COMPLIANCE OF THE THEORY OF LIMITING EQUILIBRIUM WITH THE HYPOTHESES UNDERLYING JANSEN’S THEORY OF SOIL MECHANICS

Abstract

This paper presents positions regarding approaches to using the Jansen formula for calculating stresses in vertical cylindrical and rectangular containers in relation to the positions of the theory of limiting equilibrium of soils. The equations of the theory of limit equilibrium are widely used for calculations in many applied problems. The range of problems extends from problems of limit equilibrium of slopes (slopes), problems of calculating the foundations of building structures, problems of the stressed state of bulk materials in containers with rigid walls – silos, trays, feeders, etc. Such problems often come down to the need for numerical integration of partial differential equations for the equilibrium of an infinitesimal element of an array, together with the condition for the limit state of equilibrium. The Jansen equations and similar equations are closed analytical expressions that allow one to quickly estimate the stresses in vertical containers with rigid walls filled with bulk materials. Numerous experiments show that the Jansen formula is well satisfied, especially in the asymptotic plane, but numerical calculations (for example, direct modeling using the discrete element method DEM) show that other types of stress states can be realized in high silos. These expressions are obtained through the use of a number of hypotheses, the implementation of which is not always obvious, while the use of the Jansen equation to determine the pressure on the bottom and cylindrical wall of the container does not explicitly concern the theory of limit equilibrium. The work shows that the Jansen formula corresponds to the state of limit equilibrium – it can be obtained directly from the averaged equations of statics if we additionally assume that the distribution of tangential stresses along the radius is represented by a linear function, and assuming that the radial stresses are uniformly distributed over the section, and this is completely corresponds to the solution of the corresponding limit equilibrium problem. An additional method for improving the convergence of numerical calculations is also presented.