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The results for different unit cells are presented in Figs. However, it might not be feasible from the practical point of view to request output from all integration points especially for larger unit cells or denser macroscopic meshes. Since the location of the largest principal strain is not known a priori , it is more feasible, from the practical point of view, to find it from the large-scale results in a post-processing manner; the subscale analysis for a specific unit cell can then be easily repeated.
As shown in [ 32 ], the crack widths were underestimated for the DD boundary condition in the original version of the two-scale model.
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From Fig. Moreover, no significant difference could be observed between crack widths computed by the two-scale models. There, the evolving crack width computed by the two-scale model with macroscopic slip agreed much better with the fully-resolved solution, especially for denser macroscopic meshes. It is noteworthy, that for the prescribed strain-slip—slip gradient history, more cracks developed within the RVEs. This is believed to be caused by larger strains computed by the model, as a result of the localised character of the macroscopic strain field Fig.
Furthermore, it can be noticed that the fully-resolved model with perfect bond produced a smaller crack width. This confirms that the steel—concrete bond-slip has a large impact on local results, e. Influence of the combination of RVE size and the macroscopic mesh size on the results. In the presented results, we note that the size of the RVE and the resolution of the macroscale mesh have a strong effect on the results.
This is a result stemming from incomplete separation of scales; scale-mixing. This phenomenon was observed for transient problems in [ 31 ], where it was shown that for a given macroscale mesh size, there exists an optimal size of the RVE, which results in the best possible fit to the fully-resolved solution. Also in the present study, there is a strong coupling between the size of the RVE and the macroscale mesh resolution.
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In order to capture the fluctuating fields, a coarse large-scale mesh needs a larger RVE, where the fluctuations can develop. On the other hand, the absence of fully developed fluctuation field in a smaller RVE can be compensated by a fine macroscopic mesh, allowing for fluctuations on the higher scale, see Fig. Consequently, the definition of macroscopic slip changes for different sizes of the RVE. The purpose of a two-scale formulation is that the modelling error and the discretisation error should cancel out.
As illustrated in Fig. However, for coarser meshes often of engineering interest the discretisation error may cause a larger RVE to provide a better approximation. To summarise, incorporation of macroscopic reinforcement slip results in scale mixing; a combination of the macroscopic mesh size and the unit cell size must be considered in order to obtain the optimal result.
It is then assumed that the reinforcement slip, in addition to the concrete displacements, possesses both a macroscopic and a fluctuation component. Within the RVE, the standard linear variation of the macroscopic fields was considered, which corresponds to first-order computational homogenisation. A procedure for computing the effective work conjugates, i. The subscale response of the RVE subjected to macroscopic reinforcement slip was investigated in analyses of pull-out tests.
It was shown that for small RVEs, i. For larger RVEs, i. The incorporation of an independent macroscopic slip variable resulted in slightly more involved large-scale problem. The Dirichlet—Dirichlet DD boundary conditions constituted an approximate upper bound on the structural response, and the dependency of the result on the size of the macroscopic mesh was negligible. Even though the enrichment with macroscopic slip had little influence on the effective load-deflection relations for a reinforced concrete deep beam, it had a pronounced effect on the character of the macroscopic strain field.
With the macroscopic enrichment strain localisation was exhibited by the strain field to some extent. In turn, this localised character resulted in larger strains; these increased the evolving crack width significantly. Although still underestimated by the smallest RVE, the crack width obtained with larger RVEs agreed with the single-scale solution reasonably well. The macroscopic slip field demonstrated the significance of both the size of the unit cell and the size of the large-scale mesh, for the presented problem.
A coarse macroscopic mesh with a large RVE, and a fine macroscopic mesh with a small RVE demarcate the potential range, wherein the most meaningful results can be expected. It is noteworthy that the interpretation of macroscopic slip is governed by the size of the RVE. Apart from verification of the multiscale modelling methodology, the main benefit of multiscale formulations is the potential to save computational effort without compromising the quality of local results.
For larger reinforced concrete structures, like e.
For future work, additional types of boundary conditions could be considered for the RVE problem to ensure more uniform fluctuation of slip and bond stress along longer reinforcement bars. Additionally, three-dimensional reinforced concrete RVE should also be studied in order to be able to model a greater range of large reinforced concrete structures.
Here, we assume the discretisation errors inside the RVE problem to be negligible. Skip to main content Skip to sections. Advertisement Hide. Download PDF. A multiscale model for reinforced concrete with macroscopic variation of reinforcement slip. Open Access. First Online: 12 June The remainder of the paper is organised as follows: After some preliminaries concerning the fully-resolved problem in Sect. Section 4 presents the subscale response of a reinforced concrete RVE subjected to varying reinforcement slip.
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In Sect. The paper is concluded with Sect. Open image in new window. In order to state the variational format, 3 , 4 , and 6 must be recast into weak forms. In the two-scale model setting, the local field is replaced by the homogenised field, i.
Splitting the test functions in Eq. Following the VMS ansatz , i. Table 1 Geometry of the unit cells. For the bulk of the RVE, bilinear quadrilateral elements were used to model concrete of grade C30 [ 12 ]. The Mazars model [ 22 , 23 ] is a commonly used isotropic continuum damage formulation for concrete. Furthermore, the interface constraint in Eq. Note, that in this example similar material properties for steel, concrete and the interface were chosen for all elements.
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However, they could all be varying, i. Table 2 Material parameters for the concrete, steel, and the interface. The same cannot be said about the larger RVEs, where a greater macroscopic slip was needed to be prescribed to extract the full bond-slip behaviour up to the plateau. The reason is that longer reinforcement bars did not have a constant distribution of bond stresses along the bar.
The slip and bond stress distribution along the bars in all RVEs are shown in Fig. It is noteworthy, that prescribing the macroscopic slip via Dirichlet boundary conditions, i.
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Both the amplitude and fluctuation of the slip are highly dependent on the size of the RVE which translates directly to the length of the rebar. Therefore, the interpretation of the macroscopic slip fields depends on the RVE size. A practical implication of this is that subscale results must always be consulted in order to get the correct interpretation of the large-scale results.
On the other hand, it could be argued that a better way of imposing the reinforcement slip would be to do it in an average sense, through some integrated measure along the bar. Just as for the single-scale analysis, only half of the beam was modelled, and both the support and loading platens were simulated by tying the nodes appropriately. In view of the fact that the beam was uniformly reinforced, a single type of a RVE was enough to represent the subscale composition. At the symmetry line, the degrees of freedom corresponding to the horizontal displacement and reinforcement slip were locked.
Modelling methods and material models described in Sect. A typical two-scale model is schematically shown in Fig. Furthermore, to study the sensitivity of the model to macroscopic mesh size, five different meshes were considered, see Fig. As previously alluded to in Sect. Furthermore, the size of the macroscopic mesh can also be expected to play a significant role in the analysis. Such curves were plotted for different sizes of the unit cells and different macroscopic meshes in Figs.