# Simplified Modeling Strategy for the Thermomechanical Analysis of Massive Reinforced Concrete Structures at an Early Age

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## Abstract

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## Featured Application

**This work is intended to provide the tools needed for predicting the behavior of massive concrete structures in order to better control early-age cracking.**

## Abstract

## 1. Introduction

## 2. The RG8 Test

#### Blocks Subjected to Restrained Shrinkage

- A central block (5.10 m × 0.80 m × 0.50 m), plus a 0.4-m transition zone on each side with a gradual increase in the width. The reinforcement map is given in Figure 1a; in the cross-section, it includes a combination of 32-mm diameter (10) and 12-mm diameter (4) reinforcements; stirrups are 16 mm in diameter, and they are spaced at 200 mm.
- Two heads (0.9 m × 2.2 m × 0.9 m) serve to support the struts described below;
- Two steel struts fastened on the two heads to restrain strain evolution, as shown in the picture in Figure 1.

## 3. Models

#### 3.1. Multi-Fiber Beam Modeling

#### 3.2. Models for Concrete

- Concrete maturation;
- Evolution of mechanical performance with maturation;
- Autogenous shrinkage of concrete;
- Basic creep of concrete;
- Thermal deformation of both concrete and steel;
- Elasto-damaging behavior of concrete;
- Elastoplastic behavior of reinforcements;

#### 3.2.1. Early-Age Thermal Model

- the thermal problem was treated independently from the mechanical problem;
- temperature variations were assumed to be identical in each section of the beam (independence with respect to the longitudinal axis x).

- The energy balance equation:$$C\stackrel{\u2022}{T}=\nabla (k\nabla T)+L\stackrel{\u2022}{\xi}$$

^{−1}·K

^{−1}), L the latent hydration heat (J·m

^{−3}), and C the volumetric thermal capacity (J·m

^{3}·K

^{−1}).

- ξ is the degree of hydration, as given by the Arrhenius equation:$$\stackrel{\u2022}{\xi}=A(\xi )\mathrm{exp}\left(-\frac{{E}_{a}}{RT}\right)$$

_{a}is the activation energy (J·mol

^{−1}), R the ideal gas constant (R = 8.3145 J·K

^{−1}·mol

^{−1}), and A(ξ) the chemical affinity (s

^{−1}).

_{0}, the percolation threshold (close to 0), and ξ

_{∞}, the degree of hydration upon completion of the hydration reaction. To describe the corresponding state of the medium, the maturity M is introduced [13]:

_{+}is the positive part operator. M evolves between 0 (beginning of the process) and 1 (end of hydration).

_{i}is the initial affinity, ζ governs the deceleration phase of the hydration process, and A

_{p}is the maximum value of the affinity function attained for M = M

_{p}.

- The external exchange is solved as follows:$$\varphi =h\left({T}_{s}-{T}_{ext}\right)n$$

^{−2}), n the vector normal to the surface, T

_{s}the surface temperature (K), and T

_{ext}the external temperature (K). h is the exchange coefficient whose value changes when the mold is removed from 2, calculated by taking into account the insulation produce by the formwork, to 10 W·m

^{−2}·K

^{−1}[15].

#### 3.2.2. Mechanical Model

_{t}= Max(ε

_{0t}, max ε

_{t}), with ε

_{0t}being the tensile strain threshold, Y

_{c}= Max(ε

_{0c}, max(−ε

_{c})), with ε

_{0c}being the compressive strain threshold. Generally speaking, ε

_{0t}corresponds to the tensile strain at peak (f

_{t}), which then allows writing: ε

_{0t}= f

_{t}/E.

_{f}was derived from [17,18]. ε

_{f}is the fracturing strain and

_{c}the mean crack spacing, as estimated here close to the stirrup spacing, i.e., 200 mm (this assumption has been verified by the experimental results presented in Section 4.2.3.).

