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Two-dimensional free vibration analysis of cracked FGM beams by Ritz method

English Abstract

Investigations of free vibration analysis of beams are of considerable significance in the field of engineering as well as research. The existence of crack may affect the dynamical behavior and stiffness of beams. The present thesis is committed to analyze the problem of free vibration of cracked beams. The previous beam theories are relying on some hypothesis which may not be appropriate enough when considering some cases, such that some improvements are needed. The present two-dimensional analysis does not take any assumptions in the 2D plane of deformation, more reliable results can be expected. Numerous papers were published tackling the problem of free vibration of cracked beams. However, the previous models about beams with cracks are limited or complicated. The present analysis uses a modified crack functions based on William’s crack functions and two-dimensional elasticity equilibrium equations. Meanwhile, the crack opening mode (mode I) and crack shearing mode (mode II) are taken into account, the superposition of these two modes can describe most of cases about the FGM beam with a crack. The 2D free vibration analysis of FGM cracked beams is investigated by Ritz method. Displacement fields are composed of Chebyshev polynomials and crack functions, while a coordinate transformation needs to be done between local polar coordinate located at the crack tip and global Cartisian coordinate. Minimizing the maximum energy functional with respect to each unknown coefficient, the governing eigenvalue equations in matrix form are obtained. Considering the singularity effect at the crack tip, the integral domain is divided into 4 parts. Moreover, the numerical results are studied by 2D Gaussian integration, 4 significant figures are taken throughout the convergence and parametric studies and the results converge to at least 3 significant figures. The convergence studies and comparisons are incorporated to illustrate the good performance of the present method. iv Two FG materials, Al and Al2O3, and volume fraction exponents (k = 0, 0.5, 1, 10) are taken. The first five non-dimensional frequency parameters of clamped-free, clamped-pinned, clamped-clamped and pinned-pinned cracked FGM beams with various crack positions (Lc/L = 0.2, 0.4, 0.6, 0.8 or Lc/L = 0.2, 0.4, 0.5), depth ratios (dc/h = 0.2, 0.4, 0.6) and slenderness ratios (L/h = 5, 10) are presented. Parametric studies investigate the influence of crack positions, crack depth ratios, slenderness ratios of beams and FGM volume fraction exponents, also some corresponding mode shapes are exhibited. Majority of results are presented for the first time which could be taken as a reference for other studies.

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Liu, Bo Yan


Faculty of Science and Technology


Department of Civil and Environmental Engineering




Functionally gradient materials

Vibration -- Mathematical models

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