Approximate 3-D model for analysis of laminated plates with arbitrary lay-ups, loading and boundary conditions

Available exact solution techniques of elasto-static problems entail limitations on the choice of lay-ups, loading and boundary conditions and impose restrictions on strain and stress fields as well, to overcome algebraic difficulties inherent to modeling of laminated and sandwich composites. Therefore in fact they become unsuitable for testing accuracy of modern laminated plate theories aiming to very accurately describing 3-D stress fields in real conditions of use of multilayered composites, nowadays widespread in engineering applications. To overcome the assumption of too restrictive hypotheses, an approximate 3-D solution technique is proposed and assessed that is able to automatically solve problems which due to the lay-ups, loading and boundary conditions assumed would not be solved with the exact techniques available. A quite general, accurate structural model is developed that comes to constitute a generalization of available physically-based zig-zag theories, being free from through-thickness assumptions and because zig-zag functions are not explicitly contained, the layerwise contributions being represented by the redefinition of coefficients of the through-thickness series expansion. It is based solely on the prescriptions of the theory of elasticity, i.e., displacement and stress compatibility at interfaces, fulfillment of local equilibrium equations at points across the thickness and of stress boundary constraints. A truncated expansion series of trial functions and unknown amplitudes is used to represent variables, whose coefficients are determined in exact form using a symbolic calculus tool that enforces all elasticity constraints and in conjunction with Rayleigh-Ritz and Lagrange multipliers methods.