A two-dimensional shear deformable ANCF consistent rotation-based formulation beam element

Abstract

It has been demonstrated recently that the absolute nodal coordinate formulation (ANCF) can be used to develop lower-order consistent rotation-based formulations (CRBFs) that employ finite rotation parameters as nodal coordinates without the need for interpolating the rotation field. The objective of this study is to develop new planar shear deformable ANCF/CRBF beam elements and demonstrate their use. A cubic ANCF/CRBF shear deformable beam element is first developed starting with the ANCF kinematic description that employs position vector gradients as nodal coordinates. The transverse position vector gradients at the nodes are expressed in terms of finite rotation parameters, leading to a lower-dimensional beam element model that captures the shear deformation, ensures continuity of the stresses and rotations at the nodes, allows for an arbitrary large displacement, and has a kinematic description consistent with geometry methods and suitable for systematically modeling curved structures. The results, obtained using the new planar ANCF/CRBF shear deformable beam element, are compared with the original and more general ANCF shear deformable beam element. Another lower-dimension bilinear CRBF beam element which has three coordinates at each node, two translation coordinates and one rotation parameter, is also developed in this investigation. The formulations of the three finite elements, including the ANCF finite element, considered in this investigation are compared. Numerical results are presented in order to demonstrate the use of the new formulations and test their performances in the analysis of large displacements and deformations. While the ANCF/CRBF assumptions are evaluated numerically, the results obtained show, in general, a good agreement between the elements considered in this study. The results also show that the CRBF finite elements, which have nonlinear mass matrix, can be more efficient for smaller meshes. As the mesh size and the number of finite elements increase, the original higher-order ANCF finite elements, which have constant mass matrix and zero Coriolis and centrifugal forces, become more efficient. The ANCF/CRBF approach clearly demonstrates that there is no need for introducing an independent rotation field to capture the shear effect.