A Unified Corotational Framework for 2D Beams: Bridging the Crisfield and Krenk Formulations

Abstract

This paper presentes a unified corotational framework for 2D beam elements that achieves a theoretical synthesis between the seminal formulations of Crisfield and Krenk. A central contribution of this research is the analytical demonstration that these two traditionally distinct approaches converge to an identical tangent stiffness matrix, offering a unified perspective on objective corotational kinematics. The framework is systematically applied to three fundamental kinematic theories: Euler-Bernoulli, Timoshenko, and shallow arch Euler-Bernoulli, demonstrating its versatility across different levels of structural refinement. By decomposing the motion into a global rigid-body component and a local deformational part, characterized by three degrees of freedom, the formulation effectively isolates the nonlinearities of the coordinate transformation. A robust numerical strategy utilizing a modulo function is implemented to resolve the ±2π periodicity singularity, ensuring stable convergence during extreme rigid-body rotations. Internal forces and tangent stiffness matrices are derived via the principle of virtual work, providing a transparent and modular implementation for each beam theory. The accuracy and computational efficiency of the unified formulation are validated through a series of challenging numerical benchmarks involving the geometrically nonlinear analysis of beams, frames, and arches, featuring complex equilibrium paths with limit points and multiple snap-back loops. Comprehensive mesh convergence studies confirm that the shallow arch Euler-Bernoulli element, incorporating explicit axial-bending coupling, achieves superior accuracy and convergence efficiency in problems involving significant geometric nonlinearity.