## 2 Statement of the Problem  
A model of the schematic plot and geometric parameters of a curved shallow shell based on an orthogonal curvilinear coordinate (\(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}=\xi\)) system is shown in Figure 1. The orthogonal coordinates (\(\xi_{1}\), \(\xi_{2}\), \(\xi\)) are placed on the curved shell mid-surface, \(\xi_{3}=\xi=0\), and the \(\zeta\) coordinate coincides with the vertical direction to the mid-surface. The curved shell has a principal curvature denoted as R1 and R2 located on the middle surface (\(\xi=0\)). The shell is made up of N, a finite number, of perfectly bonded layers with uniform thickness. The diverse geometries of the shell are obtained by adjusting the radius of curvature and designating a cylindrical shell (R1 = R, R2 = \(\infty\) or R1 = \(\infty\), R2 = R), spherical (R1 = R, R2 = R), and plate (R1 = \(\infty\), R2 = \(\infty\)). In this study, the analysis used is a linear static analysis. It is also assumed that there is no penetration and no delamination between the layers and that they are perfectly bonded. The displacement field addressing first order shear deformation theory may be written as [45]:  
\[\overline{u}_{1}(\xi_{1},\xi_{2},\xi)=u_{1}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 1}(\xi_{1},\xi_{2}) \tag{1}\]  
\[\overline{u}_{2}(\xi_{1},\xi_{2},\xi)=u_{2}(\xi_{1},\xi_{2})+\xi\cdot\theta_{ 2}(\xi_{1},\xi_{2}) \tag{2}\]  
\[\overline{u}_{3}(\xi_{1},\xi_{2},\xi)=u_{3}(\xi_{1},\xi_{2}) \tag{3}\]  
where \(u_{i}\) (\(i\) = 1,2,3) are the displacement field of a point on the middle-surface of the shell along the \(\xi_{1}\), \(\xi_{2}\), and \(\xi_{3}\) axes. \(\theta_{1}\) and \(\theta_{2}\) are the rotations around the \(\xi_{2}\) and \(\xi_{1}\) axis, respectively. The equivalent single-layer approach based on a first-order shear deformation theory is incorporated into the shell formulation. According to the kinematic relations from the theory of elasticity in curvilinear coordinates, the generalized strain-displacement relations for the doubly curved panels using the displacement fields in Equations (1)-(3) are expressed as below in the case of small elastic deformation:  
\[\varepsilon_{1}^{0}=\frac{\partial u_{1}}{\partial\xi_{1}}+\frac{u_{3}}{\text {R}_{1}},\varepsilon_{1}^{1}=\frac{\partial\theta_{1}}{\partial\xi_{1}} \tag{4}\]  
\[\varepsilon_{2}^{0}=\frac{\partial u_{2}}{\partial\xi_{2}}+\frac{u_{3}}{\text {R}_{2}},\varepsilon_{2}^{1}=\frac{\partial\theta_{2}}{\partial\xi_{2}} \tag{5}\]  
\[\varepsilon_{6}^{0}=\frac{\partial u_{2}}{\partial\xi_{1}}+\frac{\partial u_{ 1}}{\partial\xi_{2}}+\frac{1}{2}\bigg{(}\frac{1}{\text{R}_{2}}-\frac{1}{\text {R}_{1}}\bigg{)}\bigg{(}\frac{\partial u_{2}}{\partial\xi_{1}}-\frac{\partial u _{1}}{\partial\xi_{2}}\bigg{)},\varepsilon_{6}^{1}=\frac{\partial\theta_{1}}{ \partial\xi_{2}}+\frac{\partial\theta_{2}}{\partial\xi_{1}} \tag{6}\]  
\[\varepsilon_{4}^{0}=\theta_{1}+\frac{\partial u_{3}}{\partial\xi_{1}}-\frac{ u_{1}}{\text{R}_{1}} \tag{7}\]  
\[\varepsilon_{5}^{0}=\theta_{2}+\frac{\partial u_{3}}{\partial\xi_{2}}-\frac{ u_{2}}{\text{R}_{2}} \tag{8}\]  
Figure 1: Geometry of a laminated doubly curved panel. For the sake of brevity, the derivation of the equilibrium equations of the composite doubly curved panel using the virtual work principle is not explained here. Further explanations are given in Reddy [45].