On Saint Venant's Problem for Helicoidal Beams

Abstract

This paper proposes a novel solution strategy for SaintVenant's problem based on Hamilton's formalism. SaintVenant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 أ— 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: SaintVenant's solutions exist because rigidbody motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigidbody motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closedform solutions of the reduced problem are found and threedimensional stress and strain fields can be recovered from the closedform solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of threedimensional elasticity and threedimensional FEM analysis.