# Flexural Strength of Concrete Beams Made of Recycled Aggregates: An Experimental and Soft Computing-Based Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) equal to 0.96 for testing data suggested that their recommended predictive model was good enough. It is worth mentioning that the input parameters of their model comprised the reinforcement ratio, compressive strength of concrete, cement dosage, elastic modulus of concrete, yielding stress of steel, ultimate strength of steel, and load. The deflection was set as the model output in their ANN-based predictive model. They used 82 experimental sets of data to train their intelligent model. Perera et al. [33] utilized ANN to predict the ultimate shear strength of reinforced concrete beams.

## 2. Methods

#### 2.1. Artificial Neural Network

#### 2.2. Particle Swarm Optimization Algorithm

_{1}and c

_{2}are two acceleration constants; ${P}_{i}^{k}$ is the best position of the ith particle up to iteration k; and ${P}_{g}^{k}$ is the best position among all particles in the swarm up to iteration k. In Figure 1, the velocity updating process is shown.

#### 2.3. Imperialist-Competitive-Algorithm-Based ANN

#### 2.4. Hybrid ANNs (PSO-Based ANN and ICA-Based ANN)

## 3. Experimental-Based Dataset

#### Experimental Procedure of the Performed Tests

## 4. Soft Computing Modeling Procedure

#### 4.1. PSO-Based ANN Modeling Procedure

_{1}and C

_{2}, were set as 2 for the primary sensitivity analysis. The primary number of particles was set to 200. Figure 9 shows the influence of the number of iterations on the performance of the PSO-based ANN predictive model. This figure suggests that after 40 iterations, there was a negligible change in MSE; hence, the number of iterations used during the sensitivity analyses was set to 40.

_{1}and C

_{2}, should be identified. For this reason, the effect of the number of particles on the model performance was investigated. For this reason, several models with 100, 150, 200, 300, 400, and 500 particles were run and it was found that the number of particles had no remarkable effect on the model performance, which was in good agreement with the authors’ previous works [47]. On the other hand, to investigate the importance of PSO coefficients, the effect of considering different values for C

_{1}and C

_{2}was taken into consideration. For the purpose of brevity, details on the sensitivity results are not presented here; however, it was found that setting the values of C

_{1}and C

_{2}equal to 2 can lead to an acceptable prediction performance; hence, the aforementioned values were set to 2, as suggested in the literature [48]. It is worth mentioning that when the results of the sensitivity analyses were close to each other, the performance of the testing data was considered. It should be underlined that in all the analyses, 80% of the data were considered for training the predictive model and the remaining data were used to test the developed model. After identifying the proper values for PSO parameters, the best ANN structure should be defined. In other words, the optimal number of hidden nodes should be identified. Researchers often implement two approaches to determine the optimal number of hidden nodes: using suggested equations and/or a trial-and-error procedure. In this study, a trial-and-error procedure was implemented to determine the optimal number of hidden nodes. The effects of 7, 8, 9, 10, 11, and 12 hidden nodes on the model performance was investigated. It was found that when nine hidden nodes were used in the PSO-based ANN predictive model, the model performed the best. For the purpose of brevity, only the results of the PSO-based ANN predictive models with 8 and 9 nodes are presented in Figure 10 and Figure 11, respectively. It is worth mentioning that to reduce the likelihood of accidental results, each model was run five times. Nevertheless, as suggested in Figure 11, the second predictive model coefficient correlation of 0.997 and MSE of 0.08% for testing data outperformed other models. The overall results of different runs with nine hidden nodes showed that the PSO-based ANN predictive model worked well enough, as shown in Figure 10 and Figure 11. The nine-node predictive model outperformed the eight-node predictive model, although the results of the latter were also reliable to some extent. The best results are illustrated in a better way in the Main Results and Discussion section.

