Stresses in Bins and Hoppers, by Gabriel I. Tardos

-- 4. The Method of Slices Applied to Hoppers --

Janssen's method was extended to hoppers first by Walker [1966] and subsequently improved by many authors; a detailed account of the main results is given by Drescher [1991]. Consider the plane hopper slice in fig. 5 where taking the origin of axes at the virtual apex is appropriate.

Fig. 5 - Differential plane slices in a hopper

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The varying width of the hopper is taken as b = y tan . The equilibrium of the slice in the vertical direction becomes as before [10]      2 y tan y - 2 tan (y + dy) (y + dy) + 2 dy (n tan + t) - 2 y dy tan = 0 where the additional terms compared with Eq. 2 appear because of the orientation of the walls. Dividing the whole equation by 2 y tan dy and neglecting the term 2 tan dy dy as being much smaller compared with other terms, one obtains the differential form [11a]      dy / dy + (1 / y) (y - n - t / tan ) + = 0 where all stresses, the vertical y, the tangential wall t, and the normal wall stress n, appear. It is again easy to show that for conical and square hoppers the same equation applies with a modified coefficient on the second term on the left-hand side [11b]      dy / dy + (m / y) (y - n - t / tan ) + = 0 where again m = 1 for the 2D plane and m = 2 for the 3D slice. Taking the correlation proposed by Janssen as t = K y and relating the normal and vertical stresses as n = K y cot W (see Eq. 9), the above differential equation reduces to [12]      dy / dy + m N y / y + = 0 Here N = 1 - K (cot + cot W) is a material parameter that replaces the Janssen constant, K, in Eq. 5. There is an additional major difference between the above and the corresponding equation for the bin, Eq. 5, in that the second term is nonlinear as it contains the variable y instead of the constant parameter b. For the case in which m N -1 there is a general analytical solution of the above equation in the form [13]     y = - y / (1 + m N) + C y-mN where again C is a constant of integration. This constant can be determined by imposing a value of the vertical stress y at the upper surface of the hopper where y = H and this results in an exponential stress distribution. If the silo is mostly empty and material in the hopper exhibits a free surface, the vertical stress at the top of the fill can be taken as zero and the solution becomes [14]      y = [- y / (1 + m N)] [1 - (H / y)1+mN] If however the silo is mostly full and material is present in both the hopper and the bin, the boundary condition at the bin-hopper connection is not clear cut. It was found by many researchers that the stress at the connecting point is not continuous and in fact exhibits a large discontinuity resulting in a peak value at the connection point. It is usually incorrect therefore to equate the vertical stress at the top of the hopper with the maximum stress at the bottom of the bin to obtain a general solution for the stress distribution in Eq. 13. There are many explanations for this peculiar behavior but all relate to the way wall friction acts in the powder on the vertical bin wall and the inclined hopper wall and whether or not the material is in the filling or emptying stage of the silo. The behavior is mostly due to the switching from the active state of stress in the bin to the passive state in the hopper as explained in the previous section. Fortunately, the stress distribution at the connection point does not have a major influence on the distribution in the converging section of the hopper and this fact is used below to find an approximate solution for the lower part of the hopper. To get the approximate solution, one assumes that the hopper is tall and a solution is sought for small values of the distance y from the apex. Under these conditions the integration constant in Eq. 13 can be taken as C = 0 and one obtains the approximate results for stresses in the lower part of the hopper as [15]      y = - y / (1 + m N) ; t = - y K / (1 + m N); n = - y K cot W / (1 + m N) The plane slice in the hopper presented in fig. 5 is not unique, and different other shapes have been used in the literature, each giving a somewhat different value for the stress. Fig. 6 - Cylindrical, convex, and concave slices in hoppers For example, using the cylindrical slice as shown schematically in fig. 6a yields the approximate average radial stress as [16]      r = - M r / (1 + m N) where M = (2 - m) / sin - 2 (1 - m) / (1 + cos ), and m = 1 for the plane, cylindrical and m = 2 for the conical hopper and r is the radial distance measured form the apex as shown in fig. 6a. The above result is the so called radial solution for the stress distribution and has, beyond some historic significance in that it was proposed by A. Jenike as early as 1963, also practical value in hopper design. One has to note that the above stress, and the other two average stresses derived from it as in Eq. 15, only depend explicitly on the radial distance from the apex with the coefficients M and N being functions of only material and wall properties. Other curvilinear slices were also proposed notably by Enstad [1975 and 1977] and these were associated with the directions where the maximum stresses occur in the hopper, the so-called principal stresses, 1 and 2. The geometry of the slices aligned with the principal stresses are also shown in Fig. 6 as the convex slice for 2 that represents the passive state and the concave slice for 1 that represents the active state of stress. The important characteristic of these slices is the fact that there are only normal stresses present and all shear stresses are zero. The orientation of the slice makes an angle with the horizontal as shown in the figure. The maximum stress also known as the minor principal stress is then given, for the passive state, i.e., the convex slice (see Fig. 6b), by [17a]      2 = 1 / K = - F r / (1 + m G) where the constants F and G are given below and replace the coefficients M and N in Eq. 16. The stress 1, in the above equation is depicted schematically in Fig. 3b. A similar set of values can be obtained for the active state of stress, i.e., the concave slice shown in Fig. 6c where the coefficients F and G are slightly different.