## Abstract

Chapter 1

Chapter l contains the introduction to this thesis. The scope of the thesis is partly to investigate different numerical and analytical models based on fracture mechanical ideas, which are able to predict size effects, and partly to perform an experimental investigation on high-strength concrete.

Chapter 2

A description of the factors which influence the strength and cracking of concrete and high strength concrete is made. Then basic linear fracture mechanics is outlined followed by a description and evaluation of the models used to describe concrete fracture in tension. The chapter ends with a description of the different types of size effects. Three examples which discuss the two terms 'size effect' and 'brittleness' and the importance of a stiff test rig. Finally some brittleness numbers are defined.

Chapter 3

In chapter 3 the most well-known numerical methods which use the fictitious crack to describe fracture in concrete are presented. Two of the methods are combined into a power method which is stable for all brittleness numbers and which is able of calculating the entire load-displacement curve even for very ductile beams. This method is used extensively in the rest of the thesis.

Chapter 4

Since analytical methods are very time consuming different analytical models have been developed. Three methods for plain concrete are presented, where one of the methods is developed by the author. The method is based on three different fracture models. Also two models applicable for lightly reinforced concrete are presented.

Chapter S

An experimental investigation is performed in a closed-loop testing setup where high-strength concrete in three-point bending is tested. In order to determine the material parameters in the fictitious crack model an estimation procedure based on the solving of an optimization problem is developed. The material parameters and the load-displacement curves obtained by using this procedure are presented. Different size effects are observed, and it is concluded that edge effects play a dominant role. Dyeing experiments were performed on ordinary specimens and specimens which are saw-cut out.

General Conclusions.

One of the major problems solved in this thesis is the stability problem described in chapter 3. The direct sub-structure method is thereby complete, and is a strong tool, when analyzing structures of quasi brittle materials. The only limit of the method is now the number of nodes used in the midsection. When a fictitious crack develops, the normalized size of this zone is dependent on the brittleness of the structure. The more brittle the structure the smaller the fracture process zone, and consequently many nodes are necessary for describing this zone.

The analytical methods described in chapter 4 show very different results. The analytical method developed by the author, R. Brincker and S. Krenk, seems to be the most promising. This method is capable of predicting the size effect on the modulus of rupture. Furthermore it is based on basic principles and the goveming equations are explicit and simple. These properties of the model make it a very powerful tool, which is applicable for the designing engineer. The method is also extended to reinforced concrete, where the results look very promising.

The large experimental investigation on high-strength concrete seems to have been too small However, some general results were obtained. It is observed that the decrease in the modulus of rupture with the increase of beam depth is of the same order as that of normal strength concrete. The fracture toughness is increasing with the beam depth, making LEFM inadequate for high-strength, even though high-strength is considered to be a brittle material. The data fitting performed by finding the minimum of three different function which describes the difference between a numerical and an experimental load displacement curve turned out to be very time consuming. If the constitutive parameters are wanted for high strength concrete this method is then the only applicable. The method is, however, not recommended.

Chapter l contains the introduction to this thesis. The scope of the thesis is partly to investigate different numerical and analytical models based on fracture mechanical ideas, which are able to predict size effects, and partly to perform an experimental investigation on high-strength concrete.

Chapter 2

A description of the factors which influence the strength and cracking of concrete and high strength concrete is made. Then basic linear fracture mechanics is outlined followed by a description and evaluation of the models used to describe concrete fracture in tension. The chapter ends with a description of the different types of size effects. Three examples which discuss the two terms 'size effect' and 'brittleness' and the importance of a stiff test rig. Finally some brittleness numbers are defined.

Chapter 3

In chapter 3 the most well-known numerical methods which use the fictitious crack to describe fracture in concrete are presented. Two of the methods are combined into a power method which is stable for all brittleness numbers and which is able of calculating the entire load-displacement curve even for very ductile beams. This method is used extensively in the rest of the thesis.

Chapter 4

Since analytical methods are very time consuming different analytical models have been developed. Three methods for plain concrete are presented, where one of the methods is developed by the author. The method is based on three different fracture models. Also two models applicable for lightly reinforced concrete are presented.

Chapter S

An experimental investigation is performed in a closed-loop testing setup where high-strength concrete in three-point bending is tested. In order to determine the material parameters in the fictitious crack model an estimation procedure based on the solving of an optimization problem is developed. The material parameters and the load-displacement curves obtained by using this procedure are presented. Different size effects are observed, and it is concluded that edge effects play a dominant role. Dyeing experiments were performed on ordinary specimens and specimens which are saw-cut out.

General Conclusions.

One of the major problems solved in this thesis is the stability problem described in chapter 3. The direct sub-structure method is thereby complete, and is a strong tool, when analyzing structures of quasi brittle materials. The only limit of the method is now the number of nodes used in the midsection. When a fictitious crack develops, the normalized size of this zone is dependent on the brittleness of the structure. The more brittle the structure the smaller the fracture process zone, and consequently many nodes are necessary for describing this zone.

The analytical methods described in chapter 4 show very different results. The analytical method developed by the author, R. Brincker and S. Krenk, seems to be the most promising. This method is capable of predicting the size effect on the modulus of rupture. Furthermore it is based on basic principles and the goveming equations are explicit and simple. These properties of the model make it a very powerful tool, which is applicable for the designing engineer. The method is also extended to reinforced concrete, where the results look very promising.

The large experimental investigation on high-strength concrete seems to have been too small However, some general results were obtained. It is observed that the decrease in the modulus of rupture with the increase of beam depth is of the same order as that of normal strength concrete. The fracture toughness is increasing with the beam depth, making LEFM inadequate for high-strength, even though high-strength is considered to be a brittle material. The data fitting performed by finding the minimum of three different function which describes the difference between a numerical and an experimental load displacement curve turned out to be very time consuming. If the constitutive parameters are wanted for high strength concrete this method is then the only applicable. The method is, however, not recommended.

Original language | English |
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Place of Publication | Aalborg |
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Publisher | Dept. of Building Technology and Structural Engineering |

Number of pages | 158 |

Publication status | Published - 1992 |

## Keywords

- Concrete
- Fracture mechanics
- Numerical modeling