3.1. Artificial neural network

An artificial neural network (ANN) is an information processing model simulated according to the information processing way of biological neuron systems. It is made up of a large number of neurons that are interconnected through connections to solve a particular problem. The ANN network structure is made up of three or more layers, depicted in Fig 2, consisting of (i) an input layer that is the leftmost layer of the network, representing the input parameters, (ii) an output layer is the rightmost layer of the network, representing the results achieved, and (iii) one or more hidden layers denoting the logical inference of the network. The input layer contains the information x_i (i = 1,2,.., n) from the original data. The values x_i are then multiplied by a weighted value $$ w_{jk}^1$$ (j = 1,2,…, N) where k = 1,2,.., n. The value $$ \underset{k=1}{\overset{n}{\mathrm{\Sigma }}}{w}_{jk}^1{x}_k+{\theta }_j^1$$ becomes the input of the hidden layer, and continues to be multiplied by another weighted value $$ w_{ij}^2$$ (i = 1,2,…, m), becoming the input of the output layer. The values $$ {\theta }_j^1$$ represent the bias associated with each neuron in the hidden layer.

Fig 2

Artificial neural network model.

Finally, the output value is obtained as $$ \underset{j=1}{\overset{L}{\mathrm{\Sigma }}}{w}_{ij}^2\left(\underset{k=1}{\overset{n}{\mathrm{\Sigma }}}{w}_{jk}^1{x}_k+{\theta }_j^1\right)+{\theta }_i^2$$.

The learning process of the neural network corresponds the process in which the network continuously changes the values $$ w_{jk}^1$$ and $$ w_{ij}^2$$ in order to change the output value y_i, until the y_i value meets the requirements (to reach a certain error compared with the target output). The back-propagation algorithm (BPNN) is widely used to train the neural network [69]. The BP algorithm is a slope reduction technique that minimizes errors for a particular training pattern in adjusting the weight by a small amount each time (or each epoch, iteration). Nerve cells in each layer are linked to the front and rear nerve cells with each associated weight, as presented above. However, the standard backpropagation algorithm typically has a slow convergence rate [70]. One of the generalization-type methods, such as the Bayesian Regularization (BR) [71, 72] is commonly used to avoid such a convergence problem and obtain a lower mean square error between the output of ANN and the target [73].

Bayesian regularization is a network training function that updates the weight and bias values according to Levenberg-Marquardt optimization. It minimizes a combination of squared errors and weights, then determines the correct combination to produce a network with high prediction accuracy. From a Bayesian point of view, the regularization corresponds to a prior probability distribution over free parameters w of the model. Using the notation of MacKay [71], the regularized target function can be written as:

3.2. Monte Carlo simulation

Numerical prediction models involving Monte Carlo simulation can explain the output results’ variation through statistical analysis. The Monte Carlo method is potent to calculate the influence of the input variability on the output results using numerical AI models [40, 44, 75, 76]. In this study, the objective of the Monte Carlo method is to randomly repeat simulations, taking into account the variability in the input space, then calculate the corresponding output through a machine learning model [77]. The robustness of Monte Carlo simulation and sensitivity of input variables can be evaluated through statistical performance criteria of the output results. The statistical convergence of Monte Carlo simulation has been carried out using the normalized convergence criteria as follow [78–80]:

3.3. Performance criteria

To evaluate the effectiveness of the proposed ANN model, several evaluation criteria are proposed, including mean absolute error (MAE), root mean square error (RMSE), and the coefficient of determination (R²). Precisely, MAE represents the average amplitude of the model error but does not indicate the biased trend of the model output and actual values. When MAE = 0, the model value completely coincides with the actual value, thus the model could be considered as "ideal". The values of MAE are in the range of (0; +∞). Besides, root mean square error (RMSE) is one of the fundamental criteria and is commonly used to evaluate the predictive modeling results. It is common to use the RMSE to denote the mean magnitude of the error. In particular, RMSE is very sensitive to large error values. Therefore, the closer the RMSE is to the MAE, the more stable the model error. Criteria MAE, RMSE do not indicate the deviation between the model’s output value and the actual value, and in the range of (0; +∞). R² is the coefficient of determination that represents the suitability of the data with the algorithm, and in the range of (0; 1). The R² values close to 0 present the model’s poor performance, whereas the values close to 1 show good model accuracy. These values are represented by the following equations:

