Statically Admissible Stress Fields in Conical Sand Valleys and Heaps: A Validation of Haar–von Kármán Hypothesis

Abstract

The usage of the Haar–von Kármán hypothesis for circumferential stress in axisymmetric problems is still questionable. This study aims to reveal statically admissible stress fields in conical sand valleys and heaps by applying the Haar–von Kármán hypothesis to rigid-plastic material obeying the Mohr–Coulomb criterion. The original boundary-value problem reduces to a pair of ordinary differential equations by employing the self-similarity assumption in conjunction with the condition of incipient failure everywhere. The pair of governing equations are numerically integrated with the fifth-order adaptive Runge–Kutta method. The shooting method is employed to numerically convert the boundary-value problem to the initial-value problem. Moreover, the singularities that occurred in the integrations are coped with by numerical perturbation of the initial boundary conditions. A line of stress discontinuity together with the stress jump condition is imposed to link the stress solution between the rigid and plastic regions. Admissible stress fields in various axisymmetric geometries are successfully achieved whereby the load transfer mechanisms in the conical sand heaps and valleys are elucidated. The arch action that occurred in bulk solid is observed. The effect of the circumferential stress hypotheses, that is, Haar–von Kármán hypothesis and Lévy’s flow rule, on the results of rigid-plastic solution is demonstrated; therefore, the advantages and disadvantages of both the hypotheses are elucidated. This study rigorously justifies the validity of the Haar–von Kármán hypothesis, thus a guideline on determining the value of hoop stress using the Haar–von Kármán hypothesis is proposed.