Piecewise hierarchical p-version axisymmetric shell element for geometrically nonlinear behavior of laminated composites

Abstract

This paper presents a three node laminated composite axisymmetric finite element formulation for geometrically nonlinear (GNL) analysis where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed first by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacements at the interface between the laminas. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C 0 continuity of displacements across the interelement as well as interlamina boundaries. The individual lamina stiffness matrices and the equivalent nodal force vectors are derived using the principle of virtual work and the hierarchical displacement approximation for the laminas. Interlamina continuity conditions are used to construct the transformation matrices for the individual laminas. These matrices permit transformation of the lamina degrees of freedom to the laminate degrees of freedom. The interlamina behavior incorporated in this formulation is in total agreement with the physics of laminate behavior. In formulating the properties of the element, the complete axisymmetric state of stresses and strains is considered, hence the element is equally effective for very thin as well as extremely thick laminated shells. Incremental equations of equilibrium are derived and solved using standard Newton-Raphson method. The total load is divided in increments and for each increment of load equilibrium iterations are performed until each component of the residuals and the generalized nodal displacement vector are within preset tolerances. Numerical examples are presented to show the p-convergence characteristics, accuracy, and simplicity of modeling and overall advantages of the present formulation.