5.1. Investigation on the convergence of results

Table 4 summarizes the ANN parameters used in this study, and related information on Monte Carlo simulations. For the ANN and Monte Carlo simulation modeling, MATLAB programming language is used. Nothing that two cases of the hidden layer, consisting of 1 hidden layer and 2 hidden layers, are considered in this study. The number of neurons in each hidden layer is varied, ranging from 10 to 20 neurons, in order to cover the range of neuron as suggested in previously published works, such as in Neville (1986) [81], Gallant (1993) [82], Nagendra (1998) [83], Kannellopoulas (1997) [84]. Under the random sampling effect, the convergence of ANN model was investigated with a total of 66,000 simulations, including 121 architectures for the case of 2 hidden layers and 11 architectures for those with one hidden layer. Investigation on the convergence of results is crucial in Monte Carlo simulation, aiming at determining: (i) the appropriate number of Monte Carlo simulations, and (ii) the reliability of the simulation results. Fig 4 shows the convergence of results for all ANN architectures performed in this study. It is observed that the convergence is assured for both training, testing datasets for all cases over 500 simulations (Fig 4). Fig 4A and 4B show the normalized convergence results of R². It can be seen that a fluctuation of 1% around the mean value is obtained after about 300 simulations, whereas with the same number of Monte Carlo simulations, RMSE and MAE vary about 2% of the mean values (Fig 4C–4F). Overall, the results are stable around the average values, normalized to 1 when the number of Monte Carlo simulations is 500. Thus, it can be stated that the reliable results obtained by the proposed ANN model with 1 and 2 hidden layers are converged after 500 simulations, under the random sampling effect of data. In the next step, the optimization process of different ANN architectures is performed.

Fig 4

Convergence analysis for different proposed ANN architecture with respect to the testing parts: (a) R² of ANN with 1 hidden layer; (b) R² of ANN with 2 hidden layer; (c) RMSE of ANN with 1 hidden layer; (d) RMSE of ANN with 2 hidden layer; (e) MAE of ANN with 1 hidden layer; and (a) MAE of ANN with 2 hidden layer.

Table 4

Summary of different ANN characteristics and investigation parameters in this study.

Parameter	Parameter	Description
Fix	Neurons in input layer	9
	Neurons in output layer	1
	Hidden layer activation function	Sigmoid
	Output layer activation function	Linear
	Cost function	Mean Square Error (MSE)
	Number of epochs	1000
	Number of simulations	500
	Training algorithm	Bayesian regularization backpropagation
Parametric study	Number of hidden layers	Varying from 1 to 2
	Neurons in hidden layer	Varying from 10 to 20, step of 1

5.2. Prediction performance of different ANN architectures

5.2.1. ANN architectures with one hidden layer

In this section, the performance of all the ANN architectures is evaluated via statistical criteria such as the mean and standard deviation values of R², RMSE, MAE. The evaluation is carried out for both the training and testing parts. Fig 5 shows the performance of ANN in function of the neuron in the hidden layer, varying from 10 to 20, regarding the mean and StD values of R², RMSE, and MAE for the training and testing parts. It can be seen that the case of 17 neurons exhibits the best prediction results. In fact, the values of R² are the highest compared with the other 11 cases, with the values of 0.975 and 0.8886 for the training and testing parts, respectively (cf. Fig 5A and 5B). Similarly, the values of RMSE and MAE for this architecture are also lower compared with the remaining cases, along with better standard deviation values (cf. Fig 5C–5F). Precisely, the average values of RMSE are 13.38 and 33.77 for the training, testing datasets, respectively. The average values of MAE are 9.75 and 24.84 for the training, testing datasets, respectively. It is worth noticing that the testing part’s performance is critical when evaluating the prediction capability of a model. The criteria associated with the testing dataset reflect the ability of a model to predict new data, which are not considered in the training phase of a given model. Overall, 17 neurons in the hidden layer generate the best ANN architecture with one hidden layer.

Fig 5

Performance of the ANN with 1 hidden layer in function of the neuron in the hidden layer, with respect to (a) mean and StD of R² for the training part; (b) mean and StD of R² for the testing part; (c) mean and StD of RMSE for the training part; (d) mean and StD of RMSE for the testing part; (e) mean and StD of MAE for the training part; and (f) mean and StD of MAE for the testing part.

5.2.2. ANN architectures containing two hidden layers

Concerning the ANN architecture with 2 hidden layers, various structures are investigated, as shown in Figs 6 and 7 for both the training and testing datasets, respectively. With respect to the training dataset, it is observed that a higher number of neurons in the second hidden layer produces higher prediction accuracy (i.e., R² of 0.95, RMSE of 14, and MAE of 12). An optimal zone is observed to achieve the highest accuracy, regardless of the number of neurons in the first hidden layer. This is an excellent example to demonstrate that an appropriate ANN structure should be determined before performing any further simulations. Investigation of the effect of neurons in the two hidden layers is performed via standard deviation plots of the three error criteria (Fig 6B, 6D and 6F). The training part exhibited small standard deviation values, in general, with a higher number of neurons in both hidden layers (i.e., above 15 neurons).

