A novel view on the canonical 'stretching-due-to-bending' nonlinear effect: A slighly curved beam

The analysis of nonlinear coupling between flexural and longitudinal vibrations of ideally straight elastic beams is a classical topic in vibration theory. The boundary conditions usually applied in this analysis are formulated as immobile hinges, which generate the canonical 'stretching-due-to-bending' nonlinear effect. The key assumption in formulation of this model is neglecting the longitudinal inertia, which may be referred to as 'static condensation'. The aims of this work are, first, to re-consider this problem from the viewpoint of a nonlinear theory of curved beams, and, second, employ alternative type of boundary conditions, known as class-consistent ones. A paradoxical difference in nonlinear parts of Duffing equations obtained in the limit of vanishing curvature of an initially curved beam and in the case of an ideally straight beam is demonstrated and explained.