Pressure Distribution in Bins and Hoppers

The static pressure under a liquid, column is isotropic and is determined by the height of the column above the point of measurement, h, and the density of the liquid p 

In a column of particulate solids contained in a vertical bin, the pressure at the base will not be proportional to the height of the column because of the friction between the solids and the wall. Moreover, a complex stress distribution develops in the system, which depends on the properties of the particulate solids as well as the loading method. The latter affects the mobilization of friction, both at the wall and within the powder. Finally, arching or doming may further complicate matters. Hence, an exact solution to the problem is hard to obtain. In 1895, Janssen (18) derived a simple equation for the pressure at the base of the bin, which is still frequently quoted and used. The assumptions that he made are the vertical compressive stress is constant over any horizontal plane, the ratio of horizontal and vertical stresses is constant and independent of depth, the bulk density is constant, and the wall friction is fully mobilized, that is, the powder is in incipient slip condition at the wall. 

A force balance over a differential element (Fig. 4.5) simply using pressure P instead of the compressive stress, with shear stress at the wall xw = aw tan fiw + cw, where [lw is the angle of internal friction and cw is the coefficient of cohesion at the wall 

Clearly the pressure at the base approaches a limiting value as H goes to infinity 

most of the weight is supported frictionally by the walls of the bin. The maximum pressure is proportional to bin diameter and inversely proportional to the coefficient of friction at the wall. 

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It should be noted that the main objective in all of the foregoing analyses is to estimate the upper bounds of wall pressure distributions in bins and hoppers for both active and passive stresses. Simplicity of calculations in these methods makes them appropriate for structural design purposes of bins and hoppers. However, they are not meant to introduce a complete solution for the stress field within the bulk solids. 

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