## 4 Numerical Results and Discussion
### Validation Study  
This section is firstly intended to examine the validity of the obtained results. For that purpose, displacement results were validated with thin (\(a/h\) = 100) symmetric and antisymmetric cross-ply laminated panels with well-known Navier-type boundary conditions, as mentioned in [PERSON] and [PERSON] [18]. The stress results were then validated using the finite element method for the same geometrical and mechanical properties. In this solution, the coefficients \(b_{n}=d_{n}=0\) are taken as zero in order to regenerate the Navier-type boundary conditions. Normalized central deflections and stress results for cross-ply shells are given in Table 1. It is worth noting that almost no difference was detected changing from moderately deep (R/\(a\) = 10) to (R/\(a\) = Plate) between the BDM and GDQ under the very well-known SS3-type boundary conditions. The relatively small discrepancies between these methodologies and the finite element method in the ANSYS Package program can be predicated to the relationship between strong-form and weak-form formulations. In our proposed solution, the methodologies for BDM and GDQ methods are based on the strong form, which satisfies the compulsory and natural boundary conditions. In contrast to strong form, the finite element method uses weak-form formulation and does not apply natural or force boundary conditions which are imposed on the secondary variable, such as forces and moments.  
The proposed methodologies are also compared with the finite element method for the current boundary conditions (SS-RS) in Table 2. Comparison results are given for antisymmetric [0\({}^{\circ}\)/90\({}^{\circ}\)] laminated cross-ply spherical (R1 = R2 = R) moderately thick (\(a/h\) = 20) panels for different R/\(a\) ratios. Since the boundary conditions cannot be fully integrated using the finite element method, there is a difference between the FEM and the intended solution methods.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & [0\({}^{\circ}\)/90\({}^{\circ}\)] & & & [0\({}^{\circ}\)/90\({}^{\circ}\)/0\({}^{\circ}\)] & & & [0\({}^{\circ}\)/90\({}^{\circ}\)/90\({}^{\circ}\)/0\({}^{\circ}\)] \\ \hline
**R/\(a\)** & **BDM** & **GDQ** & **Ref. [18]** & **BDM** & **GDQ** & **Ref. [18]** & **BDM** & **GDQ** & **Ref. [18]** \\ \hline
10 & 5.5428 & 5.542 & 5.5428 & 3.644 & 3.644 & 3.6445 & 3.720 & 3.720 & 3.720 \\
20 & 11.273 & 11.27 & 11.273 & 5.547 & 5.547 & 5.5473 & 5.661 & 5.661 & 5.661 \\
50 & \(w^{*}\) & 15.714 & 15.71 & 15.714 & 6.482 & 6.482 & 6.4827 & 6.614 & 6.614 & 6.614 \\
100 & 16.645 & 16.64 & 16.645 & 6.642 & 6.642 & 6.6421 & 6.777 & 6.777 & 6.777 \\ Plate & 16.979 & 16.97 & 16.980 & 6.697 & 6.696 & 6.6970 & 6.833 & 6.833 & 6.842 \\ \hline
**R/\(a\)** & **BDM** & **GDQ** & **FEM** & **BDM** & **GDQ** & **FEM** & **BDM** & **GDQ** & **FEM** \\ \hline
10 & 420.17 & 420.2 & 418.98 & 4688.6 & 4688.6 & 4678.5 & 4961.8 & 4961.8 & 4954.4 \\
20 & 868.116 & 868.1 & 866.180 & 6962.9 & 6963 & 6950 & 7238.2 & 7238.2 & 7227 \\
50 & \(\sigma_{x}{}^{*}\) & 1196.10 & 1196.1 & 1196 & 7956.6 & 7956.7 & 7985.2 & 8175.8 & 8175.8 & 8209.2 \\
100 & 1256.62 & 1256.6 & 1255.6 & 8081.2 & 8081.3 & 8075 & 8271 & 8271 & 8265.9 \\ Plate & 1268.70 & 1268.7 & 1267.4 & 8073 & 8073 & 8064 & 8229.6 & 8229.5 & 8222.7 \\ \hline \end{tabular}
\end{table}
Table 1: Normalized central deflections and stresses of cross-ply laminated shells under distributed load (\(a/h\) = 100).