The Saint-Venant problem and principle in elasticity

Abstract

The traditional semi-inverse solution method of the Saint-Venant problem and the Saint-Venant principle, which were described in the Euclidian space under the Lagrange system formulation, are updated to be solved in the symplectic space under the conservative Hamiltonian system. Thus, the Saint-Venant problem and the Saint-Venant principle have been unified by the direct method. It is proved in the present paper that all the Saint-Venant solutions can be obtained directly via the zero eigenvalue solutions and all their Jordan normal forms of the corresponding Hamiltonian operator matrix and the Saint-Venant principle corresponds to neglect of all the non-zero eigenvalue solutions, in which the non-zero eigenvalue gives the decay rate.