# Investigating the Bond Strength of FRP Rebars in Concrete under High Temperature Using Gene-Expression Programming Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Database

_{s}), bar diameter (d

_{b}), anchorage length (L), compressive strength (${f}_{c}^{\prime}$), and cover-to-diameter ratio (c/d

_{b}) as input parameters. Mathematically,

#### 2.2. GEP Model Development

^{2}functions were used to establish relationships within ETs, while the + function was used to establish relationships between ETs. The model was trained until the optimal solution is established. In other words, it means that the best fit is the moment at which the model’s performance in terms of correlations and error indices stops improving. It may be noted that the performance of the validation data was also checked to prevent model over-fitting. Once the optimal performance was achieved, the model’s ability to produce mathematical equations was discontinued. Figure 2 represents the framework of GEP proposed for this study.

#### 2.3. Evaluation Criteria

^{2}), and Pearson’s coefficient

^{®}, were utilised to appraise the accuracy of the developed GEP model. All these parameters are well-recognised indicators for evaluating the strength of data-driven models [51,52,53,54,55].

## 3. Results and Discussion

#### 3.1. Effect of Genetic Variables

#### 3.2. Performance of the Undertaken Trials

#### 3.2.1. Statistical Indices Analysis

^{2}, MAE, and RMSE) for all the trials are summarised in Table 3. For the training data and validation data, the minimum value of R was found to be 0.691 and 0.521 for trial 7, respectively. For the training data of trial 7, the maximum value of MAE was found to be 2.617, while for the validation data, it was found to be 3.647. Moreover, the values of RMSE for both the datasets were 3.417 and 4.647, respectively. The best values, that is, 0.941, 0.885, 1.620, and 2.087, respectively, for R, R

^{2}, MAE, and RMSE were obtained for trial 10. The statistical analysis of all the trials revealed a close correlation between experimental and projected results; nevertheless, the performance of trial 10 findings outperformed its counterparts. Therefore, the model from trial 10 is more reliable for future prediction of bond strength of FRP rebars in concrete at high temperatures.

#### 3.2.2. Regression Plot Analysis and Error Analysis

#### 3.2.3. Predicted vs. Experimental Ratio

#### 3.3. GEP Formulations

_{b}is the diameter of bar; B

_{s}is the bar surface; FM is failure mode; T is the temperature; ${f}_{c}^{\prime}$ is the concrete compressive strength; L is the anchorage length; and $\frac{c}{{d}_{b}}$ is the cover-to-diameter ratio) is identified in Equation (2) as being used to predict the value of bond strength.

#### 3.4. Parametric Analysis

## 4. Conclusions

- The proposed GEP model has the potential to be used as a tool for extracting features and making predictions in very intricate nonlinear engineering systems. The GEP architecture needs to be optimised via hit-and-trial method based on the dataset’s size and complexity. Dependence on the amount and quality of the dataset is a key factor in the GEP model’s utility and accuracy.
- Measures of statistical performance including R, RMSE, and MAE were applied to both the training data and the validation data to assess the quality of the models. R = 0.941, MAE = 1.620, and RMSE = 2.087 for training, and 0.935, 2.046, and 2.370 for validation, according to the statistical indices, demonstrates that the established GEP model can accurately estimate bond strength.
- The parametric analysis shows that the developed GEP model can accurately predict the response of different input variables on the strength of FRP bars. Due to the limited data, this analysis is limited to ribbed bars.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Comparison of the regression slopes for the training and validation data of the optimised trial No. 10.

**Figure 6.**Error analysis of the optimised trial. (

**a**) Tracing of experimental results by the predictions; (

**b**) absolute errors for trial No. 10 predictions.

**Figure 9.**Parametric study of the GEP model showing variation of bond strength with change in input variable for Type I (debonding) and Type II (pull-out) failure modes corresponding to Type-II bar surface (Ribbed).

Variable | Input Variables | Target | |||||||
---|---|---|---|---|---|---|---|---|---|

Temperature | Failure Mode | Fibre Type | Bar Surface | Diameter | Anchorage Length | Compressive Strength | Cover-to-Diameter Ratio | Bond Strength | |

Identification | T | FM | FT | B_{s} | d_{b} | L | ${f}_{c}^{\prime}$ | c/d_{b} | BS |

Descriptive statistics | °C | - | - | - | mm | mm | MPa | - | MPa |

Mean | 150.99 | 1.95 | 1.43 | 2.27 | 8.66 | 135.61 | 42.52 | 7.80 | 10.32 |

Standard Error | 8.74 | 0.07 | 0.04 | 0.11 | 0.14 | 15.46 | 0.69 | 0.23 | 0.53 |

Median | 125.00 | 2.00 | 1.00 | 2.00 | 8.00 | 47.50 | 42.76 | 7.37 | 10.84 |

Mode | 20.00 | 2.00 | 1.00 | 1.00 | 8.00 | 40.00 | 33.70 | 5.75 | 3.40 |

Standard Deviation | 105.59 | 0.90 | 0.50 | 1.30 | 1.68 | 186.75 | 8.30 | 2.79 | 6.35 |

Sample Variance | 11,149.45 | 0.81 | 0.25 | 1.69 | 2.81 | 34,875.73 | 68.89 | 7.81 | 40.34 |

