Geometrically nonlinear analysis of composite beams based on global–local superposition

Abstract

A composite beam finite element is designed to capture through-thickness effects, specifically normal stress and strain and transverse shear, in the context of geometrically nonlinear analyses. The starting point for the formulation is a similar element already proposed for linear analyzes based on a global–local superposition approach, where local functions are defined in each layer of the laminate, and global functions are defined along the thickness. The consistency of the kinematic hypotheses is guaranteed by imposing the continuity equations of displacements through the thickness, the force balance equations along the thickness, directly or indirectly, by imposing the continuity of transverse stresses, and by applying the boundary conditions on the lower and upper surfaces of the elements. In the context of nonlinear analyzes, the imposition of continuity of displacements is straightforward. However, the continuity of the transverse stresses needs to be carefully imposed, as the relevant stresses are the second order Piola-Kirchhoff stresses and the strains are the Green-Lagrange strains, consistent with the total Lagrangian approach used. The constitutive equations are written in incremental form and a detailed analysis is conducted to ensure that the stresses and strains involved are physically consistent across the different reference frames employed. In order to assess the accuracy of the numerical model implemented, a unique semi-analytical technique is developed to obtain the response of asymmetrical laminated beams under compression.