The problem of the determination of the horizontal and vertical force distributions at the base of a stockpile is a famous outstanding problem in granular theory, and has been exhaustively examined. In all past studies, the shape of the stockpile is assumed to be either a two-dimensional wedge or a three-dimensional cone. Here we present the only known exact analytical solutions of the governing equations for the continuum mechanical theory of granular material for two-dimensional parabolic and three-dimensional cubic curved stockpiles. Such curved profiles are known to occur experimentally, as well as in the interiors of blast furnaces. The model assumes that the stockpile is composed of two regions, which are an inner rigid region and an outer yield region. For such infinite stockpiles we follow normal practice and determine the force distributions at a certain height and we argue that these forces should approximate those for a stockpile of finite height resting on a horizontal surface. The solutions presented are valid for granular materials which we term 'highly frictional', by which we mean that the angle of internal friction φ is such that sinφ ap1. We note that there exist many real granular materials, for example black and brown coal, possessing angles of internal friction in the range of 60 to 65 degrees, resulting in values of sinφ equal to around 0.87 to 0.91. The exact parametric solution is applied to the outer yield region and it is extended continuously into the inner rigid region. Numerical results for both two- and three-dimensional problems indicate that the magnitudes of the horizontal and the vertical forces at the base have their maximum values at the stockpile extremities.
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