# Neural Networks—Deflection Prediction of Continuous Beams with GFRP Reinforcement

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Relationship between deflections values obtained in experimental research and predicted deflections values from training and test sets.

**Table 1.**Some of the equations for calculating the deflection of continuous beams loaded with concentrated forces in the middle of the span.

Equation | Comment |
---|---|

$$\Delta =\frac{7}{768}\cdot \frac{P{L}^{3}}{{E}_{c}{I}_{e}}$$
| Equation derived from elastic analysis |

$${\Delta}_{\mathrm{max}}=\frac{P{L}^{3}}{48{E}_{c}{I}_{cr}}\left(\frac{5}{16}-\frac{15}{8}\left(1-\frac{{I}_{cr}}{{I}_{g}}\right)\cdot {\left(\frac{{L}_{g}}{L}\right)}^{3}\right)$$
| The deflection calculation according to CSA S806-12 [2] is based on the moment–curve ratio along the span |

Beam | Longitudinal Reinforcement Support—Upper Zone | Longitudinal Reinforcement Midspan—Lower Zone | Reinforcement | Cross-Section |
---|---|---|---|---|

B1 | 3Φ14 | 2Φ12 + 1Φ10 | GFRP polyester—wrapped | |

B2 | 2Φ10 + 1Φ14 | 2Φ12 + 1Φ14 | GFRP polyester—wrapped | |

B3 | 2Φ10 + 1Φ12 | 2Φ14 + 1Φ12 | GFRP polyester—wrapped | |

B4 | 3Φ12 | 3Φ10 | GFRP polyester—wrapped | |

B5 | 2Φ9 + 1Φ12 | 2Φ12 + 1Φ9 | GFRP polyester—wrapped | |

B6 | 2Φ9 + 1Φ10 | 2Φ12 + 1Φ10 | GFRP polyester—wrapped | |

B7 | 3Φ7 | 2Φ6 + 1Φ7 | GFRP polyester—wrapped | |

B8 | 2Φ6 + 1Φ7 | 2Φ7 + 1Φ6 | GFRP polyester—wrapped | |

B9 | 4Φ12 | 2Φ10 + 2Φ9 | GFRP epoxy—ribbed (deformed) | |

B10 | 3Φ10 + 1Φ12 | 2Φ12 + 2Φ9 | GFRP epoxy—ribbed (deformed) | |

B11 | 3Φ9 | 3Φ12 + 1Φ10 | GFRP epoxy—ribbed (deformed) |

Input Data Number | Input Data Description | Data Type | Units of Measure | Min | Max | Mean Value |
---|---|---|---|---|---|---|

Input 1 | Force | numerical | kN | 5 | 137.75 | 55.571 |

Input 2 | Effective moment of inertia of the cross-section in the middle of the span | numerical | mm^{4} | 328.0033 | 19,531.25 | 2011.399 |

Input 3 | Effective moment of inertia of the cross-section above the support | numerical | mm^{4} | 290.1866 | 19,531.25 | 1965.198 |

Output Data Number | Output Data Description | Data Type | Units of Measure | Min | Max | Mean Value | |
---|---|---|---|---|---|---|---|

Output 1 | Deflection | numerical | mm | 0.11 | 31.20 | 9.703 | $\left(-\infty ,+\infty \right)$ |

Function | Mark | Explanation | Range |
---|---|---|---|

Identity | $x$. | ly in the output layer | $\left(-\infty ,+\infty \right)$ |

Rectified Linear units | $max\left(0,x\right)$ | Activation of neurons is transmitted directly as an output if it is positive, and if it is negative, 0 is transmitted. It has been proven to have 6 times better convergence compared to the hyperbolic tangent function. | $\left(0,+\infty \right)$ |

Hyperbolic tangent | $\frac{2}{\left(1+{e}^{-2x}\right)}-1$ | Activation of neurons is transmitted directly as an output if it is positive, and if it is negative, 0 is transmitted. | $\left(-1,1\right)$ |

Swish | $x\ast sigmoid\left(x\right)$ | A function that is nonlinearly interpolated between a linear and a ReLu function. | $\left(0,x\right)$ |

Model Name | Model Characteristics | Activation Function of Hidden Layers | Activation Function of Output Layer | MAPE Training Set [%] | MAPE Test Set [%] |
---|---|---|---|---|---|

NND4 | MLP 3-7-14-3-1 | ReLu | Identity | 9.89 | 13.40 |

NND5 | MLP 3-7-13-3-1 | Tanh | Identity | 13.82 | 13.55 |

NND6 | MLP 3-7-9-3-1 | Swish | Identity | 13.48 | 13.45 |

NND8 | MLP 6-7-7-3-1 | ReLu | Identity | 11.75 | 12.61 |

**Table 7.**Models of artificial neural networks for continuous beam deflection estimate (StandardScalar).

Model Name | Model Characteristics | Activation Function of Hidden Layers | Activation Function of Output Layer | MAPE Training Set [%] | MAPE Test Set [%] |
---|---|---|---|---|---|

NND1 | MLP 3-7-15-3-1 | ReLu | Identity | 10.62 | 21.34 |

NND2 | MLP 3-7-12-3-1 | Tanh | Identity | 14.61 | 11.83 |

NND3 | MLP 3-7-15-3-1 | Swish | Identity | 8.56 | 9.00 |

NND7 | MLP 3-7-8-3-1 | ReLu | Identity | 8.97 | 17.84 |

**Table 8.**Models of artificial neural networks with random choice of data for continuous beam deflection estimate (kFold-CrossValidation, k = 10).

Model Name | Data Scaling Procedure | Model Characteristics | Activation Function of Hidden Layers | Activation Function of Output Layer | MAPE [%] | σ [%] |
---|---|---|---|---|---|---|

NND9 | StandardScalar | MLP 3-7-13-3-1 | ReLu | Identity | 33.61 | 15.22 |

**Table 9.**Models of artificial neural networks with random choice of data for continuous beam deflection estimate (LOOCV).

Model Name | Data Scaling Procedure | Model Characteristics | Activation Function of Hidden Layers | Activation Function of Output Layer | MAPE Training Set [%] | MAPE Test Set [%] |
---|---|---|---|---|---|---|

NND10 | Min-Max | MLP 3-7-14-3-1 | ReLu | Identity | 11.05 | 13.50 |

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**MDPI and ACS Style**

Beljkaš, Ž.; Baša, N.
Neural Networks—Deflection Prediction of Continuous Beams with GFRP Reinforcement. *Appl. Sci.* **2021**, *11*, 3429.
https://doi.org/10.3390/app11083429

**AMA Style**

Beljkaš Ž, Baša N.
Neural Networks—Deflection Prediction of Continuous Beams with GFRP Reinforcement. *Applied Sciences*. 2021; 11(8):3429.
https://doi.org/10.3390/app11083429

**Chicago/Turabian Style**

Beljkaš, Željka, and Nikola Baša.
2021. "Neural Networks—Deflection Prediction of Continuous Beams with GFRP Reinforcement" *Applied Sciences* 11, no. 8: 3429.
https://doi.org/10.3390/app11083429