# An Artificial Network-Based Prediction of Key Reference Zones on Axial Stress–Strain Curves of FRP-Confined Concrete

^{1}

^{2}

^{*}

## Abstract

**:**

_{h,rup}) and transition zone, namely transition strain (ε

_{c1}) and stress (f’

_{c1}), on axial stress–strain curves of FRP-confined concrete. These are key parameters for estimating a transition zone of stress–strain curves. In this study, accompanied with these parameters, ultimate condition parameters, including compressive strength and ultimate axial strain, were predicted using a comprehensive database. Various combinations of input data and ANN parameters were used to increase the accuracy of the predictions. A sensitivity analysis and a model validation assessment were performed to evaluate the predictability of the developed models. At the end, a comparison between the proposed models in this study and existing ANN and design-oriented models was presented. It is shown that the accuracy of the developed ANN models in this study is higher or comparable to that of existing ANN models. Additionally, the developed models in this study to predict f’

_{c1}and ε

_{c1}exhibit a higher accuracy compared to existing design-oriented models. These results indicate that the proposed ANN models capture the lateral confinement effect on ultimate and transition zones of FRP-confined concrete with a more robust performance compared to existing models.

## 1. Introduction

_{cc}) and axial transition stress (f’

_{c1}) [4,9]. However, they could not accurately estimate the strain corresponding to f’

_{cc}(ε

_{cu}) [4,9]. Furthermore, the existing expressions for ε

_{cu}prediction used experimental hoop rupture strain (ε

_{h,rup}). However, ε

_{h,rup}does not readily exist in designing procedures. Additionally, Bisby et al. [14,15] discussed that the reported experimental ε

_{h,rup}had an inconsistency. In addition, some of the experimental data could not follow the trend of ε

_{h,rup}. They discussed that this is partly due to location of shear planes and using strain gauges as a measurement method [3,14,15]. Similar to ε

_{cu}and ε

_{h,rup}, accurate estimation of axial strain at the transition zone (ε

_{c1}), as another key reference strain, was not possible using existing proposed design-oriented expressions [4]. These observations in the performed literature review indicate that more investigation is essential on the prediction of significant reference strains, namely ε

_{cu}, ε

_{h,rup}, and ε

_{c1}.

_{cu}and f’

_{cc}of confined concretes with FRP. Different studies used ANN and GP to estimate ε

_{cu}and f’

_{cc}(e.g., [16,23,24,25,26,27]). It was reported that an ANN was able to predict f’

_{cc}more accurately compared to traditional approaches [16]. Wu et al. [28] used the other type of neural network, i.e., radial basis function network, to predict f’

_{cc}. Cascardi et al. [29] determined f’

_{cc}by predicting the efficiency of a FRP jacket using an ANN technique. Recently, Isleem et al. [30] used ANN to build a confinement model to predict the ultimate condition of FRP-confined concrete. However, their model was limited to the test results obtained in their study. Jiang et al. [16] predicted ε

_{cu}in addition to f’

_{cc}using an ANN accompanied with the prediction of the stress–strain relationship. Similar to the concretes with a circular cross-section, the ultimate condition of the concretes with square and rectangular cross-sections were also examined by ANN analysis in different studies [31,32,33]. Nonetheless, the literature review revealed that no research has examined the prediction of a transition zone using an ANN. Existing models were either complex, dealt with a small number of databases, or ignored key influential input parameters on the behavior of FRP-confined concrete. The availability of the five key reference parameters (i.e., ε

_{cu}, f’

_{cc}, ε

_{c1}, f’

_{c1}, and ε

_{h,rup}) helps to model the whole curve of the FRP-confined concrete accurately, as was explained by Fallah Pour et al. [4]. An ANN can handle a complex database with large variables, identify sensitivity of input and output parameters, and establish relations between the input and output variables.

## 2. Experimental Test Database

## 3. Brief Overview of an Artificial Neural Network (ANN)

_{i}), and a bias (b) is added to this value for each neuron, as shown in Equation (1).

_{1}, x

_{2}, …, x

_{m}are input vectors, w

_{i1}, w

_{i2}, …, w

_{im}are the weight of neuron I, and b

_{i}is bias.

_{i}is needed. This nonlinear process is called transfer (activation) function and is shown in Equation (2).

## 4. Optimal ANN Selection

#### 4.1. Configuration of a Neural Network

#### 4.2. Training of Network

^{−10}, and infinitive time was fixed for maximum time for the training procedure. At the end, min-max normalization methods were used for all the ANN analyses.

#### 4.3. ANN Network

#### 4.4. Predictability Analysis

## 5. ANN-Based Prediction Models

#### 5.1. Ultimate Condition

_{cu}, f’

_{cc}, and ε

_{h,rup}. Determination of this point offers a potential instrument to designers for predicting the maximum resistance of the system [42]. It should be noted that the corresponding strain to unconfined concrete strength (f’

_{co}), i.e., ε

_{co}, has been considered as an input variable and was calculated by the proposed expression by Ref. [10] (i.e., ${\epsilon}_{c\mathrm{o}}=\frac{f{\prime}_{c0}^{0.225}}{1000}\text{}{\left(\frac{152}{D}\right)}^{0.1}{\left(\frac{2\text{}D}{H}\right)}^{0.13}$, where D and H are diameter and height of the specimen). This lowered the dependency of the input variables to experimental values and made the input variables readily available. This was the main purpose of this study to develop an accurate model, which uses readily available input data with simplicity to apply.

