, Volume 51, Issue 4, pp 951–977 | Cite as

In-plane thermal loading effects on vibrational characteristics of functionally graded nanobeams

  • Farzad EbrahimiEmail author
  • Erfan Salari
  • Seyed Amir Hosein Hosseini


In the present study, the effect of in-plane thermal loading on vibrational behaviour of functionally graded (FG) nanobeams are carried out by presenting both Navier type solution and differential transform method. Classical and first order shear deformation beam theories are adopted to count for the effect of shear deformations. Thermo-mechanical properties of FG nanobeam are supposed to vary smoothly throughout the thickness based on power-law model and material properties are assumed to be temperature-dependent. Eringen’s nonlocal elasticity theory is exploited to describe the size dependency of FG nanobeam. Using Hamilton’s principle, the nonlocal equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FG nanobeams based on Euler–Bernoulli and Timoshenko beam theories. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. In following a parametric study is accompanied to examine the effects of the several parameters such as temperature change, gradient indexes, small scale parameter, mode number and boundary conditions on the natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeam are significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.


In-plane thermal loading Eringen’s nonlocal elasticity theory vibration Functionally graded materials Elasticity Navier type solution 


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Farzad Ebrahimi
    • 1
    Email author
  • Erfan Salari
    • 1
  • Seyed Amir Hosein Hosseini
    • 2
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringImam Khomeini International UniversityQazvinIran
  2. 2.Department of Mechanical EngineeringUniversity of ZanjanZanjanIran

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