## 3 Equilibrium and Stability Equations of FG Plate Placed on Elastic Foundation  
By introducing HSDT with shape functions, numerous authors have eliminated the disadvantages of CPT and FSDT. In order to produce good results for specific dynamical and static problems, many of these shape functions have been introduced. It is important to note that the shape functions proposed by various authors (Table 1) are not generally applicable to all types of problems. This paper introduces a new shape function:  
\[\mathrm{f(z)}=\mathrm{z}\Big{(}\mathrm{cosh}\Big{(}\frac{\mathrm{z}}{\mathrm{h }}\Big{)}-1.388\Big{)} \tag{2}\]  
The starting point for developing a new shape function was a comparative analysis of the advantages and disadvantages of existing shape functions given in Table 1. The initial conditions which had to be satisfied in developing a new shape function are: the function has to be an odd function \begin{table}
\begin{tabular}{c c} \hline
**Number of Function** & **Shape Function, \(f(z)\)** \\ \hline No. 1 [41] & \((\mathrm{z}/2)\big{(}\mathrm{h}^{2}/4-\mathrm{z}^{2}/3\big{)}\) \\ No. 2 [42] & \((\mathrm{5z}/4)\big{(}1-4\mathrm{z}^{2}/3\mathrm{h}^{2}\big{)}\) \\ No. 3 [43] & \((\mathrm{h}/\pi)\sin(\pi\mathrm{z}/\mathrm{h})\) \\ No. 4 [44] & \(\sin(\pi\mathrm{z}/\mathrm{h})\mathrm{e}^{\mathrm{i}\cos\left(\frac{\pi}{2} \right)}+(\pi\mathrm{z}/2\mathrm{h})\) \\ No. 5–6 [45] & \(\tan(\mathrm{m}\mathrm{z})-\mathrm{z}\,\mathrm{sec}^{2}(\mathrm{m}/2)\), \(\mathrm{m}=[1/5\mathrm{h},\pi/2\mathrm{h}]\) \\ No. 7 [46,47] & \(\mathrm{z}\exp\left(-2(\mathrm{z}/\mathrm{h})^{2}\right)\mathrm{z}\exp\left( \frac{-2(\mathrm{z}/\mathrm{h})^{2}}{\ln\alpha}\right)\), \(\forall\alpha>0\) \\ No. 8 [48] & \(\mathrm{z}\cdot 2.85^{-2(\mathrm{z}/\mathrm{h})^{2}}+0.028z\) \\ No. 9 [49] & \(\mathrm{\xi}[(\mathrm{h}/\pi)\sin(\pi\mathrm{z}/\mathrm{h})-\mathrm{z}],\mathrm{ \xi}=[1,1/\mathrm{cosh}(\pi/2)-1]\) \\ No. 10 [50] & \(\mathrm{\mathrm{sinh}(\mathrm{z}/\mathrm{h})-\mathrm{z}\,\mathrm{cosh}(1/2)}\) \\ No. 11 [51] & \(\mathrm{z}\,\mathrm{sech}\left(\mathrm{z}^{2}/\mathrm{h}^{2}\right)-\mathrm{z} \,\mathrm{sech}\left(\pi/4\right)[1-(\pi/2)\mathrm{tanh}(\pi/4)]\) \\ No. 12 [51] & \((3\pi/2)\mathrm{htanh}(\mathrm{z}/\mathrm{h})-(3\pi/2)\mathrm{z}\,\mathrm{sech }^{2}(1/2)\) \\ No. 13 [52] & \(\frac{\mathrm{z}\cos(1/2)}{-1+\cos(1/2)}-\frac{\mathrm{h}\sin(\mathrm{z}/ \mathrm{h})}{-1+\cos(1/2)}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Different type of shape functions. Figure 3: Mathematical model of the functionally graded plate placed on elastic foundation. of the thickness coordinate, the function has to satisfy zero stress conditions for out of plane shear stresses, the function has to be analytically integrable in order to additionally increases the precision of the results obtained. Table 1 shows that the newly developed shape function belongs to the category of simple mathematic functions. This makes the process of integration easier and it consequently reduces the calculation time significantly. Since the function is analytically integrable, it is not necessary to switch to numeric integration, and that fact additionally increases the precision of the results obtained. Here, the assumed form of the displacement field [50] is:  
\[ \begin{array}{ll}\mathrm{u}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})&= \mathrm{u}_{0}(\mathrm{x},\mathrm{y},\mathrm{t})-\mathrm{z}\frac{\partial \mathrm{w}}{\partial\mathrm{x}}(\mathrm{x},\mathrm{y},\mathrm{t})+\mathrm{f} (\mathrm{z})\theta_{\mathrm{x}},\\ \mathrm{v}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})&=\mathrm{v}_{0}( \mathrm{x},\mathrm{y},\mathrm{t})-\mathrm{z}\frac{\partial\mathrm{w}}{ \partial\mathrm{y}}(\mathrm{x},\mathrm{y},\mathrm{t})+\mathrm{f}(\mathrm{z}) \theta_{\mathrm{y}},\\ \mathrm{w}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})&=\mathrm{w}_{0}( \mathrm{x},\mathrm{y},\mathrm{t}).\end{array}  \tag{3}\]  
It is necessary to apply the relations between strains and displacements based on [PERSON]'s non-linear theory of elasticity so as to define the components of unit loads [53].