In elasticity, the method of forces, wherein stress parameters are considered as the primary unknowns, is known as the Beltrami-Michell formulation. The Beltrami-Michell formulation can only solve stress boundary value problems; it cannot handle the more prevalent displacement or mixed boundary value problems of elasticity. Therefore, this formulation, which has restricted application, could not become a true alternative to the Navier displacement method, which can solve all three types of boundary value problems. The restrictions of the Beltrami-Michell formulation have been alleviated by augmenting the classical formulation with a novel set of conditions identified as the boundary compatibility conditions. This new method, which completes the classical force formulation, has been termed the completed Beltrami-Michell formulation. The completed Beltrami-Michell formulation can solve general elasticity problems, with stress, displacement, and mixed boundary conditions in terms of stresses as the primary unknowns. The completed Beltrami-Michell formulation is derived from the stationary condition of the variational functional of the integrated force method. In the completed Beltrami-Michell formulation, stresses for kinematically stable structures can be obtained without any reference to displacements either in the field or on the boundary. This paper presents the completed Beltrami-Michell formulation and its derivation from the variational functional of the integrated force method. Examples are presented to demonstrate the applicability of the completed formulation for analyzing mixed boundary value problems under thermomechanical loads. Selected examples include analysis of a composite cylindrical shell, wherein membrane and bending response are coupled, and a composite circular plate.
All Science Journal Classification (ASJC) codes
- Aerospace Engineering