## 4 Numerical Examples
### Clamped Square Plate Subjected to Uniform Load  
A clamped square plate subjected to a uniformly distributed load \(q\) is further investigated to test the performance of the present elements for clamp boundary conditions. The geometry and material parameters of the clamped plate are the same as those of the above simply supported plate. The displacement boundary conditions of the present theory along the clamped edges in Figure 10 in local coordinates are \(\overline{\theta}_{s}=0,\ \overline{\theta}_{n}=0\) and \(w=0\). The results for the normalized central deflection \(W_{0}\) of the clamped square plate are compared in Table 4. It is seen that, for the plate with the clamped boundary conditions, the predictions of FiSDTPE, DST-BL, and RDKTM based on FSDT agree well with the reference solutions [67] for plates, but the TrSDTPE and FfSDTPE based on HSDTs seem to underestimate the deflections compared with the reference solutions [67] for thick plates of \(h/L\geq 0.2\). To further illustrate the accuracy of the present shear elements, the comparisons of the predictions by different elements and the solutions by 3D elasticity FEM software ANSYS using 20-nodes hexahedron isoparametric element and an element side length of 0.05 are listed in Table 5. By taking the 3D FEM solutions as the benchmark, Table 5 indicates that the present TrSDTPE and FfSDTPE show better solution accuracy than the elements DST-BL, RDKTM, and FiSDTPE based on FSDT for the plates involving clamp boundaries. Moreover, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline _h/L_ & **0.001** & **0.01** & **0.10** & **0.15** & **0.20** & **0.25** & **0.30** \\ \hline TrSDTPE (32) & 1.3964 & 1.5225 & 1.6029 & 1.6846 & 1.798 & 1.9428 & 2.1187 \\ TrSDTPE (128) & 1.5441 & 1.5572 & 1.6262 & 1.7119 & 1.8312 & 1.9835 & 2.1682 \\ TrSDTPE (512) & 1.5607 & 1.5622 & 1.6316 & 1.7186 & 1.8394 & 1.9935 & 2.1801 \\ TrSDTPE (2048) & 1.5626 & 1.5632 & 1.6330 & 1.7203 & 1.8414 & 1.9957 & 2.1825 \\ FfSDTPE (2048) & 1.5620 & 1.5632 & 1.6315 & 1.7166 & 1.8343 & 1.9836 & 2.1637 \\ FiSDTPE (2048) & 1.5623 & 1.5633 & 1.6340 & 1.7233 & 1.8483 & 2.0090 & 2.2054 \\ DST-BL (2048) & 1.5634 & 1.5642 & 1.6452 & 1.7385 & 1.8665 & 2.0293 & 2.2273 \\ RDKTM (2048) & 1.5634 & 1.5640 & 1.6346 & 1.7239 & 1.8490 & 2.0098 & 2.2063 \\ Ref. [67] & 1.5625 & 1.5632 & 1.6339 & 1.7232 & 1.8482 & 2.0089 & 2.2054 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Normalized deflection for the clamped circular plate. Figure 14: Variation of condition number versus number of elements. the TrSDTPE with a third-order shear function \(g(z)\) shows the best solution accuracy among all the elements for the clamped plate. The DOFs of the different methods employed in Table 5 are compared by taking the case of \(h/L=0.3\) for the clamped plate. Assuming that a \(16\times 16\) regular mesh is employed for the plate element discretization and a \(16\times 16\times 9\) regular mesh for the 20-nodes hexahedron 3D element discretization, we can see that the total DOFs of the DST-BL/RDKTM element, HSDTs, and 3D 20-nodes hexahedron element are 867, 17408, and 41337, respectively. It is seen that the total DOFs and the efficiency of HSDTs are between the DST-BL/RDKTM and the 3D FEM. Although the number of DOFs only decreases to 42% of the 3D FEM method, the present 2D HSDTs have the advantage of simplicity and flexibility in the mesh generation compared with 3D FEM for the plates with different thicknesses. Compared with other 2D plate elements such as DST-BL and RDKTM, the computational DOFs of the present 2D HSDTs seem to be relatively higher, but the formulation and the numerical implementation of the high-order shear deformation theory in the present HSDTs are much simpler than those of the DST-BL/RDKTM. From the point of view of the 2D analysis, the total computational cost of the present elements is bearable and worthy in terms of its advantages in formulation and implementation.