A Proposed Soft Computing Model for Ultimate Strength Estimation of FRP-Conﬁned Concrete Cylinders

: In this paper, the feed-forward backpropagation neural network (FFBPNN) is used to propose a new formulation for predicting the compressive strength of ﬁber-reinforced polymer (FRP)-conﬁned concrete cylinders. A set of experimental data has been considered in the analysis. The data include information about the dimensions of the concrete cylinders (diameter, length) and the total thickness of FRP layers, unconﬁned ultimate concrete strength, ultimate conﬁnement pressure, ultimate tensile strength of the FRP laminates and the ultimate concrete strength of the concrete cylinders. The conﬁned ultimate concrete strength is considered as the output data, while other parameters are considered as the input data. These parameters are mostly used in existing FRP-conﬁned concrete models. Soft computing techniques are used to estimate the compressive strength of FRP-conﬁned concrete cylinders. Finally, a new formulation is proposed. The results of the proposed formula are compared to the existing methods. To verify the proposed method, results are compared with other methods. The results show that the described method can forecast the compressive strength of FRP-conﬁned concrete cylinders with high precision in comparison with the existing formulas. Moreover, the mean percentage of error for the proposed method is very low (3.49%). Furthermore, the proposed formula can estimate the ultimate compressive capacity of FRP-conﬁned concrete cylinders with a di ﬀ erent type of FRP and arbitrary thickness in the initial design of practical projects.


Introduction
A combination of high-strength fibers and matrix leads to the construction of a fiber-reinforced polymer (FRP). The primary role of the matrix is to bind these fibers together to construct structural shapes. Four common types of fibers (i.e., aramid, carbon, glass, and high-strength steel) and also two standard matrices exist (i.e., epoxies and esters) [1,2]. A new area has been opened in the civil engineering field due to the beneficial properties of FRP in the repair and rehabilitation of existing structures. The FRP can create a continuous confinement action for the concrete member, and can also increase the corrosion resistance of members [3]. Hereby, FRPs are popularly used to repair or retrofit the reinforcing frame members [4][5][6][7][8][9][10]. Studies on the behavior of FRP and FRP-confined concrete have advanced rapidly in recent years [11]. There are a lot of publications proposing a formula

Overview of Existing Models
Some published publications offer a formula to forecast the compressive strength of FRPCCC ( f cc ). In these papers, certain parameters are adopted as the input parameters. These parameters include the diameter of the concrete cylinder (d), length of the concrete cylinder (L), unconfined ultimate concrete strength ( f co ), the thickness of FRP layer (t), ultimate confinement pressure ( f l ) and ultimate tensile strength of the FRP laminate ( f f ). Table 1 shows the existing formula to compute the compressive strength of FRPCCC. Table 1. Some of the existing formulas for predicting the compressive strength of fiber-reinforced polymer-confined concrete cylinders (FRPCCC).
It should be noted that when a concrete cylinder is subjected to the axial compression force, the compressive strength is less than its value for the FRPCCC (see Figure 1). It means that P1 < P2.

