Invariant isogeometric formulation for the geometric stiffness matrix of spatial curved Kirchhoff rods

This paper presents an invariant isogeometric formulation for the geometric stiffness matrix of spatial curved Kirchhoff rods considering various end moments, i.e., the internal (member) moments and applied (conservative) moments. There are two levels of rigid-body qualification, one is on the buckling theory of the rod itself and the other on the isogeometric formulation for discretization. Both will be illustrated. Based on the updated Lagrangian formulation of three-dimensional continua, the rotational effect of end moments is naturally included in the external virtual work done by end tractions without introducing any definition of finite rotations. Both the geometric torsion and curvatures of the rod are considered closely for the centroidal axis, except with the omission of higher order terms. The geometric stiffness matrix for internal moments is consistent with that of the geometrically exact rod model with its rigid-body quality demonstrated. For structures rigorously defined for the deformed state, the geometric stiffness matrix after global assembly is always symmetric, for both the internal and external moments. By adopting the invariant isogeometric discretization following our previous work, a series of numerical examples, including the cases of external conservative moments, angled joint and complicated spatial geometry, were solved for buckling analysis, by which the reliability of the geometric stiffness matrix derived is verified via comparison with the analytical or straight beam solutions. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper presents an invariant isogeometric formulation for the geometric stiffness matrix of spatial curved Kirchhoff rods, considering various end moments. The study includes discussions on the buckling theory of the rod itself, the isogeometric formulation for discretization, and the rotational effect of end moments. The reliability of the geometric stiffness matrix derived is verified through numerical examples and comparison with analytical solutions.