An efficient through-thickness integration scheme in an unlimited layer doubly curved isoparametric composite shell element

This paper presents an efficient numerical integration scheme for evaluating the matrices (stiffness, mass, stress-stiffness and thermal load) for a doubly curved, multilayered, composite, quadrilateral shell finite element. The element formulation is based on three-dimensional continuum mechanics theory and it is applicable to the analysis of thin and moderately thick composite shells. The conventional formulation requires a 2 × 2 × 2 or 2 × 2 × 1 Gauss integration per layer for the calculation of element matrices. This method becomes uneconomical when a large number of layers is used owing to an excessive amount of computations. The present formulation is based on explicit separation of the thickness variable from the shell surface parallel variables. With the through-thickness variables separated, they are combined with the thickness dependent material properties and integrated separately. The element matrices are computed using the integrated material matrices and only a 2 × 2 spatial Gauss integration scheme. The response results using the present formulation are identical to those obtained using the conventional formulation. For a small number of layers, the present method requires slightly more CPU time. However, for a larger number of layers, numerical data are presented to demonstrate that the present formulation is an order-of-magnitude economical compared to the conventional scheme.