The paper extends the formulation of a 2D geometrically exact beam element proposed in our previous paper [1] to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The paper develops the theoretical framework based on the Navier-Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, i.e., the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the appendix.

翻译：本文扩展了我们前一篇论文[1]中提议的 2D 精确度光束元素的公式, 以曲线弹性束为缩放范围。 该公式基于综合形态的平衡方程式, 加上将内部力量与部门变形变量联系起来的动态关系和部分方程式。 由此产生的一阶差异方程式被有限差异办法所近似, 边界值问题被用射击方法转换为初始值问题 。 该文件根据纳维尔- 伯诺利假设开发了理论框架, 并有可能扩展至尖状易软梁 。 详细介绍了用于评价等同节点力量和元素变色硬度的数值方程式 。 与标准的有限要素方程式配制不同, 目前的方法可以通过在元素级别上改进集成方案来提高准确性。 开发方案的效率和准确性被7个内部示例所记录, 包括圆形和parblic 箭头、 螺旋状状形, 以及带有正弦- 伸缩形 的正弦曲线和 曲线中心力的数值评价程序 。 与标准性要素组合的变变变变变的变法关系 。 所显示的内变的内变的变的变的变的变法是不为 。 。 。 任何变的变的变式的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变法是不为 。 。,, 变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变是的变的变的变的变的变的变的变的变的变的变的变的变的变的变是的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变