Exact solution and stability of postbuckling configurations of beams

Ali H. Nayfeh, Samir A. Emam

Research output: Contribution to journalArticlepeer-review

241 Citations (Scopus)


We present an exact solution for the postbuckling configurations of beams with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed-fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.

Original languageEnglish
Pages (from-to)395-408
Number of pages14
JournalNonlinear Dynamics
Issue number4
Publication statusPublished - Dec 2008


  • Buckled beams
  • Exact solution
  • Free vibration
  • Stability

ASJC Scopus subject areas

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering


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