#### 4.2. ICA-Based ANN Modeling Procedure

## 5. Main Results and Discussion

## 6. Summary and Conclusions

_{c}, and F

_{y}. Overall, the R-value of 0.997, VAF value of 99.35, and MSE value of 0.08% for the testing data showed that the PSO-based ANN predictive model with nine hidden nodes in one hidden layer was a feasible tool for predicting the ultimate flexural strength of the RRC beams. Despite the promising results, a word of caution is required regarding generalizing the prediction performance of every predictive model. In fact, the reliability of soft computing-based predictive models depends on the quality, quantity, and range of the training data. Hence, when the range of future data is beyond the range of the implemented dataset, the predictability of intelligent models is open to question. In this regard, further research on enhancing the implemented dataset in this study is recommended.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Post-failure behavior and crack propagation of specimens: RC beam (

**left**side) and RRC beam (

**right**side).

**Figure 18.**Predicted values using the recommended predictive models versus the measured values (testing data).

Symbol * | Type | Unit | Minimum | Maximum | Average |
---|---|---|---|---|---|

RCA | Input | % | 0 | 100 | 50 |

B | Input | mm | 100 | 400 | 185 |

d | Input | mm | 160 | 525 | 245 |

a/d | Input | - | 1.92 | 5.14 | 3.57 |

L/d | Input | - | 4.81 | 17.5 | 11.16 |

ρ | Input | - | 0.28 | 2.54 | 1.06 |

f_{c} | Input | MPa | 26.8 | 105.3 | 44 |

F_{y} | Input | MPa | 318 | 640 | 460 |

Mu | Output | kN·m | 8 | 879 | 75 |

_{c}: compressive strength of concrete, F

_{y}: yield strength of steel.

Chemical Composition | L.O.I | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | CaO | SO_{3} | MgO |
---|---|---|---|---|---|---|---|

% | 1.05 | 21.5 | 5.1 | 4.4 | 63.2 | 2.1 | 1.75 |

Model No. | Parameter | Training Data | Testing Data | |||
---|---|---|---|---|---|---|

No. of Countries | No. of Imperialists | |||||

R | MSE | R | MSE | |||

1 | 125 | 10 | 0.954 | 0.049 | 0.875 | 0.049 |

2 | 125 | 5 | 0.979 | 0.0040 | 0.843 | 0.0062 |

3 | 200 | 20 | 0.969 | 0.0035 | 0.984 | 0.008 |

4 | 200 | 10 | 0.976 | 0.0024 | 0.901 | 0.0317 |

5 | 200 | 15 | 0.967 | 0.0043 | 0.968 | 0.0115 |

6 | 250 | 15 | 0.968 | 0.0034 | 0.965 | 0.0201 |

7 | 300 | 30 | 0.976 | 0.0043 | 0.857 | 0.025 |

8 | 300 | 15 | 0.972 | 0.0033 | 0.974 | 0.0082 |

9 | 300 | 20 | 0.982 | 0.0036 | 0.904 | 0.0038 |

10 | 400 | 40 | 0.979 | 0.0045 | 0.856 | 0.0052 |

11 | 400 | 30 | 0.939 | 0.0060 | 0.982 | 0.0075 |

12 | 400 | 20 | 0.981 | 0.0039 | 0.905 | 0.0114 |

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**MDPI and ACS Style**

Momeni, E.; Omidinasab, F.; Dalvand, A.; Goodarzimehr, V.; Eskandari, A.
Flexural Strength of Concrete Beams Made of Recycled Aggregates: An Experimental and Soft Computing-Based Study. *Sustainability* **2022**, *14*, 11769.
https://doi.org/10.3390/su141811769

**AMA Style**

Momeni E, Omidinasab F, Dalvand A, Goodarzimehr V, Eskandari A.
Flexural Strength of Concrete Beams Made of Recycled Aggregates: An Experimental and Soft Computing-Based Study. *Sustainability*. 2022; 14(18):11769.
https://doi.org/10.3390/su141811769

**Chicago/Turabian Style**

Momeni, Ehsan, Fereydoon Omidinasab, Ahmad Dalvand, Vahid Goodarzimehr, and Abas Eskandari.
2022. "Flexural Strength of Concrete Beams Made of Recycled Aggregates: An Experimental and Soft Computing-Based Study" *Sustainability* 14, no. 18: 11769.
https://doi.org/10.3390/su141811769