# FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk

^{1}

^{2}

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## Abstract

**:**

## Featured Application

**Gradient Material Properties in Mass Concrete to increase Durability.**

## Abstract

## 1. Introduction

- How does stress and tensile strength develop in gradient mass concrete?
- Is there a positive effect on the crack risk of concrete when using gradient concrete instead of homogeneous concrete?

## 2. Finite Element Model and Material Properties Used in Analysis (Materials and Methods)

#### 2.1. Finite Element Model

#### 2.1.1. Geometry and Elements

#### 2.1.2. Thermal Boundaries

^{2}/K [34]. The heat transfer coefficient is changed at the time of removal of the formwork to 25.5 W/m

^{2}/K [34] to account for the changing in loss of thermal energy. The ambient temperature with day and night fluctuations was simulated using a sinus function. The mean value is assumed to be 20 °C and the amplitude is 10 °C.

#### 2.1.3. Mechanical Boundaries

#### 2.1.4. Temperature Evolution (Thermo-Chemical Model)

- q … vector of local heat flux density, J s
^{−1}m^{−2} - λ … thermal conductivity, J s
^{−1}m^{−1}K^{−1} - ∇T … temperature gradient, K m
^{−1}

^{2}T) + Q/dt = ρ·c·dT/dt

- λ … thermal conductivity, J s
^{−1}m^{−1}K^{−1} - T … temperature, K
- Q … thermal energy released in hydration, J m
^{−3} - dt … time increment, s
- ρ … density, kg m
^{−3} - c … specific heat capacity, J kg
^{−1}K^{−1}

_{eff}= ∫ exp(A/R·[1/293 − 1/(273 + T(t))])·dt

- t
_{eff}… effective reaction time, s - A … temperature depended activation energy, J mol
^{−1} - R … gas constant, J mol
^{−1}K^{−1} - dt … time increment, s

_{A}; E

_{A}+ 1470·(20 − T(t))}

_{A}: base value of activation energy, J mol

^{−1}

_{∞}·c·exp(−(τ

_{e}/t

_{eff})

^{α})·(τ

_{e}/t

_{eff})

^{(α−1)})/t

_{eff}

- Q
_{∞}… total hydration energy of the cement, J m^{−3} - c … amount of cement in the mix, kg m
^{−3} - τ
_{e}… fitting parameter, s - α: fitting parameter, -

#### 2.2. Material Properties

#### 2.2.1. Material Distributions

#### 2.2.2. Time Dependency of the Material Properties

- Heat generation
- Thermal conductivity
- Young’s modulus of the concrete
- Poisson ratio
- Concrete compressive strength
- Tensile strength

#### 2.2.3. Requirements for Durable Strong Surface Layer Concrete

^{3}and the maximum w/c-ratio is 0.5.

_{ca}

- J … flux of CO
_{2}into the concrete, kg m^{−2}s^{−1} - D … diffusion coefficient (CO
_{2}), m^{2}s^{−1} - Δc … difference in CO
_{2}content between the concrete surface and the moving boundary, kg(CO_{2}) m^{−3} - x
_{ca}… distance of the moving carbonation boundary from the concrete surface, m

_{2}, as shown in Equation (7):

_{ca}/dt

- J … flux of CO
_{2}into the concrete, kg m^{−2}s^{−1} - a … CO
_{2}binding capacity of the concrete, g(CO_{2}) m^{−3} - t … time, s

_{ca}is assumed zero at time zero. Combining Equations (6) and (7) and integrating over time, the following solution is obtained, see Equation (8):

_{ca}= (2D·Δc·t/a)

^{0.5}

_{ca}= k

_{NAC}·t

^{0.5}

- t … concrete age, a
- k
_{NAC}… carbonation rate (2D Δc/a)^{0.5}, mm a^{−}^{0.5}

_{2}concentration, curing, wetting effects) additional factors are applied in [4,5,6].

