Hereunder is presented a theory of distribution of vertical stress in the powder in equilibrium in a conical vessel. The same problem has been treated by Aoki and Walker, but their theory were based on the assumption that the vertical stress was constant on a horizontal plane. On the other hand, Tanaka et. al. have shown that the vertical stress of the powder in equilibrium in a conical vessel is not constant on a horizontal plane, but changes according to the distance from the axis of the conical vessel. As far as we know, no theory has yet been developed dealing with the variation of vertical stress with some distance from the axis of the cone.
We have developed a theory of distribution of vertical stress, taking into account its variation with the distance from the axis of the cone. It was assumed at first that the vertical stress and the horizontal stress were both connected by Rankine's law. The effect of compressibility has been neglected. From the condition of eqilibrium of powder an equation has been derived to determine the vertical stress as a function of both the distance z and h from the vertex of the cone and the distance ξ from the axis of the cone. A solution has been found, giving vertical stress as one of Taylor's series in ξ. The solution contains a number of arbitary constants, and it has been found that the solution shows a tendency to fit to the experimental curves of the distribution of vertical stress by choosing appropriate values for the constants.
Thus, we have come to the conclusion that Rankine's law is not valid in the present problem. This may be understood if we remember that Rankine's law is derived from Mohr's theory of yielding. Rankine's law may be modified by considering the influence of the so-called cohesive force between the powder particles. However, it has been found that a modified Rankine's law is not sufficient to explain the disagreement of our theory with what the experiments indicate.
