Two‐scale finite element modelling of reinforced concrete structures: Effective response and subscale fracture development

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 FE² 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.