A low-order, hexahedral finite element for modelling shells

A thin, eight-node, tri-linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co-ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in-plane linear variations of selected components of the transverse shear stress. Further, a plane-stress constitutive assumption is introduced, and an explicit treatment of the finite element’s thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip-loaded twisted beam, point-loaded hemisphere, point-loaded sphere, tip-loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations.