A co-rotational formulation for arbitrarily shaped planar beams under large displacements and rotations

Abstract

A planar co-rotational beam formulation is proposed for the large displacement and large rotation analysis of beams with arbitrary initial geometries. A local frame is attached to each differential beam segment and follows its rigid body translation and rotation, enabling a consistent kinematic description suitable for higher-order basis functions. An auxiliary straight reference configuration is introduced so that both the arbitrary initial configuration and the current deformed configuration can be described relative to a common reference, enabling arbitrary initial geometries to be treated within the co-rotational framework. While the framework is general and applicable to a wide range of higher-order basis functions, Non-Uniform Rational B-Splines (NURBS) basis functions are adopted in this study to exploit their superior geometric representation and smoothness. The performance of the proposed formulation is evaluated through five numerical examples, including static, post-buckling, dynamic, and coupled static-dynamic cases. The results demonstrate excellent convergence, accurate prediction of large-deformation responses, effective mitigation of locking without additional treatment, and robust performance for beams with non-uniform cross-sections and spatially varying curvature.