On the solution of Almansi–Michell’s problem

Abstract

This paper develops a Hamiltonian formalism for the solution of Almansi–Michell’s problem that generalizes the corresponding solution of Saint-Venant’s problem. Saint-Venant’s and Almansi–Michell’s problems can be represented as homogenous and non-homogenous Hamiltonian systems, respectively. The solution of Almansi–Michell’s problem is determined by the coefficients of the Hamiltonian matrix but also by the distribution pattern of the applied loading. The solution proceeds in two steps: first, for the homogenous problem, a projective transformation is constructed based on a symplectic matrix and second, the effects of the external loading are taken into account by augmenting this projection. With the help of this projection, the three-dimensional governing equations of Almansi–Michell’s problem are reduced to a set of one-dimensional beam-like equations, leading to a recursive solution process. Furthermore, the three-dimensional displacement, strain, and stress fields can be recovered from the one-dimensional solution. Numerical examples show that the predictions of the proposed approach are in excellent agreement with exact solutions of two-dimensional elasticity and three-dimensional FEM analysis.