Nonlinear free vibration of thin functionally graded beams using the finite element method

A finite element model of large amplitude free vibrations of thin functionally graded beams with immovably supported ends is developed in this paper. The material properties of functionally graded beams are assumed to vary according to the power law distribution through the thickness direction. The finite element model is formulated in a variationally correct way based on Euler-Bernoulli beam theory and von Karman geometric nonlinearity. The linear exact displacement fields of the static case are used as the shape functions. The time response of each node is assumed to be a harmonic function, and the error residuals due to this assumption are minimized by employing the Galerkin method. Together this assumption and method transform the finite element equations to an eigenvalue equation that can be solved using a direct iterative method in tandem with the principle of energy conservation. The accuracy of the proposed method is demonstrated by comparing the frequencies and amplitudes with those of other methods presented in the literature. Finally, the relation between the frequencies of functionally graded beams and those of the homogeneous beams at various initial amplitudes is also examined.