Stresses in Bins and Hoppers, by Gabriel I. Tardos

-- 3. Janssen's Equations for Bins --

Fig. 4 - Differential slices in a bin

The slice is taken somewhere in the middle of the bin. The origin of the coordinate system is at the top of the fill with the y axis pointing downward. The figure depicts a 2D slice of width 2b and a rectangular slice of dimensions 2b x 2b (the corresponding circular slice of radius b is not shown). Two stresses, the average vertical stress within the slice y and the average tangential stress t on the wall determine the equilibrium of the slice in the vertical direction. The third stress, the average normal stress on the wall n, is related to the tangential stress by the law of friction, t = n tan W. Letting the thickness of the slice (see Fig. 4a) be dy and therefore its weight be 2 b dy and taking the average vertical force at the upper surface of the slice to be 2 b y (and the force on the lower surface to be 2 b (y + d y)), gives for the force equilibrium the expression: [2]      2 b y - 2 b (y + dy) - 2 dy t + 2 b dy = 0 Simplifying the above equation and dividing by 2b dy gives the differential form [3a]      dy / dy + t / b - = 0 It is easy to show that a similar equation is obtained for rectangular and cylindrical slices except that the second term on the left-hand side is multiplied by a factor of two. The general form of the above equation becomes (for all bins considered in Fig. 1) [3b]      dy / dy + m t / b - = 0 where, m = 1 for the plane, 2D and m = 2 for the cylindrical and rectangular slices. A further generalization of the above equation is obtained by defining the hydraulic radius of the bin for any cross section of any shape as R = A / P, where A is the area and P is the perimeter. With this notation, the equilibrium equation becomes [3c]      dy / dy + t / R - = 0 In what follows, we will employ the general form given in Eq. 3b with m being a geometrical constant, in view of the application of the method described above to converging hoppers. One cannot solve Eq. 3c directly since it contains two unknowns, the average vertical y and the average tangential stress t. To obtain a solution, additional information is required as a correlation between the two stresses. Janssen proposed such a correlation in the form [4]      t = K y where K is the so-called Janssen constant. Eq. 4 implies a constant relationship between the two average stresses, independent of the geometry of the slice and, while there is no physical reason that such a relationship should hold, it was found in practice that for routine industrial calculations the results are reasonably accurate. Combining Eqs. 4 and 3c, one gets [5]      dy / dy + m K y / b - = 0 which yields a general solution of the form [6]      y = [( b / m K) exp (-m K y / b) + C] exp (-m K y / b) and where C is a constant of integration. Using the boundary condition that at the top of the fill the stress is zero or that y =0 at y = 0, one gets the solution [7]      y = ( b / m K) [1 - exp (-m K y / b)] = ymax [1 - exp (-m K y / b)] which is an exponential function with a maximum value at large depth or large values of y, given by ymax = /(m K). At very small distances, close to the upper surface of the bin, one can expand the exponential function as [8]      exp (-m K y / b) 1 - m K y / b + . . . and using this in Eq. 7 yields the vertical stress y y = B g y. This is exactly the stress encountered in a static fluid of constant density as one moves down from its surface and is the so-called hydrostatic pressure. It is seen that the average vertical stress in a bulk powder follows the hydrostatic pressure distribution in a fluid of the same density but deviates from this value at large distances below the surface to reach a maximum constant value of ymax. This is an interesting property of bulk solids that is due to friction between grains and between grains and the container walls. In the fluid, the entire weight is supported by the bottom plate of the containing vessel. Powders, on the other hand, exhibit friction and therefore some weight is supported by the wall. The overall implication is that below a certain depth in the powder, all weight is supported by walls through friction generating either the active or passive states of stress as explained above. The average tangential wall stress is given as a function of y by Eq. 4. The average normal stress on the wall n can be calculated from the definition of the angle of wall friction (see Fig. 4) as [9]      n = t cot W = K y cot W = ( b / m) [1 - exp(-m K y / b)] cot W Unlike the vertical stress, this stress can easily be measured experimentally and is used to design the metallic structure of the bin; comparison of the above calculated values with experimental measurements is given at the end of the next section.