Dynamic behavior of nonuniform functionally graded Euler-Bernoulli beams under multiple moving forces

Le Thi Ha, Nguyen Dinh Kien, Vu Tuan Anh
Author affiliations


  • Le Thi Ha Hanoi University of Transport and Communications, Vietnam
  • Nguyen Dinh Kien Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
  • Vu Tuan Anh Hanoi University of Science and Technology, Vietnam




functionally graded beam, physically neutral surface, moving force, dynamic behavior, finite element method


The dynamic behavior of nonuniform Euler-Bernoulli beams made of transversely functionally graded material under multiple moving forces is studied by the finite element method. The beam cross-section is assumed to vary in the width direction by two different types. A simple finite element formulation, accounting for variation of the material properties through the beam thickness  and the shift in the physically neutral surface, is derived and employed in the study. The exact variation of the cross-sectional profile is employed in evaluation of the element stiffness and mass matrices. The dynamic response of the beam is computed with the aid of the implicit Newmark method. The numerical results show that the derived finite element formulation is capable to assess accurately the dynamic characteristics of the beam by using just several elements. The effect of the moving speed, material inhomogeneity and section profile on the dynamic behavior of the beams is investigated. The influence of the distance between the forces as well as the number of forces on the dynamic response is also examined and highlighted.


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How to Cite

L. T. Ha, N. D. Kien and V. T. Anh, Dynamic behavior of nonuniform functionally graded Euler-Bernoulli beams under multiple moving forces, Vietnam J. Mech. 37 (2015) 151–168. DOI: https://doi.org/10.15625/0866-7136/37/3/4075.



Research Article

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