Doctoral thesis, 2020

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.

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.

cracking

computational homogenisation

multiscale

bond-slip

reinforced concrete

Chalmers, Architecture and Civil Engineering, Structural Engineering

International Journal for Numerical Methods in Engineering,; Vol. 114(2018)p. 1074-1102

**Journal article**

Computational Mechanics,; Vol. 63(2019)p. 139-158

**Journal article**

International Journal for Numerical Methods in Engineering,; Vol. 121(2020)p. 1822-1846

**Journal article**

Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications,; (2019)p. 251-256

**Paper in proceedings**

Even though the actual physical phenomena involved in the cracking process are quite complicated, models exist which can give us accurate predictions. These models simulate what's happening to the material when forces, like e.g., gravity or traffic loads, act on it. In practice, we create computer models of the engineering structures we want to analyse. To facilitate the computations, the computer model is divided into small pieces called finite elements. Cracks can have lengths in the order of decimetres, and are thus much smaller than the structure, which usually ranges from tens to hundreds of metres. In terms of crack modelling, this means that the finite elements must also be very small, which results in very large computer models requiring a long time to produce results. Fortunately, there exist multiscale modelling techniques, which are able to provide detailed small-scale results even if the structure is modelled with fairly large finite elements.

In this thesis, steps are taken to extend the existing multiscale modelling techniques to reinforced concrete structures. This way, detailed results such as crack widths and patterns can be obtained even for very large structures such as bridges or nuclear reactor containment buildings. More specifically, this is achieved by analysing the material response at different length scales, and connecting these scales to each other in an appropriate way. Additionally, thanks to parallel computing, the methods proposed in this thesis can potentially shorten the time it takes to analyse reinforced concrete structures with computer models.

Swedish Research Council (VR), 2015-01-01 -- 2018-12-31.

Swedish Research Council (VR), 2019-01-01 -- 2023-12-31.

Applied Mechanics

Infrastructure Engineering

Other Civil Engineering

Building Technologies

Building Futures (2010-2018)

Materials Science

C3SE (Chalmers Centre for Computational Science and Engineering)

978-91-7905-238-6

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4705

Chalmers University of Technology

VDL, Chalmers Tvärgata 4C

Opponent: Adnan Ibrahimbegovic, University of Tehcnology of Compiegne/Sorbonne University