to-height ratios. Beyond a critical value, strength surges, especially under elevated confining stress. Additionally, a critical mesh size significantly affects the weakest pillar behavior. Our results confirm the 'size effect,' wherein strength generally decreases with increasing pillar volume. Thus, strength decreases with rising volume until a threshold. Particularly noteworthy is the phenomenon observed in the presence of cracks; initially, an increase in mesh size leads to a decline in strength, corresponding to an increase in the number of cracks. However, this decline stabilizes beyond a critical mesh size, after which strength experiences a resurgence echoing behavior seen in the homogeneous case. In this paper, the reproduction of the scale effect by an algorithm based on the Mathematica code was made to allow a probabilistic study to be carried out because of the random existence of discontinuities in nature – another hand to carry out stochastic modeling of fractures and its influence on the rock mass strength.