# Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates
## 5 Numerical Results and Discussion
### Convergence and Validation of Present Formulation  
#### 5.1.1 Free Vibration Analysis  
**Example 1. ** The convergence study for the non-dimensional frequency parameter was carried out for UD and FG-CNT-reinforced rhombic plate shown in Table 2. The dimensionless frequency parameter of the UD, FG-V, FG-O, and FG-X type distributed CNTRC rhombic plate was computed for different mesh sizes and clamped boundary conditions. The results were computed for \(V^{\ast}_{\rm CNT}=0.11\) and a skew angle of \(15^{\circ}\). The convergence study indicated that \(16\,\times\,16\) mesh is satisfactory for the free vibration analysis of functionally graded CNTRC rhombic plate using current nine-noded isoparametric elements. Hence, \(16\,\times\,16\) mesh size was adopted for all the cases of free vibration analysis of a functionally graded CNT-reinforced rhombic plate. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mesh** & **UD** & **FG-V** & **FG-O** & **FG-X** \\ \hline \(8\,\times\,8\) & 18.5932 & 18.0731 & 16.6424 & 19.2637 \\ \(10\,\times\,10\) & 18.5915 & 18.0719 & 16.4112 & 19.2623 \\ \(12\,\times\,12\) & 18.5906 & 18.0709 & 16.4093 & 19.2612 \\ \(14\,\times\,14\) & 18.5894 & 18.0697 & 16.4085 & 19.2606 \\ \(16\,\times\,16\) & 18.5892 & 18.0694 & 16.4084 & 19.2604 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Convergence of non-dimensional frequency of FG-CNT-reinforced clamped rhombic plate. **Example 2. ** Tables 3 and 4 show the results of the bending and free vibration analyses for an isotropic square plate (\(\
u=0.3\)), respectively. The maximum deflection and axial stress were compared with the results provided by [PERSON] [41] and the frequency parameter of the isotropic plate was compared with an exact solution [42] and HSDT results for a moderately thick plate [43]. **Example 3. ** Free vibration analyses of the FG-V CNT-reinforced composite plate for various _a/h_ ratios were presented in Tables 5 and 6. The first six non-dimensional frequencies were compared with [PERSON] et al. [5]. Three volume fractions (\(V^{\bullet}_{\textbf{CNT}}=0.11\) and \(0.14\)) and three side-to-thickness ratios (_a/h_ = 10, 20 and 50) were taken for comparison. The frequency parameter of simply supported and clamped boundary condition was found to be closer to [PERSON] et al. [5]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Source** & & **Mode** \\ \hline  & (1, 1) & (1, 2) & (1, 3) \\ Present & 0.093 & 0.221 & 0.415 \\ [PERSON] et al. [43] & 0.093 & 0.222 & 0.415 \\ [PERSON] et al. [42] & 0.093 & 0.223 & 0.417 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of non-dimensional frequency parameter of the simply supported square isotropic plate. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**_a/h_** & **Source** & \(\overline{m}\) & \(\overline{\sigma}_{\textbf{x}}\) \\ \hline
10 & Present & 4.666 & 0.289 \\  & [PERSON] [41] & 4.770 & 0.289 \\ \hline
20 & Present & 4.491 & 0.287 \\  & [PERSON] [41] & 4.570 & 0.268 \\ \hline
50 & Present & 4.408 & 0.284 \\  & [PERSON] [41] & 4.496 & 0.266 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of non-dimensional maximum deflection and normal stress of square isotropic plate subjected to uniform load.