Fig 6

Color map of ANN with 2 hidden layers in function of the neuron in the hidden layer for the training part with respect to (a) mean values of R²; (b) StD of R²; (c) mean of RMSE; (d) StD of RMSE; (e) mean of MAE; and (f) StD of MAE.

Fig 7

Colop-map of ANN with 2 hidden layers in function of the neuron in the hidden layer for the testing part with respect to (a) mean values of R²; (b) StD of R²; (c) mean of RMSE; (d) StD of RMSE; (e) mean of MAE; and (f) StD of MAE.

Similarly, the testing part results are in good agreement with those of the training part, where the higher number of neurons in the second hidden layer is required to obtain better prediction accuracy (cf. Fig 7). However, it is clearly observed that the prediction accuracy of ANN structure with 2 hidden layers is lower than that containing 1 hidden layer. Indeed, the highest values of R² are 0.7696 and 0.8886 for the cases of ANN using 2 and 1 hidden layer, respectively. Besides, the lowest values of RMSE and MAE are 51.60 and 36.50 for ANN structure using 2 hidden layers, whereas those values are 33.77 and 24.84 for ANN using 1 hidden layer, respectively.

Overall, it can be stated that ANN structure with 1 hidden layer and 17 neurons is the most effective predictor for the problem, and thus, it can be used for further investigation.

5.3. Prediction performance of typical ANN architecture

This section is dedicated to the presentation of typical results related to the best ANN architecture containing 1 hidden layer with 17 neurons, where the R² value of the testing part is highest. Fig 8 shows a comparison of experimental and predicted shear strength results in function of the sample index for the training and testing datasets. The comparison shows that the predicted shear strength values are close to the experimental ones. The predicted results of the training dataset as a function of the experimental data are shown in Fig 9A, whereas the testing part’s results are displayed in Fig 9B, respectively. The regression graph of all data is shown in Fig 9C. In these figures, the linear fits are displayed, also highlighting the R², RMSE, MAE and StD values. The R² values are 0.9634, 0.9577, and 0.9599 for the training, testing, and all dataset, respectively. It is observed that the linear regression lines are very close to the diagonal lines, which confirms the strong correlation between predicted and experimental shear strength. In terms of RMSE, MAE, and StD values, the best ANN architecture gives a high prediction performance. The RMSE values are 18.45 MPa, 23.06 MPa, and 19.97MPa; MAE values are 14.13 MPa, 16.93 MPa, and 14.98 MPa; StD values are 18.56, 23.37, and 20.05 for the training, testing, and all data, respectively. A good agreement between the predicted and the experimental shear strength of FRP concrete beam is obtained

Fig 8

Experimental and predicted shear strength results in function of the sample index for the training and testing datasets.

Fig 9

Regression graphs for the case of the best predictor ANN architecture containing 17 neurons in 1 hidden layer (a) training dataset; (b) testing dataset; and (c) all dataset.

Finally, several previously published models and the corresponding accuracy are given for comparison purposes (cf. Table 5). It could be seen that the proposed ANN model in this study exhibits good accuracy compared with other works. The accuracy of the best model (R² = 0.960) is slightly inferior to the accuracy obtained by the Multivariate adaptive regression splines (MARS) model in the work of Abbasloo et al. [54] (R² = 0.974). However, the authors used a smaller amount of samples (112 data) than this work (125 data). This might be the reason for a small difference in prediction accuracy. Second, the present study uses a smaller number of inputs (9 inputs) than in the work of Abbasloo et al. [54] (12 inputs). Without affecting too much the prediction accuracy, the reduction of 3 inputs is not significantly pronounced with a small dataset, but might be of great interest while considering a bigger dataset or industrial design context.

Table 5

Comparison with the literature for prediction of shear strength of fiber reinforcement bars concrete beams.

