Finite element analysis of smooth, folded and multi-shell structures

Abstract

The paper is concerned with the nonlinear theory and finite element analysis of shell structures with an arbitrary geometry, loading and boundary conditions. A complete set of shell field equations and side conditions (boundary and jump conditions) is derived from the basic laws of continuum mechanics. The developed shell theory includes the so-called drilling couples as well as the drilling rotation. It is shown that this property is crucial in the analysis of irregular shell structures, such as those containing folds, branches, column supports, stiffeners, etc. The relevant variational principles with relaxed regularity requirements are also presented. These principles provide the mathematical basis for the formulation of various classes of shell finite elements. The developed finite elements include a displacement/rotation based Lagrange family, a stress resultant based mixed and a semi-mixed family as well as so-called assumed strain elements. All elements have six degrees of freedom at each node, three translational and three rotational ones, including the drilling rotation formulated on the foundation of an exact (in defined sense) shell theory. As such, they are equally applicable to smooth as well as to irregular shell structures. The general applicability of the developed elements is illustrated through an extensive numerical analysis of the representative test examples. In order to obtain a still deeper insight into the problem a Lagrange family of standard degenerated shell elements with five degrees of freedom per node and an element with six degrees of freedom per node based on the von Karman plate theory are considered as well. The presented numerical results include complex plate and doubly-curved shell structures. Linear and non-linear solutions with a pre- and post-buckling analysis are discussed.