A two-scale model is derived from a fully resolved model where the response of concrete, steel reinforcement, and bond between them are considered. The pertinent "effective" large-scale problem is derived from selective homogenisation in terms of the equilibrium of reinforced concrete considered as a single-phase solid. Variational formulations of the representative volume element problem are established in terms of the subscale displacement fields for the plain concrete continuum and the reinforcement bars. Dirichlet and Neumann boundary conditions (BCs) are imposed on the concrete (pertaining to uniform boundary displacement and constant boundary traction, respectively) and on the reinforcement bars (pertaining to prescribed boundary displacement and vanishing sectional forces, respectively). Different representative volume element sizes and combinations of BCs were used in FE 2 analyses of a deep beam subjected to four-point bending. Results were compared with those of full resolution (single-scale). The most reliable response was obtained for the case of Dirichlet-Dirichlet BCs, with a good match between the models in terms of the deformed shape, force-deflection relation, and average strain. Even though the maximum crack widths were underestimated, the Dirichlet-Dirichlet combination provided an approximate upper bound on the structural stiffness.

A single-scale model for reinforced concrete, comprising the plain concrete continuum, reinforcement bars and the bond between them, is used as a basis for deriving a two-scale model. The large-scale problem, representing the "effective" reinforced concrete solid, is enriched by an effective reinforcement slip variable. The subscale problem on a Representative Volume Element (RVE) is defined by Dirichlet boundary conditions. The response of the RVEs of different sizes was investigated by means of pull-out tests. The resulting two-scale formulation was used in an FE 2 analysis of a deep beam. Load-deflection relations, crack widths, and strain fields were compared to those obtained from a single-scale analysis. Incorporating the independent macroscopic reinforcement slip variable resulted in a more pronounced localisation of the effective strain field. This produced a more accurate estimation of the crack widths than the two-scale formulation neglecting the effective reinforcement slip variable.

SUMMARY A two‐scale model for reinforced concrete, in which the large‐scale problem formulation is enriched by an effective reinforcement slip variable, is derived from the single‐scale model describing the response of plain concrete, reinforcement steel, and the bond between them. The subscale problem on the representative volume element (RVE) is correspondingly defined as finding the response of the RVE subjected to effective variables (strain, slip, and slip gradient) imposed from the large scale. A novel volumetric definition of effective reinforcement slip and its gradient is devised, and the corresponding subscale problem is formulated. The newly defined effective variables are imposed on the RVE in a weak sense using Lagrange multipliers. The response of the RVEs of different sizes was investigated by means of pull‐through tests, and the novel boundary condition type was used in FE2 analyses of a deep beam. Locally, prescribing the macroscopic reinforcement slip and its gradient in the proposed manner resulted in reduced RVE‐size dependency of effective work conjugates, which allows for more objective description of reinforcement slip in two‐scale modeling of reinforced concrete. Globally, this formulation produced more consistent amplitudes of effective slip fluctuations and more consistent maximum crack width predictions.

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