Coupled bending-torsional dynamic stiffness matrix of an axially loaded timoshenko beam element

Abstract

Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.