A Novel Strain-Continuous Finite Element Formulation for Geometrically Exact Thin-Walled Beam on Lie Algebra

Abstract

Most existing geometrically exact thin-walled beam formulations on Lie group SE(3)$\times$$ℝ$ considering warping are $C$${\hspace{0.17em}}^0$-continuous. In this study, a novel strain-continuous element for geometrically exact thin-walled beam on Lie algebra $s$$e$(3)$\times$$ℝ$ is originally proposed, in which the torsion-related Wagner effect and warping are considered in the constitutive relations. The proposed beam element is not only locking-free intrinsically, but also is $C$${\hspace{0.17em}}^1$-continuous by using the proposed geometrical Hermite interpolation on Lie algebra. This interpolation based on Magnus expansion can simplify the description of interpolated element stress tensor and strain energy. Then, the simplified discrete strain operator can be obtained by interpolating the virtual twist vector totally independently of the beam element formulation, which can avoid calculating complex matrix and further reduce geometric nonlinearity greatly. Furthermore, the computational complexity of the discrete equilibrium equation and the corresponding Jacobi matrix for static problems can be reduced. Finally, five static examples without topological loops are taken to validate the fourth-order convergence rate, strain-continuity, path independence, and singularity-free properties of the proposed beam element formulation.