Multiscale Modelling of Reinforced Concrete Structures

Concrete cracks at relatively low tensile stresses; cracks open up for ingress of harmful substances, negatively affecting the durability of reinforced concrete structures. Crack widths are thus limited in the design codes, and accurate predictions are needed, especially for large reinforced concrete structures such as bridges or nuclear reactor containment buildings. On the one hand, cracking of concrete, constitutive behaviour of steel, and the bond between them must be accounted for in order to properly describe crack growth. On the other hand, explicitly resolving these features in large structures could prove computationally intractable.This thesis concerns multiscale modelling of reinforced concrete structures. More specifically, different two-scale models, based on Variationally Consistent Homogenisation (VCH), are developed. In these models, the response of a Representative Volume Element (RVE) is upscaled to a few popular structural models: a homogenised solid in plane stress, the effective Euler-Bernoulli beam and the effective Kirchhoff-Love plate. The effective response of the RVE is defined through a boundary value problem, for which different types of boundary conditions are developed and discussed. Furthermore, in order to allow for reinforcement slip transfer across the large-scale elements, a novel macroscopic reinforcement slip field is introduced.The developed two-scale models are used to analyse reinforced concrete deep beams subjected to membrane loads, reinforced concrete beams subjected to uniaxial tension and bending, and reinforced concrete panels subjected to combinations of membrane and bending loads. The results show that the general structural behaviour is reflected well by the multiscale models compared to single-scale analyses.By enriching the model with a macroscopic reinforcement slip field prescribed at the boundary of the RVE, the crack width predictions given by the two-scale models are improved and localisation of effective strain is observed at the large-scale. However, the results were dependent on the large-scale mesh and RVE sizes. In order to improve the objectivity of the model, a novel boundary condition type, prescribing the effective slip in the volume of the RVE, was developed. The macroscopic reinforcement slip became no longer RVE-size dependent, and the maximum crack width predictions were more consistent and showed a smaller variance for different large-scale meshes and sizes of RVEs.In conclusion, the developed two-scale models allow for the analysis of a wide range of reinforced concrete structures, and show potential in saving computational time in comparison to single-scale analyses

Two-scale modelling of reinforced concrete deep beams: Choice of unit cell and comparison with single-scale modelling

Two-scale and single-scale models are used to analyse the response of reinforced concrete deep beams with different reinforcement layouts. To this end, a novel approach of modelling non-uniformly reinforced structures in a multiscale manner is developed. Parameterised generation of suitable unit cells is described, and the subdivision of problem domain into regions with different substructures is presented. Three different reinforced concrete deep beams with available experimental data are analysed. Mid-span deflections are slightly underestimated by both models, while the maximum load is captured reasonably well

Experiments and calibration of a bond-slip relation and efficiency factors for textile reinforcement in concrete

Textile reinforcement yarns consist of many filaments, which can slip relative each other. At modelling of the global structural behaviour, interfilament slip in the yarns, and slip between the yarns and the concrete can be considered by efficiency factors for the stiffness and strength of the yarns, and by applying a bond-slip relation between yarns and concrete. In this work, an effective and robust method for calibration of such models was developed. Two-sided asymmetrical pull-out tests were carried out, with varying embedment lengths designed to obtain both pull-out and rupture of the textile as failure mode. The efficiency factors for strength and stiffness of the textile were very similar, 34% and 35% respectively. This indicates the stress distribution within a yarn to be uneven in a similar manner for small and large stress levels, and that interfilament slip has a larger influence than variation of filaments’ strength

Multiscale Modelling of Reinforced Concrete

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

Upscaling of three-dimensional reinforced concrete representative volume elements to effective beam and plate models

Two-scale models for reinforced concrete, where the large-scale problems are defined in terms of Euler-Bernoulli beam and Kirchhoff-Love plate models, are constructed. The subscale problem on the Representative Volume Element (RVE) is correspondingly outlined as finding the response of the three-dimensional RVE comprising plain concrete continuum, reinforcement bars and the bond between them. The boundary region of the periodic mesh is modelled with special solid elements, which allow for prescribing the macroscopic input via strongly periodic boundary conditions in an effective way. The effective response of the reinforced concrete RVEs of different sizes subjected to tension and pure bending is investigated for both effective beam and plate models. A series of experiments on reinforced concrete panels subjected to bending and membrane loads is simulated, and the effective moment–curvature response is studied. Within the developed framework, an arbitrary macroscopic loading in terms of membrane strains and curvatures can be prescribed on the RVE, and the corresponding effective response is obtained, making the proposed formulation feasible for future use in an FE2 scheme

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

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, as well as 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.\ua0 A novel volumetric definition of effective reinforcement slip and its gradient is devised, and the corresponding subscale problem is formulated.\ua0 The newly-defined effective variables are imposed on the RVE in a weak sense via 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 FE^2 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 modelling of reinforced concrete. Globally, this formulation produced more consistent amplitudes of effective slip fluctuations, as well as more consistent maximum crack width predictions

On Periodic Boundary Conditions in Variationally Consistent Homogenisation of Beams and Plates

A computationally efficient strategy to prescribe periodic boundary conditions on three- dimensional Representative Volume Elements (RVEs) is outlined. In particular, the cases of having an Euler-Bernoulli beam and a Kirchhff-Love plate problem at the macroscale are considered within a computational homogenisation framework. Special solid elements for the boundary region of the periodic mesh have been developed, in which some of the degrees of freedom depend on those of their periodic counterparts, the macroscopic data and the size of the RVE

On Periodic Boundary Conditions in Variationally Consistent Homogenisation of Beams and Plates

A computationally efficient strategy to prescribe periodic boundary conditions on three- dimensional Representative Volume Elements (RVEs) is outlined. In particular, the cases of having an Euler-Bernoulli beam and a Kirchhff-Love plate problem at the macroscale are considered within a computational homogenisation framework. Special solid elements for the boundary region of the periodic mesh have been developed, in which some of the degrees of freedom depend on those of their periodic counterparts, the macroscopic data and the size of the RVE

On Periodic Boundary Conditions in Variationally Consistent Homogenisation of Beams and Plates

A computationally efficient strategy to prescribe periodic boundary conditions on three- dimensional Representative Volume Elements (RVEs) is outlined. In particular, the cases of having an Euler-Bernoulli beam and a Kirchhff-Love plate problem at the macroscale are considered within a computational homogenisation framework. Special solid elements for the boundary region of the periodic mesh have been developed, in which some of the degrees of freedom depend on those of their periodic counterparts, the macroscopic data and the size of the RVE

On the micro-to-macro transition of reinforcement slip in two-scale modelling

A two-scale model for reinforced concrete, in which the macroscopic problem formulation is enriched by an effective reinforcement slip variable is considered. The corresponding subscale problem on the Representative Volume Element (RVE) is defined in terms of finding the response of the RVE subjected to effective variables (strain, slip, slip gradient) imposed from the macroscale. In this contribution, the two possible approaches of prescribing the effective reinforcement slip are discussed. Namely, a boundary definition of the macroscopic slip can be employed and the variable is thus prescribed only at boundary of the RVE, which corresponds to Dirichlet boundary conditions. Alternatively, a volumetric averaging measure can be used to define the effective reinforcement slip. In this case, the effective variables are imposed on the RVE in a weak sense via Lagrange multipliers. It is shown that the weak enforcement of reinforcement slip and its gradient resulted in objective interpretation of the effective variable (and its work conjugates), which was not pathologically dependent on the size of the RVE