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Structural Engineering and Mechanics
  Volume 61, Number 2, January25 2017 , pages 209-220
DOI: https://doi.org/10.12989/sem.2017.61.2.209
 

Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation
AliReza Setoodeh and Mohammad Rezae

 
Abstract
    The purpose of this paper is to study the geometrically nonlinear free vibration of functionally graded nano/micro beams (FGNBs) based on the modified couple stress theory. For practical applications, some analytical expressions of nonlinear frequencies for FGNBs on a nonlinear Pasternak foundation are developed. Hamilton\'s principle is employed to obtain nonlinear governing differential equations in the context of both Euler-Bernoulli and Timoshenko beam theories for a comprehensive investigation. The modified continuum theory contains one material length scale parameter to capture the size effect. The variation of two-constituent material along the thickness is modeled using Reddy\'s power-law. Also, the Mori-Tanaka method as an accurate homogenization technique is implemented to estimate the effective material properties of the FGNBs. The results are presented for both hinged-hinged and clamped-clamped boundary conditions. The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method and then the powerful method of homotopy analysis is utilized to obtain the semi-analytical solutions. Eventually, the presented analytical expressions are used to examine the influences of the length scale parameter, material gradient index, and elastic foundation on the nonlinear free vibration of FGNBs.
 
Key Words
    modified couple stress theory; nonlinear vibration; functionally graded nano/micro-beam; homotopy analysis method
 
Address
AliReza Setoodeh and Mohammad Rezae: Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz 71555, Iran
 
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