_{NAC}= 2.5 mm/year

^{0.5}and a carbonation rate for CEM III/B of k

_{NAC}= 4.5 mm/year

^{0.5}. Additionally, an execution tolerance of Δc = 10 mm, according to [4,5,6] and standard test conditions (k

_{e}= k

_{a}= k

_{c}= W(t) = 1) for carbonation rate were assumed. The following equation (10) was used:

_{nom}= Δc + γ

_{f}·k

_{NAC}·t

^{0.5}·(k

_{e}·k

_{c}·k

_{a})·W(t)

- c
_{nom}… nominal concrete cover, mm - Δc … permitted tolerance, mm
- k
_{NAC}… carbonation rate under natural conditions (std. test conditions), mm year^{−0.5} - t … service life, a
- k
_{e}… factor to account for relative humidity - k
_{a}… factor to account for CO_{2}concentration - k
_{c}… factor to account for curing/execution - W(t) … factor to account for wetting events
- γ
_{f}… partial safety factor

#### 2.2.4. Requirements for Low Hydration Energy Core Concrete

#### 2.2.5. Used Material Properties

_{eff}) = E

_{∞}·exp( −τ/(max(t

_{eff}− t

_{solidify}; E

_{∞}·10

^{−10}))

^{α}

- E(t
_{eff}) … elastic modulus depending on effective time, N/mm^{2} - E
_{∞}… elastic modulus at end of hydration, N/mm^{2} - τ … fitting parameter, h
- α … fitting parameter, -
- t
_{solidify}… time of solidification, h

_{eff}) = 0.18·sin(π/2·Q(t

_{eff})/Q

_{∞}) + 0.48·exp(−10·Q(t

_{eff})/Q

_{∞})

- ν(t
_{eff}) … Poisson ratio depending on effective time, - - Q(t
_{eff}) … thermal energy release depending on effective time, J - Q
_{∞}… thermal energy release at end of hydration, J

## 3. Results

- (1)
- homogeneous durable concrete (a)
- (2)
- two zones, durable concrete in the surface layer and low hydration energy concrete in the center (b)
- (3)
- three zones; gradient concrete in between the surface layer and the core (c)
- (4)
- homogeneous low hydration energy concrete (d)

_{cr}= σ/f

_{ct,m}

- I
_{cr}… crack index, - - σ … stress, N/mm
^{2} - f
_{ct,m}…concrete tensile strength, N/mm^{2}

**d**). In this model the whole part consists of hydration energy optimized concrete. The high crack risk is caused by the slow evolution of the tensile strength in this concrete. Tensile stresses in the first five days exceed the tensile strength of the concrete. The slow evolution of tensile strength might be an interesting aspect if crack width should be controlled because a reduced value for the tensile strength f

_{ct,eff}can be used, which leads to lower crack controlling reinforcement (see also [48,49]). The good characteristics to control cracks are abolished if a performance-based approach for concrete cover is applied, since the increasing thickness of the concrete cover leads to higher crack-controlling reinforcement.

## 4. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Material distribution in the models; (

**a**) homogeneous material property distribution of durable concrete; (

**b**) zoned material property distribution; (

**c**) gradient material property distribution; (

**d**) homogeneous material property distribution of hydration energy optimized concrete.

**Figure 6.**Distribution of maximum crack index (I

_{cr}) at a side (mirrored quarter model) for (

**a**) homogeneous durability optimized material distribution; (

**b**) zoned material property distribution; (

**c**) gradient material property distribution and (

**d**) homogeneous hydration energy optimized concrete.

**Figure 7.**Time development of the tensile strength (f

_{ct}) and maximum tensile stress (σ

_{max,principal}) at the reference point (see Figure 6) for (

**a**) homogeneous durability optimized material distribution; (

**b**) zoned material property distribution; (

**c**) gradient material property distribution and (

**d**) homogeneous hydration energy optimized concrete.

**Figure 8.**Time development of the crack index (I

_{cr}) at the reference point (see Figure 6) for (

**a**) homogeneous durability optimized material distribution; (

**b**) zoned material property distribution; (

**c**) gradient material property distribution and (

**d**) homogeneous hydration energy optimized concrete.

**Table 1.**Recommended limiting values for composition and properties of concrete to meet the desired material properties for exposure classes.

