A series of locking-free beam element models in absolute nodal coordinate formulation

To alleviate the locking problem in the ANCF beam elements, sufficient transverse gradient vectors are incorporated in the cross section to enrich the distribution of transverse strain along the cross section of the beam. Building upon this novel concept, this paper utilizes Pascal trigonometric polynomial to determine the position interpolation field of beam elements, and the distribution of transverse gradient vectors along the beam section is clarified through the collocation of boundary points and Chebyshev interpolation nodes, and then a series of locking-free beam models, based on the absolute nodal coordinate formulation, are developed. Additionally, it reveals the inherent mechanical mechanism of higher-order beam models in alleviating locking through strict mathematical analysis. Furthermore, to demonstrate the effectiveness of the new elements, six numerical simulation examples are designed, namely, three static examples and three dynamic examples, which involve small deformation statics, large deformation statics, small-scale elastic deformation, large-scale elastic deformation problems. Finally, the simulation results of the first four order beam models, Patel–Shabana model, and ECM approach are compared and analyzed in detail. The results indicate that the proposed higher-order beam models have high accuracy and can effectively eliminate the unnecessary influence caused by locking in complex mechanical problems, involving statics and dynamics problems.