Exact natural frequencies of a three-dimensional shear–torsion beam with doubly asymmetric cross-section using a two-dimensional approach

This paper presents an exact analytical approach to the calculation of the natural frequencies of structures comprising contiguous, three-dimensional shear–torsion beams with doubly asymmetric cross-section. Such component members have the unusual theoretical property that they allow for coupled torsional and shearing deformation, but not bending deformation. Initially, exact dynamic member stiffness matrices (exact finite elements) are developed for planar shear and torsional motion. These matrices can be combined in the usual way to model stepped and continuous beams whose uncoupled frequencies can then be determined exactly. It is then shown how the corresponding coupled frequencies can be established easily from the uncoupled values through an exact relationship. This enables coupled, three-dimensional vibration problems to be solved very efficiently using a two-dimensional approach. The paper is concluded with an example that clarifies the theory, together with a parametric study that enables guidance to be given as to when lateral–torsional coupling may safely be ignored.