# Uniaxial Static Stress Estimation for Concrete Structures Using Digital Image Correlation

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## Abstract

**:**

## 1. Introduction

## 2. Stress Estimation Method

_{FEM}and V

_{DIC}are the displacement from the DIC and the FEM, (x′,y′) is the new coordinate system, (x,y) and (u,v) are the positions within the original coordinate systems, and (x

_{0},y

_{0}) and (u

_{0},v

_{0}) are the coordinates of the hole center in the coordinate system of the FEM and the DIC, respectively. Consequently, the static stress can be estimated by comparing those two displacement fields in the same coordinate system.

_{DIC}(0,0) and V

_{FEM}(0,0) represent the displacement at the hole center, σ is the static stress to be estimated, and σ

_{0}is a reference stress. The displacements from the DIC and the FEM are subtracted by V

_{DIC}(0,0) and V

_{FEM}(0,0), in order to disregard the rigid body motion of the whole specimen, as the displacement at the hole center represents the motion of the whole structural body. When the rigid body motion is neglected, V

_{DIC}equals V

_{FEM}but is amplified by the amount of static stress. Using the two-dimensional displacement fields (V

_{DIC}and V

_{FEM}), the static stress can be estimated as follows:

^{+}is the Moore-Penrose inverse, (x

_{k}’,y

_{k}’) are the coordinates of the k-th point, and n is the number of points used. The optimal static stress of the concrete structure, denoted by σ, can hence be obtained.

## 3. Experimental Validation

_{estimated}and σ

_{applied}are the estimated and applied stress, respectively. The MSE represents the regression quality between the DIC and the FEM displacement fields, with smaller MSE values indicating more accurate stress estimations. Hence, the static stress can be estimated by using the proposed method and the estimation reliability can also be checked by evaluating the MSE.

_{FEM}and V

_{DIC}are adjusted using the estimated stress and rigid body motion in Equation (4), as follows:

_{FEM}and V’

_{DIC}with respect to their data index for Tests 1 and 5. During Test 1, the stress is more accurately estimated (the lowest MSE value) and the V’

_{FEM}and V’

_{DIC}datasets agree quite well with each other (Figure 7a). On the contrary, during Test 5, the V’

_{DIC}dataset deviates considerably from the V’

_{FEM}dataset, particularly between data indices 50 and 250; this deviation results in a large MSE value and in an inaccurate stress estimation (Figure 7b). The large deviation in the 50–250 index interval is caused by non-uniaxial stress conditions, due to the misalignment of the specimen. In fact, the top and bottom faces of the specimen are not perfectly parallel to each other, possibly due to the faulty fabrication, which causes biaxial stresses. In Figure 8, considerable displacement is observed on the red dotted line, whereas the corresponding displacement in the FE model is negligible. Thus, the actual stress distribution in the concrete specimen is expected to be quite different from V

_{FEM}, which is obtained by assuming uniaxial stress, resulting in a large MSE value for Test 5. Therefore, the quality of the stress estimation can be indirectly assessed using the MSE.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

(x,y) | Position in the original coordinate system obtained by the FEM. |

(u,v) | Position in the original coordinate system obtained by the DIC. |

(x′,y′) | Position in the new coordinate system obtained by the FEM and DIC. The origin is the hole center. |

(x_{0},_{y}_{0}) | Position of the hole center in the original coordinate system obtained by the FEM. |

(u_{0},v_{0}) | Position of the hole center in the original coordinate system obtained by the DIC. |

V_{DIC}(x,y) | Vertical displacement obtained by the DIC at (x,y). |

V_{FEM}(x,y) | Vertical displacement obtained by the FEM at (x,y). |

V’_{FEM}(x,y) | Adjusted vertical displacement obtained by the DIC at (x,y). |

V’_{DIC}(x,y) | Adjusted vertical displacement obtained by the FEM at (x,y). |

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**Figure 5.**Displacement fields calculated using the FE model: (

**a**) Displacement field of the intact and the damaged concrete, respectively denoted by V

_{before}and V

_{after}, obtained by the FE model; (

**b**) The displacement field due to the hole-drilling.

**Figure 7.**A pointwise comparison between the FEM results, scaled by the estimated stress and the DIC results, subtracted by the rigid body motion: (

**a**) Test 1; (

**b**) Test 5.

Concrete Specimen Size (Width × Height × Depth) | Type A: 100 mm × 400 mm × 100 mm |
---|---|

Type B: 150 mm × 300 mm × 150 mm | |

Elastic modulus | 24.4 GPa |

Poisson’s ratio | 0.17 |

Test Number | Specimen Type | Applied Stress (MPa) | Estimated Stress (MPA) | Estimation Error (%) | MSE (×10^{−8}) |
---|---|---|---|---|---|

1 | B | 14.70 | 15.53 | 5.67 | 99.3 |

2 | A | 14.10 | 15.24 | 8.07 | 163.1 |

3 | B | 14.48 | 10.45 | 27.83 | 234.9 |

4 | A | 13.80 | 15.92 | 15.36 | 739.8 |

5 | B | 14.00 | 9.92 | 29.13 | 781.2 |

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**MDPI and ACS Style**

Lee, J.; Kim, E.J.; Gwon, S.; Cho, S.; Sim, S.-H. Uniaxial Static Stress Estimation for Concrete Structures Using Digital Image Correlation. *Sensors* **2019**, *19*, 319.
https://doi.org/10.3390/s19020319

**AMA Style**

Lee J, Kim EJ, Gwon S, Cho S, Sim S-H. Uniaxial Static Stress Estimation for Concrete Structures Using Digital Image Correlation. *Sensors*. 2019; 19(2):319.
https://doi.org/10.3390/s19020319

**Chicago/Turabian Style**

Lee, Junhwa, Eun Jin Kim, Seongwoo Gwon, Soojin Cho, and Sung-Han Sim. 2019. "Uniaxial Static Stress Estimation for Concrete Structures Using Digital Image Correlation" *Sensors* 19, no. 2: 319.
https://doi.org/10.3390/s19020319