## 5 Numerical Illustrations
### Validation and Convergence Studies of the Present Formulations  
It should be pointed again that the present formulations can be easily degraded to the straight beam when the radius \(R_{0}\) approaches infinity. Accordingly, it yields \(\lambda=0\) and thus \(\eta=1\), hence Equations (16) and (31) reduce to the corresponding ones for straight beams with various end boundary conditions [34]. As the first example, FGM straight beams (\(\lambda=0\)) with various boundary conditions are considered as particular cases for validation of presented method, and the non-dimensional frequency parameter \(\Omega=\omega H\sqrt{\rho_{0}^{1}/c_{550}^{1}}\) for different length-to-depth ratios are tabulated in Tables 1 and 2, in which the material properties of FGMs vary as an exponential relation \(\kappa=\kappa_{1}^{0}\exp[a\Big{(}\frac{1}{2}+\zeta\Big{)}]\) for S-S straight beams, and a power way \(\kappa=\kappa_{1}^{0}\Big{(}\frac{1}{2}-\zeta\Big{)}^{\alpha}+\kappa_{2}^{0} \Big{[}1-\Big{(}\frac{1}{2}-\zeta\Big{)}^{\alpha}\Big{]}\) for C-C and C-F straight beams along the thickness direction, respectively. Here, \(\alpha\) is the gradient index, and \(\kappa_{1}^{0}\) and \(\kappa_{2}^{0}\) denote the material properties of two type materials as follows:  
\[ \begin{array}{ll}\kappa_{1}^{0}:&c_{11}^{0}=139\;\mathrm{GPa},\;c_{13}^{0}=14 \;\mathrm{GPa},\;c_{33}^{0}=336.4\;\mathrm{GPa},\;c_{55}^{0}=162.5\;\mathrm{GPa},\;\rho^{0}=7.5\times 10^{3}\;\mathrm{kg/m^{3}}\\ \kappa_{2}^{0}:&c_{11}^{0}=209.7\;\mathrm{GPa},\;c_{13}^{0}=105.1\;\mathrm{GPa},\;c_{33}^{0}=210.9\;\mathrm{GPa},\;c_{55}^{0}=42.5\;\mathrm{GPa},\;\rho^{0}= 5.676\times 10^{3}\;\mathrm{kg/m^{3}}\end{array} \]  
From the tables, it is seen easily that the present results, including the analytical and semi-analytical ones, agree quite well with those in [34]. In particular, the semi-analytical results based on DQM are completely identical to the analytical results for S-S beam. This illustrates that the approximate laminate model has a very good convergence characteristic. It should be pointed out that more separate layers for approximate laminate model and sampling point number for DQM are needed to yield an accurate result, and \(p=20\) and sampling point number \(N=30\) are adopted in the calculations here. For further verification, the natural frequencies for laminated circular arches are predicted by the present method and compared with those in [37]. The calculations are performed for various values of the modular ratio \(E_{1}/E_{2}\) and the other mechanical properties of each lamina are assumed to be \(G_{12}/E_{2}=G_{13}/E_{2}=G_{23}/E_{2}=0.5\), and \(\mu_{12}=\mu_{13}=\mu_{23}=0.25\). The first three natural frequencies of laminated arches with S-S and C-C boundary conditions are listed in Table 3. It is also obvious that the results are in good agreement with the existing ones. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**H/L = 1/15**} & \multicolumn{4}{c}{**H/L = 1/7**} \\ \cline{2-7}  & **Analytical** & **Semi-Analytical** & **Ref. [34]** & **Analytical** & **Semi-Analytical** & **Ref. [34]** \\ \hline
1 & 0.011572 & 0.011572 & 0.011572 & 0.052470 & 0.052470 & 0.052470 \\
2 & 0.045801 & 0.045801 & 0.045801 & 0.200727 & 0.200727 & 0.200727 \\
3 & 0.101319 & 0.101319 & 0.101319 & 0.414204 & 0.414204 & 0.414204 \\
4 & 0.176130 & 0.176130 & 0.176130 & 0.423816 & 0.423816 & 0.423816 \\
5 & 0.193297 & 0.193297 & 0.193297 & 0.700022 & 0.700022 & 0.700022 \\
6 & 0.267916 & 0.267916 & 0.267916 & 0.828370 & 0.828370 & 0.828370 \\
7 & 0.374308 & 0.374308 & 0.374308 & 1.012457 & 1.012457 & 1.012457 \\
8 & 0.386591 & 0.386591 & 0.386591 & 1.242456 & 1.242456 & 1.242456 \\
9 & 0.493076 & 0.493076 & 0.493076 & 1.349458 & 1.349458 & 1.349458 \\
10 & 0.579877 & 0.579877 & 0.579877 & 1.656399 & 1.656399 & 1.656399 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Non-dimensional frequencies \(\Omega\) for S-S FGM straight beams (\(\alpha=0.5\)).