# Estimation of the Ultimate Strength of FRP Strips-to-Masonry Substrates Bond
## 3 Prediction with New Models
### The ANN Model  
Artificial Neural Networks are a good predictive tool used for engineering problems [70,71]. Their operation is based on the principle of the prosperity of the nervous system. The data are divided into three sets: training, validation, and testing. ANN is composed of layers, neurons, and weights. The main task of the network is to find a pattern of connections and relationships between inputs and outputs [71]. There are many types of networks [71,72]. A feed-forward neural network involves three types of layers: input, hidden, and output layers (see Figure 2).  
Here, the log-sigmoid transfer function was utilized in the ANN model. Furthermore, one hidden layer was used in this model. At first, all collected data were normalized using the below equation [2]. Next, the normalized data were accepted for three stages, including training, validating, and testing. Since ANN uses the log-sigmoid transfer function to avoid the saturation problem, collapse on the border values, and, consequently, a low training rate, the mapping function converted the real input values to corresponding values in the range of 0.1-0.9. It should be noted that this range w usual in the pre-processing of datasets in engineering applications [71].  
\[\begin{split}& g_{scaled}=(0.9-0.1)\big{(}\frac{g-g_{\min}}{g_{ \max}-g_{\min}}\big{)}+0.1\\ & 0.1\leq g_{scaled}\leq 0.9\end{split} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Quantity** & \(\mathbf{t_{p}}(\mathbf{mm})\) & \(\mathbf{E_{p}}(\mathbf{kN/mm^{2}})\) & \(\mathbf{b_{p}}(\mathbf{mm})\) & \(\mathbf{l_{b}}(\mathbf{mm})\) & \(\mathbf{b_{m}}(\mathbf{mm})\) & \(\mathbf{f_{nt}}(\mathbf{MPa})\) & \(\mathbf{P_{max}}(\mathbf{kN})\) \\ \hline Mean & 0.877 & 120.719 & 32.781 & 170.440 & 347.127 & 2.232 & 15.630 \\ \hline Minimum & 0.12 & 22.3 & 6.35 & 50 & 200 & 0.780739 & 2.13 \\ \hline Maximum & 6.35 & 230 & 50 & 420 & 740 & 3.57 & 84.5 \\ \hline Standard deviation & 1.058 & 83.490 & 13.842 & 105.624 & 197.375 & 0.965 & 20.013 \\ \hline Coefficient of variation & 1.202 & 0.689 & 0.421 & 0.617 & 0.566 & 0.431 & 1.276 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistical properties of experimental data.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Parameters** \\ \hline Input & \(a=t_{p}\times E_{p}(\mathrm{kN/mm})\) \\ \cline{3-6}  & Input & \(b=b_{p}/b_{m}\) (\(\mathrm{mm/mm}\)) \\ \cline{3-6}  & & \(c=l_{b}(\mathrm{mm})\) \\ \cline{3-6}  & & \(d=f_{ut}(\mathrm{MPa})\) \\ \hline \hline \multicolumn{4}{c}{Output} & \multicolumn{3}{c}{\(F=P_{\max}/b_{p}\) (\(\mathrm{kN/mm}\))} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Newly proposed parameters.  
Figure 2: A schematic model of a three-layer neural network: input signals (\(a\)–\(d\)) and output signal \(F\).  
where \(g\), \(g_{\min}\), \(g_{\max}\), and \(g_{scalal}\) are the selected data, the minimum value of data, the maximum value of data, and the scaled value for the data, respectively. This study adopted the Levenberg-Marquardt algorithm for training, validating, and testing datasets. Firstly, various sets were examined to find the optimum distribution of the dataset (training, validating, and testing sets are 70%, 15%, and 15% of all datasets, respectively). A 4:n:1 network was chosen. This network had four inputs, \(n\) hidden neurons, and one output (see Figure 2). A criterion should be supposed to check and stop the training step of the network.  
Figures 5-7 show the results for the training of the selected network (4-10-1). Moreover, Figure 5 exhibits the excellent performance of the selected network. Furthermore, in Figure 7, the training state parameters, such as performance gradient, mu (i.e., the momentum constant), and validation failing (i.e., val fail) epoch, are illustrated based on the Levenberg-Marquardt algorithm [71].