Abstract The analysis of three-dimensional (3D) stress states can be complex and computationally expensive, especially when large deflections cause a nonlinear structural response. Slender structures are conventionally modelled as one-dimensional beams but even these rather simpler analyses can become complicated, e.g. for variable cross-sections and planforms (i.e. non-prismatic curved beams). In this paper, we present an alternative procedure based on the recently developed Unified Formulation in which the kinematic description of a beam builds upon two shape functions, one for the beam’s axis, the other for its cross-section. This approach predicts 3D displacement and stress fields accurately and is computationally efficient in comparison with 3D finite elements. However, current modelling capabilities are limited to the use of prismatic elements. As a means for further applicability, we propose a method to create beam elements with variable planform and variable cross-section, i.e. of general shape. This method employs an additional set of shape functions which describes the geometry of the structure exactly. These functions are different from those used for describing the kinematics and provide local curvilinear basis vectors upon which 3D Jacobian transformation matrices are produced to define non-prismatic elements. The model proposed is benchmarked against 3D finite element analyses, as well as analytical and experimental results available in the literature. Significant computational efficiency gains over 3D finite elements are observed for similar levels of accuracy, for both linear and geometrically nonlinear analyses.

中文翻译：

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具有一般非棱柱形弯曲几何形状的梁状结构的有效建模

摘要 三维 (3D) 应力状态的分析可能很复杂且计算量很大，特别是当大挠度导致非线性结构响应时。细长结构通常被建模为一维梁，但即使是这些相当简单的分析也会变得复杂，例如对于可变截面和平面（即非棱柱形弯曲梁）。在本文中，我们提出了一种基于最近开发的统一公式的替代程序，其中梁的运动学描述建立在两个形状函数上，一个用于梁的轴，另一个用于其横截面。这种方法可以准确地预测 3D 位移和应力场，并且与 3D 有限元相比计算效率更高。然而，当前的建模能力仅限于使用棱柱元素。作为进一步适用的手段，我们提出了一种方法来创建具有可变平面和可变横截面，即具有一般形状的梁单元。这种方法使用了一组额外的形状函数，它准确地描述了结构的几何形状。这些函数不同于用于描述运动学的函数，并提供局部曲线基向量，在此基础上生成 3D 雅可比变换矩阵以定义非棱镜元素。所提出的模型以 3D 有限元分析以及文献中可用的分析和实验结果为基准。对于类似的精度水平，观察到比 3D 有限元显着的计算效率提升，