Piecewise hierarchical p-version curved shell element for geometrically nonlinear behavior of laminated composite plates and shells

Abstract

This paper presents a nine-node three-dimensional laminated composite curved shell finite element formulation for geometrically nonlinear (GNL) analysis of laminated plates and shells 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 interfaces 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 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 which 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 for composite plates and shells. In formulating the properties of the element, complete three-dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick laminated shells and plates. Incremental equations of equilibrium are derived and solved using standard Newton's 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 accuracy, p-convergence characteristics and overall advantages of the present formulation.