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.

本文针对空间弯曲基尔霍夫杆的几何刚度矩阵，提出了一种不变的等几何公式，其中考虑了各种端力矩，即内部（构件）力矩和应用（保守）力矩。刚体鉴定有两个级别，一个是关于杆本身的屈曲理论，另一个是关于离散化的等几何公式。两者都将说明。基于更新的三维连续性的拉格朗日公式，在不引入有限旋转的任何定义的情况下，通过外部引力自然将端点力矩的旋转效应包括在外部虚拟功中。杆的几何扭转和曲率均以质心轴为准，除非省略了高阶项。内部力矩的几何刚度矩阵与几何精确的杆模型一致，并证明了其刚体质量。对于为变形状态严格定义的结构，整体组装后的几何刚度矩阵对于内部和外部力矩始终是对称的。通过在我们之前的工作中采用不变等几何离散化，解决了一系列数值示例，包括外部保守矩，斜角节点和复杂空间几何的情况，以进行屈曲分析，从而验证了导出的几何刚度矩阵的可靠性通过与解析或直光束解决方案进行比较。对于为变形状态严格定义的结构，整体组装后的几何刚度矩阵对于内部和外部力矩始终是对称的。通过在我们之前的工作中采用不变等几何离散化，解决了一系列数值示例，包括外部保守矩，斜角节点和复杂空间几何的情况，以进行屈曲分析，从而验证了导出的几何刚度矩阵的可靠性通过与解析或直光束解决方案进行比较。对于为变形状态严格定义的结构，整体组装后的几何刚度矩阵对于内部和外部力矩始终是对称的。通过在我们之前的工作中采用不变等几何离散化，解决了一系列数值示例，包括外部保守矩，斜角节点和复杂空间几何的情况，以进行屈曲分析，从而验证了导出的几何刚度矩阵的可靠性通过与解析或直光束解决方案进行比较。