Proposing a New Formulation to Predict the Compressive Strength of FRP-Confined Concrete Cylinder
In this paper, firstly, a set of experimental data is collected from the published literature [17,58,60,[65][66][67][68][69][70][71][72][73][74][75][76][77][78] (see Table A1 in Appendix A). Then, the collected data are divided into input and output parameters (see Table 2). The values for minimum, maximum, mean, standard deviation, and coefficient of variation for the collected data are depicted in Table 3. ANNs are among the computational software methods used. The neural networks can find the existing patterns between the input and output data of experiments or simulations via training [79]. It is noteworthy that layers, neurons and weights can compose the neural networks. Here, the primary role of the weights is to relate every neuron in each layer to the neurons in other layers. Every neuron is associated with neurons in other layers by the weights. Every layer processes the input data and transfers them to the next layer. Additionally, an input layer, two or more hidden layers and an output layer compose the feed-forward neural network. A three-layer neural network is depicted in Figure 2. As mentioned in Section 3, the number of collected data is 281. These data are used for the learning, validating, and testing of ANNs. In the neural network modeling, log-sigmoid transfer functions are used and one hidden layer is selected. Firstly, all selected data are normalized based on the following equation: (1) where f , f min , f max and f scaled are the selected parameters, their minimum and maximum values are based on Table 3 and the value of the scaled parameters, respectively. Based on Equation (1), the scaled parameters place in the range between 0.1 and 0.9, as recognized by the log-sigmoid transfer functions. (1), the scaled parameters place in the range between 0.1 and 0.9, as recognized by the log-sigmoid transfer functions. The Levenberg-Marquardt algorithm is used to train randomly divided input and output vectors, which are called training (also learning), validating (also verifying) and testing datasets. Since improving the performance of the ANN model can be done by finding the optimal distribution of the datasets, various sets were analyzed. Finally, the best division was chosen, in which 70% of all data were training sets, while 15% of all data were validating and testing sets, respectively.
For this purpose, a 6:n:1 network is considered with six inputs, n hidden neurons and one output, respectively (see Figure 2). Moreover, the flowchart of the utilized ANN is depicted in Figure 3.   The Levenberg-Marquardt algorithm is used to train randomly divided input and output vectors, which are called training (also learning), validating (also verifying) and testing datasets. Since improving the performance of the ANN model can be done by finding the optimal distribution of the datasets, various sets were analyzed. Finally, the best division was chosen, in which 70% of all data were training sets, while 15% of all data were validating and testing sets, respectively.
For this purpose, a 6:n:1 network is considered with six inputs, n hidden neurons and one output, respectively (see Figure 2). Moreover, the flowchart of the utilized ANN is depicted in Figure 3.    The mean squared error (MSE) is considered as a criterion to stop the training of the networks. The MSE is defined as the average squared difference and is an important value that indicates an error between the network output and the actual value obtained from research. Therefore, when the quantity for the desired network has a minimum value, this network has a better performance. In addition, in a network, the correlation between outputs and targets is measured by regression values (R-values). The R-value is a parameter to measure the correlation between targets and outputs. These two criteria are selected to recognize which network has a better performance. Figure 4 shows the regression values of the networks versus the different numbers of neurons in hidden layers. Furthermore, Figure 5 presents the maximum absolute value for the error of each network. From the above description and considering Figures 4 and 5, it can be concluded that a network with 15 hidden neurons had the best performance.
(R-values). The R-value is a parameter to measure the correlation between targets and outputs. These two criteria are selected to recognize which network has a better performance. Figure 4 shows the regression values of the networks versus the different numbers of neurons in hidden layers. Furthermore, Figure 5 presents the maximum absolute value for the error of each network. From the above description and considering Figures 4 and 5, it can be concluded that a network with 15 hidden neurons had the best performance.   (R-values). The R-value is a parameter to measure the correlation between targets and outputs. These two criteria are selected to recognize which network has a better performance. Figure 4 shows the regression values of the networks versus the different numbers of neurons in hidden layers. Furthermore, Figure 5 presents the maximum absolute value for the error of each network. From the above description and considering Figures 4 and 5, it can be concluded that a network with 15 hidden neurons had the best performance.        It should be noted that the ANN technique cannot propose a formulation to predict the compressive strength of FRPCCC. Therefore, in the next section of this paper, the K-fold crossvalidation technique is used to obtain a new formulation. Then, the efficiency of the proposed formula is examined.

Using a Model with a K-Fold Cross-Validation Technique in FFBPNN
In this section of the paper, a K-fold cross-validation (KFCV) technique is applied for the optimization and evaluation of the perfected ANN [80,81]. In the KFCV technique, the data are divided randomly into K folds. Then, the K-1 folds are used for training, and the last fold is used to test the neural network. In the parametric study conducted, the values for K, changing from two to five and K = 4, are considered. The process of learning and testing is conducted for all the K sections. Therefore, all the K sections contribute to the learning and testing of the ANN. This process is iterated three times for the reduction and variation of KFCV and similar distribution of data in each K. The performance of the neural network for each iteration can be computed by the percentage of correct predictions in the neural network for K folds.
In every epoch, the performance evaluation of the neural network is calculated. The curve is the correct classification factor (CCF), it is drawn for three iterations and, finally, it is averaged. In the CCF curve, after a specified epoch, the curve is saturated. Then, the optimal epoch is defined using 10% of the curve plateau. In this study, a neural network with three layers is selected for the sake of simplicity. For optimization of the ANN structure, some neurons in the hidden layer are optimized. For this purpose, the selecting criteria are considered to be the area under the CCF curve (AUCCF). Therefore, the AUCCF is measured until it reaches the optimal epoch. Hence, different neurons, from two to 13 neurons in the hidden layer, are selected and the KFCV process is repeated for the structures. Finally, the structure with the maximum efficiency can be determined by drawing the CCF and calculating the AUCCF. Figure 9 shows the AUCCF curve. As can be seen from this figure, the 6:11:1 structure with 11 neurons in the hidden layer has the highest performance with 86.6%. Figure 10 shows the CCF curve for the optimized ANN structure. As shown in this figure, the optimum epoch is 224. It should be noted that the ANN technique cannot propose a formulation to predict the compressive strength of FRPCCC. Therefore, in the next section of this paper, the K-fold cross-validation technique is used to obtain a new formulation. Then, the efficiency of the proposed formula is examined.