#### 3.2.3. Thermo-Chemo-Mechanical Coupling

_{0}, the percolation threshold (assumed to lie close to 0) M = 0; moreover, when ξ = ξ

_{∞}, the degree of hydration upon completion of the reaction (as calculated from the w/c ratio of the concrete mix thanks to the relation proposed by [19]) M = 1.

- Young’s modulus: E(M) = ME
_{∞}, with E_{∞}being the value when M = 1 (mature concrete); - Concrete tensile strength: f
_{t}(M) = E(M). ε_{0t}(ε_{0t}is assumed to be constant regardless of the maturity).

#### 3.2.4. Scale Effect

_{0}is a material parameter. The ratio between the tensile threshold value stemming from a laboratory test (volume V

_{td}) and the tensile threshold required to be used on the RG8 test is then given by:

_{c}) with the Weibull parameter was proposed in [21]. In the present case, in which f

_{c}= 63.5 MPa, m is estimated to be equal to 13.5. As deduced from (10) and the tensile strength value (f

_{td}= ε

_{0t}E) obtained on a conventional cylindrical sample (i.e., diameter: 0.16 m, height: 0.32 m), the ratio between the tensile strengths (f

_{td}/f

_{tRG}

_{8}) is therefore equal to roughly 1.55. The tests employed in the present study, however, are splitting tests on same-sized cylindrical samples. A recent study [22] using the concept of high-strength volume (HSV, [23]) has shown that the equivalent tensile volume in such a splitting test amounts to close to 4% of the total sample volume. From the Weilbull approach, it can thus be deduced that the tensile strength, which must be introduced into the RG8 test, equals approximately half the nominal splitting strength (2.2 MPa vs. 4.6 MPa).

#### 3.2.5. Global Description of the Problem

_{th}and ε

_{au}are the thermal strain and the autogenous shrinkage respectively:

^{−1}) and κ

_{∞}being the final value of shrinkage (μm/m).

_{bc}is the basic creep. ${\tau}_{bc}(M)={\eta}_{bc}/{k}_{bc}(M)$ is the characteristic time of a given Kelvin-Voigt model, with:

_{bc_∞}is the final stiffness, when M = 1.

## 4. Application to the RG8 Test

#### 4.1. Thermo-Chemo-Mechanical Loading

- From 0 to 120 h, the temperature variation is mostly due to hydration effects. Thanks to the energy balance Equation (1), this evolution has been evaluated; Figure 2a shows good agreement with the in situ experimental measurements both at the center and near the surface of the beam section;
- From 120 h to 700 h (i.e., the time corresponding to strut removal), the temperature variation is mainly due to external temperature evolution (day-night changes). Due to the massiveness of the beam, this change only concerns material close to the surface and, for the sake of simplification, the calculation has been performed with a constant temperature of 12 °C. At the same time, the struts are sensitive to the day-night temperature differential. The cycle lasts 24 h, and in accordance to the measured temperature, the discrepancy is assumed to lie between 9° and 15 °C.

#### 4.1.1. Shrinkage and Creep

- Autogenous shrinkage was calibrated in order to obtain κ
_{∞}(Equation (14)). - From compression tests loaded at 2 days and 7 days, the basic creep model (Equations (15) and (16)) was also calibrated (see results in Appendix A, Table A1). In order to optimize coverage of the entire test duration, the characteristic times of the three K-V models were: τ
_{1}= 0.1 day, τ_{2}= 1 day, and τ_{3}= 10 days.

#### 4.1.2. Maturity and Mechanical Performance of Concrete

#### 4.1.3. Mechanical Loading

#### 4.2. Results

#### 4.2.1. From 0 to 120 h: Maturation and Cracking Period

_{rb}absorbed by the set of reinforcements within a given section can be deduced. Strain gauges were also placed on the struts. With knowledge of the strut surface area, the force F

_{st}being exerted on them can also be deduced. At a given time, each cross-section is in equilibrium. From this balance, the force F

_{c}supporting the concrete section can be deduced as:

_{s}is the Young’s modulus for steel, A

_{i}and ε

_{i}are, respectively, the area and strain on rebar i I, and A

_{st}and ε

_{st}are, respectively, the cross-sectional area of each strut and corresponding strain.

