The proposed two-dimensional geometrically exact beam element extends our previous work by including the effects of shear distortion, and also of distributed forces and moments acting along the beam. The general flexibility-based formulation exploits the kinematic equations combined with the inverted sectional equations and the integrated form of equilibrium equations. The resulting set of three first-order differential equations is discretized by finite differences and the boundary value problem is converted into an initial value problem using the shooting method. Due to the special structure of the governing equations, the scheme remains explicit even though the first derivatives are approximated by central differences, leading to high accuracy. The main advantage of the adopted approach is that the error can be efficiently reduced by refining the computational grid used for finite differences at the element level while keeping the number of global degrees of freedom low. The efficiency is also increased by dealing directly with the global centerline coordinates and sectional inclination with respect to global axes as the primary unknowns at the element level, thereby avoiding transformations between local and global coordinates. Two formulations of the sectional equations, referred to as the Reissner and Ziegler models, are presented and compared. In particular, stability of an axially loaded beam/column is investigated and the connections to the Haringx and Engesser stability theories are discussed. Both approaches are tested in a series of numerical examples, which illustrate (i) high accuracy with quadratic convergence when the spatial discretization is refined, (ii) easy modeling of variable stiffness along the element (such as rigid joint offsets), (iii) efficient and accurate characterization of the buckling and post-buckling behavior.

翻译：本文提出的二维几何精确梁单元扩展了我们先前的工作，通过纳入剪切变形效应以及沿梁分布的力和力矩的作用。该基于柔度的一般性公式利用了运动学方程、反演截面方程以及积分形式的平衡方程。所得三个一阶微分方程组通过有限差分法进行离散化，并利用打靶法将边值问题转化为初值问题。由于控制方程的特殊结构，即使一阶导数采用中心差分近似，该格式仍保持显式，从而获得高精度。所采用方法的主要优势在于，通过在单元层面细化用于有限差分的计算网格，同时保持全局自由度数量较低，可以高效地减少误差。通过在单元层面直接处理全局中心线坐标和截面相对于全局轴的倾角作为主要未知量，从而避免了局部与全局坐标之间的转换，也提高了效率。本文提出并比较了两种截面方程公式，分别称为Reissner模型和Ziegler模型。特别地，研究了轴向加载梁/柱的稳定性，并讨论了其与Haringx和Engesser稳定性理论的联系。两种方法均通过一系列数值算例进行了测试，这些算例说明了：(i) 空间离散细化时具有二次收敛的高精度，(ii) 易于模拟沿单元变化的刚度（如刚性节点偏移），(iii) 高效且准确地表征屈曲和后屈曲行为。