On the linear normal modes of planar pre-stressed curved beams

The natural frequencies and mode shapes of planar shear undeformable beams around their curved pre-stressed post-buckling configurations are investigated neglecting rotary inertia effects. Two mechanical models are considered depending on the assumed boundary conditions in the buckling and post-buckling phases. With the first model, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a sliding hinge. In the second case, the beam is assumed inextensible in the pre-stressed phase (same boundary conditions as above), whereas it is extensible in the subsequent free linear dynamic phase because the sliding hinged boundary is changed into a stationary hinged end. Linear vibrations are governed by partial-differential equations with non-constant coefficients and the solutions for the frequencies and mode shapes are found employing two approximate approaches: a fully numerical method based on a finite element formulation and a semi-analytical method based on a weak formulation (Galerkin method). The main results are compared and a close agreement in the outcomes is found. The leading mechanical differences in the linear normal modes of the two pre-stressed curved beam models are discussed.