Two-scale models for reinforced concrete, where the large-scale problems are defined in terms of Euler–Bernoulli beam and Kirchhoff–Love plate models, are constructed. The subscale problem on the Representative Volume Element (RVE) is correspondingly outlined as finding the response of the three-dimensional RVE comprising plain concrete continuum, reinforcement bars and the bond between them. The boundary region of the periodic mesh is modelled with special solid elements, which allow for prescribing the macroscopic input via strongly periodic boundary conditions in an effective way. The effective response of the reinforced concrete RVEs of different sizes subjected to tension and pure bending is investigated for both effective beam and plate models. A series of experiments on reinforced concrete panels subjected to bending and membrane loads is simulated, and the effective moment–curvature response is studied. Within the developed framework, an arbitrary macroscopic loading in terms of membrane strains and curvatures can be prescribed on the RVE, and the corresponding effective response is obtained, making the proposed formulation feasible for future use in an FE2 scheme.

中文翻译：

---

将三维钢筋混凝土代表性体积元素放大到有效的梁和板模型

构建了钢筋混凝土的两尺度模型，其中根据 Euler-Bernoulli 梁和 Kirchhoff-Love 板模型定义了大规模问题。代表性体积元素 (RVE) 上的子尺度问题相应地概述为寻找三维 RVE 的响应，包括素混凝土连续体、钢筋和它们之间的结合。周期性网格的边界区域使用特殊的实体元素建模，这允许以有效的方式通过强周期性边界条件规定宏观输入。对于有效梁和板模型，研究了不同尺寸的钢筋混凝土 RVE 在受拉和纯弯曲时的有效响应。对受弯曲和膜荷载作用的钢筋混凝土板进行了一系列试验，并研究了有效弯矩-曲率响应。在开发的框架内，可以在 RVE 上规定任意宏观载荷的膜应变和曲率，并获得相应的有效响应，使所提出的公式在未来用于 FE2 方案中可行。