Free vibration analysis of FGM framed nanostructures using variational-consistent boundary conditions

Duong The Hung, Tran Van Lien, Tran Binh Dinh, Nguyen Tat Thang
Author affiliations


  • Duong The Hung Thainguyen University of Technology, Vietnam
  • Tran Van Lien Hanoi University of Civil Engineering, Vietnam
  • Tran Binh Dinh Hanoi University of Civil Engineering, Vietnam
  • Nguyen Tat Thang Hanoi University of Civil Engineering, Vietnam



FGM, nanobeam, DSM, variational-consistent boundary conditions, nondimensional frequency


This paper analyses free vibrations of framed nanostructures made of Functionally Graded Material (FGM) based on the Nonlocal Elastic Theory (NET) and the Dynamic Stiffness Method (DSM). FGM characteristics vary nonlinearly throughout the height of the beam element. The NET considers the nonlocal parameter that perfectly captured the size effect of nanostructures. However, the NET makes nonlocal paradoxes in the bending and vibration behaviour of framed nanostructures with the free ends. To overcome these phenomena, the nanostructure is modelled according to the Euler–Bernoulli beam theory and the variational-consistent nonlocal boundary conditions have been derived. The exact solutions of differential equations of motion and variational-consistent nonlocal boundary conditions are found using the DSM. The influences of the nonlocal, material, geometry parameters and Pasternak’s foundation on the free vibration are then analyzed. It is shown that the study can be applied to other FGMs as well as more complicated framed structures.


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How to Cite

D. T. Hung, T. V. Lien, T. B. Dinh and N. T. Thang, Free vibration analysis of FGM framed nanostructures using variational-consistent boundary conditions, Vietnam J. Mech. 45 (2023) 145–163. DOI:



Research Article


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