Coupled axial-bending dynamic stiffness matrix and its applications for a Timoshenko beam with mass and elastic axes eccentricity

Abstract

The dynamic stiffness matrix of a coupled axial-bending Timoshenko beam is developed to investigate the free vibration behaviour of such beams and their assemblies. Applying Hamilton's principle, the governing differential equations of motion of a Timoshenko beam in free vibration is derived by considering the axial-bending coupling effect arising from the mass axis eccentricity with the elastic axis of the beam cross-section. The differential equations are then solved in an exact sense, giving expressions for the axial and bending displacements as well as the bending rotation. The expressions for axial force, shear force and bending moment are formed using the natural boundary conditions which resulted from the Hamiltonian formulation. Next, the frequency-dependent dynamic stiffness matrix of the coupled axial-bending Timoshenko beam is derived by relating the amplitudes of the axial force, shear force and bending moment to the corresponding amplitudes of axial displacement, bending displacement and bending rotation. The resulting dynamic stiffness matrix is effectively applied to investigate the free vibration behaviour of axial-bending coupled Timoshenko beams by making use of the Wittrick-Williams algorithm as solution technique. The results with emphasis on the axial-bending coupling effects and the importance of the shear deformation and rotatory inertia in free vibration behaviour of coupled axial-bending Timoshenko beams and frameworks are discussed with significant conclusions drawn.