# Early Shrinkage Modeling of Complex Internally Confined Concrete Based on Capillary Tension Theory

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Design

#### 2.1. Concrete Mixing Ratio

#### 2.2. Specimen Design

- Steel strength is Q235B, and thickness is 10 mm;
- The diameter of the bolts is 16 mm, the length is 80 mm, and it is welded on both sides of the steel plate in a square arrangement, as shown in Figure 1d;
- Horizontal distribution reinforcement is Φ10@200, and longitudinal horizontal reinforcement is Φ10@200. These are bidirectionally arranged along the test specimen in two layers, with the same diameter of reinforcement in both directions. The studs are passed through the distribution reinforcement mesh, as shown in Figure 1c,e.

#### 2.3. Performance Testing

#### 2.3.1. Mechanical Properties Testing

#### 2.3.2. Concrete Restraint Shrinkage Test

#### 2.3.3. Concrete Pore Structure Test

^{3}. Subsequently, the hydration of the concrete was prevented using acetone, and the specimens were dried in a vacuum dryer. Finally, the internal pore structure of the concrete was tested using an AUTOIV9510 fully automated mercury porosimeter.

## 3. Shrinkage Modeling of Concrete with Complex Internal Constraints Based on Capillary Tension Theory

#### 3.1. Shrinkage Modeling of Concrete with Complex Internal Constraints

^{−2}N/m at 20 °C; θ denotes the contact angle at the liquid–solid interface (0 for concrete); and r refers to the most accretive pore size of the pores.

_{sc}is the modulus of elasticity of the concrete with respect to the capillary pore stresses; ${\sigma}_{t-i}$ is the restrained tensile stresses on the concrete in section i, i = 1, 2, 3; and μ is the Poisson’s ratio.

## 4. Hole Structures in Complex Internally Confined Concrete

#### 4.1. Confinement Factor of Concrete by Steel Plates, Pins, and Reinforcement Bars

_{s}

_{c}).

_{sc}of concrete relative to the capillary stress can be converted from the static compressive elastic modulus E

_{c}of concrete. The ratio of E

_{c}to E

_{s}

_{c}varies between 2.5 and 3.5 as the concrete strength increases. In this study, the ratio was set to 3.2. The static compressive modulus of elasticity of concrete at various ages is listed in Table 3, and the Poisson’s ratio of concrete is taken to be 0.2 as per the relevant specification. Accordingly, the constraints on the concrete in each specimen are as follows: S5 < S1 < S4 < S6 < S3 < S2.

#### 4.2. Influence of Steel Plates, Pins, and Reinforcement on the Structure of Concrete Holes

## 5. Shrinkage of Complex Internally Confined Concrete

#### 5.1. Effect of Constraint Factor on Concrete Shrinkage

^{4}MPa, 4.27 × 10

^{4}MPa, and 4.44 × 10

^{4}MPa at the ages of 3, 7, and 28 d, respectively, which were higher than the constraint coefficients of specimen S1 with a steel plate thickness of 10 mm at the same ages, corresponding to 0.4 × 10

^{4}MPa, 0.4 × 10

^{4}MPa, and 0.4 × 10

^{4}MPa, respectively. The shrinkage values of the S2 specimens at 3, 7, and 28 d were 66 με, 97 με, and 149 με, which were 46.4%, 39.8%, and 35.8% lower than the shrinkage values of the S1 specimens, respectively.

^{4}MPa, 3.91 × 10

^{4}MPa, and 4.09 × 10

^{4}MPa, respectively, which were higher than those of specimen S1 at the same ages by 0.04 × 10

^{4}MPa, 0.04 × 10

^{4}MPa, and 0.05 × 10

^{4}MPa, respectively. Regarding the constraint coefficients of specimen S4 at ages of 3, 7, and 28 d, the shrinkage values were 117 με, 160 με, and 206 με, respectively, which were lower than those of specimen S1 by 6.33 με, 0.67 με, and 25 με.

