Janssen's problem

Abstract

1. Consider a vessel filled with liquid. The problem of finding the pressure of the liquid on the bottom and sides of the vessel has an elementary solution. This solution follows from the equations of equilibrium, Pascal's law, and the absence of tangential stresses at the boundary. Now suppose that the vessel is filled with an ideally loose material consisting of separate solid particles. In this case the problem of determining the pressure is much more difficult. This is because there is no analog of Pascal's law for loose materials, and because, in contrast with liquids, the pressure of a loose material depends not only on the depth of filling of the vessel but also on the method of filling.Determining the pressure of a loose material on the bottom and sides of a vessel is a classical problem in mechanics. Interest in it is prompted by the need to analyze various bunkers for storing loose and powdered materials, chemical reactors filled with granulated components, and so on.The first solution to this problem for vertical side walls was derived by Janssen [1]. His solution is based on two hypotheses: 1) The coefficient of lateral thrust k, equal to the ratio of the normal pressure on the wall to the mean normal pressure in a horizontal cross section, is constant (an analog of Pascal's law); 2) the tangential stress on the walls is completely developed and is proportional to the normal pressure.Subsequent investigations of this problem were essentLally based on these hypotheses [2]. The constructional norms and specifications at present in force are also based on Janssen's solution, to which a number of empirical correction coefficients are added [3].To make use of Janssen' s solution we must have data on the coefficient of lateral thrust. The existing recommendednorms for k range over very wide intervals. For example, for grain stores the USSR norm s give k ~ 0.4 [3]; the USA norms give k ~ 0.6; in West Germany the value k ~ 0.3 was used for manyyears, then the value k ~ 0.5 was intrvduced; these values have been described as "purely conjectural~ [4]. Thus in Janssen's scheme the question of the value of the coefficient of lateral thrust is still open. Moreover, while adhering to this scheme it is impossible to estimate its error. Experimental estimates of the error encounter various theoretical difficulties associated with the dependence of the pressure on the yield of the transducers and vessel walls and on other factors.In this connection solving the problem by the methods of mechanics of a continuous medium without involving hypotheses (1) and (2) is of consLderable interest. Solving the problem by exact methods enables us to find the distributions of all the stresses and to make a quantitative estLmate of the influence of various factors. With this approach, in contrast with Janssen's scheme, generalization to the case of inclined or curved side walls presents no new difficulties. From the known solution we can calculate the coefficient of lateral thrust and estimate the error of the engineering solution b...

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