Exact dynamic stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section

The dynamic member stiffness matrix of a three-dimensional shear beam with doubly asymmetric cross-section is derived exactly from the governing, sixth-order differential equation of motion. Such a formulation accounts for the uniform distribution of mass in the member and necessitates the solution of a transcendental eigenvalue problem. This is achieved using the Wittrick–Williams algorithm, where the necessary parameters are developed using a generalised procedure. An example is given to clarify the theory, together with a small parametric study that indicates when lateral–torsional coupling may safely be ignored. The work also holds considerable potential in its application to the approximate analysis of asymmetric, multi-storey, three-dimensional frame structures.