## 4 Results
### Rectangular Plate Clamped at the Contour  
Following boundary conditions must be satisfied (see Figure 19)  
\[ \begin{split} w(x_{1},x_{2})\Big{|}_{x_{1}=\pm a_{1}}& =0;\quad\varphi_{1}(x_{1},x_{2})\Big{|}_{x_{1}=\pm a_{1}}=0;\\ w(x_{1},x_{2})\Big{|}_{x_{2}=\pm a_{2}}&=0;\quad \varphi_{2}(x_{1},x_{2})\Big{|}_{x_{2}=\pm a_{2}}=0.\end{split}  \tag{34}\] \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline
**Magnitude** & **Edge** & **Unit** & **Present Method** & **Timoshenko** & **Abaqus** \\ \hline \(w\) & Simply supported 1 & m & 0 & 0 & 0 \\ \(M_{11}\) & Simply supported 1 & N m & 0 & 0 & 416.46 \\ \(w\) & Simply supported 2 & m & 0 & 0 & 0 \\ \(M_{22}\) & Simply supported 2 & N m & 0 & 0 & 1170.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Compliance with boundary conditions at the midpoints of the edges. Figure 18: The bending moments \(M_{22}\) in section \(x_{1}=0\) of a simply supported rectangular plate. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & & & **Present Method** & \multicolumn{2}{c}{**Timoshenko**} & \multicolumn{2}{c}{**Abaqus**} & \multicolumn{2}{c}{**RE**} \\
**Magnitude** & **Unit** & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** \\ \hline \(w\) & m & 0.00 & 0.0008 & 0.00 & 0.0008 & 0.00 & 0.0008 & 0\% & 0\% \\ \(M_{11}\) & N m & 0.00 & 4668.88 & 0.00 & 4668.88  
In this example, numerical studies showed that boundary conditions were met with high accuracy \(-\)\(1\times 10^{-8}\) for deflection and \(1\times 10^{-7}\) for slopes. It is seen that symmetry and antisymmetry conditions are performed. The deflection of the center of considered plate is less than previous plate. At the middle cross-section, the bending moment in the center of the plate is smaller than at the edge. Results of calculations are presented in Table 5. As in previous examples the maximum values of theoretical results are close to the maximum values of results obtained in the ABAQUS package (Appendix A.3), but the minimum ones for bending moments do not coincide. The numerical results of bending problem of clamped rectangular thin plates presented in [77] are in excellent agreement between the exact numerical-analytical solution obtained by the generalized integral transform technique (GITT) and FEM obtained by the commercial program ABAQUS, but during the analysis, the plates were discretized using sufficiently refined S4R elements \(-\) 200 elements in the \(x\) direction and 200 \(\times\)\(c\) elements in the \(y\) direction. This means that sufficient accuracy was obtained for 40,000 elements, while in the examples presented in this article, the model consists of 512 elements. Compliance with boundary conditions at the midpoints of the edges is presented in Table 6. Boundary conditions are satisfied exactly. \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & & \multicolumn{2}{c}{**Present Method**} & \multicolumn{2}{c}{**Abaqus**} & \multicolumn{2}{c}{**RE**} \\
**Magnitude** & **Unit** & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** \\ \hline \(w\) & m & 0.00 & 0.0002 & 0.00 & 0.0002 & 0\% & 0\% \\ \(M_{11}\) & N m & \(-\)4174.73 & 1909.82 & \(-\)3326 & 1877 & 25.52\% & 1.75\% \\ \(M_{22}\) & N m & \(-\)9305.64 & 5180.73 & \(-\)7727 & 5080 & 20.43\% & 1.98\% \\ \(M_{12}\) & N m & \(-\)1190.87 & 1190.87 & \(-\)1120 & 1120 & 6.33\% & 6.33\% \\ \(Q_{1}\) & N & \(-\)6806.21 & 6806.21 & — & — & — & — \\ \(Q_{2}\) & N & \(-\)12,667.36 & 12,667.36 & — & — & — & — \\ \(V_{1}\) & N & \(-\)7268.18 & 7268.18 & — & — & — & — \\ \(V_{2}\) & N & \(-\)12,667.93 & 12,667.93 & — & — & — & — \\ \(\varphi_{1}\) & rad & 0.00 & 0.00 & 0.00 & 0.00 & 0\% & 0\% \\ \(\varphi_{2}\) & rad & \(-\)0.0002 & 0.0002 & \(-\)0.0002 & 0.0002 & 0\% & 0\% \\ \hline \end{tabular}
\end{table} 
Table 5: Minimal and maximal values of magnitudes with their relational error (RE). Figure 25: The slope \(\varphi_{2}\) in section \(x_{1}=0\) of a rectangular plate with clamped edges. Plots are constructed for all approximations from 1 to 5.