Analysis of laminated plates using boundary element method
Formulations and implementations of the Boundary Element Method (BEM) for bending, membrane stress, buckling and post-buckling analyses of laminated plates are presented. Symmetrically laminated plates are assumed for which the bending-stretching coupling is absent. From the generalized Rayleigh-Green identity corresponding to the plate-bending problem, boundary integral equations are derived using the appropriate fundamental solution. Integral equations are transformed into a system of equations in matrix form by introducing boundary element interpolation models. Linear and quadratic discontinuous boundary elements are employed combined with special schemes for the approximation of jump terms at corners. Singular integrals over elements containing the source point are evaluated from closed-form expressions derived through analytical integration. Using the stress function concept, it is shown that the membrane stress analysis due to arbitrary in-plane loading is mathematically equivalent to plate bending problem. Based on this similarity a new boundary element formulation is developed for the prediction of membrane stresses in a laminated plate. The same fundamental solution, which was used for plate bending problem, is used with the replacement of flexural coefficients with extensional compliance coefficients. A new formulation for the buckling analysis, which is similar to that for the plate bending problem, leads to integral equations with an irreducible domain integral depending on the plate deflection. Boundary modelling is combined with deflection modelling over the plate so that three integral equations are approximated as a discrete system of equations forming an eigenvalue problem from which the critical load is evaluated. This approach removes the need for integral equations involving the domain curvatures yielding directly the buckling mode of the plate. Formulations of membrane stress and bending analyses are expanded by including the nonlinear terms arising from large deflections and combined for the development of an incremental algorithm predicting the post-buckling behaviour of laminate plate. The C codes implementing the solution algorithms are applied to several benchmark problems involving orthotropic and general anisotropic plates and BEM predictions are compared with solutions available from the literature or obtainable through a general-purpose finite element package.