## 3 Results
### Ultimate Strains Prediction  
Strain predictions for the RMSE, MARE and \(R^{2}\) values of the applied models are compared in Table 4.  
This tables shows that the ANFIS-SC models outperform other models. Here also RBNN and ANFIS-FCM have similar accuracy, and they are slightly worse than the ANFIS-SC model. A radii value of 0.1 in ANFIS-SC corresponds to 13 clusters. Similar to the previous application, here too ANFIS-SC has many more membership functions and parameters compared to ANFIS-FCM. Table 5 gives the rule base of the optimal ANFIS-SC model. The estimated results of the optimal RBNN, ANFIS-FCM and ANFIS-SC models are shown in Figure 6.  
This figure shows that the estimates of the ANFIS-SC model are closer to the corresponding measured values, especially for high strain values. The variation of strength versus strain ratio and confinement stiffness ratio for the optimal ANFIS-SC model is shown in Figure 7. It is clear from the figure that strain takes its maximum value when the strain ratio and confinement stiffness ratio are both at their maximum values. The linear relationship between strain and strain ratio is clearly seen in the range of confinement-stiffness-ratio between 0.4 and 0.6. The effects of strain ratio and the confinement stiffness ratio inputs on strength and strain were also investigated using the ANFIS-SC method because it has less control parameters than RBNN. For the RBNN models, the optimal radii and hidden node numbers should be determined \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{
\begin{tabular}{c} **Control** \\ **Parameters** \\ \end{tabular} } & \multicolumn{4}{c}{**Training**} & \multicolumn{4}{c}{**Test**} \\ \cline{3-8}  & & **RMSE** & **MARE** & \(\mathbf{R^{2}}\) & **RMSE** & **MARE** & \(\mathbf{R^{2}}\) \\ \hline RBNN & 0.640 & 2.43 & 29.9 & 0.767 & 2.51 & 31.6 & 0.752 \\ ANFIS-SC & 0.1 & 2.31 & 29.4 & 0.790 & 2.47 & 30.5 & 0.766 \\ ANFIS-FCM & 6 & 2.33 & 31.3 & 0.786 & 2.57 & 31.9 & 0.742 \\ MSTree & & 1.90 & 21.8 & 0.858 & 2.72 & 33.3 & 0.711 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical performance of RBNN, ANFIS-SC, ANFIS-FCM and MSTree models in strain predictions. Figure 5: Variation of strength versus strain ratio and confinement stiffness ratio for the optimal ANFIS-FCM model. whereas, in obtaining ANFIS-SC model, the radii value which indicates the number of clusters is only distinguished. The simulation results of the ANFIS-SC models are reported in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Inputs**} & \multirow{2}{*}{**Control**} & \multicolumn{3}{c}{**Training**} & \multicolumn{3}{c}{**Test**} \\ \cline{3-8}  & **Parameters** & **RMSE** & **MAE** & \(R^{2}\) & **RMSE** & **MARE** & \(R^{2}\) \\ \hline \multicolumn{8}{c}{Strength} \\ \hline \(\rho_{c}\) & 0.1 & 0.852 & 32.8 & 0.249 & 0.804 & 32.8 & 0.054 \\ \(\rho_{K}\) & 0.1 & 0.646 & 21.3 & 0.512 & 0.648 & 25.1 & 0.445 \\ \hline \multicolumn{8}{c}{Strain} \\ \hline \(\rho_{c}\) & 0.2 & 4.081 & 63.6 & 0.343 & 4.406 & 59.2 & 0.244 \\ \(\rho_{K}\) & 0.1 & 4.453 & 71.8 & 0.218 & 4.330 & 63.2 & 0.265 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Effect of strain ratio (\(\rho_{c}\)) and the confinement stiffness ratio (\(\rho_{K}\)) inputs on strength and strain with respect to ANFIS-SC. Figure 6: Strain estimates of the RBNN, ANFIS-SC, ANFIS-FCM and M5 Tree models (dimensionless: \(\epsilon_{cc}/\epsilon_{co}\)).