On the bi-planar motion of a Timoshenko beam with shear resistant in-fill

An exact dynamic stiffness matrix is developed to describe the motion of a uniform beam in which the flexural and torsional motion is coupled in three dimensions by its doubly asymmetric cross-section. 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 can be of open or closed section and flexure is modeled according to Timoshenko theory. In addition, it provides warping, shear and Saint-Venant rigidity, while the core material provides Saint-Venant and shear rigidity. The partial differential equations governing the free vibration of the beam, together with the associated natural boundary conditions, are derived using Hamiltonʹs principle. This gives rise to five coupled equations that are subsequently combined into a single, twelfth 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, examples are given to confirm the accuracy of the theory presented.