#### 5.1.1. f’_{cc}

_{cc}. This was accompanied with the investigation of the correlation between input variables to avoid multicollinearity.

_{co}and ε

_{co}. As ε

_{co}is a function of f’

_{co}, only f’

_{co}was used to predict f’

_{cc}in this study.

_{cc}for various sets of input variables. As shown, the Levenberg–Marquardt and Bayesian regularization algorithms showed the lowest AAE compared to the scaled conjugate gradient. Moreover, the hidden layer with 25 or 30 neurons generally developed the lowest AAE value, although some exceptions can be observed, especially for the Levenberg–Marquardt algorithm. It is also observed that the Bayesian regularization algorithm had the lowest AAE and RMSE among all the studied cases, and the accuracy of the prediction by this algorithm increased by increasing the input variables number. Additionally, a third set of input variables offered lower values of AAE compared to other analyzed sets. As a result, the third set of variables was selected as best combination of variables.

_{l}/f’

_{co}, as normalized lateral stiffness, was also considered as an input variable, although f’

_{co}and K

_{l}(lateral stiffness) existed in the input variables set. As can be seen in Table 2, the correlation coefficient for K

_{l}and K

_{l}/f’

_{co}was 0.747. K

_{l}/f’

_{co}was added to the variable set as an independent variable due to its significance in the mechanical characteristic of FRP-confined concrete. Equation (5) presents the best composition of input variables to obtain the highest accuracy for prediction of f’

_{cc}.

_{f}is the FRP total thickness, ε

_{fu}is the fiber ultimate tensile strain, E

_{f}is FRP elastic modulus, and f

_{f}is fiber ultimate tensile strength.

_{cc}prediction. It should be noted that all the analyzed cases were not presented in this table due to the similarity of results. Furthermore, a scaled conjugate gradient algorithm was not studied in this step due its lower accuracy compared to the other algorithms. According to Table 4, two hidden layers with 15 neurons in first hidden layer and 10 neurons in second hidden layer using the Bayesian regularization algorithm developed the most accurate prediction for f’

_{cc}. Comparing the most accurate model with one hidden layer (presented in Table 3) to that with two hidden layers (presented in Table 4) revealed that a more accurate prediction was obtained by two hidden layers; however, the accuracy improvement was not notably significant.

_{cc}prediction is shown in Table 5. Due to the highest accuracy obtained by the Bayesian regularization algorithm, the influence of the transfer function was examined using this algorithm. Various transfer functions, such as tansig, logsig, pureline, Elliot sigmoid (elliotsig), and triangular basis (tribas), were studied. It should be noted that various combinations of a transfer function were initially studied to find the highest accuracy for f’

_{cc}prediction. However, only a few of them are displayed in Table 5 due to the similarity of results. According to the table, tansig was the transfer function developing the most accurate prediction for f’

_{cc}. Table 6 displays the final ANN parameters for f’

_{cc}model.

_{cc}is illustrated in Figure 4. According to Figure 4a, experimental and predicted f’

_{cc}had a close consistency. In this figure, a 45° line was added, which is a representative of the perfect agreement. Based on Figure 4b, the best match with the imposed goal for accuracy (MSE) was observed at epoch’s number 833. K-fold cross-validation was applied on the final ANN model for prediction of f’

_{cc}. This analysis was performed to assess the machine learning performance on unseen data. In this analysis, the database was randomly split in a K division. The popular value for K ranges was 5 to 10. One of the K portions of the database was considered as a test dataset, and all remaining K-1 portions were considered as a training dataset. The ANN analysis was performed and the obtained statistical indicators, which showed the model performance, were kept. This procedure was repeated for K times, and a comparison between the obtained statistical indicators was made. By this procedure, each individual dataset could be used at least one time as a test dataset, and K-1 time as a training dataset. It should be noted that, in this study, AAE was used to compare the performance of the models using various training and test datasets, i.e., K-fold cross-validation. Figure 4c illustrates the obtained AAE by K-fold cross-validation. In this figure, lower limit, upper limit, and average of the obtained AAE were shown. Maximum AAE was obtained at 10.2%, and the minimum value was obtained at 6.4%. This indicates that dataset selection caused a variation of accuracy in the developed model.

#### 5.1.2. ε_{cu}

_{cc}was performed to find the best ANN architecture for predicting ε

_{cu}. The obtained outcome for independent variables variations and the number of layers is presented in Table 7. Based on the table, the combination of variables with the Bayesian regularization algorithm offered the highest accuracy compared to the other algorithms. Equation (6) presents the best composition of input variables to obtain the highest accuracy for the prediction of ε

_{cu}.

_{2}is strain enhancement coefficient proposed by Fallah Pour et al. [4]. As shown in Table 2, simultaneous use of f’

_{co}and ε

_{co}could lead to multicollinearity in ε

_{cu}model. However, the obtained results of ε

_{cu}showed that using both parameters in predicting the model resulted in a more accurate prediction.

_{cu}, are presented in Table 6. According to the table, the Levenberg–Marquardt algorithm and the tansig transfer function with two hidden layers of six and eight neurons offered the most accurate and simple ANN model for predicting ε

_{cu}. Cross-validation analysis was carried out on the final ANN map. The developed model performance for ε

_{cu}is shown in Figure 5. Based on Figure 5a, the predicted ε

_{cu}values were close to experimental ε

_{cu}values. Figure 5b exhibits that the considered goal as limit for BP neural network analysis of ε

_{cu}reached at epoch of 95. Additionally, it is shown in Figure 5c that, similar to f’

_{cc}, the selection of the dataset exhibited a major influence on the ε

_{cu}model accuracy.