Exposure Class Requirement | XC4 | XF3 |
---|---|---|

Maximum w/c | 0.50 | 0.50 |

Minimum strength class | C30/37 | C30/37 |

Minimum cement content (kg/m^{3}) | 300 | 320 |

Minimum air content (%) | - | 4.0 |

Other requirements | - | -^{1} |

^{1}Aggregate in accordance with EN 12620 [38] with sufficient freeze-thaw resistance.

Material Property | Requirement |
---|---|

Thermal energy release | <200 kJ/m^{3} |

Concrete compressive strength | C25/30 |

Consistence class ^{1} | >40 cm flow spread |

Maximum w/c | - |

Minimum cement content (kg/m^{3}) | - |

Minimum air content (%) | - |

^{1}To achieve processability.

Component | Durability Optimized Surface Layer Concrete kg/m^{3} | Hydration Energy Optimized Core Concrete kg/m^{3} |
---|---|---|

Aggregate | 2090 | 2060 |

Cement Type CEM I | - | 320 |

Cement Type CEM III/B | 270 | - |

Water | 135 | 144 |

Superplasticiser ^{1} (BASF Glenium SKY 707) | 2.7 | - |

Air-Entraining Admixture (BASF Master Air 9030) | 2.7 | - |

^{1}To achieve processability.

Material Property ^{1} | Unit | Durability Optimized Surface Layer Concrete ^{2} | Hydration Energy Optimized Core Concrete |
---|---|---|---|

Cement type | - | CEM I | CEM III/B |

Cement content | kg m^{−3} | 320 | 270 |

Initial temperature of the concrete | °C | 20 | |

Activation energy | kJ mol^{−1} | 33.5 | 41.0 |

Max. thermal hydration energy of 1 kg cement | kJ kg^{−1} | 340 | 200 |

Maximum thermal hydration energy of 1 kg concrete | kJ kg^{−1} | 37.5 | 23.5 |

Maximum temperature rise under adiabatic conditions | °C | 34.0 | 21.5 |

E_{c,m,28} | GPa | 35.1 | 33.0 |

f_{c,c,m,28} | MPa | 62.0 | 41.0 |

f_{c,t,m,28} | MPa | 4.4 | 3.0 |

Thermal capacity | kJ kg^{−1} K^{−1} | 1.1 | |

Thermal conductivity at t_{0} | Js^{−1} m^{−1} K^{−1} | 3 | |

Thermal conductivity at t_{∞} | Js^{−1} m^{−1} K^{−1} | 2 | |

Poisson’s ratio at t_{0} | - | 0.5 | |

Poisson’s ratio at t_{∞} | - | 0.2 | |

Thermal expansion coefficient | K^{−1} | 10^{−5} | |

Density | kg m^{−3} | 2300 |

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## Share and Cite

**MDPI and ACS Style**

Strieder, E.; Hilber, R.; Stierschneider, E.; Bergmeister, K.
FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk. *Appl. Sci.* **2018**, *8*, 246.
https://doi.org/10.3390/app8020246

**AMA Style**

Strieder E, Hilber R, Stierschneider E, Bergmeister K.
FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk. *Applied Sciences*. 2018; 8(2):246.
https://doi.org/10.3390/app8020246

**Chicago/Turabian Style**

Strieder, Emanuel, Raimund Hilber, Elisabeth Stierschneider, and Konrad Bergmeister.
2018. "FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk" *Applied Sciences* 8, no. 2: 246.
https://doi.org/10.3390/app8020246