Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid-plane
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RESEARCH PAPER
Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid‑plane Formulations Hadi Babaei1 · Yaser Kiani2 · Mohammad Reza Eslami3 Received: 3 May 2020 / Accepted: 20 August 2020 © Shiraz University 2020
Abstract The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surfacebased formulation, the governing equations of motion are established. For the case of beams with immovable edges in inplane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters. Keywords Nonlinear free vibration · FGM beam · Shear deformation theory · Nonlinear elastic foundation · Two-step perturbation method · Thermal environment
1 Introduction In the last several years, elastic beams and columns made of advanced materials are widely used in structural design problems. Functionally graded materials (FGMs) are a new advanced class of non-homogeneous materials. Typically, an FG material is made from a mixture of metal and ceramic, in which properties vary continuously from one surface to the other. * Yaser Kiani [email protected] Hadi Babaei [email protected] Mohammad Reza Eslami [email protected] 1
Mechanical Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran
2
Faculty of Engineering, Shahrekord University, Shahrekord, Iran
3
Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran
Many theoretical investigations on the small and large amplitude vibrations of thin beams with/without elastic foundation and with different end conditions have received considerable attention in the following studies. For example, Emam and Nayfeh (2009) investigated nonlinear postbuckling and free vibration of composite thin-walled beams under axial compressive load. They proposed an exact solution for the governing differential equations associated with the post-buckling and free vibration problems. In this study, post-buckling path and natural frequencies of pre/post-buckled composite beams with several boundary conditi
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