Using a Model with a K-Fold Cross-Validation Technique in FFBPNN
In this section of the paper, a K-fold cross-validation (KFCV) technique is applied for the optimization and evaluation of the perfected ANN [80,81]. In the KFCV technique, the data are divided randomly into K folds. Then, the K-1 folds are used for training, and the last fold is used to test the neural network. In the parametric study conducted, the values for K, changing from two to five and K = 4, are considered. The process of learning and testing is conducted for all the K sections. Therefore, all the K sections contribute to the learning and testing of the ANN. This process is iterated three times for the reduction and variation of KFCV and similar distribution of data in each K. The performance of the neural network for each iteration can be computed by the percentage of correct predictions in the neural network for K folds.
In every epoch, the performance evaluation of the neural network is calculated. The curve is the correct classification factor (CCF), it is drawn for three iterations and, finally, it is averaged. In the CCF curve, after a specified epoch, the curve is saturated. Then, the optimal epoch is defined using 10% of the curve plateau. In this study, a neural network with three layers is selected for the sake of simplicity. For optimization of the ANN structure, some neurons in the hidden layer are optimized. For this purpose, the selecting criteria are considered to be the area under the CCF curve (AUCCF). Therefore, the AUCCF is measured until it reaches the optimal epoch. Hence, different neurons, from two to 13 neurons in the hidden layer, are selected and the KFCV process is repeated for the structures. Finally, the structure with the maximum efficiency can be determined by drawing the CCF and calculating the AUCCF. Figure 9 shows the AUCCF curve. As can be seen from this figure, the 6:11:1 structure with 11 neurons in the hidden layer has the highest performance with 86.6%. Figure 10 shows the CCF curve for the optimized ANN structure. As shown in this figure, the optimum epoch is 224.  Data that are predicted by the optimized ANN neural network and the training data are plotted in Figure 11. As shown in this figure, the correlation coefficient is equal to 0.9809, which confirms the performance of the optimized ANN structure.  Data that are predicted by the optimized ANN neural network and the training data are plotted in Figure 11. As shown in this figure, the correlation coefficient is equal to 0.9809, which confirms the performance of the optimized ANN structure. Data that are predicted by the optimized ANN neural network and the training data are plotted in Figure 11. As shown in this figure, the correlation coefficient is equal to 0.9809, which confirms the performance of the optimized ANN structure.
Appl. Sci. 2020, 10, 1769 11 of 21 Figure 11. The correlation coefficient of the predicted data by optimized ANN structure and training data.
The Tansig and Pureline activation functions are selected for the hidden layer and the output layer, respectively. Considering the optimum structure of the neural network, weights, biases, and activation functions, a relation, such as Equation (2), could be extracted: Input, Output, IW, LW, b1, and b2 in Equation (2)

Comparison of the Proposed Strength Model with Existing Empirical Ones
Five known models are selected [12,13,15,16,18,56] to verify the proposed formula. It must be noted that no formula has been proposed in the most recent available publication [55]. The formula Predicted data by optimized neural network Training data Figure 11. The correlation coefficient of the predicted data by optimized ANN structure and training data.
The Tansig and Pureline activation functions are selected for the hidden layer and the output layer, respectively. Considering the optimum structure of the neural network, weights, biases, and activation functions, a relation, such as Equation (2), could be extracted: Input, Output, IW, LW, b 1, and b 2 in Equation (2) are constant coefficients, which are defined as follows:

