In-plane and out-of-plane vibration modes of laminated composite beams with arbitrary lay-ups

This paper mainly presents the in-plane and out-of-plane vibration analysis of thin-to-moderately thick composite beams with arbitrary lay-ups using the first order shear deformation theory. The material couplings, i.e. bending–stretching, bending–twisting and stretching–twisting couplings, along with the effects of shear deformation, rotary inertia and Poisson's effect are taken into account. In order to obtain the free response of the structure, a semi-analytical solution is adopted based on the variational formulation and solving the weak form of the governing equations by extremizing the objective functional with respect to unknown displacement components and Lagrange multipliers. We also propose a higher order beam element which is a convenient tool for numerical implementation in one-dimensional finite element analysis of the problem. Semi-analytical and finite element solutions are utilized and verified for various numerical case studies. Numerical examples are presented for both the semi-analytical and finite element solutions. The results show excellent agreement with the solutions obtained by the full three-dimensional finite element model in commercial package ANSYS. These numerical examples also demonstrate that ignoring the role of out-of-plane displacement in the conventional one-dimensional models can lead to significant error in calculating the torsional modes of laminated composite beams.