Ref.	Machine learning algorithm	Inputs	Sample size	Performance measure
Nehdi et al. [2]	Genetic algorithm	6 inputs: the beam’s effective depth (d) and with (bw); the shear span to depth ratio (a/d); the longitudinal reinforcement ratio (ρfl); the compressive strength of concrete (f’c); the ratio of the modulus of elasticity of FRP to that of steel (Efl/Es); the ultimate capacity of FRP shear reinforcement (ffυ).	168 data include: 100 data with shear reinforcement, 68 data without shear reinforcement	AAE = 22.42% Vm = Vmeasured Vcal = Vcalculated $$
Abbasloo et al. [54]	Multivariate adaptive regression splines (MARS) - M5’	12 inputs: the average concrete compressive strength (fcm); the ratio of shear span to the effective depth (a/d); the cross-section width (b); the effective depth of the cross-section (d); the area of longitudinal and transversal reinforcement (AF and AFw); the ultimate tensile strength of longitudinal and transversal reinforcement (fFu and fFwu); Young’s modulus of longitudinal and transversal reinforcement (EF and EFw); the longitudinal and transversal reinforcement ratio (ρF and ρFw).	112 data with shear reinforcement	R2 = 0.9735 (MARS) RMSE = 9.03 (MARS) R2 = 0.9004 (M5’) RMSE = 32.52 (M5’)
This study	ANN—Bayesian Regularization (BR)	9 input: Beam width (bw); Effective depth (d); The ratio of the shear span to the effective depth (a/d); Compressive strength of concrete (fc); Longitudinal FRP reinforcement ratio (ρf); Modulus of elasticity of longitudinal FRP reinforcement (Ef); FRP shear reinforcement ratio (ρs); Tensile strength of FRP shear reinforcement (fs); Modulus of elasticity of FRP shear reinforcement(Es).	125 data with shear reinforcement	Training data R2 = 0.9634 RMSE = 18.45 MAE = 14.13 Testing data R2 = 0.9577 RMSE = 23.06 MAE = 16.93 All data R2 = 0.9599 RMSE = 19.97 MAE = 14.98

5.4. Analysis of prediction accuracy based on reinforcement type

In this section, the prediction capability of the best ANN architecture (i.e., ANN-9-17-1) is analyzed considering different types of FRP reinforcements. The performance of the best ANN architecture in function of flexural reinforcement type is investigated by mean and StD values of R² (Fig 10A), mean and StD values of RMSE (Fig 10B), and mean and StD values of MAE (Fig 10C) for 500 Monte Carlo simulations. In fact, the accuracy of ANN model in predicting the shear strength is strongly dependent on the types of flexural reinforcement, namely steel, AFRP, CFRP, and GFRP. It is observed that the shear strength prediction by the proposed ANN architecture for AFRP has the lowest accuracy comparing with other types. In this case, the mean value of R² is the lowest (i.e., 0.90), and those of RMSE (i.e., 24.40), MAE (i.e., 18.13) are the highest. Besides, the prediction accuracy remains similar compared with other cases of flexural reinforcement. It is worth noticing that the prediction accuracy of ANN for AFRP type flexural reinforcement is the lowest, even though the AFRP flexural reinforcement type possesses the highest number of samples (41 samples as indicated in Table 3). This might come from the possible interaction between both types of reinforcement (flexural and shear), or an inappropriate range of inputs and output compared with those of the full dataset. Thus, further investigation of this behavior should be considered.

Fig 10

Performance of the best ANN architecture containing 17 neurons with 1 hidden layer in function of flexural reinforcement type, with respect to (a) mean and StD values of R²; (b) mean and StD values of RMSE; and (c) mean and StD values of MAE.

Fig 11 shows the performance of the best ANN architecture containing 17 neurons with 1 hidden layer in function of shear reinforcement type. Similar to the previous investigation, four types of reinforcement, including VFRP, AFRP, CFRP, and GFRP, significantly influence the ANN model accuracy in predicting the shear strength. The mean value of R² in predicting the shear strength for the case of VFRP is the lowest (Fig 11A). However, the mean values of RMSE and MAE are the smallest (cf. Fig 11B and 11C), and the StD values, in this case, are significantly higher compared with other types of shear reinforcement. It could be concluded that the proposed ANN model is sensitive in predicting such type of reinforcement, which might come from the insufficient number of samples (only 10 samples, see Table 3). Moreover, the mean value of R² in predicting the shear strength using AFRP, CFRP, and GFRP shear reinforcement is higher than that of VFRP, but the mean values are higher than those with AFRP, CFRP, and GFRP. In this case, it is difficult to evaluate the model’s accuracy based on the mean values of these statistical measurements. However, based on the StD values of R², RMSE, and MAE, it could be stated that the model accuracy is very sensitive in predicting VFRP as shear reinforcement.

Fig 11

Performance of the best ANN architecture containing 17 neurons with 1 hidden layer in function of shear reinforcement type, with respect to: (a) mean and StD of R²; (b) mean and StD of RMSE; (c) mean and StD of MAE.