^{4}MPa, 3.87 × 10

^{4}MPa, and 4.04 × 10

^{4}MPa, respectively, which were higher than those of specimen S5 at the same ages by 0.03 × 10

^{4}MPa, 0.03 × 10

^{4}MPa, and 0.03 × 10

^{4}MPa, respectively. The constraint coefficients of specimen S1 at ages of 3, 7, and 28 d for the shrinkage values were 124 με, 161 με, and 232 με, respectively, which were 12 με, 48 με, and 53 με less than those of the S5 specimens.

^{4}MPa, 4.08 × 10

^{4}MPa, and 4.26 × 10

^{4}MPa, respectively, which were higher than those of specimen S1 at the same ages by 0.21 × 10

^{4}MPa, 0.21 × 10

^{4}MPa, and 0.22 × 10

^{4}MPa, respectively. For the constraint coefficients of specimen S3 at ages of 3, 7, and 28 d, the shrinkage values were 88 με, 136 με, and 174 με, respectively, which were 28.8%, 15.6%, and 25.1% lower than those of the S4 specimens.

^{4}MPa, 3.97 × 10

^{4}MPa, and 4.14 × 10

^{4}MPa, respectively, which were higher than those of specimen S1 at the same ages by 0.1 × 10

^{4}MPa, 0.1 × 10

^{4}MPa, and 0.1 × 10

^{4}MPa, respectively. Moreover, the constraint coefficients of specimen S6 at 3, 7, 28 d corresponded to shrinkage values of S6 103 με, 136 με, and 191 με, respectively, which were 16.7%, 15.4%, and 17.7% less than those of the S1 specimen.

#### 5.2. Effect of Constraint Coefficient on Capillary Pore Stresses

^{4}MPa, 3.91 × 10

^{4}MPa, and 4.09 × 10

^{4}MPa, respectively, which were higher than those of specimen S2 at the same ages by 0.04 × 10

^{4}MPa, 0.04 × 10

^{4}MPa, and 0.05 × 10

^{4}MPa, and the corresponding capillary pore stresses were reduced from 0.99 MPa, 1.44 MPa, and 2.44 MPa in specimen S1 to 0.89 MPa, 1.36 MPa, and 2.31 MPa in specimen S4.

^{4}MPa, 3.87 × 10

^{4}MPa, and 4.04 × 10

^{4}MPa, respectively, which were higher than the confinement coefficients of the S5 specimens configured with a stud height of 60 mm at the same ages, which corresponded to 0.03 × 10

^{4}MPa, 0.03 × 10

^{4}MPa, and 0.03 × 10

^{4}MPa, respectively. Moreover, the capillary pore stresses of the S1 specimen at 3, 7, and 28 d were 0.99 MPa, 1.44 MPa, and 2.44 MPa, respectively, which were, respectively, 0.12 MPa, 0.27 MPa, and 0.21 MPa lower than those of the S5 specimen.

^{4}MPa, 4.08 × 10

^{4}MPa, and 4.26 × 10

^{4}MPa, respectively, which were higher than the constraint coefficients of S1 specimens configured with a stud spacing of 200 mm at the same ages, corresponding to 0.21 × 10

^{4}MPa, 0.21 × 10

^{4}MPa, and 0.22 × 10

^{4}MPa, respectively. Moreover, the capillary pore stresses of the S3 specimens at 3, 7, and 28 d were 0.76 MPa, 1.05 MPa, and 1.80 MPa, respectively, which in turn were 0.22 MPa, 0.40 MPa, and 0.64 MPa lower than the capillary pore stresses of the S1 specimens.

^{4}MPa, 3.97 × 10

^{4}MPa, and 4.09 × 10

^{4}MPa, respectively, which were higher than the confinement coefficients of specimen S1 at the same ages by 0.1 × 10

^{4}MPa, 0.1 × 10

^{4}MPa, and 0.1 × 10

^{4}MPa, respectively, and those of the S6 specimen at the ages of 3, 7, and 28 d. The capillary pore stresses of the S6 specimen at 3, 7, and 28 d of age were 0.82 MPa, 1.14 MPa, and 2.02 MPa, respectively, which were 0.16 MPa, 0.30 MPa, and 0.42 MPa lower than those of the S1 specimen.

