Efficient modeling and order reduction of new 3D beam elements with warping via absolute nodal coordinate formulation

To describe the particular mechanical behaviors of beams with both uniform and non-uniform cross sections, such as the bidirectional bending, torsion-bending coupling, the torsion-related warping, the cross-sectional stretch, and Wagner effects, a series of efficient higher-order beam elements (HOBEs) is proposed in the frame of the absolute nodal coordinate formulation (ANCF). In the proposed HOBEs, a new mixed kinematic description of beam elements is introduced via the warping functions and slope vectors. Compared with the existing HOBEs using Lagrange polynomials, the additional degrees of freedom per element proposed to accurately describe the warping deformation are dramatically reduced. Moreover, the tremendous Von-Mises stress on the cross sections in the existing HOBEs does not occur in the proposed new HOBEs. Compared with the classical nonlinear finite elements formulations, the complete 3D strain state with the higher-order terms allows the cross-sectional stretch and avoids the expensive calculations of the extra warping and Wagner strain measures and their derivatives. Moreover, the transverse integration allows an arbitrary section shape to vary along the beam axial direction. Thus, these new HOBEs benefit from the efficient warping description in the classical FE and inherit the advantages of 3D-continuum theory in the ANCF. In addition, the shear locking is alleviated due to the ability to capture the non-uniform distribution of shear stress, and the Poisson locking is addressed via the enhanced continuum mechanics approach. Finally, the proposed HOBEs are validated and compared using statics and dynamics undergoing complex significant deformations on various benchmarks, FEs, commercial codes, and experimental data.