In this paper, an instability analysis of three-dimensional (3D) beams supported with 3D hinges under axial and torsional loadings is presented in the large displacement and rotation regime. An exact displacement field is proposed based on the central line and orientation of the cross section consisting of nine parameters corresponding to 3D centroid movements and rotations. The Cauchy–Green deformation tensor is derived in the local coordinate system according to the proposed displacement field. The deformation tensor and a normal-shear constitutive model with highly polynomial nonlinearity are developed based on continuum mechanics. A finite element formulation is then established based on the higher-order shape functions to avoid shear and membrane locking issues. The elemental governing equations of equilibrium as well as 3D nodal forces and moments are obtained using the Hamiltonian principle. To solve the final nonlinear equilibrium equations, Newton–Raphson and Riks techniques by an incremental-iterative scheme are implemented. The numerical results are presented to assess instability behaviors of beams with different cross sections and various 3D boundary conditions. The effects of 3D hinge joints on the stability of beams under axial and torsional loadings are studied for the first time. The numerical results reveal instability in bending and lateral-torsional buckling for beams supported by 3D hinge joints. This phenomenon is proved by both the finite strain model and its linearization for small deformations. The numerical results show that the present finite element formulation is robust, reliable as well as simple and easy to model instability of 3D beams in the large displacement regime.

在本文中，在大位移和旋转状态下，对在轴向和扭转载荷下由3D铰链支撑的三维（3D）梁的不稳定性进行了分析。根据中心线和横截面的方向（由对应于3D重心运动和旋转的9个参数组成），提出了一个精确的位移场。根据提议的位移场，在局部坐标系中导出了柯西-格林变形张量。基于连续力学建立了变形张量和具有高多项式非线性的正剪本构模型。然后基于高阶形状函数建立有限元公式，以避免剪切和膜锁定问题。使用汉密尔顿原理获得平衡的基本控制方程以及3D节点力和力矩。为了解决最终的非线性平衡方程，通过增量迭代方案实施了牛顿-拉夫森和里克斯技术。给出了数值结果，以评估具有不同横截面和各种3D边界条件的梁的不稳定性行为。首次研究了3D铰链节点对梁在轴向和扭转载荷下的稳定性的影响。数值结果表明，在由3D铰链关节支撑的梁的弯曲和横向扭转屈曲中，存在不稳定性。有限应变模型及其对小变形的线性化都证明了这一现象。数值结果表明，目前的有限元公式是鲁棒的，