#### 5.1.3. ε_{h,rup}

_{cc}and ε

_{cu}was followed to determine the most accurate BP neural network model for ε

_{h,rup}prediction. Based on the obtained results, input variables presented in Equation (7) with the Bayesian regularization algorithm and 25 neurons developed the most accurate predictions of ε

_{h,rup}.

_{cc}and ε

_{cu}, applying tansig as the transfer function led to the development of the most accurate prediction of ε

_{h,rup}when compared to other transfer functions. Figure 6 exhibits the developed ε

_{h,rup}model performance. Based on Figure 6a, predicted ε

_{h,rup}had close values to experimental ε

_{h,rup}values. In Figure 6b, the best match between accuracy criteria for the ANN model and the least difference between train and validation datasets was at epoch of 395. Figure 6c displays that, similar to f’

_{cc}and ε

_{cu}, the performance of the ANN depended significantly on the selection of different types of datasets, i.e., train, test, and validation.

#### 5.2. Transition Zone

_{c1}, ε

_{c1}) determines the point in which the first ascending nonlinear segment shifts toward the second ascending quasi-linear segment on the curve. As explained for the ultimate condition, using a similar database to the previous work of this research group offers an ability to compare the developed models with existing best performing models.

#### 5.2.1. f’_{c1}

_{c1}using an ANN analysis.

_{co}and ε

_{co}had maximum correlation, and the best prediction was obtained when only f’

_{co}was considered. As shown in Table 6, one hidden layer having 25 neurons with the Bayesian regularization algorithm and tansig as a transfer function was considered as the final ANN prediction parameters for f’

_{c1}. Although an increasing hidden layers number could cause increasing the f’

_{c1}prediction accuracy, the improvement of the accuracy was not significant.

_{c1}is illustrated in Figure 7. Figure 7a exhibits that the values recorded by experiments and model predictions had a good agreement. Figure 7b illustrates that the accuracy goal in f’

_{c1}analysis was met at epoch of 136. Figure 7c illustrates the K-fold cross-validation for f’

_{c1}prediction. As can be seen, the variation of test datasets did not significantly influence the accuracy of the f’

_{c1}model.

#### 5.2.2. ε_{c1}

_{c1}by variation of input data was obtained by using 30 neurons in one hidden layer and the Levenberg–Marquardt algorithm with the tansig transfer function. An in-depth investigation on the correlation between input variables was performed similar to the other prediction models, and the obtained result was similar to that of ε

_{cu}and ε

_{h,rup}. The obtained results revealed that the Pearson’s correlation for f’

_{co}and ε

_{co}had the highest correlation coefficient, and only one of these input variables should be used to avoid multicollinearity, as illustrated in Table 8. However, the maximum accuracy was obtained when both f’

_{co}and ε

_{co}were used. Equation (9) presents the input parameters, which offered the highest accuracy for ε

_{c1}prediction.

_{c1}. Furthermore, analysis of the transfer function showed that tansig offered the most accurate ε

_{c1}. The performance of the developed model for ε

_{c1}is shown in Figure 8. Based on Figure 8a, similar to other key reference points, the predicted values of ε

_{c1}were in good consistency with the recorded values in the experiments. Based on Figure 8b, the imposed criteria on MSE in the analysis was respected early in the calculation. Finally, as per Figure 8c, AAE varied significantly by the variation of test dataset of ε

_{c1}.

## 6. Model Verification

#### 6.1. Sensitivity Analysis

_{i}is ith of the kth dependent variable, and $\overline{\mu}$ is average of kth dependent variable. The results of sensitivity analysis for assessing the impact of each input on the ultimate condition predictions are observed in Figure 9. Based on Figure 9a, K

_{l}and f’

_{co}had highest impact on predicting f’

_{cc}. Moreover, all the input variables exhibited a positive influence on f’

_{cc}, except f

_{f}and ε

_{fu}. Conversely, ε

_{cu}prediction was significantly influenced by ε

_{fu}, according to Figure 9b. Additionally, K

_{l}/f’

_{co}, t

_{f}, and k

_{2}developed a positive influence, the same as ε

_{fu}, but other input variables showed a negative effect on ε

_{cu}prediction. It should be mentioned that D had a negative influence on the ε

_{cu}prediction. According to Figure 9c, the only parameter with a positive influence on ε

_{h,rup}prediction was t

_{f}, which was physically expected. Other input data, including K

_{l}, E

_{f}, and f’

_{co}, showed a negative influence. Moreover, height and diameter slightly influenced ε

_{h,rup}prediction, and their influence was negative.

_{c1}prediction. Moreover, f’

_{co}, K

_{l}, and t

_{f}were the three parameters with the highest influence on f’

_{c1}prediction, and their influences were positive. Additionally, E

_{f}, f

_{f}, and K

_{l}/f’

_{co}negatively influenced the f’

_{c1}prediction. According to Figure 10b, there were two input variables of E

_{f}and f

_{f}, which negatively influenced the prediction of ε

_{c1}, and other variables had a positive influence. It should be noted again that D as a geometry parameter slightly influenced ε

_{c1}prediction, similar to f’

_{c1}and ε

_{h,rup}and opposite to ε

_{cu}.

#### 6.2. Model Validation

_{c1}where one of the criteria, i.e., R

_{m}, was not satisfied. Figure 11 shows the box plot of f’

_{c1}and ε

_{c1}of this study. These parameters were selected as representative of the results. As can be seen in the figure, there were only a few samples where their predictions were in out-layers.