#### 4.2.2. From 0 to 700 h: Restrained Shrinkage

- 3 main cracks are observed during the experiment. The calculation exhibits an initial cracking zone in the beam center at 76 h, as shown in Figure 4c, which is consistent with the first experimental crack appearing at 71 h. At 103 h, the cracking zone extends on both sides at the same time, while the experimental cracks appear on the left side at 167 h and then on the right side at 239 h. It can thus be concluded that the calculation indeed provides for these two cracking phases (left and right of the central zone) but prematurely with respect to the experiment. In the case of the present loading (tension + bending), as demonstrated in [7] for multi-fiber beams, the damage is diffuse and the resultant mesh is insensitive. This diffuse characteristic and the loading mesh symmetry are probably at the origin of the simultaneous appearance of the left and right zones.
- Figure 7a,b display the global responses of all 3 systems (struts, concrete, and rebar). In the central section, once the concrete is cracked, the rebar and struts are the main active systems. In the left section outside the cracking zone, the main active systems are the struts and concrete, as the load supported by the rebar is close to zero. After 120 h, the mean values of these loads are practically constant, and the small cyclic evolution that appears is due to the cyclic day-night temperature variations, which generate a quasi-cyclic load in the struts that is subsequently transmitted to the beam.

#### 4.2.3. Final Stage: 4-Point Bending Loading

- The experimental curve (dashed black line): the beam was only loaded up to the serviceability limit state (SLS).
- The calculated curve for the beam after the restrained shrinkage loading program (blue curve).
- The calculated curve for the same beam considered to be structurally sound (red curve). This loading triggers plasticity on the rebar; the steel behavior considered here is an elastoplastic model with linear hardening (f
_{e}= 500 MPa).

- The calculated RG8 curve is in good compliance with the experimental curve.
- The two calculated curves (RG8 and structurally sound beams) are very close during most of the loading evolution.
- A tremendous difference is observed between the initial stiffness values. The initial stiffness is strongly reduced (about 50%) due to concrete cracking produced by the restrained shrinkage. An additional calculation has shown that the first vibration mode is equal to 31.7 Hz instead of 59.2 Hz for the sound beam, which leads to a very different dynamic response (Figure 8c). This decrease must be taken into account in cases of dynamic loading, such as an earthquake.

- At a low loading level (205 kN), cracking appears in the center of the beam between cracks due to restrained shrinkage, while the diffuse damage, as developed during the restrained shrinkage phase, is located in the same area and remains relatively unchanged during the first loading phase.
- For the following phases (685 kN and 1100 kN), cracks and damage develop on the upper part of the beam (loading occurs from the bottom up); moreover, the damage contour, which is always diffuse, is quite consistent with the crack pattern.

- The initial stiffness is heavily influenced by the restrained shrinkage treatment. The ultimate beam strength however remains the same, with or without this treatment.
- The calculations have yielded consistent results in terms of both global behavior (load-displacement) and local indicators (damage vs. crack pattern).

## 5. Conclusions

- On structural durability, such as steel corrosion or the alkali-aggregate reaction, sulfate attack, and freeze-thaw [26].
- On its mechanical response, especially in a dynamic loading, such as an earthquake (in the present case, the first natural frequency was reduced by nearly 50%).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Concrete at Early-Age | C(KJ/°C·m ^{3}) | k | L | h | κ | α | k^{1}_{bc_}_{∞} | k^{2}_{bc_}_{∞} | k^{3}_{bc_}_{∞} |

(W/mK) | (KJ/m^{3}) | (W/m^{2}K) | (μm/m) | (μm/m°C) | (Gpa) | (Gpa) | (Gpa) | ||

