# Back-Propagation Neural Network Optimized by K-Fold Cross-Validation for Prediction of Torsional Strength of Reinforced Concrete Beam

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## Abstract

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^{2}) are among the evaluation metrics used to assess the performance of the trained model. To elaborate on the superiority of the proposed network models in predicting the torsional strength of RC beams, a parametric study is conducted by comparing the proposed model to three commonly used empirical formulae from existing design codes. The comparative findings of this research study demonstrate that the performance of the BP neural network is highly similar to that of design codes; however, its accuracy is inadequate. After improving the weights and thresholds by k-fold cross-validation and GA, the prediction of the BP neural network shows higher consistency with the actual measured values. The outcome of this study can be used as a theoretical reference for the optimal design of RC beams in practical applications.

## 1. Introduction

## 2. Data Collection and Analysis

## 3. Methodology

#### 3.1. Design Code

#### 3.2. K-Fold Cross-Validation

- Select one of the training folds as the testing dataset.
- The remaining K−1 groups are used as the training set.
- Use the selected training dataset to train the model and evaluate it with the testing dataset.

#### 3.3. BP Neural Network and Genetic Algorithm

#### 3.4. Model Parameter Setting

#### 3.5. Evaluation Metrics

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Historical distributions of parameters (

**a**) RC beam section’s width ($b$); (

**b**) RC beam section’s depth ($h$); (

**c**) Closed stirrup width (${b}^{\prime}$); (

**d**) Closed stirrup depth (${h}^{\prime}$); (

**e**) Compressive strength (${f}_{c}{}^{\prime}$); (

**f**) Longitudinal reinforcement ratio (${\rho}_{l}$); (

**g**) Yield strength of the longitudinal reinforcement (${f}_{yl}$); (

**h**) Transverse reinforcement ratio (${\mathsf{\rho}}_{t}$); (

**i**) Yield strength of transverse reinforcement (${f}_{yt}$); (

**j**) Closed stirrup spacing (s); (

**k**) Torsional strength (${T}_{n}$).

**Figure 8.**Error distribution between different models. (

**a**) Error comparison between BPNN and GA-BPNN. (

**b**) Error comparison between BPNN and optimized BPNN. (

**c**) Error comparison between optimized BPNN and GA-BPNN. (

**d**) Error comparison between GA-BPNN and optimized GA-BPNN. (

**e**) Error comparison between optimized BPNN and optimized GA-BPNN.

**Figure 15.**Radar diagram of calculation results. (

**a**) BP neural network; (

**b**) GA-BP neural network; (

**c**) Optimized BP neural network; (

**d**) Optimized GA-BP neural network; (

**e**) ACI-318-14; (

**f**) BS-8110; (

**g**) TBC-500-2000.

Parameters | Input/Output | Unit | Minimum | Maximum | Average | σ |
---|---|---|---|---|---|---|

Section details | b | mm | 85 | 600 | 265.943 | 124.295 |

h | mm | 178 | 600 | 391.155 | 134.699 | |

b′ | mm | 56.5 | 546 | 219.021 | 112.81 | |

h′ | mm | 149.5 | 549 | 336.241 | 123.514 | |

Concrete | ${f}_{c}$ | MPa | 14.3 | 109.8 | 45.309 | 20.175 |

Longitudinal bar | ${f}_{yl}$ | MPa | 310 | 724 | 437.871 | 121.795 |

${\mathsf{\rho}}_{l}$ | Percentage | 0.18 | 3.89 | 1.370 | 0.980 | |

Transvers bar | ${f}_{yt}$ | MPa | 265 | 715 | 430.422 | 130.735 |

${\mathsf{\rho}}_{t}$ | Percentage | 0.13 | 3.2 | 1.034 | 0.539 | |

s | mm | 41 | 300 | 104.095 | 39.595 | |

Test strength | ${\mathrm{T}}_{\mathrm{u}}$ | kN·m | 2.18 | 239 | 265.943 | 124.295 |

Parameters | PC1 | PC2 | PC3 | PC4 | PC5 | PC6 | PC7 |
---|---|---|---|---|---|---|---|

$b$ | 0.4513 | −0.0733 | −0.1332 | −0.0737 | −0.2801 | 0.1528 | 0.3637 |

$h$ | 0.4105 | −0.1879 | −0.1967 | 0.2998 | 0.2935 | −0.0503 | −0.1717 |

${b}^{\prime}$ | 0.4447 | −0.0665 | −0.14140 | −0.1248 | −0.2906 | 0.2082 | 0.3884 |

${h}^{\prime}$ | 0.4029 | −0.1520 | −0.2235 | 0.3162 | 0.3667 | −0.0109 | −0.2833 |

${f}_{c}{}^{\prime}$ | 0.1861 | 0.4078 | 0.14240 | −0.4710 | 0.6869 | 0.1614 | 0.2243 |

