On the solution of Almansi-Michell's problem

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.