An analytical approach for the free vibration analysis of generally laminated composite beams with shear effect and rotary inertia

In this paper, the free vibration analysis of generally laminated composite beam (LCB) based on Timoshenko beam theory are presented using the method of Lagrange multipliers where in the free vibration problem is posed as a constrained variational problem. The effect of material couplings (bending–tension, bending–twist, and tension–twist couplings) with the effects of shear deformation, rotary inertia and Poisson's effect are taken into account. Analytical expression for the natural frequencies and mode shapes are presented. The calculated natural frequencies are verified against some available results in the literature and very good agreement is observed. Furthermore, the effects of some parameters such as slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration and boundary conditions on the natural frequency and mode shape are examined.

► An analytical solution for the vibrational characteristics of the laminated composite beam has been presented.
► The Poisson's effect and all of the material couplings have been considered.
► The analysis can be applied for a laminated beam with general lay-ups.
► By using the orthogonal Legendre polynomials, the rapid convergent has been observed.