Based on the continuum mechanics principles, a rigorous formulation is presented for the linearized stiffness equation of three-dimensional beam elements with account taken of the joint moment equilibrium in the deformed configuration $C_2$. By sticking to the Bernoulli–Euler hypothesis of plane sections and elasticity definitions for stress resultants, the bending moments and torque of the element are shown to be quasi- and semi-tangential, respectively, in the updated Lagrangian formulation. Further, by invoking the moment equilibrium conditions for structural nodes at $C_2$, the induced moment matrix that first appears to be antisymmetric on the element level turns out to be symmetric upon assembly of all elements on the structural level. The joint equilibrium conditions at $C_2$, as represented by the induced moment matrix, are central not only to the out-of-plane buckling analysis of angled frames, but also to the simulation of curved beams by the straight-beam elements. Examples on the buckling of angled frames and curved beams are provided to support the theory presented.

基于连续力学原理，考虑了变形状态下的弯矩平衡，给出了三维梁单元线性刚度方程的严格表示。 $C_2$。通过坚持平面截面的伯努利-欧拉假设和应力合成的弹性定义，在更新的拉格朗日公式中，单元的弯矩和扭矩分别显示为准切线和半切线。此外，通过调用结构节点的弯矩平衡条件$C_2$，最初在单元层上看起来是反对称的感应矩矩阵在结构层上所有单元的组装上都是对称的。的联合平衡条件$C_2$如感应矩矩阵所表示的，不仅对有角度框架的平面外屈曲分析至关重要，而且对于直光束元素对弯曲光束的仿真也很重要。提供了有关倾斜框架和弯曲梁屈曲的示例，以支持所提出的理论。