## 3 Equilibrium and Stability Equations of FG Plate Placed on Elastic Foundation
 \tag{14}\]  
where \(N_{\rm xx}^{0},N_{\rm yy}^{0}\) and \(N_{\rm xy}^{0}\) are the resultants of pre-buckling forces:  
\[N_{\rm xx}^{0}=N_{\rm yy}^{0}=-\int\limits_{-h/2}^{h/2}\frac{E(z)\alpha(z)T(z )}{1-v}dz,\ \ N_{\rm xy}^{0}=0. \tag{15}\]  
Analytical solutions are obtained by using assumed solution forms and boundary conditions in accordance with [PERSON]'s methods [54]. Boundary conditions along the edges of the rigidly fixed-simply supported plate rectangular plate are the following:  
\[ \begin{array}{l}u_{1}=v_{1}=w_{1}=\theta_{\rm y1}=N_{\rm xx}^{1}=M_{\rm xx }^{1}=P_{\rm xx}^{1}=0\ \mbox{along sides}\ \ x=0\ \mbox{and}\ x=a,\\ u_{1}=v_{1}=w_{1}=\theta_{\rm x1}=N_{\rm yy}^{1}=M_{\rm yy}^{1}=P_{\rm yy}^ {1}=0\ \mbox{along sides}\ \ y=0\ \mbox{and}\ y=b.\end{array}  \tag{16}\]  
In order to satisfy the previously defined boundary conditions, the following [PERSON]'s solution is assumed [54]:  
\[ \begin{array}{l}u_{1}(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{ \infty}U_{\rm mn}^{1}\cos\frac{m\pi n}{a}\sin\frac{n\pi y}{b},\ \ v_{1}(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}V_{\rm mn}^{ 1}\sin\frac{m\pi x}{a}\cos\frac{n\pi y}{b},\\ w_{1}(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}W_{\rm mn}^{ 1}\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b},\\ \theta_{\rm x1}(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}T_{ \rm xm}^{1}\cos\frac{m\pi y}{a}\sin\frac{n\pi y}{b},\ \ \ \theta_{\rm y1}(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}T_{\rm ymn }^{1}\sin\frac{m\pi x}{a}\cos\frac{n\pi y}{b}.\end{array}  \tag{17}\]  
where \(U_{\rm mn}^{1},V_{\rm mn}^{1},W_{\rm mn}^{1},T_{\rm xmn}^{1},T_{\rm ymn}^{1}\) are parameters which are to be determined. Based on [PERSON]'s solution, the stability equation becomes:  
\[[\mathbf{L}-\Omega\mathbf{I}]\mathbf{U}=0, \tag{18}\]  
where \(\mathbf{U}=\left\{ \begin{array}{cccc}U_{\rm mn}^{1}&V_{\rm mn}^{1}&W_{\rm mn }^{1}&T_{\rm xmn}^{1}&T_{\rm ymn}^{1}\end{array} \right\}^{\rm T}\) and \(\Omega\) buckling parameter. Coefficients \(L_{\rm ij}\), \((i,j=1\div 5)\) are defined in the following way:  
\[ \begin{array}{l}L_{11}=\alpha^{2}A_{11}+\beta^{2}A_{66},\ \ L_{12}=\alpha\beta(A_{12}+A_{66}),\ \ L_{13}=0,\ \ L_{14}=\alpha^{2}D_{11}+\beta^{2}D_{66},\\ L_{15}=\alpha\beta(D_{12}+D_{16}),\ \ L_{22}=\alpha^{2}A_{66}+\beta^{2}A_{22},\ \ L_{23}=0,\ \ L_{24}=\alpha\beta(D_{12}+D_{16}),\\ L_{25}=\alpha^{2}C_{66}+\beta^{2}C_{22},\ \ L_{33}=\alpha^{4}E_{11}+2\alpha^{2}\beta^{2}E_{12 }+4\alpha^{2}\beta^{2}E_{66}+\beta^{4}E_{22},\\ L_{34}=-\alpha^{3}F_{11}-\alpha\beta^{2}F_{12}-2\alpha\beta^{2}E_{66},\ \ L_{35}=-\alpha^{2}\beta F_{12}-2\alpha^{2}\beta F_{66}-\beta^{3}F_{22},\\ L_{44}=\mathrm{H}_{44}+\alpha^{2}G_{11}+\alpha^{2}G_{66},\ \ L_{45}=\alpha\beta(G_{12}+G_{66}),\ \ L_{55}=H_{55}+\alpha^{2}G_{66}+\alpha^{2}G_{22},\end{array}  \tag{19}\]  
while the matrix \(I_{\rm ij},\ (i,j=1\div 5)\) is determined as:  
\[ \begin{array}{l}\mathbf{I}=\left\{\begin{array}{l}0,\\ \alpha^{2}N_{\rm x}^{0}+\beta^{2}N_{\rm y}^{0}+k_{0}+k_{1}\Big{(}\alpha^{2}+ \beta^{2}\Big{)},\ \ (i,j=3),\ \alpha=m\pi/a,\ \beta=n\pi/b\end{array}\right.\end{array}  \tag{20}\]  
The determinant in (18) must be equal to zero value in order to get nontrivial solutions:  
\[|\mathbf{L}-\Omega\mathbf{I}|=0. \tag{21}\]