Free Vibration Analysis of Functionally Graded Nanobeams Based on Different Order Beam Theories Using Ritz Method

This paper presents the fundamental frequency analysis of functionally graded (FG) nanobeams using Ritz method subjected to different sets of boundary conditions. The vibration analysis is based on the classical, the first-order and different higher-order shear deformation beam theories while including rotary inertia. The material properties of FG nanobeams are assumed to vary through the thickness according to the power-law exponent form. Based on the nonlocal constitutive relations of Eringen, the frequencies equations are obtained by the weak forms of the governing differential equations. In this study, the effects of material distribution, nonlocal parameter, beam theories, slenderness ratios and boundary conditions on the fundamental frequency are discussed. The analysis is validated by comparing the obtained results with the available results from the existing literature

Analyses of a composite functionally graded material beam with a new transverse shear deformation function

In the present paper, we offer a higher-order shear deformation theory for bending of functionally graded beam. A new polynomial shear function is used which satisfies the stress-free boundary conditions (exact boundary conditions on the stress) at both, top and bottom surfaces of the beam. Hence, the shear correction factor is not necessary. Additionally, the present theory has strong similarities with Timoshenko beam theory in some concepts such as equations of movement, boundary conditions and stress resultant expressions. The governing equations and boundary conditions are derived from the principle of minimum potential energy. Functionally graded material FGM beams have a smooth variation of material properties due to continuous (unbroken) change in micro structural details. The variation of material properties is along the beam thickness and assumed to follow a power-law of the volume fraction of the constituents. Finite element numerical solutions obtained with the new polynomial shear function are presented and the obtained results are evaluated versus the existing solutions to verify the validity of the present theory. At last, the influences of power law indicator and the new shear deformation polynomial function on the bending of functionally graded beams are explored

Free Vibration Analysis of Functionally Graded Beams

Free vibration analysis of functionally graded beams is carried out for various classical boundary conditions. Two separate finite element formulations, one based on Euler-Bernoulli beam theory and other based on Timoshenko beam theory are developed. Principle of virtual work is used to obtain the finite element system of equations. Numerical results are provided to demonstrate the effect of transverse shear on the natural frequencies and mode shapes for different length-to-thickness ratios and volume fraction exponents of functionally graded material (FGM) beams for the boundary conditions considered. It was observed that transverse shear significantly affects the fundamental frequency and mode shape for lower length to thickness ratios of FGM beams. Further, the effect was observed to be more prominent at higher modes for all the volume fraction exponents of FGM beam.Defence Science Journal, 2012, 62(3), pp.139-146, DOI:http://dx.doi.org/10.14429/dsj.62.132