Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

Abstract

Geometrically non-linear vibrations of functionally graded Euler-Bernoulli beams with multi-cracks, subjected to a harmonic distributed force, are examined in this paper using a theoretical model based on Hamilton's principle and spectral analysis. The homogenisation procedure is performed, based on the neutral surface approach, and reduces the FG beams analysis to that of an equivalent homogeneous multi-cracked beam. The so-called multidimensional Duffing equation obtained and solved using a simplified method (second formulation) previously applied to various non-linear structural vibration problems. The curvature distributions associated to the multi-cracked beam forced deflection shapes are obtained for each value of the excitation level and frequency. The parametric study performed in the case of a beam and the detailed numerical results are given in hand to demonstrate the effectiveness of the proposed procedure, and in the other hand conducted to analyse many effects such as the beam material property, the presence of crack, the vibration amplitudes and the applied harmonic force on the non-linear dynamic behaviour of FG beams.

Geometrically non-linear free and forced vibration of a shallow arch

“…In Figs. 10, the curvature distributions are plotted, associated to the first linear and non-linear deflection, of a C-C shallow arch in free vibration case, for various initial rise (2,4,8) and a maximum non-dimensional amplitudes ( * = 1.5). The corresponding values are summarized in Table 6, in which the percentage correction at the clamps and at the middles are given for various values of initial rise.…”

“…Investigating the geometrical non-linearity is one of the major consideration on the design process of the arches. The study of the geometrical non-linearity of a beams, plates ,shells and circular arches were investigated by the authors of [1][2][3][4][5][6][7][8]. The authors classified the arches following their shallowness ratio into two classes; shallow arches and deep or no-shallow arches.…”

Analysis of Geometrically Non-Linear Free Vibrations of Functional Graded Beams in a Thermal Environment

“…El khouddar et al [8] studied the free and forced non-linear vibrations of laminated composite beams under different boundary conditions, using the Euler-Bernoulli beam theory and the Green-Lagrang nonlinearity. From the Euler-Bernoulli beam theory, Chajdi et al [9] examined the forced nonlinear vibrations of an FGM beam with multiple cracks. Based on the Euler-Bernoulli beam theory, Outassafte et al [10], [11] contributed to the geometrically non-linear free vibration of a fixed-ended arch.…”