#### 5.3. Effect of Capillary Pore Stress on Concrete Shrinkage

#### 5.4. Shrinkage Prediction of Complex Internally Confined Concrete

## 6. Conclusions

- The degree of the restraining of concrete by the steel plate, studs, and reinforcement is expressed by the restraining coefficient λ. The larger the restraining coefficient, the stronger the restraining effect. The constraint coefficient increases with the increase in the steel plate thickness, reinforcement diameter, stud diameter, and stud height, and increases with the decrease in stud spacing.
- Steel plates, studs, and reinforcement have an important effect on concrete shrinkage, which decreases with the increase in the thickness of the steel plate, diameter of the reinforcement, diameter of the studs, and the height of the studs and increases with the increase in the stud spacing. This effect is not only due to the coordination of the deformation of the steel plate, studs, and reinforcement but also because the increase in the corresponding parameter increases the constraint coefficient. Moreover, the constraint factor will lead to a reduction in the capillary pore stress, which leads to a decrease in the shrinkage of the concrete.
- Comparing the measured shrinkage strain values of restrained concrete at the ages of 3, 7, and 28 d with the predicted values of the shrinkage model revealed that the average error predicted during each age was below 15%. This proves that the shrinkage model is feasible for predicting the shrinkage strain of concrete under the joint constraints of steel plates, bolts, and steel bars.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Size and construction of specimens: (

**a**) Elevation of the test piece; (

**b**) 1-1 Cross-section; (

**c**) 2-2 Cross-section; (

**d**) Test piece S1 bolt arrangement drawing; (

**e**) Test piece S1 rebar layout drawing.

**Figure 2.**Arrangement of measuring points for shrinkage of restrained concrete: (

**a**) elevation plan; (

**b**) side view.

**Figure 6.**Three typical sections in restrained concrete: (

**a**) Type 1 cross-section; (

**b**) Type 2 cross-section; (

**c**) Type 3 cross-section.

**Figure 7.**Variation in concrete pore structure with constraint coefficient: (

**a**) optimal pore size; (

**b**) porosity; (

**c**) average pore size; (

**d**) median aperture.

**Figure 8.**Diagram of shrinkage of concrete with constraint coefficient: (

**a**) 3 days of age; (

**b**) 7 days of age; (

**c**) 28 days of age.

**Figure 9.**Relationship between capillary stress and constraint coefficient of concrete at different ages: (

**a**) 3 days of age; (

**b**) 7 days of age; (

**c**) 28 days of age.

**Figure 10.**Effect of capillary stress on shrinkage of concrete at different ages: (

**a**) 3 days of age; (

**b**) 7 days of age; (

**c**) 28 days of age.

**Figure 11.**Schematic of the effect of steel plate, stud, and steel bar on the shrinkage of concrete.

**Figure 12.**Comparison of test results and calculated values at different ages: (

**a**) 3 days of age; (

**b**) 7 days of age; (

**c**) 28 days of age.

Water to Binder Ratio | Cement | Course Aggregate | Sand | Water | Fly Ash | Additive |
---|---|---|---|---|---|---|

0.29 | 482 | 1064 | 680 | 154 | 48 | 12.7 |

No. | Section Size (mm) | Plate Thickness (mm) | Distribution Rebar Straight Diameter (mm) | Bolt Diameter (mm) | Bolt Height (mm) | Bolt Spacing (mm) | Curing Condition |
---|---|---|---|---|---|---|---|

