Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM , exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.

中文翻译：

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bROM：通过模型降阶对梁理论的扩展

梁分析在整个现代历史的结构设计中发挥着至关重要的作用。经典梁理论依赖于某些先验运动学假设下的分析推导（并且计算成本低且准确），但它们不适用于正交各向异性材料和任意横截面等。另一方面，通过有限元 (FE) 方法求解的完整 3D 线弹性理论为各种工程问题提供了解决方案，但代价是较高的计算成本。降阶模型试图通过假设解以低维基础表示来提供与 3D 线性弹性一样通用但与经典梁理论一样高效的理论。在文献中提供的几个选项中，通过经验体积法 (ECM) 进行域分解和超还原（我们将其命名为 ddROM）展现了这一引人注目的功能。然而，应用于梁结构的 ddROM 可以为每个节点提供比大多数仅考虑横截面刚体运动的标准商业 FE 代码更多的自由度 (DoF)。本文提出了一种称为梁降阶模型 (bROM) 的方法，通过压缩过程和回归过程将 ddROM 集成到标准有限元代码中。Condensation 给出了 FE 商业规范所需的给定长度的超级单元的刚度矩阵。这是通过组装多个元件并找到所施加的位移和反应（拉格朗日乘数）之间的关系来实现的。然后使用压缩过程创建一个刚度矩阵数据库，该数据库是执行回归的梁长度的函数。首先给出了一些数值示例，以在各向同性梁的情况下对照经典梁理论验证所提出的方法，最后，两个正交各向异性梁与有限元相比，显示了该方法的实用性。