TY - JOUR
T1 - Comparison of variational, differential quadrature, and approximate closed-form solution methods for buckling of highly flexurally anisotropic laminates
AU - Wu, Zhangming
AU - Raju, Gangadharan
AU - Weaver, Paul M.
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Buckling
KW - Differential quadrature method
KW - Flexural-twist coupling
KW - Hellinger-Reissner variational principle
KW - Lagrangian multiplier
UR - http://www.scopus.com/inward/record.url?scp=84881351891&partnerID=8YFLogxK
U2 - 10.1061/(ASCE)EM.1943-7889.0000468
DO - 10.1061/(ASCE)EM.1943-7889.0000468
M3 - Article
AN - SCOPUS:84881351891
SN - 0733-9399
VL - 139
SP - 1073
EP - 1083
JO - Journal of Engineering Mechanics
JF - Journal of Engineering Mechanics
IS - 8
ER -