## 4 Applications
### Effect of Through-the-Thickness Distribution of CNTs  
The following examples show the variation of the natural frequencies as a function of the volume fraction distribution along the thickness of the lamina by varying the parameter \(p^{(1)}\) of a five-parameter \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Uniform CNT Reinforcement Distribution**} \\ \hline \(f\)**[Hz]** & **FSDT\({}_{RS}^{x=5/6}\)** & **ED2\({}^{x=5/6}\)** & **ED3** & **3D FEM** \\ \hline
1 & 120.787 & 120.982 & 121.091 & 121.155 \\
2 & 241.448 & 241.821 & 242.199 & 242.324 \\
3 & 241.448 & 241.821 & 242.199 & 242.324 \\
4 & 349.728 & 350.238 & 350.992 & 351.162 \\
5 & 421.114 & 421.732 & 422.774 & 422.987 \\
6 & 423.751 & 424.381 & 425.403 & 425.619 \\
7 & 520.239 & 520.961 & 522.499 & 522.739 \\
8 & 520.239 & 520.961 & 522.499 & 522.739 \\
9 & 652.917 & 653.821 & 656.100 & 656.432 \\
10 & 652.917 & 653.821 & 656.100 & 656.432 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the first ten natural frequency variations of a CCCC square plate made of a single CNT-reinforced lamina of constant thickness \(h=0.1\) m given by GDQ method and FEM. The 3D FEM solution was obtained by commercial software Strand7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{5}{c}{**Lamination Scheme:**} \\ \hline \(\left(\mathbf{FG-CNT}_{CNT(\mathbf{SP})(\mathbf{e}^{(2)}=1/\mathbf{b}^{(2)}=0/ \mathbf{c}^{(2)}=0/\mathbf{d}^{(2)}=1/p^{(2)}=0}\right)/\mathbf{PM}/F\mathbf{G- CNT}_{CNT(\mathbf{SP})(\mathbf{e}^{(2)}=1/\mathbf{b}^{(2)}=0/\mathbf{c}^{(2)}=0/ \mathbf{d}^{(2)}=1/p^{(2)}=0)}^{\mathbf{(2)}=0}\right)\) \\ \hline \(f\)**[Hz]** & **FSDT\({}_{RS}^{x=1}\)** & **FSDT\({}_{RS}^{x=5/6}\)** & **EDZ2\({}^{x=1}\)** & **EDZ2\({}^{x=5/6}\)** & **EDZ3** & **3D FEM** \\ \hline
1 & 105.243 & 103.892 & 105.559 & 104.188 & 105.482 \\
2 & 204.169 & 200.091 & 204.815 & 200.691 & 204.637 & 204.608 \\
3 & 204.169 & 200.091 & 204.815 & 200.691 & 204.637 & 204.608 \\
4 & 289.471 & 282.386 & 290.400 & 283.241 & 290.096 & 290.038 \\
5 & 343.586 & 334.061 & 344.730 & 335.112 & 344.322 & 344.246 \\
6 & 346.354 & 336.858 & 347.512 & 337.923 & 347.104 & 347.037 \\
7 & 418.461 & 405.747 & 419.848 & 407.016 & 419.308 & 419.190 \\
8 & 418.461 & 405.747 & 419.848 & 407.016 & 419.308 & 419.190 \\
9 & 514.520 & 496.736 & 516.310 & 498.371 & 515.554 & 515.393 \\
10 & 514.520 & 496.736 & 516.310 & 498.371 & 515.554 & 515.393 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the first ten natural frequency variations of a CCCC square plate made of a 3-layered material (CNT/matrix/CNT) with various thickness \(h=0.02\) m/\(0.06\) m/\(0.02\) m given by GDQ method and FEM. The 3D FEM solution was obtained by commercial software Strand7. exponential law. A functionally graded CNT-reinforced fully clamped square plate (Figure 4a) made of a single ply of constant thickness (\(h=0.1\) m) is assumed. The agglomeration parameters and mass fraction value of the CNTs are given in Tables 4 and 5. Material was described with the five-parameter exponential law defined in Figure 5a. At the bottom of the plate, elastic foundations were applied, the parameters of which can be found in Tables 4 and 5. In this example, the following theories were used: \(\text{FSDT}_{RS}^{\text{r}=5/6}\), \(\text{ED2}^{\text{r}=5/6}\) and ED3. Since the lamination scheme is characterized by only one layer, the Murakami's function was not needed. Tables 4 and 5 show the first ten natural frequencies for each theory for different values of the parameter \(p^{(1)}\) with two different parameters of elastic foundations. The overall behavior of the through-the-thickness distribution variation can be observed more easily from the graphs depicted in Figures 6 and 7.