Vibration Analysis of Third-Order Shear Deformable FGM Beams with Elastic Support by Chebyshev Collocation Method

In this paper, we present new results of natural frequencies for the functionally graded beams based on Chebyshev collocation method and the third-order shear deformation theory (TSDT), without requiring any shear correction factors. The beams are assumed to be elastically supported by translational and rotational springs, or simply known as elastically restrained ends. The material compositions of the beams across the gradient direction are described by different mathematical models including the simple power law, exponential and Mori–Tanaka models, and their effects on the response of beams are analyzed. We first present the Chebyshev collocation formulation of the coupled differential equations of motion for free vibration of FGM beams considering different boundary conditions, and then verify the results obtained by the proposed approach against reference ones. A parametric study is also performed for parameters such as thickness, spring constant factor, material volume fraction index, etc. The present numerical results reveal that the proposed method can offer accurate frequency results for the FGM beams as compared with those available in the literature. The results also indicate that the spring constant factors have a significant effect on the frequencies of the beams.