Kurtosis | 1.70 | 2.05 | −1.95 | −1.65 | 0.66 | 3.04 | −0.75 | −0.91 | −1.36 |

Skewness | 1.06 | 1.20 | 0.28 | 0.31 | 0.52 | 2.07 | 0.45 | 0.54 | −0.01 |

Range | 580.00 | 4.00 | 1.00 | 3.00 | 6.70 | 764.00 | 30.93 | 9.35 | 22.87 |

Minimum | 20.00 | 1.00 | 1.00 | 1.00 | 6.00 | 20.00 | 32.00 | 3.15 | 0.42 |

Maximum | 600.00 | 5.00 | 2.00 | 4.00 | 12.70 | 784.00 | 62.93 | 12.50 | 23.29 |

Count | 146.00 | 146.00 | 146.00 | 146.00 | 146.00 | 146.00 | 146.00 | 146.00 | 146.00 |

Confidence Level (95.0%) | 17.27 | 0.15 | 0.08 | 0.21 | 0.27 | 30.55 | 1.36 | 0.46 | 1.04 |

Categorical Input Variable | Property | Code |
---|---|---|

Bar Surface (Bs) | Sand-coated (SC) | 1 |

Ribbed (RB) | 2 | |

Fibre-wounded (SW) | ||

SC + SW | 3 | |

SC + RB | 4 | |

Failure mode (FM) | Debonding (D) | 1 |

Pull-out (P) | 2 | |

Shear failure of concrete (SF) | 3 | |

FRP rupture (R) | 4 | |

Splitting of concrete (S) | 5 | |

Type of FRP | GFRP | 1 |

BFRP | 2 |

Trial No. | Used Variables | No. of Chromosomes | Head Size | Number of Genes | Constants per Gene | No. of Literals | Program Size | Training Dataset | Validation Dataset | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Best Fitness | RMSE | MAE | R^{2} | R | Best Fitness | RMSE | MAE | R^{2} | R | ||||||||

1 | 8 | 30 | 8 | 3 | 10 | 17 | 41 | 249.7 | 3.004 | 2.326 | 0.764 | 0.874 | 198.550 | 4.036 | 3.197 | 0.626 | 0.791 |

2 | 8 | 50 | 8 | 3 | 10 | 15 | 42 | 245.3 | 3.076 | 2.442 | 0.749 | 0.865 | 229.980 | 3.348 | 2.750 | 0.754 | 0.868 |

3 | 8 | 100 | 8 | 3 | 10 | 15 | 35 | 241.9 | 3.133 | 2.430 | 0.741 | 0.861 | 204.770 | 3.883 | 3.136 | 0.653 | 0.808 |

4 | 7 | 150 | 8 | 3 | 10 | 13 | 41 | 307.9 | 2.248 | 1.580 | 0.866 | 0.931 | 265.820 | 2.762 | 1.876 | 0.823 | 0.907 |

5 | 8 | 200 | 8 | 3 | 10 | 16 | 37 | 236.8 | 3.223 | 2.649 | 0.725 | 0.851 | 205.790 | 3.859 | 3.152 | 0.693 | 0.832 |

6 | 8 | 150 | 9 | 3 | 10 | 20 | 47 | 261.5 | 2.820 | 2.188 | 0.790 | 0.889 | 207.110 | 3.820 | 3.040 | 0.767 | 0.876 |

7 | 8 | 150 | 10 | 3 | 10 | 19 | 45 | 226.4 | 3.417 | 2.617 | 0.691 | 0.831 | 177.080 | 4.647 | 3.647 | 0.521 | 0.722 |

8 | 8 | 150 | 11 | 3 | 10 | 16 | 45 | 270.8 | 2.692 | 2.071 | 0.807 | 0.898 | 203.860 | 3.905 | 3.186 | 0.647 | 0.804 |

9 | 8 | 150 | 12 | 3 | 10 | 16 | 55 | 330.2 | 2.028 | 1.567 | 0.892 | 0.944 | 286.760 | 2.487 | 2.148 | 0.862 | 0.928 |

10 | 7 | 150 | 8 | 4 | 10 | 25 | 65 | 323.9 | 2.087 | 1.620 | 0.885 | 0.941 | 296.750 | 2.370 | 2.046 | 0.875 | 0.935 |

11 | 7 | 150 | 8 | 5 | 10 | 20 | 83 | 261.0 | 2.830 | 2.314 | 0.787 | 0.887 | 230.950 | 3.329 | 2.750 | 0.763 | 0.873 |

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**MDPI and ACS Style**

Amin, M.N.; Iqbal, M.; Althoey, F.; Khan, K.; Faraz, M.I.; Qadir, M.G.; Alabdullah, A.A.; Ajwad, A.
Investigating the Bond Strength of FRP Rebars in Concrete under High Temperature Using Gene-Expression Programming Model. *Polymers* **2022**, *14*, 2992.
https://doi.org/10.3390/polym14152992

**AMA Style**

Amin MN, Iqbal M, Althoey F, Khan K, Faraz MI, Qadir MG, Alabdullah AA, Ajwad A.
Investigating the Bond Strength of FRP Rebars in Concrete under High Temperature Using Gene-Expression Programming Model. *Polymers*. 2022; 14(15):2992.
https://doi.org/10.3390/polym14152992

**Chicago/Turabian Style**

Amin, Muhammad Nasir, Mudassir Iqbal, Fadi Althoey, Kaffayatullah Khan, Muhammad Iftikhar Faraz, Muhammad Ghulam Qadir, Anas Abdulalim Alabdullah, and Ali Ajwad.
2022. "Investigating the Bond Strength of FRP Rebars in Concrete under High Temperature Using Gene-Expression Programming Model" *Polymers* 14, no. 15: 2992.
https://doi.org/10.3390/polym14152992