Hereunder is presented a consideration on the theory of distribution of vertical stress in powder piled in a cylindrical vessel in a state of equilibrium. The same problem was treated by Janssen, but his theory was based on the assumption that vertical stress was constant on a horizontal plane. On the other hand, experimental studies show that vertical stress in powder in equilibrium in a cylindrical vessel does not remain constant on a horizontal plane, but changes according to the distance from the axis of the cylinder. As far as we are informed, no theory has yet been developed dealing with the variation of vertical stress with the distance from the axis of the cylinder.
We have developed a theory on distribution of vertical stress, taking into account its variation with the distance from the axis of the cylinder. It is assumed that both the vertical stress and the horizontal stress are connected by Rankine's law. But in the theory proposed under this assumption the possible effect of compression is neglected. From the condition of powder in equilibrium an equation has been derived to determine the vertical stress as a function of both the distance h from the free surface of powder and the distance ξ=r/R from the axis of the cylinder. A solution has been found in taking the vertical stress as expressed in Taylor's series in ξ. The boundary condition at the free surface is so chosen that the total pressure over the free surface is equal to zero. It is shown that our theory agrees fairly well with the experimental data by taking the product of Rankine's coefficient and the frictional coefficient of the wall as appropriate value.
