This work develops a simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod. The cross-section undergoes a rigid body motion. A rotation tensor with the Rodrigues formula is used to describe the rotation, which makes the updating of the rotational variables very simple. A formula for the Rodrigues parameters in function of the displacements derivative and the torsion angle is for the first time settled down. The consistent connection between elements is thoroughly discussed, and an appropriate approach is developed. Cubic Hermitian interpolation for the displacements together with linear Lagrange interpolation for the torsion incremental angle were employed within the usual Finite Element Method, leading to adequate C 1 continuity. A set of numerical benchmark examples illustrates the usefulness of the formulation and numerical implementation.

中文翻译：

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用于伯努利-欧拉杆几何精确分析的简单有限元

这项工作为伯努利-欧拉杆的几何精确分析开发了一个简单的有限元。没有考虑横向剪切变形。定义了能量共轭横截面应力和应变。假定杆为直线参考配置。横截面进行刚体运动。使用罗德里格斯公式的旋转张量来描述旋转，这使得旋转变量的更新变得非常简单。首次确定了作为位移导数和扭转角函数的罗德里格斯参数公式。彻底讨论了元素之间的一致联系，并制定了适当的方法。位移的三次厄米插值以及扭转增量角的线性拉格朗日插值在通常的有限元方法中使用，导致足够的C 1 连续性。一组数值基准示例说明了公式和数值实现的有用性。