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A numerical method is developed for the analysis of general quadrilateral, moderately thick orthotropic plates having arbitrary boundary conditions. The procedure is based on the application of the Rayleigh–Ritz method in conjunction with the Reissner–Mindlin thick plate theory. A set of complete polynomials in terms of natural co-ordinates comprising of boundary and domain terms, which satisfy the boundary conditions, is deployed as the basic functions to approximate the real displacement field. The generalized displacements and the eigenvalues are determined by imposing the principle of stationary potential energy. Although the procedure has the ability to solve problems involving thick quadrilateral plates, the numerical examples presented are mostly for skew plates. The results herein are compared with their counterparts determined by other investigators. Copyright © 2002 John Wiley & Sons, Ltd.

In a companion paper, the effects of approximations in the flexural-torsional stability analysis of beams was studied, and it was shown that a second-order rotation matrix was sufficiently accurate for a flexural-torsional stability analysis. However, the second-order rotation matrix is not necessarily accurate in formulating finite element model for a 3-D non-linear analysis of thin-walled beams of open cross-section. The approximations in the second-order rotation matrix may introduce ‘self-straining’ due to superimposed rigid-body motions, which may lead to physically incorrect predictions of the 3-D non-linear behaviour of beams. In a 3-D non-linear elastic–plastic analysis, numerical integration over the cross-section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross-section that is not consistent with the strain distributions may lead to incorrect predictions of the 3-D non-linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3-D non-linear analysis of beams. It is found that a finite element model for 3-D non-linear analysis based on the second-order rotation matrix leads to over-stiff predictions of the flexural-torsional buckling and postbuckling response and to an overestimate of the maximum load-carrying capacities of beams in some cases. To perform a correct 3-D non-linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross-section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3-D non-linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.

In formulating a finite element model for the flexural–torsional stability and 3-D non-linear analyses of thin-walled beams, a rotation matrix is usually used to obtain the non-linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin-walled beams of open cross-section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin-walled beams, while a companion paper investigates the effects of approximations in the 3-D non-linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin-walled beams, modification of the model is needed, or a finite element model based on a second-order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.

Unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction. For plate structures, this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core. This paper investigates the use of simple and orthogonal polynomials in the Rayleigh–Ritz method for unilateral plate buckling. The orthogonal polynomials used are the classical Chebyshev types 1 and 2, Legrende, Hermite and Laguerre. The study presents a comparison between the efficiency of the polynomial-based displacement functions with regard to elastic bilateral and unilateral plate buckling, where efficiency is measured as a function of their convergence characteristics. Some buckling solutions for plates with varying boundary conditions and in-plane shear loads are also provided as an illustration. Copyright © 1999 John Wiley & Sons, Ltd.