# Interpretable Machine Learning-Based Prediction Model for Concrete Cover Separation of FRP-Strengthened RC Beams

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## Abstract

**:**

## 1. Introduction

## 2. Workflow

_{s}), beam width (b), concrete strength (f′

_{c}), area of the longitudinal reinforcement (A

_{st}), yield strength of the reinforcement (f

_{sy}), cross-sectional area of the FRP (A

_{f}), Young’s modulus of FRP (E

_{f}), ratio of the design moment to the design shear at the end of the FRP (M*/V*), contribution of the hoop reinforcement to the shear force (V

_{us}), and contribution of the concrete to the shear force (V

_{uc}). The output parameter is the shear force at the end of the FRP (V*) when CCS failure occurs.

## 3. Dataset Construction

#### 3.1. Parameter Selection Criteria

- (1)
- The failure mode of all beams is CCS, and there are no other modes.
- (2)
- The geometrical characteristics and parameters of the beams are described in detail.
- (3)
- The FRP sheets were not pre-stressed.

#### 3.2. Inputs and Outputs

_{s}), beam width (b), concrete strength (f′

_{c}), area of the longitudinal reinforcement (A

_{st}), yield strength of the reinforcement (f

_{sy}), cross-sectional area of the FRP (A

_{f}), Young’s modulus of FRP (E

_{f}), ratio of the design moment to the design shear at the end of the FRP (M*/V*), contribution of the hoop reinforcement to the shear force (V

_{us}), and contribution of the concrete to the shear force (V

_{uc}). The output parameter is the shear force at the end of the FRP (V*) when CCS failure occurs.

#### 3.3. Description of the Dataset

_{st}, A

_{f}, M*/V*, V

_{us}, and V

_{uc}are more concentrated, while the distributions of the other parameters are more discrete. From Figure 2, it can be seen that the correlation between most of the indicators is weak, except that there is a large correlation between A

_{1}and A

_{9}and A

_{10}. Therefore, the exclusion of A

_{1}(d

_{s}) is taken into account in the modeling.

## 4. Machine Learning Models

#### 4.1. Linear Regression

_{0}is the intercept, β

_{1}through β

_{k}are the coefficients of the independent variables, and ε is the error term.

#### 4.2. Support Vector Regression

#### 4.3. Backpropagation Neural Network

#### 4.4. Decision Tree

#### 4.5. Random Forest

#### 4.6. XGBoost

#### 4.7. Shapley Additive Explanation

## 5. Results and Discussion

#### 5.1. Machine Learning Model Construction

#### 5.2. Performance Criteria

^{2}), and the root mean square error (RMSE), and the expressions for R

^{2}and RMSE are as follows:

#### 5.3. Machine Learning Model Evaluation

^{2}, and RMSE of each machine learning model on the training and test sets using Taylor diagrams.

^{2}and the lowest RMSE on both the training and test sets. LR, SVR, and RF have better performance on the training set, but perform poorly on the test set, and the generalization ability needs to be improved.

#### 5.4. Existing Model Evaluation

^{2}and the coefficient of variation (CV), and the relationship between the calculated and experimental values of the code-suggested models is shown in Figure 6, and the relationship between the calculated and experimental values of the researcher-suggested models is shown in Figure 7.

^{2}(0.95) and the lowest CV (16%), which is better than the models proposed by the researchers and codes in Figure 6 and Figure 7.

## 6. Parametric Study

_{c}and V

_{us}have a large effect on V*, and A

_{st}and b have a smaller effect on V*. The sensitivities between each parameter and V* are also shown in SHAP, which is shown in Figure 10.

_{uc}is small (<15), V

_{uc}and V* have an approximately linear relationship, but as V

_{uc}increases, it does not cause a significant change in V*. There is an inverse relationship between V* and f

_{sv}, but the trend is not significant. In addition, when f′

_{c}is small (<50 MPa), the relationship between V* and f′

_{c}is not obvious, and when f′

_{c}is large, as f′

_{c}increases, V* decreases. There is no obvious relationship between V

_{us}and V*.

## 7. Conclusions

- (1)
- Of all the machine learning models, XGBoost is the best at predicting CCS, with a better distribution of deviations on both the training and test sets. In addition, the XGBoost model also has the maximum goodness-of-fit, the minimum standard deviation, and the minimum root mean square error on both the training and test sets.
- (2)
- The models proposed by AS and TR55 overestimated the shear force during CCS, while the models of ACI, fib, and most researchers are conservative. In addition, the R
^{2}and CV of these models are not satisfactory. Compared to the above models, XGBoost has a higher R^{2}(0.95) and a lower CV (16%). - (3)
- The parameters that have a greater influence on V* are the contribution of the concrete to the shear force, the yield strength of the reinforcement, the concrete strength, and the contribution of the hoop reinforcement where V* is approximately proportional to the contribution of the concrete to the shear force and approximately inversely proportional to the yield strength of the reinforcement and the concrete strength.
- (4)
- In this study, the parameters affecting CCS failure were statistically analyzed based on SHAP. However, mechanism-based analyses are scarce and further research is needed in the future.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Parameter | d_{s} (A_{1}) | B (A_{2}) | f′_{c} (A_{3}) | A_{st} (A_{4}) | f_{sy} (A_{5}) | A_{f} (A_{6}) | E_{f} (A_{7}) | M*/V* (A_{8}) | V_{us} (A_{9}) | V_{uc} (A_{10}) |
---|---|---|---|---|---|---|---|---|---|---|

Min. | 69 | 100 | 19 | 57 | 350 | 13 | 10 | 0 | 3 | 4 |

Max. | 375 | 400 | 80 | 1272 | 611 | 912 | 271 | 550 | 491 | 182 |

Average | 176 | 139 | 42 | 224 | 481 | 120 | 185 | 128 | 89 | 34 |

Standard deviation | 47% | 35% | 53% | 18% | 79% | 13% | 68% | 23% | 18% | 19% |

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**MDPI and ACS Style**

Zheng, S.; Hu, T.; Yu, Y.
Interpretable Machine Learning-Based Prediction Model for Concrete Cover Separation of FRP-Strengthened RC Beams. *Materials* **2024**, *17*, 1957.
https://doi.org/10.3390/ma17091957

**AMA Style**

Zheng S, Hu T, Yu Y.
Interpretable Machine Learning-Based Prediction Model for Concrete Cover Separation of FRP-Strengthened RC Beams. *Materials*. 2024; 17(9):1957.
https://doi.org/10.3390/ma17091957

**Chicago/Turabian Style**

Zheng, Sheng, Tianyu Hu, and Yong Yu.
2024. "Interpretable Machine Learning-Based Prediction Model for Concrete Cover Separation of FRP-Strengthened RC Beams" *Materials* 17, no. 9: 1957.
https://doi.org/10.3390/ma17091957