An efficient Finite Shear Deformable Beam Element for Extensional-Bending Coupled Vibration of Antisymmetric Laminated Beams

Abstract

An exact one-dimensional finite beam element for extensional-bending coupled vibration analyses for antisymmetric composite laminated beams under various harmonic axial bending forces is developed in this study. The dynamic coupled equations and related boundary conditions are derived from Hamilton’s variational principle. The formulation is based on Timoshenko beam theory and accounts for the effects of shear deformation caused by bending and translational and rotary inertia. It is also captured the effects of Poisson’s ratio and structural extensional-bending coupling coming from composite material anisotropy. From the resulting coupled field equations, the closed form solutions are exactly obtained. A set of shape functions is then developed based on the exact solutions of the coupled equations and is utilized to formulate a finite beam element. The new beam element has two nodes with six degrees of freedom per node and successfully captures the coupled extensional-bending static and steady-state dynamic responses of antisymmetric composite laminate beams under harmonic forces. Several examples are performed for antisymmetric cross-ply and angle-ply laminated composite beams to investigate the effects of transverse shear deformation, and fiber orientation angle on coupled natural frequencies, quasi-static and steady state dynamic responses. Results based on the present finite element formulation are assessed and validated against other well-established finite element and exact solutions available in the literature.