# Free Shrinkage Strains of Box Girders with Concrete Overlays

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometry and Mesh Size

#### 2.2. Humidity Profiles

^{2}/day) as a function of ${h}_{d}$, ${t}_{0}$ is the number of curing days, and $U(t-{t}_{0})$ is the Heaviside unit step function. Since the diffusion coefficient D is a function of the internal drying humidity, Equation (2) is a non-linear differential equation and iterations are needed to calculate its solution. D can be estimated with Equation (3):

^{2}/day, ${f}_{ck0}=10$ MPa, and ${f}_{ck}={f}_{cm}-8$ MPa. The characteristic compressive strength of the concrete is ${f}_{ck}$, and ${f}_{cm}$ represents the mean compressive strength of the specimen at 28 days, which is taken in this research to be equal to 38 MPa.

#### 2.2.1. Initial Drying Humidity Profile

#### 2.2.2. Drying Humidity: Solution Method

**J**is the Jacobian matrix. The element mass matrix is calculated with Equation (7), where $\rho $ is the mass density, then the global stiffness and mass matrices are assembled using element connectivity information [24].

#### 2.3. Free Shrinkage Strains

^{3}. The magnitudes of ${\epsilon}_{s\infty}$ and ${\epsilon}_{s}^{0}$ are 680 $\mathsf{\mu}\mathsf{\u03f5}$ and 882 $\mathsf{\mu}\mathsf{\u03f5}$, respectively.

## 3. Results

#### 3.1. Humidity Profiles

#### 3.2. Free Shrinkage Strain Profiles

## 4. Discussion

#### 4.1. Humidity Profiles: Discussion of the Results

#### 4.2. Free Shrinkage Strains: Discussion of the Results

## 5. Conclusions

- The humidity gradient between the overlay and the substrate generates differential volume changes between them. The substrate deformations are negligible while the overlay is subjected to high shrinkage strains.
- The internal humidity gradient must be considered because it influences the drying shrinkage and it may result in increased tensile stress, which may lead to early age cracking.
- The predicted free shrinkage strain values suggest that the overlay might be quite susceptible to cracking.
- The computation of shrinkage strains increases with time, and after 50 years, full equilibrium is reached; however, an analysis of about 10 years seems to be appropriate for practical applications and standard designs conducted in engineering firms, leading to a reduction in computational work for the prediction of stresses, probability of cracking, or displacements due to restrained shrinkage of new concrete overlays.

## 6. Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Diffusion coefficient ratio $D/{D}_{1}$ as a function of the drying humidity ${h}_{d}$: comparison between published research [20] and this study.

**Figure 5.**Humidity profiles in the bridge deck at midspan. The horizontal line shows the position of the overlay.

**Figure 7.**Normalized free shrinkage profiles at discrete times along one web of the box girder bridge.

**Figure 8.**Normalized free shrinkage strain results of the box girder bridge cross-section at different time steps.

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**MDPI and ACS Style**

Parmiani, M.G.; Orta, L.
Free Shrinkage Strains of Box Girders with Concrete Overlays. *Infrastructures* **2023**, *8*, 96.
https://doi.org/10.3390/infrastructures8050096

**AMA Style**

Parmiani MG, Orta L.
Free Shrinkage Strains of Box Girders with Concrete Overlays. *Infrastructures*. 2023; 8(5):96.
https://doi.org/10.3390/infrastructures8050096

**Chicago/Turabian Style**

Parmiani, Maria Giulia, and Luis Orta.
2023. "Free Shrinkage Strains of Box Girders with Concrete Overlays" *Infrastructures* 8, no. 5: 96.
https://doi.org/10.3390/infrastructures8050096