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

Wu, Zhangming and Raju, Gangadharan and Weaver, Paul M. (2013) A comparison of variational, differential quadrature, and approximate closed-form solution methods for buckling of highly flexurally anisotropic laminates. Journal of Engineering Mechanics, 139 (8). pp. 1073-1083. ISSN 0733-9399 (https://doi.org/10.1061/(ASCE)EM.1943-7889.0000468)

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Abstract

The buckling response of symmetric laminates that possess strong exural-twist coupling are studied using different methodologies. Such plates are dfficult to analyse due to localised 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 exural-twist anisotropy effect on buckling solutions. Specifically, the Hellinger-Reissner variational principle is used to study the effect of exural-twist coupling on buckling and also to explore the potential for developing closed form 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 Lagrangian multiplier theory and variational principle in evaluating buckling loads are discussed. In addition, a new simple closed form solution is shown for the case of a exurally anisotropic plate with three sides simply supported and one long edge free.

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