A multiscale model for reinforced concrete with macroscopic variation of reinforcement slip

Abstract

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.

On a volume averaged measure of macroscopic reinforcement slip in two‐scale modeling of reinforced concrete

“…Second, the large‐scale mesh dependence of the global response was rather low. Moreover, as the macroscopic strain was prescribed on the RVEs using classical Dirichlet boundary condition, the structural stiffness (and load‐deflection response) was overestimated, which was already observed by the authors in References and . As bond‐slip mechanism is very important only locally (eg, for cracking), it could be expected that its influence on the global structural response will be marginal.…”

“…Because the arrangement of reinforcement in the deep beam corresponded to the one used in RVEs described in Section 4, they were directly used to represent the substructure of the material. It is noteworthy that not only the size of the RVE, but also the size of the macroscopic mesh has influence on the results as shown by the authors in Reference . To this end, five different large‐scale meshes were constructed.…”

“…The so‐called pull‐through tests are structural simulations performed on the RVEs, where the macroscopic slip is gradually increased, so that the reinforcement bars are eventually pulled out from the concrete. Such tests were already performed by the authors in Reference , with the effective reinforcement slip being prescribed at the RVE boundary, but nothing about the subscale variation was specified. It is noteworthy that such formulation did not require any volumetric/subscale definition of the effective variable.…”

“…The mid‐span deflection was measured at the top of the beam, according to Figure , while the external force was computed as the reaction force at the loading platen. For convenience, a result for a single‐scale analysis in full resolution is also presented, cf Reference for more information about the single‐scale model. From the graphs, a few things can be concluded.…”

“…There are many different multiscale methods suitable for this task. Many of them have already been studied and used in modeling of plain concrete and even reinforced concrete . One method that shows potential is the FE 2 method, which couples the scales in a nested way, that is, the macroscopic response is obtained by computational homogenisation performed on the fine‐scale representative volume elements (RVEs), cf Figure .…”