Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method

Abstract

The free vibration analysis of axially loaded composite Timoshenko beams is carried out by using the dynamic stiffness matrix method. This is accomplished by developing an exact dynamic stiffness matrix of a composite beam with the effects of axial force, shear deformation and rotatory inertia taken into account, i.e. it is for an axially loaded composite Timoshenko beam. The theory includes the (material) coupling between the bending and torsional modes of deformations which is usually present in laminated composite beams due to ply orientation. An analytical expression for each of the elements of the dynamic stiffness matrix is derived by rigorous application of the symbolic computing package reduce. Use of such expressions leads to substantial savings in computer time when compared with numerical methods usually adopted in the absence of such expressions. The application of the dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of an example composite beam for which some comparative results are available. The solution technique used to yield the natural frequencies is that of the Wittrick–Williams algorithm. The effects of axial force, shear deformation and rotatory inertia on the natural frequencies are demonstrated. The theory developed has applications to composite wings and helicopter blades.