2400 | 2.8 | 157,840 | 2 & 10 | 110 | 12 | 3 × 10^{11} | 9 × 10^{10} | 2.5 × 10^{10} | |

A_{i} (s^{−1}) | A_{p} (s^{−1}) | M_{p} | ζ | ξ_{0} | ξ_{∞} | ||||

64.85 | 1660 | 0.172 | 75 | 0 | 0.8 | ||||

Concrete damage model | E_{∞} (Gpa) | ε_{P} | G_{f} (N/m) | ε_{f} | A_{t} | B_{t} | ε_{/} | A_{c} | B_{c} |

40 | 0.53 × 10^{−4} | 60 | 3.5 × 10^{−4} | 0.815 | 19,250 | 3.75 × 10^{−4} | 1.5 | 355 | |

Steel model | E_{s} (GPa) | σ_{y} (MPa) | E_{t} (GPa) | α (μm/m°C) | |||||

210 | 500 | 5 | 7.5 |

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**Figure 1.**Overview of the RG8 beam and reinforcement detail: (

**a**) reinforcement map of the beam; (

**b**) multi-fiber beam mesh and boundary conditions.

**Figure 2.**(

**a**) Temperature evolution at the center (black curves) and near the surface (red curves) of the beam (experiment-calculation comparison); (

**b**) uniaxial behavior for concrete exhibiting the unilateral effect.

**Figure 3.**Evolution in early-age concrete properties; test (circles)-model (curves) comparison: (

**a**) tensile strength ft; (

**b**) Young’s modulus.

**Figure 4.**Evolution during the 0–120 h time frame: (

**a**) the 3 section elements: struts (red), rebar (blue), concrete (green); (

**b**) global forces supported by the 3 systems in the central beam section (dashed line: experiment—solid line: calculation); and (

**c**) damage contour at 76 h (time of initial cracking) and 103 h (time of extended cracking).

**Figure 5.**Stress evolution on rebar (dashed line: experiment—solid line: calculation): (

**a**) in a section outside the cracking zone (left section), and (

**b**) in the central beam section.

**Figure 6.**Strain evolution in concrete, as measured by optic fibers, compared to the strain calculated at the same location (dashed line: experiment—solid line: calculation). Crack openings, appearing in the table, are deduced directly from the strain jumps.

**Figure 7.**0–700 h evolution: (

**a**) global forces supported by the 3 systems in the left section, 1 m from the beam center, outside the cracking zone (dashed line: experiment—solid line: calculation); (

**b**) global forces supported by the 3 systems in the central section (inside the cracking zone); and (

**c**) cracks observed and tensile damage contour at the end of the process (700 h).

**Figure 8.**The four-point bending RG8 test: (

**a**) total force-displacement responses (experiment, RG8 beam, and structurally sound beam); (

**b**) close-up on the first 3-mm displacement showing the large difference between the two original stiffnesses (RG8 beam and sound beam); and (

**c**) the corresponding free vibrations responses.

**Figure 9.**The four-point bending RG8 test. (

**Left**) load-displacement responses; (

**Right**) cracking and damage evolution at various loading stages.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mazars, J.; Grange, S.; Briffaut, M.
Simplified Modeling Strategy for the Thermomechanical Analysis of Massive Reinforced Concrete Structures at an Early Age. *Appl. Sci.* **2018**, *8*, 448.
https://doi.org/10.3390/app8030448

**AMA Style**

Mazars J, Grange S, Briffaut M.
Simplified Modeling Strategy for the Thermomechanical Analysis of Massive Reinforced Concrete Structures at an Early Age. *Applied Sciences*. 2018; 8(3):448.
https://doi.org/10.3390/app8030448

**Chicago/Turabian Style**

Mazars, Jacky, Stéphane Grange, and Matthieu Briffaut.
2018. "Simplified Modeling Strategy for the Thermomechanical Analysis of Massive Reinforced Concrete Structures at an Early Age" *Applied Sciences* 8, no. 3: 448.
https://doi.org/10.3390/app8030448