# A Proposed Soft Computing Model for Ultimate Strength Estimation of FRP-Confined Concrete Cylinders
## 4 Proposing a New Formulation to Predict the Compressive Strength of FRP-Confined Concrete Cylinder
### The Artificial Neural Network Model  
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:  
\[ \begin{split}& f_{scaled}=(0.9-0.1)\Big{(}\frac{f-f_{\min}}{f_{ \max}-f_{\min}}\Big{)}+0.1\\ & 0.1\leq f_{scaled}\leq 0.9\end{split}  \tag{1}\]  
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. 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.  
Figure 2: A three-layer artificial neural network. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Quantity** & \(d\) **(mm)** & \(L\) **(mm)** & \(t\) **(mm)** & \(f^{\prime}_{\alpha}\)**(MPa)** & \(f_{l}\) **(MPa)** & \(f_{r}\) **(MPa)** & \(f^{\prime}_{\alpha}\)**(MPa)** \\ \hline Mean & 133.854 & 272.014 & 0.835 & 42.642 & 15.857 & 2123.174 & 80.448 \\ Minimum & 51 & 102 & 0.089 & 19.4 & 2.33 & 229.762 & 33.8 \\  
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. Figure 3: Flowchart of the utilized artificial neural network (ANN) for predicting the confined ultimate concrete strength. After selecting a desirable network (6:15:1), the results for the training of this network are shown in Figures 6-8. It can be seen from Figure 6 that the network is well established and learned, since the values for MSE of the network begin at a large value and stop at a smaller one.