Finite rotation quadrilateral element for multi-layer beams

Abstract

SUMMARYThe paper presents the ÿnite rotations beam equations derived on use of the generalized Reissner hypothesis with a scalar parameter for the transverse extension. The beam strain and change of curvature measures are obtained from the right stretch strain, and the virtual work is given for Biot-type stress and couple resultants. The strain energy for the ÿrst-order isotropic elastic material is assumed in terms of the right stretch strain, and constitutive equations for the beam stress and couple resultants are derived.Two ÿnite rotation elements are developed from the derived beam equations: a beam element with the transverse stretch and a quadrilateral element. First, the beam element with the uniformly under-integrated tangent operator is developed. Next, the formula linking the middle-line variables and the interface variables of the beam is introduced consistently with the generalized Reissner kinematics. Linearization of this formula is performed, and the derived tangent operator is used to convert the two-node beam element to a four-node quadrilateral.Both the ÿnite elements have been tested on several numerical examples, some of highly non-linear characteristics, and their accuracy is very good. It has been established that the quadrilateral element, which is intended for applications to multi-layer beams, performs very well for high elemental aspect ratios, and can therefore be applied to modelling of very thin layers.