Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic fou
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Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations Hassen Ait Atmane . Abdelouahed Tounsi . Fabrice Bernard
Received: 25 March 2015 / Accepted: 19 July 2015 / Published online: 28 July 2015 Springer Science+Business Media Dordrecht 2015
Abstract The novelty of this paper is the use of an efficient beam theory for bending, free vibration and buckling analysis of functionally graded material (FGM) beams on two-parameter elastic foundation. The present theory accounts for both shear deformation and thickness stretching effects by a parabolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the beam without requiring any shear correction factor. Due to porosities, possibly occurring inside FGMs during fabrication, it is therefore necessary to consider the vibration, bending and buckling behaviors of beams having porosities in this work. The equation of motion for FGM beams is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The validity of the present theory is investigated H. Ait Atmane Civil Engineering Department, Faculty of Civil Engineering and Architecture, University Hassiba Benbouali, Chlef, Algeria A. Tounsi Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department, Sidi Bel Abbes, Algeria F. Bernard (&) INSA Rennes, Rennes, France e-mail: [email protected]
by comparing some of the present in literature. It can be concluded that the proposed theory is accurate and simple in solving the bending, free vibration and buckling behaviors of FGM sandwich beams. Keywords Functionally graded materials Free vibration Sandwich beam Refined beam theory Navier solution
1 Introduction Functionally graded materials (FGMs) are novel kinds of composites, made from a mixture of ceramic and metallic constituents (Koizumi 1993). The mixture ratio of the constituents varies smoothly, and the material characteristics change continually across the thickness. This leads largely to avoid stress concentration induced by discontinuity of material properties, typically observed in laminate and fiber-reinforced composites. In the recent years, many works have been performed on the dynamic and bending, vibration and buckling of FGM structures. The wave propagation analysis in beams made of FGM is discussed by Chakraborty and Gopalakrishnan (2003) by employing the spectral finite element method. Using the Euler–Bernoulli beam theory, Sankar (2001) presented an elasticity solution for bending of functionally graded beams (FG beams). Poisson’s ratio was considered to be constant, while Young’s modulus
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was supposed to change as an exponential function. By employing the Airy stress function, Zhong and Yu (2007) developed an analytical solution
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