Dynamic Stiffness Method for Free Vibration of Beams and Frameworks Using Higher Order Shear Deformation Theory

Abstract The dynamic stiffness method for free vibration of beams and frameworks is developed using a higher order shear deformation theory. Starting with the displacement field, the potential and kinetic energies of the beam in flexural vibration, are first formulated. Then, Hamilton’s principle is applied to derive the governing differential equations and associated natural boundary conditions. Next, the differential equations are solved to obtain the expressions for flexural displacement, bending rotation and the first derivative of the flexural displacement. The expressions for the shear force, bending moment and the higher-order moment are obtained from the natural boundary conditions resulting from the Hamiltonian formulation. Finally, the force vector comprising the amplitudes of the shear force, bending moment and the higher-order moment is related to the amplitudes of the displacement vector comprising the flexural displacement, bending rotation and the first derivative of the flexural displacement through the frequency-dependent dynamic stiffness matrix. The dynamic stiffness matrix for axial motion which is uncoupled from the flexural motion is now implemented to the dynamic stiffness matrix in flexural motion to analyse individual beams and frameworks for their free vibration characteristics by applying the Wittrick-Willaims algorithm. Illustrative examples are given, and significant conclusions are drawn.