## 7. Comparison of the Proposed and Existing Models

_{cc}, and only two research studies focused on ε

_{cu}. In this section, developed predictions of the ultimate condition by an ANN were compared using R

^{2}. The accuracy of the transition zone prediction was evaluated by comparing the proposed model with other design-oriented models.

#### 7.1. Ultimate Condition

_{cc}prediction by the developed model and that by existing ANN models (e.g., [48,49]). As can be seen, although a larger number of datasets was used in this study, the ANN model developed in this study offered a higher accuracy than the existing ANN models.

_{cu}is presented in Table 11. Based on the table, the proposed model offered a comparable accuracy while the number of analyzed datasets was largely compared to existing ANN models.

_{cc}(e.g., [50,51,52,53,54,55]) and ε

_{cu}(e.g., [56,57,58,59]), respectively. Based on the tables, the model by Fallah Pour et al. [4] had the highest accuracy among existing models in predicting f’

_{cc}. In addition, the model by Lim and Ozbakkaloglu [9] had the highest accuracy among existing models in predicting ε

_{cu}. However, the proposed ANN models in this study had a higher accuracy than these best performing models in predicting f’

_{cc}and ε

_{cu}.

#### 7.2. Transition Zone

_{c1}and ε

_{c1}. Based on these tables, the models by Fallah Pour et al. [4] had the highest accuracy among existing models in predicting f’

_{c1}and ε

_{c1}. However, the proposed ANN models in this study offered a higher accuracy than these best performing models in predicting the transition zone.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

E_{f} | elastic modulus of fiber |

f_{f} | ultimate tensile strength of fiber |

D | diameter of FRP-confined concrete |

Ɛ_{f} | ultimate tensile strain of fiber |

f’_{co} | compressive strength of unconfined concrete |

Ɛ_{co} | axial strain of unconfined concrete at f’_{co} |

t_{f} | thickness of FRP tube |

K_{l} | lateral stiffness |

K_{l}/f’_{co} | normalized lateral stiffness |

f’_{cc} | ultimate strength of FRP-confined concrete |

Ɛ_{cu} | ultimate strain of FRP-confined concrete |

Ɛ_{h,rup} | strain of FRP tube at rupture |

f’_{c1} | axial strength of FRP-confined concrete at transition zone |

Ɛ_{c1} | axial strain of FRP-confined concrete at transition zone |

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**Figure 1.**Frequency distribution of f’

_{co}and K

_{l}in prepared database for: (

**a**) f’

_{cc}, (

**b**) ε

_{cu}, (

**c**) ε

_{h,rup}, (

**d**) f’

_{c1}, and (

**e**) ε

_{c1}.

**Figure 4.**Performance of a developed model for compressive strength (f’

_{cc}): (

**a**) Comparison of model prediction and experimental values, (

**b**) performance accuracy, and (

**c**) cross-validation analysis.

**Figure 5.**Performance of a developed model for ultimate axial strain (ε

_{cu}): (

**a**) Comparison of model prediction and experimental values, (

**b**) performance accuracy, and (

**c**) cross-validation analysis.

**Figure 6.**Performance of a developed model for hoop rupture strain (ε

_{h,rup}): (

**a**) Comparison of model prediction and experimental values, (

**b**) performance accuracy, and (

**c**) cross-validation analysis.

**Figure 7.**Performance of a developed model for axial strength at a transition point (f’

_{c1}): (

**a**) Comparison of model prediction and experimental values, (

**b**) performance accuracy, and (

**c**) cross-validation analysis.

**Figure 8.**Performance of a developed model for axial strain at a transition point (ε

_{c1}): (

**a**) Comparison of model prediction and experimental values, (

**b**) performance accuracy, and (

**c**) cross-validation analysis.

**Figure 9.**Obtained relevance factor for ultimate condition: (

**a**) f’

_{cc}, (

**b**) ε

_{cu}, and (

**c**) ε

_{h,rup}.

Key Point | Number of Total Data | Number of Data after Evaluation |
---|---|---|

f’_{cc} | 1063 | 836 |

ε_{cu} | 1063 | 571 |

ε_{h,rup} | 506 * | 443 |

f’_{c1} | 260 | 260 |

ε_{c1} | 260 | 230 |

_{h,rup}was considered in the calculation.

E_{f} | t_{f} | D | f_{f} | ε_{co} | f’_{co} | K_{l} | K_{l}/f’_{co} | ε_{fu} | |
---|---|---|---|---|---|---|---|---|---|

f’_{cc} | |||||||||

E_{f} | 1.0 | ||||||||

t_{f} | −0.486 | 1.0 | |||||||

D | −0.081 | 0.266 | 1.0 | ||||||

f_{f} | 0.780 | −0.619 | −0.122 | 1.0 | |||||

ε_{co} | 0.082 | 0.045 | −0.213 | 0.010 | 1.0 | ||||

f’_{co} | 0.081 | 0.045 | −0.202 | 0.016 | 0.995 | 1.0 | |||

K_{l} | 0.276 | 0.230 | −0.123 | 0.082 | 0.388 | 0.398 | 1.0 | ||

K_{l}/f’_{co} | 0.265 | 0.221 | 0.002 | 0.110 | −0.207 | −0.194 | 0.747 | 1.0 | |

ε_{fu} | −0.531 | 0.010 | −0.031 | −0.106 | −0.130 | −0.124 | −0.357 | −0.303 | 1.0 |

