## 3 Solution Methodology
### Generalized Differantial Quadrature Method  
In this study, the GDQ method introduced by [PERSON] [48] which is the improved form of DQM, is used to convert the partial derivatives of the field variables in equilibrium equations as well as boundary conditions at the grid points into algebraic equations. Before the application of GDQ, the laminated shell is discretized with grid points in order to apply the equilibrium equations and boundary conditions. Gauss-Lobatto points, which increase the accuracy of derivatives in the GDQ method [39], are selected as the grid points for the solution. The Gauss-Lobatto points formulized in Equations (42) and (43) are distributed more extensively at edge points, and they create grid points on the boundaries.  
\[x_{i}=\frac{a}{2}(1-\cos\bigg{(}\frac{(i-1)\pi}{(M-1)}\bigg{)},\ \ i=1,2,\ldots,\ n_{x}+1 \tag{42}\]  
\[y_{j}=\frac{b}{2}(1-\cos\bigg{(}\frac{(j-1)\pi}{(N-1)}\bigg{)},\ \ j=1,2,\ldots,\ n_{y}+1 \tag{43}\]  
In the GDQ method, the derivative of a solution function with respect to a variable at a given discrete point can be calculated as a weighted linear sum of the function values at all discrete points in the mesh line. The stability and accuracy of the proposed numerical method is highly affected by the calculation of the weight coefficients, location, and count of grid points [40]. According to the GDQ method, the \(r\)-th or \(s\)-th-order derivative of a function \(f\) at \(x\) and \(y\) or both directions with \(n_{x}\) and \(n_{y}\) discrete grid points can be computed by:  
\[\left(\frac{\partial f^{r}\big{(}x_{i},y_{j}\big{)}}{\partial x^{r}}\right)= \sum_{k=1}^{n_{x}}C_{ik}^{(r)}f(x_{k},y_{j}),\ r=1,2,\ldots,\ n_{x}-1 \tag{44}\]  
\[\left(\frac{\partial f^{s}\big{(}x_{i},y_{j}\big{)}}{\partial y^{s}}\right)= \sum_{m=1}^{n_{y}}C_{jm}^{(r)}f(x_{i},y_{m}),\ s=1,2,\ldots,\ n_{y}-1 \tag{45}\]  
\[\left(\frac{\partial f^{(r+s)}\big{(}x_{i},y_{j}\big{)}}{\partial x^{r} \partial y^{p}}\right)=\sum_{k=1}^{n_{x}}C_{ik}^{(r)}\sum_{m=1}^{n_{y}}C_{jm} ^{(s)}f(x_{k},y_{m}),\ \ r=1,2,\ldots,\ n_{x}-1, \tag{46}\]  
\(f(x,y)\) and \(C_{ij}^{(r)}\) are the function values and the weight coefficients for the \(r\)-th order derivatives at \((x_{i},y_{j})\) grid point. Lagrange polynomial functions are chosen to determine the weight coefficients \(C_{ij}^{(r)}\). Weight coefficients for the first-order derivative, i.e., \(r=1\), can be written as \[C_{ik}^{(1)}=\frac{\phi(x_{i})}{\big{(}x_{i}-x_{j}\big{)}\phi(x_{j})},\ i,\ j=1,\ 2,\ldots,\ n_{x}\ \text{and}\ i\
eq j \tag{47}\]  
where  
\[C_{ik}^{(1)}=\frac{\phi(x_{i})}{\big{(}x_{i}-x_{j}\big{)}\phi(x_{j})},\ i,\ j=1,\ 2,\ldots,\ n_{x}\ \text{and}\ i\
eq j \tag{48}\]  
Recursive relations for higher-order derivatives can be written as  
\[C_{ij}^{(r)}=r\left[C_{ii}^{(r-1)}C_{ij}^{(1)}-\frac{C_{ij}^{(r-1)}}{\big{(}x _{i}-x_{j}\big{)}}\right],\ \ (i\
eq j) \tag{49}\]  
\[C_{ii}^{(r)}=-\sum_{j=1,\ i\
eq j}^{n_{x}}C_{ij}^{(r)} \tag{50}\]  
Following the procedure as explained above, the governing equilibrium equations subjected to mixed-type simply supported and roller skate boundary conditions are written in terms of the function values at the grid points. All of the algebraic equation system can be expressed simply in the matrix form as  
\[\text{K}U=\text{F} \tag{51}\]  
The total number of unknown coefficients in terms of unknown displacement values is \(5(n_{x}+1)\ (n_{y}+1)\). The number of equations written for the governing equilibrium equations at the internal grid points is \(5(n_{x}-1)\ (n_{y}-1)\). The number of equations written for the boundary conditions given in Equations (20) and (21) is \(10(n_{x}+1)+10(n_{y}-1)\). The total number of equations is equal to the total number of unknown coefficients. Equation (51) is subsequently solved in order to find unknown displacement values.