Comparison of the Proposed Strength Model with Existing Empirical Ones
Five known models are selected [12,13,15,16,18,56] to verify the proposed formula. It must be noted that no formula has been proposed in the most recent available publication [55]. The formula proposed in this paper can be implemented in a calculator, while, in the case of the neuro-fuzzy, neural network, multivariate adaptive regression splines and M5 model tree techniques (all considered in [55]), a computer and professional programs should be used. Figure 12 shows the values of the compressive strength of the FRPCCC obtained by the proposed and existing formula versus the experimental values. Table A1 in the Appendix section shows the experimental data that have been used to judge the ability of different methods. In fact, for all formulas, the same data are applied to forecast the compressive strengths of the FRPCCC. Figure 12 shows that the presented formula can estimate the compressive strengths of the FRPCCC with a higher precision compared to the existing formulas.
is very accurate compared to other existing ones, for which the accuracy is lower than 85%. Based on Figures 11 and 12, as well as Table 4, it is evident that the proposed formula has a good agreement with the actual values. Therefore, it can be used in the practical projects to evaluate the amount of column compressive capacity reinforced by FPR sheets in the initial design. It should be The mean percentage of error, correlation coefficient, root mean square error (RMSE), and average absolute error (AAE) for the studied methods are shown in Table 4 to verify the efficiency of the proposed method. Based on this table, it should be noted that the mean percentage of error and the correlation coefficient for the proposed method are equal to 3.49% and 0.9809, respectively. Meanwhile, the corresponding values for other existing methods are equal to over 13% and 0.41, respectively. This means that, for the proposed formula, more than 96% of the simulated results are entirely consistent with the experimental ones. Furthermore, the minimum values of RMSE and AAE are obtained for the proposed formula. Therefore, it should be pointed out that the proposed formula is very accurate compared to other existing ones, for which the accuracy is lower than 85%. Based on Figures 11 and 12, as well as Table 4, it is evident that the proposed formula has a good agreement with the actual values. Therefore, it can be used in the practical projects to evaluate the amount of column compressive capacity reinforced by FPR sheets in the initial design. It should be noted that the collected data (see Appendix A) are for different types of FRP sheets (carbon, aramid, and glass) and the FFBPNN method has been trained and tested with these data. Therefore, the proposed formula can estimate the ultimate compressive capacity of FRP-confined concrete cylinders with a different type of FRP and arbitrary thickness.

Concluding Remarks
A soft computing model for the ultimate strength estimation of FRPCCC has been proposed in this paper. A set of experimental data from the published literature has been collected and divided into input and output parameters. Firstly, the ANN model has been created and analyzed. The mean squared error and R-values have been used to verify the efficiency of the network.
The results of the analysis indicate that a network with 15 hidden neurons has the best performance. However, it should be noted that the basic ANN technique cannot propose a formulation to forecast the compressive strength of FRPCCC. Therefore, in the next step of the study, the author's improvement approach has been presented. A model with a K-fold cross-validation technique in the feed-forward backpropagation neural network has been presented. The correlation coefficient, root mean square error, mean percentage of error and average absolute error have been used to check its efficiency. The structure with 11 neurons in the hidden layer has been found to give the best performance. Finally, a comparison between the proposed formula and existing empirical ones has been conducted. To verify the proposed formula, five known models described in this paper have been selected. The results of the study show that the proposed method can estimate the compressive strengths of the FRPCCC with higher precision compared to the existing formulas. Moreover, it can be used to predict the compressive strength of FRPCCC with different types and arbitrary thicknesses of FRP (carbon, aramid, and glass). It should be noted that the mean percentage of error and the correlation coefficient for the proposed method are equal to 3.49% and 0.9809, respectively. Meanwhile, the corresponding values for other existing methods are equal to over 13% and 0.41, respectively. It means that, for the proposed formula, more than 96% of the simulated results are entirely consistent with the experimental results. Furthermore, the minimum values of RMSE and AAE have been obtained for the proposed formula. Therefore, it should be pointed out that the proposed formula is very accurate compared to other existing methods, for which the accuracy is usually lower than 85%. It should also be added that the proposed method can be easily employed using a calculator with high precision while, in the case of the neuro-fuzzy network, neural network and other known methods, a computer and sophisticated software is usually needed. Therefore, our model can be used to estimate the ultimate compressive capacity of FRP-confined concrete cylinders in the initial design of practical projects.
Finally, it should be noted that there is a lack of experimental tests on concrete cylinders made of seawater and sea sand retrofitted with FRP sheets in order to propose a formula that covers the entire region. This should be a focus in future studies.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The collected data are indicated in Table A1.