ε_{cu} | |||||||||

E_{f} | 1.0 | ||||||||

t_{f} | −0.431 | 1.0 | |||||||

D | −0.137 | 0.368 | 1.0 | ||||||

f_{f} | 0.709 | −0.594 | −0.238 | 1.0 | |||||

ε_{co} | 0.153 | −0.085 | −0.428 | 0.163 | 1.0 | ||||

f’_{co} | 0.122 | −0.007 | −0.228 | 0.092 | 0.945 | 1.0 | |||

K_{l} | 0.342 | 0.169 | −0.134 | 0.175 | 0.378 | 0.365 | 1.0 | ||

K_{l}/f’_{co} | 0.319 | 0.208 | 0.026 | 0.148 | −0.143 | −0.158 | 0.781 | 1.0 | |

ε_{fu} | −0.613 | 0.032 | −0.027 | −0.170 | −0.132 | −0.153 | −0.364 | −0.324 | 1.0 |

ε_{h,rup} | |||||||||

E_{f} | 1.0 | ||||||||

t_{f} | −0.418 | 1.0 | |||||||

D | −0.010 | 0.103 | 1.0 | ||||||

f_{f} | 0.655 | −0.594 | −0.135 | 1.0 | |||||

ε_{co} | 0.041 | 0.007 | −0.361 | 0.053 | 1.0 | ||||

f’_{co} | 0.023 | 0.055 | −0.160 | 0.001 | 0.965 | 1.0 | |||

K_{l} | 0.344 | 0.165 | −0.167 | 0.129 | 0.338 | 0.324 | 1.0 | ||

K_{l}/f’_{co} | 0.342 | 0.161 | −0.089 | 0.151 | −0.206 | −0.224 | 0.772 | 1.0 | |

ε_{fu} | −0.650 | 0.049 | −0.084 | −0.177 | −0.046 | −0.053 | −0.375 | −0.345 | 1.0 |

Levenberg–Marquardt | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) |

f’_{co}, K_{l}, ε_{fu} | 11.7 | 102.0 | 12.9 | 10.6 | 100.0 | 11.6 | 10.3 | 101.1 | 11.3 | 10.0 | 101.1 | 12.8 | 9.8 | 101.7 | 15.9 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co} | 10.8 | 101.7 | 11.3 | 10.6 | 101.4 | 11.3 | 12.0 | 104.7 | 12.3 | 10.7 | 101.1 | 11.9 | 10.1 | 101.4 | 10.6 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f} | 12.6 | 103.2 | 12.4 | 16.1 | 105.6 | 16.0 | 10.7 | 97.5 | 11.3 | 9.1 | 100.8 | 8.8 | 10.8 | 100.8 | 9.9 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f}, D | 13.5 | 106.2 | 12.6 | 10.8 | 103.0 | 11.2 | 9.0 | 101.0 | 10.5 | 9.9 | 102.7 | 10.2 | 8.7 | 100.7 | 9.3 |

Bayesian Regularization | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) |

f’_{co}, K_{l}, ε_{fu} | 10.7 | 102.0 | 11.2 | 10.0 | 101.1 | 11.5 | 9.5 | 101.3 | 10.2 | 9.2 | 102.2 | 11.3 | 8.7 | 101.5 | 9.9 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co} | 10.5 | 101.8 | 10.8 | 10.7 | 101.8 | 12.6 | 10.5 | 101.8 | 11.0 | 9.8 | 101.7 | 11.0 | 10.4 | 102.1 | 11.4 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f} | 7.7 | 100.7 | 9.7 | 7.7 | 101.0 | 8.4 | 6.6 | 100.6 | 7.0 | 6.6 | 100.3 | 6.3 | 6.5 | 100.3 | 7.8 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f}, D | 8.4 | 101.1 | 9.2 | 7.7 | 101.0 | 8.4 | 6.7 | 101.0 | 7.2 | 7.4 | 101.3 | 10.1 | 7.2 | 99.6 | 9.7 |

Scaled Conjugate Gradient | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) |

f’_{co}, K_{l}, ε_{fu} | 14.4 | 104.0 | 14.7 | 15.1 | 104.2 | 15.1 | 13.7 | 103.5 | 13.8 | 14.8 | 103.5 | 15.1 | 17.2 | 104.2 | 18.2 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co} | 15.1 | 105.0 | 14.8 | 10.8 | 102.4 | 11.6 | 18.6 | 105.6 | 18.2 | 15.7 | 103.1 | 17.5 | 19.2 | 104.4 | 20.4 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f} | 14.8 | 101.7 | 15.6 | 15.9 | 105.3 | 15.8 | 14.2 | 103.2 | 14.4 | 13.6 | 102.8 | 14.9 | 14.9 | 102.4 | 16.3 |

f’_{co}, K_{l}, ε_{fu}, K_{l}/f’_{co}, E_{f}, f_{f}, t_{f}, D | 13.7 | 102.7 | 14.1 | 11.2 | 102.2 | 12.0 | 16.3 | 104.7 | 16.9 | 17.6 | 103.4 | 17.2 | 21.6 | 103.5 | 21.3 |

Levenberg–Marquardt | Bayesian Regularization | |||||
---|---|---|---|---|---|---|

