A Rigid-body-qualified Isogeometric Formulation For Buckling Analysis Of Spatial Kirchhoff Beams

Invariance of isogeometric formulations for three-dimensional Kirchhoff beams in the sense that the classical rigid-body modes associated to translations and infinitesimal rotations can be exactly represented gives them more physical meaning.Invariance preserves the equilibrium of forces and moments at fixed ends and improve the performances considerably for finite element size.We observe the standard isogeometric formulation is non-invariant for general beam geometry,except for circular and straight beams.Motivated by this,this paper presents an invariant isogeometric formulation for the geometric stiffness matrix of spatial curved Kirchhoff beams 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 beam 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 beam are considered,almost without any approximation to the centroidal axis except for omitting the higher order terms.The geometric stiffness matrix for internal moments is consistent with that of the geometrically exact beam 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.Lateral buckling of cantilevered circular arches under various boundary conditions and end moments is studied by using an analytical approach,which can be used as the benchmarks for calibration of the finite element approaches.By adopting the invariant isogeometric discretization,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.