Boundary element modelling of non-linear buckling for symmetrically laminated plates

Abstract

The non-linear buckling of composite laminates, triggered by geometric imperfections, is here analysed adopting a boundary element methodology. The non-linear theory for thin anisotropic plates couples in-plane forces causing buckling with the consequent bending deformation. The adopted formulation for in-plane forces in terms of the stress function is mathematically identical to that for the bending problem, thus boundary integral equations and fundamental solutions of the same form are used. Differential equations governing increments of the stress function and the deflection are obtained; the resulting integral equations include irreducible domain integrals depending on powers or products of the second derivatives of the stress function and the deflection. The latter are calculated through complementary integral equations obtained from the original ones by differentiating the field variables in the domain. The solution process accounts for the domain integrals through an iterative scheme; thus there is no need for modelling any field variables in the plate domain although domain meshing is necessary for performing numerical integrations. The boundary is meshed into quadratic discontinuous elements while the domain is divided into triangular cells with linear discontinuous variation of the relevant field variables for integration purposes. The analysis is implemented through a suit of C codes and applied to a rectangular symmetrically laminated plate. Its predictions are compared with published results obtained by other methods. The accuracy and limitations of the formulation are discussed and alternative approaches for expanding its scope pointed out.

Stability analysis for laminates with general anisotropy using boundary elements

“…This approach was complemented with a mathematically similar procedure for the in-plane extension formulated in terms of the stress function; this generated accurate buckling load predictions even in cases of non-uniform in-plane loading [9]. The same solution methodology was extended to non-linear buckling through an incremental and iterative procedure [10]. A BEM solution to the buckling of general, unbalanced laminates was recently developed in terms of the stress function and transverse deflection using for both variables the same fundamental solution associated with the forth order differential operator governing anisotropic plate flexure [11].…”

A boundary element approach to buckling of general laminates

“…Initial BEM work on buckling was restricted to orthotropic plates [7,8]. Linear and nonlinear buckling analyses of anisotropic balanced laminates under any in-plane force distribution were subsequently developed and implemented [9,10]. Fundamental solutions have recently been obtained and boundary integral equations formulated for the coupled extension-flexure of general laminates [11,12].…”