Coupled flexural–torsional dynamic stiffness matrix of an elastically supported axially loaded beam with shear resistant in‐fill

Abstract

SummaryThe dynamic stiffness matrix of an axially loaded elastically supported uniform beam with doubly asymmetric cross‐section that exhibits coupling between flexural and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The beam comprises a thin‐walled outer section that encloses, and works compositely with, a core of shear resistant in‐fill material. The outer layer provides flexure, warping and Saint–Venant rigidity, while the inner layer provides both Saint–Venant and shear rigidity. A three‐parameter Winkler model is used to describe the distributed elastic support. Hamilton's principle is used to derive the partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions. This gives rise to three coupled equations that are subsequently combined into a single, 12th order, ordinary differential equation. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick–Williams algorithm, which enables the required natural frequencies to be converged upon to any required accuracy with the certain knowledge that none have been missed. Finally, in order to verify the accuracy of the presented theory, the numerical solutions are given and compared with the results that are available in the literature and finite element solutions using abaqus software. Copyright © 2014 John Wiley & Sons, Ltd.