${\mathsf{\rho}}_{l}$ | −0.1359 | 0.4595 | −0.0914 | 0.6491 | 0.1079 | −0.1887 | 0.5298 |

${f}_{yl}$ | 0.2955 | 0.4469 | 0.2218 | 0.0426 | −0.2801 | −0.1148 | −0.1564 |

${\mathsf{\rho}}_{t}$ | −0.2355 | 0.2810 | −0.5069 | 0.0827 | −0.0536 | 0.7372 | −0.1903 |

${f}_{yt}$ | 0.2680 | 0.4601 | 0.2442 | 0.0754 | −0.2099 | 0.0434 | −0.4526 |

$s$ | 0.0066 | −0.2531 | 0.6923 | 0.3522 | 0.0862 | 0.5572 | 0.1033 |

Building Standard | Expression for Torsional Strength | Reference |
---|---|---|

ACI-318-14 | ${T}_{n}=\frac{2{A}_{o}{A}_{t}{f}_{yt}}{s}cot\mathsf{\theta}$ | ${A}_{t}$—the area of one leg of a closed stirrup resisting torsion. ${A}_{o}$—the gross area enclosed by shear flow path ${f}_{yt}$—characteristic strength of the links ${A}_{e}$—the cross-sectional area of the surrounding stirrups $\mathsf{\theta}$—the torsional angel |

BS-8110 | ${T}_{n}=\frac{0.8{b}^{\prime}{h}^{\prime}\left(0.87{f}_{yt}\right){A}_{t}}{s}$ | |

TBC-500-2000 | ${T}_{n}=\frac{2{A}_{o}{A}_{e}{f}_{yt}}{2\left({b}^{\prime}+{h}^{\prime}\right)}$ |

Evaluation Metric | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

MAE (kN·m) | 8.430 | 14.536 | 10.930 | 16.924 | 14.549 | 7.059 | 4.511 | 6.072 | 5.727 | 4.737 |

MSE $({\mathrm{kN}}^{2}\xb7{\mathrm{m}}^{2})$ | 126.362 | 333.467 | 230.512 | 1144.994 | 625.964 | 82.497 | 542.109 | 289.448 | 63.630 | 51.052 |

RMSE (kN·m) | 11.241 | 18.261 | 15.183 | 33.838 | 25.019 | 9.083 | 23.283 | 17.013 | 7.977 | 7.145 |

MAPE (%) | 33.530 | 33.431 | 38.752 | 30.469 | 19.198 | 27.160 | 16.307 | 17.697 | 45.789 | 18.477 |

${\mathrm{R}}^{2}$ | 0.945 | 0.896 | 0.756 | 0.840 | 0.776 | 0.968 | 0.979 | 0.952 | 0.952 | 0.979 |

Models | MAE (kN·m) | MSE $(\mathrm{k}{\mathrm{N}}^{2}\xb7{\mathrm{m}}^{2})$ | RMSE (kN·m) | MAPE (%) | ${\mathrm{R}}^{2}$ |
---|---|---|---|---|---|

BP neural networks | 11.548 | 315.363 | 17.758 | 40.117 | 0.846 |

GA-BP neural networks | 9.109 | 240.046 | 15.493 | 19.798 | 0.887 |

Optimized BP neural networks | 7.063 | 103.100 | 10.154 | 18.957 | 0.943 |

Optimized GA-BP neural networks | 6.742 | 103.988 | 10.197 | 16.251 | 0.950 |

ACI-318-14 | 17.832 | 320.016 | 17.889 | 20.205 | 0.867 |

BS-8110 | 19.700 | 344.436 | 18.559 | 34.507 | 0.856 |

TBC-500-2000 | 24.154 | 842.799 | 29.031 | 54.252 | 0.756 |

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## Share and Cite

**MDPI and ACS Style**

Lyu, Z.; Yu, Y.; Samali, B.; Rashidi, M.; Mohammadi, M.; Nguyen, T.N.; Nguyen, A.
Back-Propagation Neural Network Optimized by K-Fold Cross-Validation for Prediction of Torsional Strength of Reinforced Concrete Beam. *Materials* **2022**, *15*, 1477.
https://doi.org/10.3390/ma15041477

**AMA Style**

Lyu Z, Yu Y, Samali B, Rashidi M, Mohammadi M, Nguyen TN, Nguyen A.
Back-Propagation Neural Network Optimized by K-Fold Cross-Validation for Prediction of Torsional Strength of Reinforced Concrete Beam. *Materials*. 2022; 15(4):1477.
https://doi.org/10.3390/ma15041477

**Chicago/Turabian Style**

Lyu, Zhaoqiu, Yang Yu, Bijan Samali, Maria Rashidi, Masoud Mohammadi, Thuc N. Nguyen, and Andy Nguyen.
2022. "Back-Propagation Neural Network Optimized by K-Fold Cross-Validation for Prediction of Torsional Strength of Reinforced Concrete Beam" *Materials* 15, no. 4: 1477.
https://doi.org/10.3390/ma15041477