The present work is concerned with the study of the geometrically non-linear steady state periodic forced response of a clamped-clamped beam containing an open crack. The model based on Hamilton's principle and spectral analysis, previously used to investigate various non-linear vibration problems, is used here to determine the effect of the excitation frequency and level of the applied harmonic force, concentrated at the cracked beam middle span, on its dynamic response at large vibration amplitudes. The formulation uses the "cracked beam functions", denoted as 'CBF', previously defined in recent works, obtained by combining the linear theory of vibration and the linear fracture mechanics theory. The crack has been modelled as a linear spring which, for a given depth, the spring constant remains the same for both directions. The results obtained may be used to detect cracks in vibrating structures, via examination of the qualitative and quantitative changes noticed in the non-linear dynamic behaviour, which is commented in the conclusion.

The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.

The purpose of the present paper is to show that the problem of geometrically non linear free vibration of symmetrically and asymmetrically laminated composite beams with immovable ends can be reduced to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness parameters. This simple formulation is developed using the governing axial equation of the beam in which the axial inertia and damping are ignored. The theoretical model is based on Hamilton’s principle and spectral analysis. Iterative form solutions are presented to calculate the fundamental nonlinear frequency parameters which are found to be in a good agreement with the published results. The non-dimensional curvatures associated to the fundamental mode are also given in the case of clamped-clamped symmetrically and asymmetrically laminated composite beams.

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