Stress profiles for tapered cylindrical cavities in granular media

Abstract

The formation of almost vertical cylindrical tunnels known as piping or rat holes in stockpiles and hoppers cause serious disruptions to the reclaiming of material. The authors have recently shown that the classical rat-hole theory proposed by Jenike and his coworkers involving the so-called “stable rat-hole equation” is not as accurate as it might be. Specifically, it is shown that the function appearing in the stable rat-hole equation which is conventionally denoted by G( φ) and referred to as the rat-holing function, is not a good approximation of the exact numerical solution. Jenike's original theory assumes a symmetrical stress distribution which is independent of height. In practice, however, rat holes tend to exhibit some tapering with height, and the purpose of this paper is to determine the stress profiles corresponding to a symmetrical but slightly tapered circular cavity. Stress distributions are found which are a perturbation of those arising from classical theory, and separable solutions involving exponential functions in the height are used to “mimic” a slightly tapered cavity. Four numerical examples are presented, and departures from the standard theory are shown graphically. For slightly tapered rat holes occurring in stockpiles, the work presented here constitutes the first rigorous mathematical analysis of this important problem.