AAE (%) | M (%) | RMSE (MPa) | AAE (%) | M (%) | RMSE (MPa) | |

Two layers (4,5) | 10.1 | 100.9 | 10.5 | 10.1 | 101.0 | 10.7 |

Two layers (5,4) | 11.1 | 102.8 | 11.7 | 9.5 | 101.6 | 10.1 |

Two layers (6,8) | 8.7 | 100.9 | 10.1 | 9.0 | 101.4 | 9.8 |

Two layers (8,6) | 10.9 | 99.3 | 12.9 | 8.2 | 101.4 | 9.3 |

Two layers (8,9) | 9.2 | 101.7 | 9.7 | 9.3 | 101.4 | 10.0 |

Two layers (9,8) | 8.5 | 101.0 | 8.9 | 9.2 | 101.4 | 10.0 |

Two layers (8,10) | 14.6 | 102.8 | 15.7 | 9.3 | 101.6 | 10.1 |

Two layers (10,8) | 7.7 | 101.1 | 8.6 | 10.4 | 101.8 | 11.3 |

Two layers (10,15) | 9.9 | 101.5 | 10.9 | 9.4 | 102.1 | 10.0 |

Two layers (15,10) | 10.2 | 101.4 | 12.0 | 6.4 | 100.4 | 7.4 |

Two layers (15,20) | 7.5 | 101.0 | 8.3 | 9.2 | 102.1 | 10.1 |

Two layers (20,15) | 9.4 | 101.3 | 10.4 | 8.8 | 101.6 | 9.4 |

Two layers (20,25) | 8.9 | 102.6 | 9.1 | 6.6 | 100.6 | 7.9 |

Two layers (25,20) | 9.4 | 102.2 | 10.2 | 9.2 | 101.6 | 9.9 |

Transfer Function | AAE (%) | M (%) | RMSE (MPa) |
---|---|---|---|

tansig-tansig | 6.4 | 100.4 | 7.4 |

logsig-logsig | 8.4 | 101.2 | 9.9 |

logsig-tansig | 10.2 | 105.0 | 11.2 |

pureline-pureline | 14.0 | 103.8 | 14.1 |

tansig-pureline | 8.5 | 101.1 | 9.1 |

elliotsig-elliotsig | 9.9 | 101.8 | 10.8 |

tribas-tribas | 20.1 | 105.8 | 22.9 |

purelin-logsig | 11.2 | 105.0 | 12.2 |

Parameter | Number of Hidden Layers | Number of Beurons | Training Function | Transfer Function | Learning Rate | Objective Function | |
---|---|---|---|---|---|---|---|

First | Second | ||||||

f’_{cc} | 2 | 15 | 10 | Bayesian regularization | tansig | 0.005 | MSE |

ε_{cu} | 2 | 6 | 8 | Levenberg–Marquardt | tansig | 0.001 | MSE |

ε_{h,rup} | 2 | 10 | 15 | Bayesian regularization | tansig | 0.005 | MSE |

f’_{c1} | 1 | 25 | - | Bayesian regularization | tansig | 0.005 | MSE |

ε_{c1} | 2 | 25 | 20 | Levenberg–Marquardt | tansig | 0.001 | MSE |

Levenberg–Marquardt | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) |

f’_{co}, K_{l}/f’_{co}, ε_{fu} | 32.6 | 96.3 | 0.005 | 31.8 | 99.3 | 0.005 | 31.3 | 101.1 | 0.005 | 35.0 | 98.5 | 0.006 | 37.0 | 96.5 | 0.007 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2} | 35.2 | 102.2 | 0.006 | 30.5 | 101.1 | 0.006 | 31.0 | 103.1 | 0.005 | 30.7 | 99.9 | 0.005 | 39.4 | 113.3 | 0.007 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f} | 32.3 | 100.4 | 0.005 | 29.4 | 98.0 | 0.005 | 28.6 | 102.0 | 0.005 | 37.4 | 86.7 | 0.006 | 25.8 | 100.1 | 0.005 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f} | 32.3 | 100.4 | 0.0055 | 32.6 | 101.0 | 0.005 | 25.6 | 101.8 | 0.005 | 40.2 | 90.1 | 0.006 | 25.6 | 101.8 | 0.005 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f} | 28.6 | 99.3 | 0.005 | 25.0 | 106.2 | 0.004 | 24.4 | 101.0 | 0.005 | 23.7 | 101.3 | 0.005 | 22.4 | 104.9 | 0.004 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f}, ε_{co}, D | 31.4 | 64.5 | 0.006 | 24.5 | 102.9 | 0.004 | 22.4 | 103.4 | 0.004 | 20.1 | 109.1 | 0.004 | 20.9 | 56.3 | 0.004 |

Bayesian Regularization | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) |

f’_{co}, K_{l}/f’_{co}, ε_{fu} | 30.6 | 100.5 | 0.005 | 29.3 | 101.8 | 0.005 | 34.0 | 100.0 | 0.006 | 30.5 | 99.1 | 0.005 | 29.5 | 97.1 | 0.006 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2} | 29.7 | 99.8 | 0.005 | 29.7 | 100.8 | 0.005 | 29.6 | 99.7 | 0.005 | 31.6 | 106.2 | 0.005 | 30.7 | 99.5 | 0.005 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f} | 26.5 | 101.2 | 0.005 | 28.5 | 103.2 | 0.005 | 21.4 | 103.2 | 0.004 | 19.9 | 102.7 | 0.004 | 19.0 | 100.2 | 0.004 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f} | 26.5 | 101.8 | 0.005 | 23.9 | 100.0 | 0.004 | 21.8 | 100.9 | 0.004 | 18.2 | 99.9 | 0.003 | 21.8 | 100.9 | 0.004 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f} | 22.9 | 102.1 | 0.004 | 21.2 | 115.0 | 0.004 | 18.2 | 100.9 | 0.004 | 16.9 | 100.3 | 0.003 | 21.4 | 84.9 | 0.004 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f}, ε_{co}, D | 21.5 | 103.8 | 0.004 | 17.9 | 102.4 | 0.004 | 16.3 | 99.5 | 0.004 | 14.8 | 100.9 | 0.003 | 14.9 | 99.9 | 0.004 |