P | 1000 × 200 | 0 | 0 | 0 | 0 | 0 | Sealed |

S1 | 1000 × 200 | 10 | 10 | 16 | 80 | 200 | Sealed |

S2 | 1000 × 200 | 12 | 10 | 16 | 80 | 200 | Sealed |

S3 | 1000 × 200 | 10 | 10 | 16 | 80 | 100 | Sealed |

S4 | 1000 × 200 | 10 | 10 | 19 | 80 | 200 | Sealed |

S5 | 1000 × 200 | 10 | 10 | 16 | 70 | 200 | Sealed |

S6 | 1000 × 200 | 10 | 12 | 16 | 80 | 200 | Sealed |

Basic Properties | Age (Days) | |||
---|---|---|---|---|

3 | 7 | 14 | 28 | |

Cube compressive strength (MPa) | 42.8 | 48.2 | 56.4 | 61.0 |

Splitting tensile strength (MPa) | 3.8 | 4.1 | 4.5 | 5.2 |

Elastic modulus (Gpa) | 30.0 | 32.3 | 35.0 | 35.5 |

No. | Constraint Factor (×10^{4} MPa) | ||
---|---|---|---|

3 Days | 7 Days | 28 Days | |

S1 | 3.75 | 3.87 | 4.04 |

S2 | 4.15 | 4.27 | 4.44 |

S3 | 3.96 | 4.08 | 4.26 |

S4 | 3.79 | 3.91 | 4.09 |

S5 | 3.72 | 3.84 | 4.01 |

S6 | 3.85 | 3.97 | 4.14 |

No. | Capillary Stress (MPa) | ||
---|---|---|---|

3 Days | 7 Days | 28 Days | |

S1 | 0.99 | 1.44 | 2.44 |

S2 | 0.56 | 0.89 | 1.38 |

S3 | 0.76 | 1.05 | 1.80 |

S4 | 0.89 | 1.36 | 2.31 |

S5 | 1.11 | 1.71 | 2.65 |

S6 | 0.82 | 1.14 | 2.02 |

No. | 3 Days | 7 Days | 28 Days | ||||||
---|---|---|---|---|---|---|---|---|---|

Tested Values | Calculated Values | Relative Error | Tested Values | Calculated Values | Relative Error | Tested Values | Calculated Value | Relative Error | |

S1 | 124 | 105 | 15.0% | 161 | 143 | 10.9% | 232 | 220 | 5.3% |

S2 | 66 | 60 | 9.5% | 97 | 89 | 8.4% | 149 | 125 | 16.3% |

S3 | 88 | 81 | 7.7% | 136 | 104 | 23.4% | 174 | 162 | 6.8% |

S4 | 117 | 95 | 19.1% | 160 | 135 | 15.7% | 207 | 208 | 0.9% |

S5 | 136 | 118 | 13.2% | 209 | 169 | 19.2% | 285 | 239 | 16.4% |

S6 | 103 | 88 | 14.9% | 136 | 113 | 16.8% | 191 | 182 | 4.7% |

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## Share and Cite

**MDPI and ACS Style**

Zhou, F.; Jiang, H.; Huang, L.; Hu, Y.; Xie, Z.; Zeng, Z.; Liu, M.; Wang, B.; Zhou, X.
Early Shrinkage Modeling of Complex Internally Confined Concrete Based on Capillary Tension Theory. *Buildings* **2023**, *13*, 2201.
https://doi.org/10.3390/buildings13092201

**AMA Style**

Zhou F, Jiang H, Huang L, Hu Y, Xie Z, Zeng Z, Liu M, Wang B, Zhou X.
Early Shrinkage Modeling of Complex Internally Confined Concrete Based on Capillary Tension Theory. *Buildings*. 2023; 13(9):2201.
https://doi.org/10.3390/buildings13092201

**Chicago/Turabian Style**

Zhou, Fengbin, Hao Jiang, Lepeng Huang, Ying Hu, Zhuolin Xie, Zhikai Zeng, Maoyi Liu, Bo Wang, and Xingyang Zhou.
2023. "Early Shrinkage Modeling of Complex Internally Confined Concrete Based on Capillary Tension Theory" *Buildings* 13, no. 9: 2201.
https://doi.org/10.3390/buildings13092201