## 6 Results and Discussion  
### Accuracy of Proposed ABC-ANN Model  
Table 6 shows the comparison between different models with experimental data for estimating \(S_{j,ini}\) and \(M_{n}\). Concerning the calculated average (\"Avg.\") and standard deviation (\"STD\") values, the results in Table 6 indicate that the ABC-ANN model provides more reliable predictions for both, compared to the EC3 or KK formulations described earlier. Using the existing empirical models, in particular, the \(S_{j,ini}\) and \(M_{n}\) predictions for some specimens are either underestimated or overestimated, and this suggests the limitation of mechanical models to capture the underlying mechanism that governs both the parameters. The ABC-ANN predictions, conversely, are characterized by minimum deviation.  
Figures 8 and 9 show the scatter graph that provided the relationship between test results and the proposed ABC-ANN model for estimating the \(S_{j,ini}\) and \(M_{n}\) parameters, respectively. In this case, the comparative results also indicate that the ABC-ANN model offers a reliable value for the ratio of experimental to computational predictions (\(R^{2}\)), for both the examined mechanical parameters, and thus confirming further the high potential and accuracy of the proposed model.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(S_{j,ini}\)} & \multicolumn{5}{c}{\(M_{n}\)} \\  & \multicolumn{3}{c}{**(kNm/rad)**} & \multicolumn{5}{c}{**(kNm)**} \\ \hline
**Test** & **Test/EC3** & **Test/KK** & **Test/ABC-ANN-S** & **Test** & **Test/EC3** & **Test/KK** & **Test/ABC-ANN-M** \\ \hline
6000 & 0.62 & 0.81 & 1.32 & 43.6 & 1.11 & 1.22 & 0.9  
Figure 8: ABC-ANN calculated versus experimentally observed values for the parameter \(S_{j,ini}\).  
Figure 9: ABC-ANN calculated versus experimentally observed values for the parameter (\(M_{u}/M_{p,beam}\)).  
By substituting the normalized values of Table 3 for each one of the six input parameters, they are represented by a \(6\times 1\) vector that is herein labeled as \(a^{(1)}\). Then, by using the following equations, the values of \(S_{j,ini}\) and \((M_{n}/M_{p,beam})\) can be calculated from:  
\[a^{(2)}=tanh\Big{(}IW\times a^{(1)}+b_{1}\Big{)} \tag{15}\]  
\[a^{(3)}=tanh\Big{(}LW1\times a^{(2)}+b_{2}\Big{)} \tag{16}\]  
\[Y_{\text{S or M}}^{Predict(Normalize)}=\tanh\Big{(}LW2\times a^{(3)}+b_{3} \Big{)} \tag{17}\]  
\[Y_{\text{S or M}}^{Predict(Actual)}=\frac{Y_{\text{S or M}}^{Predict(Normalize )}+1}{2}\times(Y_{max}-Y_{min})+Y_{min} \tag{18}\]  
The parameters \(IW\), \(LW1\), \(LW2\), \(b_{1}\), \(b_{2}\), and \(b_{3}\) are shown as vector matrices in Tables 7 and 8. Furthermore, \"\(S\)\" and \"\(M\)\" in Equations (15)-(18) stand for \(S_{j,ini}\) and \((M_{n}/M_{p,beam})\), respectively.  
Another visual measure that can be taken into account for comparing the performance of the ABC-ANN model against the component-based mechanical models (EC3 or KK) is the Taylor diagram, see Figures 10 and 11. This diagram depicts a graphical illustration of the adequacy of each investigated model, based on the root mean square-centered difference, the correlation coefficient, and the standard deviation.  
The results shown in Figures 10 and 11 clearly indicate that the closest prediction for both the \(S_{j,ini}\) and \(M_{n}\) input parameters, to the point representing the experimental data of the literature, are provided by the herein developed ABC-ANN model. The EC3 component-based model, as shown, also results in high values of root mean square-centered difference and standard deviation, thus further suggesting a good accuracy of the formulation over the selected experimental data. The same comparative parameters, conversely, are rather low regarding the application of the KK model to the selected experimental specimens.