, Volume 49, Issue 8, pp 1795–1819 | Cite as

Nonlinear nonplanar vibration of a functionally graded box beam

  • E. C. Carvalho
  • P. B. Gonçalves
  • G. Rega
  • Z. J. G. N. Del Prado
Nonlinear Dynamics and Control of Composites for Smart Engi design


A functionally graded material (FGM) is a type of material designed to change continuously within the solid. It can be designed for specific applications such as thermal barrier coatings, corrosion protection, biomedical materials, space/aerospace industries, automotive applications, compliant mechanisms etc. In these applications, many primary and secondary structural elements can be idealized as beams. So, the aim of the present work is to study the nonlinear nonplanar vibration of a clamped-free slender box beam made of a FGM. More specifically, the cross section consisting of two isotropic materials, connected by a FG layer, is considered. To correctly describe the dynamic characteristics of the system, the nonlinear integro-differential equations used in this work, which consider the flexural–flexural–torsional couplings that occur in the nonplanar motions of the beam, include both geometric and inertial nonlinearities. In addition, the Galerkin method is applied to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge–Kutta method. A detailed parametric analysis using several tools of nonlinear dynamics, unveils the complex dynamics of the FG beam in the main resonance region. The FG beam displays a complex nonlinear dynamic behavior with several coexisting planar and nonplanar solutions, leading to an intricate bifurcation scenario. Special attention is given to the symmetry breaking of beam dynamics and its influence on the bifurcations and instabilities. The results show that even small variations in cross section and material gradation have profound influence on the bifurcation diagrams and the dynamic behavior of the structure.


Functionally graded material Box beam Flexural–flexural–torsional coupling Nonplanar vibration Bifurcation analysis Symmetry breaking 



The authors gratefully acknowledge the support of Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • E. C. Carvalho
    • 1
  • P. B. Gonçalves
    • 1
  • G. Rega
    • 2
  • Z. J. G. N. Del Prado
    • 3
  1. 1.Civil Engineering DepartmentPontifical Catholic University of Rio de JaneiroRio de JaneiroBrazil
  2. 2.Sapienza University of RomeRomeItaly
  3. 3.Federal University of GoiásGoiâniaBrazil

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