Exact dynamic stiffness matrix for a thin‐walled beam of doubly asymmetric cross‐section filled with shear sensitive material

An exact dynamic stiffness matrix is developed for the flexural motion of a three‐dimensional, bi‐material beam of doubly asymmetric cross‐section. The beam comprises a thin walled outer layer that encloses and works compositely with its shear sensitive core material. The outer layer may have the form of an open or closed section and provides flexural, warping and Saint‐Venant rigidity, while the core material provides Saint‐Venant and shear rigidity. 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. Such a formulation enables the powerful modelling features associated with the finite element technique to be utilized when establishing structural models. Three examples are included to validate and illustrate the method. The work also holds considerable potential in its application to the approximate analysis of asymmetric, multi‐storey, three‐dimensional wall‐frame structures. Copyright © 2006 John Wiley & Sons, Ltd.