Scaled Conjugate Gradient | |||||||||||||||

10 Neurons | 15 Neurons | 20 Neurons | 25 Neurons | 30 Neurons | |||||||||||

Input Variables | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) | AAE (%) | M (%) | RMSE (%) |

f’_{co}, K_{l}/f’_{co}, ε_{fu} | 40.6 | 101.1 | 0.006 | 47.6 | 104.5 | 0.007 | 35.7 | 99.1 | 0.006 | 42.1 | 103.1 | 0.007 | 44.1 | 101.1 | 0.008 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2} | 42.0 | 98.7 | 0.007 | 40.2 | 100.4 | 0.006 | 44.8 | 100.6 | 0.007 | 36.7 | 105.0 | 0.006 | 35.4 | 99.7 | 0.006 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f} | 41.9 | 102.8 | 0.007 | 39.6 | 104.1 | 0.006 | 33.1 | 99.4 | 0.005 | 47.7 | 106.5 | 0.007 | 49.5 | 120.5 | 0.008 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f} | 48.6 | 127.3 | 0.007 | 40.9 | 100.0 | 0.007 | 35.4 | 97.5 | 0.006 | 41.2 | 107.0 | 0.007 | 35.4 | 97.5 | 0.006 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f} | 41.8 | 95.6 | 0.007 | 38.6 | 97.6 | 0.006 | 41.0 | 101.4 | 0.007 | 40.1 | 103.3 | 0.006 | 40.4 | 93.4 | 0.007 |

f’_{co}, K_{l}/f’_{co}, ε_{fu}, k_{2}, E_{f}, t_{f}, f_{f}, ε_{co}, D | 39.1 | 112.6 | 0.007 | 42.9 | 102.0 | 0.006 | 34.4 | 107.5 | 0.006 | 36.1 | 103.2 | 0.006 | 36.7 | 96.1 | 0.006 |

E_{f} | t_{f} | D | f_{f} | ε_{co} | f’_{co} | K_{l} | K_{l}/f’_{co} | ε_{fu} | |
---|---|---|---|---|---|---|---|---|---|

f’_{c1} | |||||||||

E_{f} | 1.0 | ||||||||

t_{f} | −0.453 | 1.0 | |||||||

D | 0.047 | 0.230 | 1.0 | ||||||

f_{f} | 0.314 | −0.431 | 0.021 | 1.0 | |||||

ε_{co} | −0.234 | 0.548 | −0.332 | −0.166 | 1.0 | ||||

f’_{co} | −0.291 | 0.712 | −0.054 | −0.242 | 0.888 | 1.0 | |||

K_{l} | 0.401 | 0.455 | −0.057 | 0.024 | 0.471 | 0.471 | 1.0 | ||

K_{l}/f’_{co} | 0.648 | −0.066 | 0.003 | 0.149 | −0.275 | −0.281 | 0.594 | 1.0 | |

ε_{fu} | −0.909 | 0.556 | 0.009 | −0.557 | 0.249 | 0.342 | −0.307 | −0.567 | 1.0 |

ε_{c1} | |||||||||

E_{f} | 1.0 | ||||||||

t_{f} | −0.523 | 1.0 | |||||||

D | 0.063 | 0.197 | 1.0 | ||||||

f_{f} | 0.922 | −0.435 | 0.116 | 1.0 | |||||

ε_{co} | −0.121 | 0.497 | −0.374 | −0.135 | 1.0 | ||||

f’_{co} | −0.218 | 0.679 | −0.071 | −0.208 | 0.872 | 1.0 | |||

K_{l} | 0.140 | 0.643 | −0.056 | 0.197 | 0.618 | 0.622 | 1.0 | ||

K_{l}/f’_{co} | 0.372 | 0.197 | −0.008 | 0.435 | −0.039 | −0.128 | 0.661 | 1.0 | |

ε_{fu} | −0.969 | 0.503 | −0.030 | −0.821 | 0.104 | 0.215 | −0.144 | −0.377 | 1.0 |

Index | Threshold | Transition Zone | Ultimate Condition | |||
---|---|---|---|---|---|---|

f’_{c1} | ε_{c1} | f’_{cc} | ε_{cu} | ε_{h.rup} | ||

$R=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({h}_{i}-\overline{{h}_{i}}\right)\left({t}_{i}-\overline{{t}_{i}}\right)}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{({h}_{i}-\overline{{h}_{i}})}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({t}_{i}-\overline{{t}_{i}}\right)}^{2}}}$ | R > 0.8 | 0.99 | 0.84 | 0.98 | 0.94 | 0.96 |

$k=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({h}_{i}{t}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{h}_{i}{}^{2}}$ | $0.85<k<1.15$ | 1.00 | 0.98 | 0.99 | 0.97 | 0.98 |

${k}^{\prime}=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({h}_{i}{t}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{t}_{i}{}^{2}}$ | $0.85<{k}^{\prime}<1.15$ | 0.99 | 1.02 | 1.01 | 1.03 | 1.02 |

${R}_{m}={R}^{2}\left(1-\sqrt{\left|{R}^{2}-{R}_{O}^{2}\right|}\right)$ ${R}_{O}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({t}_{i}-{h}_{i}^{O}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{({t}_{i}-\overline{{t}_{i}})}^{2}},$ ${h}_{i}^{O}=k{t}_{i}$ | R_{m} > 0.5Should be close to 1 | 0.87 0.99 | 0.33 0.99 | 0.76 0.99 | 0.60 0.99 | 0.67 0.99 |

