Analytical Method to Determine Vertical Stresses within a Granular Material Contained in Right Vertical Prisms

In the design of large and tall storage containers/structures such as silos, hoppers, and mine stopes, it is often required to determine the stresses within the container, especially at the bottom where stresses are the highest. Due to arching, where a substantial fraction of the self-weight of the granular material is carried by the wall, the vertical stress at the bottom of the container is significantly less than what is given by the product of the height and unit weight. Few analytical expressions published in the literature can be used to determine the vertical stresses taking into account the arching effect and on the basis of equilibrium considerations. The objective of this paper is to propose a new analytical method for determining the vertical stresses in a long container, assuming plane strain conditions. The method is extended to containers with rectangular and circular cross sections and is used to accommodate a surcharge at the top of the granular material. The values of vertical normal stresses computed for long-strip square, and circular cross sections are in very good agreement with those computed from Marston’s theory, which was recently validated against numerical and laboratory models for providing satisfactory estimates of average vertical stresses within mine stopes. The results from the proposed model also compare well with elastoplastic numerical model results, provided $K$ and $\delta$ are assumed to be as follows: (a) $K=K_0$ and $\delta =2/3\varphi$ or (b) $K=K_a$ and $\delta =\varphi$. More than the expressions, the method itself would pave the way to future applications in similar and generalized problems related to storage structures with sloping walls.