## 4 Numerical Investigation
### Parametric Study  
We refer to a BD-FG cylindrical structure with constituent phases of properties listed in Table 3, where the following dimensionless parameters are considered to compute the dimensionless structural excitation frequencies  
\[\Omega=\omega R\sqrt{\frac{\rho_{m}}{E_{m}}},K_{g}=\frac{k_{g}R^{2}}{E_{m}h^{3}},K_{w}=\frac{k_{w}R^{4}}{E_{m}h^{3}}. \tag{38}\]  
We determine the DIR, and highlight the effects of different parameters such as the thickness-to-radius ratio (\(h/R\)), the length-to-radius ratio (\(L/R\)), the static load factor, the boundary conditions, the power law indexes (\(n_{x}\),\(n_{z}\)), the type and volume fraction of porosity, and the foundation parameters, on the dynamic buckling behavior of the BD-FG cylindrical shell. In Figure 3 we plot the variation of the DIR for different thickness-to-radius ratios (\(h/R\)), where a clear shift of the DIR is observed for increasing \(h/R\) ratios. This means that the DIR becomes wider for a certain value of the dynamic load factor, and occurs with a sort of delay. An increased \(h/R\) ratio from 0.01 to 0.1 yields a global shift of the DIR origin point towards high excitation frequencies. Figure 4 shows the sensitivity of the DIR for varying \(L/R\) ratios, while keeping fixed \(h/R=0.01\). In detail, for \(L/R=1\) the structure has a wider DIR in comparison with the other values, whereby an increasing \(L/R\) ratio yields the DIR to take place at lower excitation frequencies. Based on the plots in Figure 4, we can observe a reduction of about 41.96% in the excitation frequencies corresponding to the origin of the instability region, for a \(L/R\) ranging between 1 and 10. When the \(L/R\) ratio features higher magnitudes, the bending resistance gradually reduces and yields an increased bending deformation. \begin{table}
\begin{tabular}{c c c c} \hline
**Z** & **h/R** & **[PERSON] et al. [48]** & **Present** \\ \hline
50 & 0.01 & 79.9296 (4,5) & 79.9295 \\  & 0.025 & 79.48684 (4,3) & 79.4868 \\  & 0.05 & 78.79842 (4,3) & 78.7984 \\ \hline
300 & 0.01 & 479.5066 (10,5) & 479.5065 \\  & 0.025 & 476.3834 (10,3) & 476.3834 \\  & 0.05 & 470.8775 (11,1) & 470.8775 \\ \hline
900 & 0.01 & 1438.157 (18,3) & 1438.1576 \\  & 0.025 & 1428.611 (18,2) & 1428.6108 \\  & 0.05 & 1412.380 (19,1) & 1412.3802 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparative evaluation of the dimensionless critical axial buckling load for a FG cylindrical shell with \(h=0.001\) m, \(E_{m}=70\) GPa, \(\
u=0.3\), \(n_{z}=2\). \begin{table}
\begin{tabular}{c c c c c} \hline
**Constituent Phases** & **Material** & \multicolumn{3}{c}{**Properties**} \\ \cline{3-5}  & & **E (GPa)** & \(\rho\) **(Kg/m3)** & \(\
u\) \\ \hline c & SiC & 427 & 3100 & 0.17 \\ m & Al & 70 & 2702 & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 3: Material properties of the BD-FG cylindrical shell. In Figure 5, we evaluate the effect of the static load factor on the instability region of the BD-FG cylindrical shell. As expected, in absence of a static load on the structure, the width of DIR gets smaller, whereby for an increased static load factor, it becomes gradually greater for a fixed value of dynamic load factor (i.e., \(\beta=1\)), and its origin tends to move on the left side. This proves the sensitivity of the structural instability to the static load factor. Figure 4: Effect of the length-to-radius ratio on the DIR for a BD-FG cylindrical shell with \(R=0.5\) m, \(h/R=0.02\), \(n_{x}=n_{z}=1\), \(\alpha=0.3\). Figure 3: Effect of the thickness-to-radius ratio on the dynamic instability region (DIR) for a BD-FG cylindrical shell with \(R=0.5\) m, \(L/R=1\), \(n_{x}=n_{z}=1\), \(\alpha=0.3\). As also visible in Figure 6, we evaluate the impact of different boundary conditions on the DIR of the cylindrical shell. Here, we consider three different boundary conditions, namely, S-S, C-S, and C-C boundary conditions.