## 2 Statement of the Problem
With the use of the virtual work principle, the characteristic equations of the panel, defined by five highly coupled partial differential equations in five unknowns, i.e., three displacements, and two rotations are given as below:  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]0  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]1  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]2  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]3  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]4  
where \(q\) is the distributed transverse load and \(\mathrm{N}_{i}\), \(\mathrm{M}_{i}\), \(\mathrm{P}_{i}\) (\(i=1,\,2,\,6\)) denote stress resultants, stress couples (moment resultants), and second stress couples (resultant from the second moment of stress), are as defined by [PERSON] and [PERSON] [46]. Qi (i = 4, 5), represents the transverse shear stress resultants. The forces and moments are further elaborated upon to show the individual components as follows:  
\[ \begin{cases}\mathrm{N}_{1}\\ \mathrm{N}_{2}\\ \mathrm{N}_{6}\\ \mathrm{M}_{1}\\ \mathrm{M}_{2}\\ \mathrm{M}_{6}\end{cases} = \begin{bmatrix}\mathrm{A}_{11}&\mathrm{A}_{12}& \mathrm{A}_{16}&\mathrm{B}_{11}&\mathrm{B}_{12}&\mathrm{B}_{16}\\ \mathrm{A}_{21}&\mathrm{A}_{22}&\mathrm{A}_{26}&\mathrm{B}_{21}&\mathrm{B}_{22 }&\mathrm{B}_{26}\\ \mathrm{A}_{61}&\mathrm{A}_{62}&\mathrm{A}_{66}&\mathrm{B}_{61}&\mathrm{B}_{62 }&\mathrm{B}_{66}\\ \mathrm{B}_{11}&\mathrm{B}_{12}&\mathrm{B}_{16}&\mathrm{D}_{11}&\mathrm{D}_{12 }&\mathrm{D}_{16}\\ \mathrm{B}_{21}&\mathrm{B}_{22}&\mathrm{B}_{26}&\mathrm{D}_{21}&\mathrm{D}_{22 }&\mathrm{D}_{26}\\ \mathrm{B}_{61}&\mathrm{B}_{62}&\mathrm{B}_{66}&\mathrm{D}_{61}&\mathrm{D}_{62 }&\mathrm{D}_{66}.\end{bmatrix}\begin{cases}\epsilon_{1}^{0}\\ \epsilon_{1}^{0}\\ \epsilon_{0}^{0}\\ \epsilon_{1}^{1}\\ \epsilon_{0}^{1}\\ \epsilon_{1}^{1}\\ \epsilon_{0}^{1}\end{cases}  \tag{14}\]  
\[ \begin{cases}\mathrm{Q}_{1}\\ \mathrm{Q}_{2}\end{cases} = \begin{bmatrix}\mathrm{A}_{55}&\mathrm{A}_{54}\\ \mathrm{A}_{45}&\mathrm{A}_{44}\end{bmatrix}\begin{cases}\epsilon_{4}^{0}\\ \epsilon_{5}^{0}\end{cases}  \tag{15}\]  
where \(\mathrm{A}_{ij}\), \(\mathrm{B}_{ij}\), \(\mathrm{D}_{ij}\) are the laminate rigidities. These are given as follows:  
\[\left(\mathrm{A}_{ij},\mathrm{B}_{ij},\mathrm{D}_{ij}\right)=\sum_{k=1}^{N} \int_{\xi_{k-1}}^{\xi_{k}}\mathrm{Q}_{ij}^{(k)}\left(1,\xi,\xi^{2}\right)d\xi, \quad(\mathrm{i},\;\mathrm{j=1},\;2,\;6) \tag{16}\]  
The obtained governing equations are expressed as follows:  
\[\mathrm{K}_{\mathrm{ij}}\delta_{j}=f_{\mathrm{i}}(\mathrm{i},\mathrm{j=1},\; 2,\;3,\;4,\;5)\;\mathrm{and}\;\left(\mathrm{K}_{\mathrm{ij}}=\mathrm{K}_{ \mathrm{ji}}\right) \tag{17}\]  
where  
\[\left\{\delta_{j}\right\}^{T}=\left\{u_{1}\;u_{2}\;u_{3},\theta_{1},\theta_{2}\right\} \tag{18}\]  
\[\left\{f_{\mathrm{j}}\right\}^{T}=\left\{0\;0-q,0,0\;\right\} \tag{19}\]  
The problem is solved in conjunction with the mixed type of simply supported and roller skate boundary conditions, and they are prescribed at the edges as given below:  
Roller Skate:  
\[\xi_{1}=0..a,\quad u_{1}=u_{2}=\theta_{2}=\mathrm{M}_{1}=\mathrm{Q}_{1}=0 \tag{20}\]  
Simply Supported:  
\[\xi_{2}=0..b,\quad u_{1}=u_{3}=\theta_{1}=\mathrm{M}_{2}=\mathrm{N}_{2}=0 \tag{21}\]Following section explains the solution procedure for the governing partial differential equations given in Equations (9)-(13) subjected to the above boundary conditions.