## 2 Formulation of the Problem  
Consider a composite laminated annular circular plate with an arbitrary number of layers, as shown in Figure 1. Consider \(r_{a}=a\) is the inner radius and \(r_{b}=b\) is the outer radius of the annular circular plate and \(\ell=b-a\) is the width of the annular circular plate. The curvilinear coordinate system \((r,\theta,z)\) is fixed at its reference surface, which is taken to be its middle surface. According to [23], the displacement components are assumed to be  
\[ \begin{array}{c}u(r,\theta,z,t)=u_{0}(r,\theta,t)+z\;\psi_{r}(r,\theta,t)\\ v(r,\theta,z,t)=v_{0}(r,\theta,t)+z\;\psi_{\theta}(r,\theta,t)\\ w(r,\theta,z,t)=w_{0}(r,\theta,t)\end{array}  \tag{1}\]  
The equations of stress-resultants and displacements are of the form  
\[\left( \begin{array}{c}N_{r}\\ N_{\theta}\\ N_{r\theta}\\ M_{r}\\ M_{\theta}\\ M_{r\theta}\end{array} \right)=\left( \begin{array}{cccccc}A_{11}&A_{12}&A_{1 6}&0&0&0\\ A_{12}&A_{22}&A_{26}&0&0&0\\ A_{16}&A_{26}&A_{66}&0&0&0\\ 0&0&0&D_{11}&D_{12}&D_{16}\\ 0&0&0&D_{12}&D_{22}&D_{26}\\ 0&0&0&D_{16}&D_{26}&D_{66}\end{array} \right)\left( \begin{array}{c}\varepsilon _{r}\\ \varepsilon_{\theta}\\ \gamma_{r\theta}\\ \kappa_{r}\\ \kappa_{\theta}\\ \kappa_{r\theta}\end{array} \right)\left( \begin{array}{c}Q_{\theta}\\ Q_{r}\\ \kappa_{\theta}\\ \kappa_{r\theta}\\ \kappa_{r\theta}\end{array} \right)=K\left( \begin{array}{cc}A_{44}&A_{45} \\ A_{45}&A_{55}\end{array} \right)\left( \begin{array}{c}\gamma_{\theta z}\\ \gamma_{r\tau z}\end{array} \right) \tag{2}\]  
Figure 1: Geometry of annular circular plate. where  
\[ \begin{array}{c}\varepsilon_{r}=\ \frac{\partial u_{0}}{\partial r}+z\frac{ \partial\psi_{r}}{\partial r}\,\ \varepsilon_{\theta}=\ \frac{1}{r}\ \frac{\partial\psi_{0}}{ \partial\theta}+\frac{u_{0}}{r}\ +\ z\Big{(}\frac{1}{r}\ \frac{\partial\psi_{\theta}}{ \partial\theta}+\frac{\psi_{r}}{r}\Big{)}\ \,\\ \gamma_{r\theta}=\ \frac{1}{r}\frac{\partial u_{0}}{\partial\theta}+\frac{ \partial v_{0}}{\partial r}\ -\frac{v_{0}}{r}\ +\ z\Big{(}\frac{1}{r}\ \frac{\partial\psi_{r}}{ \partial\theta}\ +\ \frac{\partial\psi_{r}}{\partial r}-\frac{\psi_{\theta}}{r}\Big{)},\\ \kappa_{r}\ =\ \frac{\partial\psi_{r}}{\partial r}\,\ \ \ \kappa_{\theta}=\frac{1}{r}\ \frac{ \partial\psi_{\theta}}{\partial\theta}+\frac{\psi_{r}}{r}\,\ \ \ \kappa_{r\theta}=\frac{ \partial\psi_{\theta}}{\partial r}+\frac{1}{r}\ \frac{\partial\psi_{r}}{ \partial\theta}-\frac{\psi_{\theta}}{r}\,\\ \gamma_{rz}=\ \psi_{r}\ +\ \frac{\partial\psi}{\partial r}\ \ \text{and}\ \ \gamma_{\theta z}=\ \psi_{\theta}+\frac{1}{r}\ \frac{ \partial\psi}{\partial\theta}\end{array}  \tag{3}\]  
The thickness variation of the \(k^{th}\) layer of the plate is assumed in the form which can be seen in Figure 2:  
\[h_{k}(r)=h_{0k}\ g(r) \tag{4}\]  
where \(g(r)=1+C_{\ell}\big{(}\frac{r-r_{\theta}}{\ell}\big{)}+C_{e}\ \exp\big{(}\frac{r-r_{\theta}}{\ell}\big{)}+C_{s}\sin\big{(}\pi\big{(}\frac{r- r_{\theta}}{\ell}\big{)}\big{)}\) and \(h_{0k}\) is a constant thickness of the \(k^{th}\) layer. The thickness of the plate becomes uniform when \(g(r)=1\). Since the thickness is assumed to be varying along the radial direction, one can define the elastic coefficients \(A_{ij}\), \(B_{ij}\) and \(D_{ij}\) (extensional and bending-extensional coupling and bending stiffness), corresponding to layers of uniform thickness with superscript '\(c\)', by  
\[A_{ij}=\ A_{ij}^{c}\ g(r),\ B_{ij}=B_{ij}^{c}\ g(r),\ D_{ij}=D_{ij}^{c}\ g(r) \tag{5}\]  
\[A_{ij}^{c}=\sum_{k}\overline{Q}_{ij}^{(k)}(z_{k}-z_{k-1}),\ B_{ij}^{c}=\frac{ 1}{2}{\sum_{k}}\overline{Q}_{ij}^{(k)}(z_{k}^{2}-z_{k-1}^{2}),D_{ij}^{c}=\frac {1}{3}{\sum_{k}}\overline{Q}_{ij}^{(k)}(z_{k}^{3}-z_{k-1}^{3})\ \ \ \text{for}\ i,j=1,2,6 \tag{6}\]  
and  
\[A_{ij}^{*}=K{\sum_{k}}\overline{Q}_{ij}^{(k)}(z_{k}-z_{k-1})\ \text{for}\ i,j=4,5 \tag{7}\]  
where \(K\) is the shear correction coefficient and \(z_{k-1}\) and \(z_{k}\) are the boundaries of the \(k\)-th layer. The value of \(K\) for a general laminate depends on lamina properties and lamination scheme.