${R}^{2}$ | Should be close to 1 | 0.98 | 0.71 | 0.95 | 0.89 | 0.92 |

$RMSE=\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({h}_{i}-{t}_{i}\right)}^{2}}$ | Should be minimum (based on output range) | 3.47 | 0.0005 | 7.45 | 0.003 | 0.001 |

$MAE=\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{h}_{i}-{t}_{i}\right|$ | Should be minimum (based on output range) | 2.36 | 0.0003 | 4.67 | 0.002 | 0.001 |

Model | Year | Number of Data | Prediction Parameter | R^{2}-All |
---|---|---|---|---|

Naderpour et al. [25] | 2010 | 213 | f’_{cc} | 0.89 |

Jiang et al. [16] | 2020 | 169 | f’_{cc} | 0.98 |

Keshtegar et al. [48] | 2021 | 780 | f’_{cc}/f’_{co} | 0.88 |

Cevik and Cabalar [23] | 2008 | 110 | f’_{cc} | 0.97 |

Cevik [49] | 2011 | 180 | f’_{cc}/f’_{co} | 0.94 * |

Jalal and Ramezanianpour [24] | 2012 | 128 | f’_{cc} | 0.95 |

Proposed model | - | 836 | f’_{cc} | 0.99 |

^{2}is only for the test dataset.

Model | Year | Number of Data | Prediction Parameter | R^{2}-All |
---|---|---|---|---|

Jiang et al. [16] | 2020 | 169 | ε_{cu} | 0.95 |

Keshtegar et al. [48] | 2021 | 780 | ε_{cu}/ε_{co} | 0.86 |

Proposed model | - | 836 | ε_{cu} | 0.89 |

Model | Test Data | AAE (%) | M (%) | SD (%) | RMSE (MPa) |
---|---|---|---|---|---|

Fallah Pour et al. [4] | 836 | 12.3 | 100.0 | 16.3 | 14.9 |

Lim et al. [27] | 836 | 12.7 | 105.0 | 17.1 | 15.2 |

Lim and Ozbakkaloglu [10] | 836 | 12.9 | 102.9 | 16.8 | 14.6 |

Berthet et al. [50] | 836 | 13.0 | 104.1 | 18.6 | 17.9 |

Wu and Zhou [51] | 836 | 13.3 | 107.2 | 18.9 | 17.6 |

Pham and Hadi [33] | 836 | 14.0 | 99.2 | 18.2 | 17.3 |

Al-Salloum [52] | 836 | 14.1 | 108.7 | 21.0 | 22.0 |

Wei and Wu [53] | 836 | 14.3 | 108.6 | 20.6 | 21.1 |

Wu and Wang [54] | 836 | 14.3 | 108.6 | 20.6 | 21.1 |

Cevik [49] | 836 | 20.2 | 107.0 | 30.1 | 19.8 |

Cevik et al. [55] | 836 | 22.1 | 104.1 | 33.6 | 27.6 |

Proposed model | 836 | 6.4 | 100.4 | 9.8 | 7.4 |

Model | Test Data | AAE (%) | M (%) | SD (%) | RMSE (%) |
---|---|---|---|---|---|

Lim and Ozbakkaloglu [10] | 571 | 20.1 | 97.5 | 23.9 | 0.50 |

Tamuzs et al. [56] | 571 | 20.7 | 106.5 | 28.7 | 0.50 |

Fallah Pour et al. [4] | 571 | 21.0 | 98.1 | 25.9 | 0.50 |

Teng et al. [57] | 571 | 22.2 | 122.1 | 34.1 | 0.66 |

Lim et al. [27] | 571 | 22.3 | 98.2 | 27.9 | 0.52 |

Binici [58] | 571 | 22.6 | 124.1 | 37.7 | 0.85 |

Youssef et al. [8] | 571 | 22.7 | 112.8 | 34.8 | 0.71 |

Berthet et al. [50] | 571 | 23.2 | 121.5 | 41.9 | 0.75 |

Wei and Wu [53] | 571 | 25.4 | 103.9 | 31.3 | 0.62 |

Pham and Hadi [33] | 571 | 26.1 | 129.8 | 40.5 | 0.85 |

Miyauchi et al. [59] | 571 | 27.4 | 122.1 | 40.9 | 0.65 |

Proposed model | 571 | 14.5 | 100.8 | 20.2 | 0.30 |

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## Share and Cite

**MDPI and ACS Style**

Fallah Pour, A.; Gholampour, A.
An Artificial Network-Based Prediction of Key Reference Zones on Axial Stress–Strain Curves of FRP-Confined Concrete. *Appl. Sci.* **2023**, *13*, 3038.
https://doi.org/10.3390/app13053038

**AMA Style**

Fallah Pour A, Gholampour A.
An Artificial Network-Based Prediction of Key Reference Zones on Axial Stress–Strain Curves of FRP-Confined Concrete. *Applied Sciences*. 2023; 13(5):3038.
https://doi.org/10.3390/app13053038

**Chicago/Turabian Style**

Fallah Pour, Ali, and Aliakbar Gholampour.
2023. "An Artificial Network-Based Prediction of Key Reference Zones on Axial Stress–Strain Curves of FRP-Confined Concrete" *Applied Sciences* 13, no. 5: 3038.
https://doi.org/10.3390/app13053038