Multiscale Modelling of Reinforced Concrete
Licentiate thesis, 2018
Since concrete cracks at relatively low tensile stresses, the durability of reinforced concrete structures is highly influenced by its brittle nature. Cracks open up for ingress of harmful substances, e.g. chlorides, which in turn cause corrosion of the reinforcement. Crack widths are thus limited in the design codes, and accurate prediction methods are needed. For structures of more complex shapes, current computational methods for crack width predictions lack precision. Hence, the development of new simulation tools is of interest. In order to properly describe the crack growth in detail, cracking of concrete, constitutive behaviour of steel, and the bond between them must be accounted for. These physical phenomena take place at length scales smaller than the dimensions of large reinforced concrete structures. Thus, multiscale modelling methods can be employed to reinforced concrete. This thesis concerns multiscale modelling of reinforced concrete. More specifically, a two-scale model, based on Variationally Consistent Homogenisation (VCH), is developed. At the large-scale, homogenised (effective) reinforced concrete is considered, whereas the underlying subscale comprises plain concrete, resolved reinforcement bars, and the bond between the two. Each point at the large-scale is associated with a Representative Volume Element (RVE) defining the effective response through a pertinent boundary value problem. In a numerical framework, the procedure pertains to a so-called FE 2 (Finite Element squared) algorithm, where each integration point in the discretised large-scale problem inherits its response from an underlying RVE problem. In order to properly account for the concrete–reinforcement bond action, the large-scale problem is formulated in terms of a novel effective reinforcement slip variable in addition to homogenised displacements. In a series of FE 2 analyses of a plane problem pertaining to a reinforced concrete deep beam with distributed reinforcement layout, the influence of boundary conditions on the RVE, as well as the sizes of the RVE and the large-scale mesh, are studied. The results of the two-scale analyses with and without incorporation of the effective reinforcement slip are compared to fully-resolved (single-scale) analysis. A good agreement with the single-scale results in terms of structural behaviour, in particular load-deflection relation and average strain, is observed. Depending on the sub-scale boundary conditions, approximate upper and lower bounds on structural stiffness are obtained. The effective strain field gains a localised character upon incorporation of the effective reinforcement slip in the model, and the predictions of crack widths are improved. The two-scale model can thus describe the structural behaviour well, and shows potential in saving computational time in comparison to single-scale analyses.