An analytical method for vibration and buckling behaviours of Functionally Graded (FG) beams with various boundary conditions under mechanical and thermal loads is presented. Based on linear straindisplacement relations, equations of motion and essential boundary conditions are derived from Hamilton's principle. In order to account for thermal effects, three cases of the temperature rise through the thickness, which are uniform, linear and nonlinear, are considered. The exact solutions are derived using the state space approach. Numerical results are presented to investigate the effects of boundary conditions, temperature distributions, material parameters and slenderness ratios on the critical temperatures, critical buckling loads, and natural frequencies as well as load-frequencies curves, temperature-frequencies curves of FG beams under thermal/mechanical loads. The accuracy and effectiveness of proposed model are verified by comparison with previous research.
Northumbria University has developed Northumbria Research Link (NRL) to enable users to access the University's research output. Copyright © and moral rights for items on NRL are retained by the individual author(s) and/or other copyright owners. Single copies of full items can be reproduced, displayed or performed, and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata page. The content must not be changed in any way. Full items must not be sold commercially in any format or medium without formal permission of the copyright holder. The full policy is available online: http://nrl.northumbria.ac.uk/policies.html This document may differ from the final, published version of the research and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher's website (a subscription may be required.)Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory AbstractThis study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam's depth and calculated by using classical rule of mixture and Mori-Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton's principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson's ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined.
In this paper, the free vibration behaviour of bi-dimensional functionally graded (BDFG) microbeams under arbitrary boundary conditions (BCs) is studied. Based on the frame work of the modified couple stress theory and Hamilton's principle, governing equations of motion are developed for the BDFG microbeams using a quasi-3D theory. The formula then can be reduced to a higher-order beam theory (HOBT) of conventional functionally graded (FG) microbeams with the material properties varying along the thickness direction only. Two types of BDFG microbeams with different patterns of material volume distribution are considered. The material properties used in this study are assumed to vary exponentially along both longitudinal and thickness directions of microbeams. Based on the state-space concept, the governing equations are solved for natural frequencies and vibration mode shapes of microbeams under various BCs. The effects of material distribution, geometric parameters and BCs are also investigated to examine the size-dependent behaviour of BDFG microbeams.
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