State space elastostatics of prismatic structures

Abstract

This paper provides an exposition of the problem of a prismatic elastic rod or beam subject to static end loading only, using a state space formulation of the linear theory of elasticity. The approach, which employs the machinery of eigenanalysis, provides a logical and complete resolution of the transmission (Saint-Venant's) problem for arbitrary cross-section, subject to determination of the Saint-Venant torsion and flexure functions which are cross-section specific. For the decay problem (Saint-Venant's principle), the approach is applied to the plane stress elastic strip, but in the transverse rather than the axial direction, leading to the well-known Papkovitch–Fadle eigenequations, which determine the decay rates of self-equilibrated loading; however, extension to other cross-sections appears unlikely. It is shown that only a repeating zero eigenvalue can lead to a non-trivial Jordan block; thus degenerate decay modes cannot exist for a prismatic structure.