Coupled axial-bending dynamic stiffness matrix for beam elements

Abstract

The dynamic stiffness matrix for beams which exhibit coupling between axial and bending deformations is developed from first principle so that their free vibration analysis can be carried out in an accurate and efficient manner. The coupling between the axial and bending motion essentially arises due to non-coincident centroid and shear centre of the beam cross-section. The dynamic stiffness theory is developed first by deriving the governing differential equations of motion of the axial-bending coupled beam using Hamilton’s principle. The differential equations are then solved in closed analytical form giving expressions for axial and bending displacements as well as the bending rotation. The expressions for axial force, shear force and bending moment are obtained from the natural boundary conditions yielded by the Hamiltonian formulation. Finally, the dynamic stiffness matrix is developed 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 ensuing dynamic stiffness matrix is applied to investigate the free vibration characteristics of a carefully selected axial-bending coupled beam. To this end, the Wittrick-Williams algorithm is used as solution technique. The results are discussed and validated with significant conclusions drawn.