## 4 Results and Discussion  
In this section, the results of several numerical examples for different span-to-height (_L/h_) ratios are presented. The specific material properties of the FG beam constructed of ceramic (Al\({}_{2}\)O\({}_{3}\)) and metal (aluminum) in this study are given as follows:  
Alumina (Al\({}_{2}\)O\({}_{3}\)): E\({}_{\rm c}\) = 380 GPa, \(\rho_{c}\) = 3960 kg/m\({}^{3}\), \(\
u\) = 0.3  
The dimensional frequency can be calculated using the following formula [25]:  
\[\overline{\omega}=\left(\frac{\omega L^{2}}{h}\right)\sqrt{\frac{\rho_{m}}{E_ {m}}} \tag{37}\]  
In the case of the ANN solution, the following parameters are considered as input data: span-to-height ratio, _L/h_ = 5 and _L/h_ = 20; beam type, simply supported (S-S), clamped-clamped (C-C), and clamped-fixed (C-F); beam theory, higher-order shear deformation beam theory or shear deformation theory (HSDT or HSDT), CBT, TBT, ESDT, etc. (see Figure 4); and the power law varying for values 0-10. All of the results were calculated and plotted only for the HSDT and CBT in simply supported FG beams by only considering the first mode (\(m\) = 1). To validate the accuracy of the analytical method and ANN predictions, the non-dimensional frequencies (\(\overline{\omega}\)) obtained by the presented approaches were compared to a reference study conducted by [PERSON] and [PERSON] [5]. Table 1 indicates the comparison of non-dimensional frequency values of simply supported (S-S) FG beams for different span-to-height ratios (_L/h_), and the power law index (\(p\)) using HSDT and CBT. As can be seen from Table 1, while the value obtained with the analytical solution yields the same results as the reference study, it also converges with the ANN prediction values with a very small margin of error. These results show us the originality and quality of the trained ANN model. However, for \(p\) = 10, a higher margin of error is observed in the ANN estimation of CBT and HSDT compared to other values. The reason for this margin of error is that dimensionless natural frequency values for \(p\) = 10 for both approaches were not attained in desired level studies in the literature when the ANN dataset was created. In order to understand the results clearly, a considerable amount of 3D and 2D diagrams (see Figures 4-12) were obtained and visualized from the analytical solution and ANN method, showing the variations of the non-dimensional frequency value according to different parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Inputs** & & **Ref. [5]** & **Present** & **ANN Prediction** & **Error (\%)** \\ \hline
**Beam Type** & **Beam Theory** & _L/h_ & \(p\) & & & **(\%)** \\ \hline \multirow{8}{*}{S-S} & \multirow{4}{*}{CBT} & & 0 & 5.3953 & 5.3953 & 5.3703 & 0.46 \\  & & & 0.5 & 4.5931 & 4.5931 & 4.6144 & 0.46 \\  & & 5 & 1 & 4.1484 & 4.1484 & 4.1775 & 0.70 \\  & & 2 & 3.7793 & 3.7793 & 3.7545 & 0.66 \\  & & & 5 & 3.5949 & 3.5949 & 3.6508 & 1.55 \\  & & & 10 & 3.4921 & 3.4921 & 3.4816 & 0.30 \\  & & & 0 & 5.4777 & 5.4777 & 5.3975 & 1.46 \\  & & & 0.5 & 4.6641 & 4.6641 & 4.5927 & 1.53 \\  & & & 1 & 4.2163 & 4.2163 & 4.1843 & 0.76 \\  & & & 2 & 3.8472 & 3.8472 & 3.8234 & 0.62 \\  & & & 5 & 3.6628 & 3.6628 & 3.6532 & 0.26 \\  & & & 10 & 3.5547 & 3.5547 & 3.4890 & 1.85 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._Figure 8: Non-dimensional frequency comparison obtained from HSDT and ANN for L/\(h\) = 5. Figure 6: Non-dimensional frequency prediction (\(\overline{\omega}\)) of ANN for CBT. Figure 7: Non-dimensional frequency prediction (\(\overline{\omega}\)) of ANN for HSDT. Figure 11: Non-dimensional frequency comparison obtained from CBT and HSDBT for \(L/h=20\). Figure 10: Non-dimensional frequency comparison obtained from CBT and HSDBT for \(L/h=5\). Figure 9: Non-dimensional frequency comparison obtained from HSDT and ANN for \(L/h=20\). Second, an artificial neural network method (ANN) was developed in order to predict the natural frequencies of FGM beams.