Spatial stress analysis in an elastic parallelepiped

The direct integration method is extended onto the 3D analysis of an elastic rectangular parallelepiped subject to arbitrary force loadings on its sides. By making use of the equilibrium equations, the integral-form expressions are derived for the stress-tensor components through the introduced Vihak functions. These expressions were efficiently used to reduce the original sets of the local boundary conditions to the equivalent sets of the integral conditions for the Vihak functions. In such a manner, the original problems are managed to be reduced to the auxiliary boundary value problems for the governing integro-differential equations with accompanying integral conditions for the Vihak functions. For solving the auxiliary problems for the key functions, special semi-analytical algorithms are suggested in engaging a specific approach for the separation of variables by making use of the complete systems of orthogonal eigen- and associated functions. This allows for determining the Vihak key functions and, consequently, the stress-tensor components in the form of explicit analytical dependencies on the applied force loadings. The solution is quite beneficial for both theoretical and practical implementations. It was shown by the numerical evidence that the solutions are efficient for the analysis of stress fields in the entire domain including edges and corners.