Linear vibrations of planar prestressed elastica arches

The natural frequencies and mode shapes of planar shear undeformable beams around their curved prestressed post-buckling configurations are investigated. Two mechanical models are considered depending on the assumed boundary conditions in the buckling and post-buckling phases. Namely, 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 roller support. In the second case, the beam is assumed inextensible in the buckling phase (same boundary conditions as above), however, it is considered extensible in the subsequent post-buckling phase because the roller support is changed into a hinged end. The post-buckling solution is obtained with three approaches: asymptotic, numerical via evaluation of an elliptic integral, and numerical via finite-element formulation (FEM). Due to the non-constant curvature, the equations of motion governing linear vibrations around the post-buckling configuration are linear 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 FEM approach and a semi-analytical method based on a weak formulation (Galerkin method) implemented in an in-house built code. 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 prestressed elastica arch models and their potential influence on nonlinear vibrations are discussed. Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.