Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

J. Awrejcewicz, A. V. Krysko, S. P. Pavlov, M. V. Zhigalov, V. A. Krysko

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge-Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the sizedependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

Original languageEnglish
Article number041018
JournalJournal of Computational and Nonlinear Dynamics
Issue number4
Publication statusPublished - 1 Jul 2017


  • functionally graded material
  • Lyapunov spectrum
  • modified couple stress theory
  • nonlinear curvilinear Timoshenko beam
  • setup/relaxation method
  • stability loss

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Mechanical Engineering
  • Applied Mathematics

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