Comparison of variational, differential quadrature, and approximate closed-form solution methods for buckling of highly flexurally anisotropic laminates

Zhangming Wu, Gangadharan Raju, Paul M. Weaver*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

17 Citations (Scopus)


The buckling response of symmetric laminates that possess strong flexural-twist coupling is studied using different methodologies. Such plates are difficult to analyze because of localized gradients in the mode shape. Initially, the energy method (Rayleigh-Ritz) using Legendre polynomials is employed, and the difficulty of achieving reliable solutions for some extreme cases is discussed. To overcome the convergence problems, the concept of Lagrangian multiplier is introduced into the Rayleigh-Ritz formulation. The Lagrangian multiplier approach is able to provide the upper and lower bounds of critical buckling load results. In addition, mixed variational principles are used to gain a better understanding of the mechanics behind the strong flexural-twist anisotropy effect on buckling solutions. Specifically, the Hellinger-Reissner variational principle is used to study the effect of flexural-twist coupling on buckling and also to explore the potential for developing closedform solutions for these problems. Finally, solutions using the differential quadrature method are obtained. Numerical results of buckling coefficients for highly anisotropic plates with different boundary conditions are studied using the proposed approaches and compared with finite-element results. The advantages of both the Lagrangian multiplier theory and the variational principle in evaluating buckling loads are discussed. In addition, a new simple closed-form solution is shown for the case of a flexurally anisotropic plate with three sides simply supported and one long edge free. © 2013 American Society of Civil Engineers.

Original languageEnglish
Pages (from-to)1073-1083
Number of pages11
JournalJournal of Engineering Mechanics
Issue number8
Publication statusPublished - 15 Aug 2013


  • Buckling
  • Flexural-twist coupling
  • Lagrangian multiplier
  • Hellinger-Reissner variational principle
  • Differential quadrature method

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