The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant, which leads to high computational efficiency. The element has been implemented into an open-source finite element code. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

中文翻译：

---

几何非线性梁单元的有效有限差分公式

这篇文章的重点是伯努利梁单元的二维几何非线性公式，它可以容纳任意大的横截面旋转。该公式基于平衡方程的积分形式，结合运动学方程和广义材料方程，得到一组三个一阶微分方程。然后通过有限差分将这些方程离散化，并使用一种受射击方法启发的技术将边界值问题转换为初始值问题。通过在单元级细化积分方案，同时保持全局自由度的数量不变，可以方便地提高数值近似的精度，从而提高计算效率。该元素已实现为开源有限元代码。数值例子显示了与在有限应变框架中制定的标准梁单元和解析解的有利比较。