Stress components and loading restrictions at the vertices of regular triangular and quadrangular pyramids

A special point of structural element (the vertex of a polyhedron) is considered as an ordinary point of deformable body representing an infinitely small particle obtained by contracting elementary volume to a point. Using this concept, the stress state at the vertices of regular triangular and quadrangular pyramids is studied in the case of a surface loading of the lateral faces of pyramids. It is shown that the stress state at the vertices of polyhedra is fully known for any loading. This fact leads to a non-classical formulation of the problem of solid mechanics for such structural elements. The conditions for load vector components are proposed, which provide the correct problem statements within the solid mechanics. The particular cases of the loading of considered structural elements are introduced. The obtained solutions are found to be in a good agreement with known analytical results.
The reported results will find application in the formulation of solid mechanics problems containing vertices (recesses) in the shape of polyhedra, in particular, when studying the interaction of the Berkovich and Vickers indenters with samples.