## 4 Numerical Investigation
### Validation  
Due to the general lack of works in the literature on the dynamic buckling behavior of BD-FG porous cylindrical shells, the proposed model is validated, herein, for an axial buckling problem of a simply supported conventional FG cylindrical shell based on two different theories. Thus, for comparative purposes, we select the same material properties and shell geometry as reported in [48; 61], and neglect the inertia terms, foundation parameters and porosity effects, while assuming a null value for \(n_{x}\). In Table 1 we summarize the results in terms of critical axial buckling load \(F_{cr}\) for a FG cylindrical shell with \(h=0.001\) m, \(L/R=0.5\), \(E_{c}=380\) GPa, \(E_{m}=70\) GPa, \(\
u=0.3\), \(\tilde{c}=0\), with a clear excellent agreement between our results and predictions in Ref. [61] based on the first order shear deformation theory. This first numerical example could be considered as limit case, where the TSDT reverts to the FSDT, since it refers to a thin shell structure, just for validation purposes. Moreaccurate results, however, are always expected under a TSDT assumption for increased values of the shell thickness, as done in the next parametric investigation.  
As further comparative study, Table 2 compares the dimensionless critical buckling load (\(P_{cr}=F_{cr}L^{2}/\pi^{2}D_{m}\); \(D_{m}=E_{m}h^{3}/12\Big{(}1-\
u_{m}^{2}\Big{)}\)) for a FG cylindrical shell with \(h=0.001\) m, \(E_{c}=380\) GPa, \(E_{m}=70\) GPa, \(\
u=0.3\), \(\bar{c}=4/3h^{2}\). Based on Table 2, it is worth noticing the high precision between our results and predictions by [PERSON] et al. (2018), which confirms the accuracy of the GDQ method. This method is thus proposed in the following parametric study, as efficient numerical tool to solve the problem.  
\begin{table}
\begin{tabular}{c c c c} \hline
**R/h** & & **Khazaeinejad and Najafizadeh (2016)** & **Present** \\ \hline
5 & Alumina & 1.598 (1,1) & 1.5975 \\  & n\({}_{\rm z}=1\) & 0.853 (1,1) & 0.8532 \\  & n\({}_{\rm z}=2\) & 0.662 (1,1) & 0.6624 \\  & n\({}_{\rm z}=5\) & 0.520 (1,1) & 0.5197 \\  & n\({}_{\rm z}=10\) & 0.450 (1,1) & 0.4500 \\  & Aluminum & 0.294 (1,1) & 0.2942 \\ \hline
10 & Alumina & 1.403 (1,1) & 1.4029 \\  & n\({}_{\rm z}=1\) & 0.759 (1,1) & 0.7589 \\  & n\({}_{\rm z}=2\) & 0.589 (1,1) & 0.5885 \\  & n\({}_{\rm z}=5\) & 0.456 (1,1) & 0.4557 \\  & n\({}_{\rm z}=10\) & 0.393 (1,1) & 0.3931 \\  & Aluminum & 0.258 (1,1) & 0.2584 \\ \hline
20 & Alumina & 1.594 (1,1) & 1.5936 \\  & n\({}_{\rm z}=1\) & 0.903 (1,1) & 0.9029 \\  & n\({}_{\rm z}=2\) & 0.698 (1,1) & 0.6977 \\  & n\({}_{\rm z}=5\) & 0.514 (1,1) & 0.5140 \\  & n\({}_{\rm z}=10\) & 0.430 (1,1) & 0.4295 \\  & Aluminum & 0.293 (1,1) & 0.2935 \\ \hline
30 & Alumina & 1.566 (2,1) & 1.5664 \\  & n\({}_{\rm z}=1\) & 0.826 (2,1) & 0.8262 \\  & n\({}_{\rm z}=2\) & 0.642 (2,1) & 0.6419 \\  & n\({}_{\rm z}=5\) & 0.511 (2,1) & 0.5108 \\  & n\({}_{\rm z}=10\) & 0.449 (2,1) & 0.4486 \\  & Aluminum & 0.289 (2,1) & 0.2885 \\ \hline
100 & Alumina & 1.443 (3,1) & 1.4428 \\  & n\({}_{\rm z}=1\) & 0.782 (3,1) & 0.7822 \\  & n\({}_{\rm z}=2\) & 0.606 (3,1) & 0.6064 \\  & n\({}_{\rm z}=5\) & 0.469 (3,1) & 0.4681 \\  & n\({}_{\rm z}=10\) & 0.404 (3,1) & 0.4008 \\  & Aluminum & 0.266 (3,1) & 0.2657 \\ \hline
300 & Alumina & 1.443 (5,1) & 1.4431 \\  & n\({}_{\rm z}=1\) & 0.787 (5,1) & 0.7841 \\  & n\({}_{\rm z}=2\) & 0.610 (5,1) & 0.6079 \\  & n\({}_{\rm z}=5\) & 0.468 (5,1) & 0.4683 \\  & results and predictions by [PERSON] et al. (2018), which confirms the accuracy of the GDQ method. This method is thus proposed in the following parametric study, as efficient numerical tool to solve the problem.