## Appendix A Convergence Study  
In this appendix, the results of the convergence study performed to determine the mesh size for the numerical comparisons are presented. They were obtained using meshes with seven-parameter elements of order \(p=8\). For conical and cylindrical shells with uniform thickness, the geometrical parameters considered are those reported in [22], and for parabolic and hemispherical shells they are those presented in [23].  
In the following tables, the label Size indicates the number of elements used for each mesh analyzed in the convergence study, and the mesh sizes considered were 4, 16, 25, and 36. Note that the first mesh size corresponds to a coarse mesh, and the last one corresponds to a fine mesh.  
Table 1 presents the convergence study for conical shells with uniform thickness (UT) and linear variable thickness (LVT), and both truncated cones are under CF boundary conditions and their volume distribution is type FG-II; see Equation (13). From the results, it is observed that meshes of 25 elements give a good convergence for the first ten frequencies.  
The results of the convergence study for cylindrical shells with UT and LVT are presented in Table 2, both of them under CF boundary conditions. From this study, in general, the convergence was observed for a mesh size of 25 elements.  
Table 3 shows the convergence of the natural frequencies for parabolic shells under SF boundary conditions, and for several values of the power-law index \(n\). For uniform thickness, a good convergence was obtained using a mesh size of 16 elements. However, for a parabolic shell with LVT, the convergence was observed at meshes of 25 elements.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Size**} & \multirow{2}{*}{\(f_{1}\)} & \multirow{2}{*}{\(f_{2}\)} & \multirow{2}{*}{\(f_{3}\)} & \multirow{2}{*}{\(f_{4}\)} & \multirow{2}{*}{\(f_{5}\)} & \multirow{2}{*}{\(f_  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & \\ \cline{3-13} \(n\) & Size & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) \\ \hline \multicolumn{13}{c}{UT} \\ \hline \multirow{4}{*}{0.6} & 4 & 115.0771 & 115.0783 & 162.5836 & 162.5836 & 163.816 & 163.8276 & 190.6016 & 205.3549 & 205.3549 & 205.7548 \\  & 16 & 115.0375 & 115.0375 & 162.5828 & 162.5828 & 162.6739 & 162.6739 & 190.6009 & 194.7506 & 194.7545 & 205.3518 \\  & 25 & 115.0374 & 115.0374 & 162.5828 & 162.5828 & 162.6739 & 162.6739 & 190.6009 & 194.7454 & 194.7454 & 205.3518 \\  & 36 & 115.0373 & 115.0373 & 162.5828 & 162.5828 & 162.6732 & 162.6732 & 190.6009 & 194.7448 & 194.7448 & 205.3518 \\ \hline \multirow{4}{*}{1} & 4 & 114.1884 & 114.1895 & 161.5536 & 161.5536 & 162.9672 & 162.9773 & 189.1796 & 203.9445 & 203.9445 & 204.8127 \\  & 16 & 114.1494 & 114.1494 & 161.5521 & 161.5521 & 161.8309 & 161.8309 & 189.1789 & 193.8818 & 193.8857 & 203.9420 \\  & 25 & 114.1493 & 114.1493 & 161.5521 & 161.5521 & 161.8301 & 161.8301 & 189.1789 & 193.8772 & 193.8772 & 203.9420 \\  & 36 & 114.1492 & 114.1492 & 161.5521 & 161.5521 & 161.8301 & 161.8301 & 189.1789 & 193.8766 & 193.8766 & 203.9420 \\ \hline \multirow{4}{*}{5} & 4 & 114.1847 & 114.1854 & 157.9431 & 157.9431 & 162.9485 & 162.9594 & 186.2609 & 202.1293 & 202.1293 & 206.7551 \\  & 16 & 114.1467 & 114.1467 & 157.9423 & 157.9423 & 161.8215 & 161.8215 & 186.2609 & 195.7845 & 195.7884 & 202.1268 \\  & 25 & 114.1466 & 114.1466 & 157.9423 & 157.9423 & 161.8207 & 161.8207 & 186.2602 & 195.7800 & 195.7800 & 202.1268 \\  & 36 & 114.1465 & 114.1465 & 157.9423 & 157.9423 & 161.8207 & 161.8207 & 186.2602 & 195.7793 & 195.7793 & 202.1262 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Convergence of natural frequencies for parabolic shells with UT and LVT under SF boundary conditions (FG-I).