Approximation of the critical buckling factor for

Authors: Dimitri Bettebghor Nathalie Bartoli

Publish Date: 2012/03/15

Volume: 46, Issue: 4, Pages: 561-584

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Abstract

This article is concerned with the approximation of the critical buckling factor for thin composite plates A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions This method allows accurate approximation of the critical buckling factor for nonorthotropic laminates under complex combined loadings including shear loading The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior eg concavity over tensor D or outofplane lamination parameters Moreover the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case Based on the numerically observed behavior a new scheme for the approximation is applied that separates each buckling mode and builds linear polynomial or rational regressions for each mode Results of this approach and applications to structural optimization are presentedThe authors are deeply grateful to the reviewers and Pr Nagendra Somanath for their meticulous reading about the manuscript making several useful remarks They would like to also thank Pr A Henrot and Pr Z Gürdal for their useful comments and discussions regarding buckling issuesWe briefly recall here the Rayleigh–Ritz method for buckling computations Note that Rayleigh–Ritz methods are also widely used in structural dynamics vibrations quantum chemistry The main idea is to write the solution of a partial differential equation system or eigenproblem as a linear combination of test functions Unlike finite element methods the test functions are defined over the whole domain and satisfy boundary conditions These basis functions are usually based on a closedform solutions of a close unperturbed problem This means that Rayleigh–Ritz methods are used when solving a pde or eigenproblem close in some sense simple geometry simple differential operator to a simple one and are not as general as finite element methods They are also close to spectral methods in the sense the test functions are defined over the whole domain and their convergence properties are usually quite good whenever the problem remains close to an original simple problem In addition to that as in spectral methods the discretization step leads to dense matrices and high accuracy can be achieved with much less degreesoffreedom than in finite elements methodsIn Fig 6 we see that some values of N x N y N xy do not give rise to buckling In both plots we observe a ’hole’ for which buckling analysis gives either a negative critical buckling factor or an infinite value This is related to the assumption that the bilinear form b N has at least one positive direction ie a displacement w such that b N w w  0 This assumption ensures that there exists at least one eigenvalue λ  0 the positive part of the spectrum is not empty In the literature concerned with linear buckling in the frame of threedimensional linear elastiticy Dauge and Suri 2006 Szabó and Királyfalvi 1999 this condition is written over the stress tensor and is referred to as nonnegativity property The aim of that section is to show that this assumption naturally explains the ’nonbuckling’ area in the loading conditions space

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