Abstract This paper presents a finite element formulation of curved thin beams, useful for modelling structures made of filiform elements. The proposed element is intended to model structures formed by several wires, subjected to very large bending displacements so that their final shapes can be completely different from the original ones. The model is based on the description of the planar wire geometry through the integration of the radius of curvature, which is approximated by means of a cubic polynomial. The solution of an overdetermined system is necessary to compute the coefficients of the polynomial. This approach allows determining the stiffness matrix of the curved wire in closed form, through the application of Castigliano's Theorem. A technique for automatic remeshing during large deformations, based on the curvature change, is also discussed in the paper. To validate the model refined finite element analyses and an experimental test have been carried out. The solution is performed analytically, and it allows to identify the actual stiffness matrix of a curved wire, considering only the degrees of freedom at the ends.

中文翻译：

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由丝状梁制成的结构建模：导线弯曲有限元的开发

摘要 本文提出了弯曲细梁的有限元公式，可用于对由丝状单元制成的结构进行建模。提议的元素旨在模拟由几根线形成的结构，承受非常大的弯曲位移，因此它们的最终形状可以与原始形状完全不同。该模型基于通过对曲率半径进行积分来描述平面线几何形状，该曲率半径通过三次多项式近似。超定系统的解对于计算多项式的系数是必要的。这种方法允许通过应用 Castigliano 定理来确定闭合形式的弯曲线的刚度矩阵。一种基于曲率变化在大变形期间自动重新网格划分的技术，论文中也有讨论。为了验证模型的精炼有限元分析和实验测试已经进行。该解决方案是通过分析执行的，它允许识别弯曲线的实际刚度矩阵，仅考虑末端的自由度。