## 3 Analytic Solution of Bending Deflection  
The Levy solution is used in the derivation of bending deflection through plate thickness in which the Levy solution applies to any type of boundary conditions under the effect of in-plane compression mechanical load and shear force. Based on the classical laminate plate theory, the stress-strain relationship is:  
Figure 6: Bending deflection against in-plane compression mechanical load for plate with one and two stiffeners. Figure 7: Bending deflection against in-plane compression mechanical load for un-stiffened plate and plate with one and two stiffeners. \[\left\{ \begin{array}{c}\sigma_{\rm xx}\\ \sigma_{\rm yy}\\ \sigma_{\rm xy}\end{array} \right\}^{\rm k}=\left[ \begin{array}{cc}\overline{ \underline{\Omega}}_{11}&\overline{\underline{\Omega}}_{12}&\overline{ \underline{\Omega}}_{16}\\ \overline{\underline{\Omega}}_{12}&\overline{\underline{\Omega}}_{22}& \overline{\underline{\Omega}}_{26}\\ \overline{\underline{\Omega}}_{16}&\overline{\underline{\Omega}}_{26}& \overline{\underline{\Omega}}_{66}\end{array} \right]*\left\{ \begin{array}{c} \varepsilon_{\rm xx}\\ \varepsilon_{\rm yy}\\ \gamma_{\rm xy}\end{array} \right\} \tag{2}\]  
where:  
\[\left\{ \begin{array}{c}\varepsilon_{\rm xx}\\ \varepsilon_{\rm yy}\\ \gamma_{\rm xy}\end{array} \right\}=\left\{ \begin{array}{c}\varepsilon_{\rm xx }^{(0)}\\ \varepsilon_{\rm yy}^{(0)}\\ \gamma_{\rm xy}^{(0)}\end{array} \right\}+\rm z*\left\{ \begin{array}{c} \varepsilon_{\rm xx}^{(1)}\\ \varepsilon_{\rm xy}^{(1)}\\ \gamma_{\rm xy}^{(1)}\end{array} \right\}\]  
Additionally,  
\[\left\{\varepsilon^{(0)}\right\}=\left\{ \begin{array}{c}\varepsilon_{\rm xx }^{(0)}\\ \varepsilon_{\rm yy}^{(0)}\\ \gamma_{\rm xy}^{(0)}\end{array} \right\}=\left\{ \begin{array}{c}\frac{ \partial\rm u_{\rm u}}{\partial\rm x}\\ \frac{\partial\rm u_{\rm u}}{\partial\rm y}\\ \frac{\partial\rm u_{\rm u}}{\partial\rm y}+\frac{\partial\rm v_{\rm u}}{ \partial\rm x}\end{array} \right\}\]  
\[\left\{\varepsilon^{(1)}\right\}=\left\{ \begin{array}{c}\varepsilon_{\rm xx }^{(1)}\\ \varepsilon_{\rm yy}^{(1)}\\ \gamma_{\rm xy}^{(1)}\end{array} \right\}=\left\{ \begin{array}{c}-\frac{ \partial^{2}\rm w_{\rm u}}{\partial\rm x^{2}}\\ -\frac{\partial^{2}\rm w_{\rm u}}{\partial\rm y^{2}}\\ -2*\frac{\partial^{2}\rm w_{\rm u}}{\partial\rm x\partial y}\end{array} \right\}\]  
The plate equation in terms of bending moments is  
\[\frac{\partial^{2}\rm M_{xx}}{\partial\rm x^{2}}+2*\frac{\partial^{2}\rm M_{ xy}}{\partial\rm x\partial y}+\frac{\partial^{2}\rm M_{yy}}{\partial\rm y^{2}}= \rm I_{0}*\frac{\partial^{2}\rm w_{\rm o}}{\partial\rm t^{2}} \tag{3}\]  
The bending moments of the rectangular plate are  
\[\left\{ \begin{array}{c}\rm M_{xx}\\ \rm M_{yy}\\ \rm M_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}\rm B_{11}&\rm B_{12}& 0\\ \rm B_{12}&\rm B_{22}&0\\ 0&0&\rm B_{66}\end{array} \right]*\left\{ \begin{array}{c}\frac{\partial\rm u _{\rm o}}{\partial\rm x}\\ \frac{\partial\rm u_{\rm o}}{\partial\rm y}\\ \frac{\partial\rm u_{\rm o}}{\partial\rm y}+\frac{\partial\rm v_{\rm o}}{ \partial\rm x}\end{array} \right\}-\left[ \begin{array}{ccc}\rm D_{11}&\rm D_{12 }&0\\ \rm D_{12}&\rm D_{22}&0\\ 0&0&\rm D_{66}\end{array} \right]*\left\{ \begin{array}{c}\frac{\partial^{2} \rm w_{\rm o}}{\partial\rm x^{2}}\\ \frac{\partial\rm u_{\rm o}}{\partial\rm y^{2}}\\ 2*\frac{\partial^{2}\rm w_{\rm o}}{\partial\rm x\partial y}\end{array} \right\} \tag{4}\]  
where  
\[\rm B_{ij}=\frac{1}{2}*\sum\limits_{k=1}^{N}\overline{\rm Q_{ij}}^{(k)}*(z_{k+ 1}^{2}-z_{k}^{2})=0\]  
, since the stacking sequences of plate orientation are the same above and below the z-axis, and this case is called special orthotropic.