A 2-dimensional model of stress distribution in a wedge hopper has been developed. This is a co-ordinate-specific version of the Lamé-Maxwell equations in a space frame dictated by the assumption of circular arc, principal stress orientation. A set of orthogonal, independent variables has been defined as x-ψo space. x is the vertical height of intersection of the circular principal stress arc with the wedge wall and the radius of the circular arc is proportional to x. ψo is the angle that the radius makes to the vertical at the lower arc in the system - lower boundary condition. The second principal stress follows ψ-lines through the vessel from ψo at the lower boundary, eventually passing through the vessel wall and leaving the system. The model has been used to integrate the stress equations along lines of principal stress using numerical techniques. An analytical solution has been found at ψo = 0 of the same mathematical form as the Enstad/Walker/Walters equations. The model can be used to predict the location of the stable, cohesive arch and to predict unviable stress states in terms of the Mohr-Coulomb yield criterion. There is a requirement for experimental data of internal stress distributions within bulk solids in hoppers and silos to validate this and other models.
Matchett, A. J., O'Neill, J. C., & Shaw, A. P. (2008). Stress distributions in 2-dimensional, wedge hoppers with circular arc stress orientation - A co-ordinate-specific Lamé-Maxwell model. Powder Technology, 187(3), 298-306. https://doi.org/10.1016/j.powtec.2008.03.013