An Overview of Higher-Order Beam Elements Based on the Absolute Nodal Coordinate Formulation

Abstract

In this paper, beam elements with particular emphasis on higher-order elements based on the absolute nodal coordinate formulation (ANCF) are thoroughly investigated from the perspective of interpolation procedure and numerical performance. A straightforward and modularized procedure to construct the shape function is proposed. Based on the unified shape function formulation, the research examines how axial and transverse interpolation strategies impact element performance. Two beams in the pure bending scenario are analyzed. The comparison study reveals that higher-order interpolation in the axial and transverse directions is necessary to represent the highly curved deformation modes and alleviate Poisson locking. The Princeton beam and a thicker beam are then studied to assess the accuracy, convergence, and numerical stability of different beam elements. Conclusions are: (1) Higher-order beam elements are generally more accurate but converge more slowly. (2) To guarantee high accuracy, a complete set of transverse quadratic gradients must be adopted in the quadratic elements, and a higher-order transverse interpolation is necessary to capture the warping effect. (3) To avoid slow convergence, the axial order should not be lower than the transverse order. (4) Higher-order beam elements lead to a stiffness matrix with a larger condition number. With an inappropriate length to cross section ratio, the transverse cubic element results in an ill-conditioned stiffness matrix that brings numerical instability. (5) The numerical stability of higher-order beam elements are more sensitive to the length to cross section ratio of the meshed beam.