# The Ultimate Strength of Fully Transparent Pressure-Resistant Spherical Cabins
## 3 Ultimate Strength of PMMA Spherical Cabin
### Intact Spherical Cabin
#### 3.1.2 Effect of Initial Geometrical Imperfection  
A series of finite element models was used to simulate the eigenvalue buckling and nonlinear buckling of PMMA spherical cabins. Eigenvalue buckling analysis was performed first to obtain the initial geometry imperfection mode as a geometrical shape input required for post-buckling analysis. If the inner radius of the spherical pressure hull was \(R=735\) mm and the thickness was \(t=29.4\) mm, which was applied in the present analysis, then \(t/R=0.04\), which was within the consideration of thin shells [29].  
In theory, the pressure shell was not subject to any constraints, but according to the regulations of the classification society [30], three-point constraints are used to restrict the model's displacement in six directions in order to get rid of its stiff displacement. The three-point constraints were used to restrict the stiff displacement of the model, and a pressure of 10 MPa was applied to the spherical shell, as shown in Figure 6. In the material properties section, the material curve was input based on Figure 4, and the element type C3D8R, with a mesh size of 22 mm and hexahedral meshing is adopted for analysis, as shown in Figure 6. Usually, the first-order buckling mode is used as the initial geometrical imperfection of the structure [31].  
Figure 5: Strength theory and buckling theory formula curve of PMMA spherical shell.  
Although the first-order mode may not necessarily be the worst-case defect situation, it is still a very simple and effective method to determine the defect sensitivity of the structure. Some researchers have also used random methods [32] or \(N\)-order characteristic defect mode methods [33] to identify the worst-case defect instability mode and calculate the minimum unstable load for structural instability. Nevertheless, rather than concentrating on the worst-case scenario, the current study examines how faults affect the complete spherical shell structure. Therefore, the first order mode is still selected as the initial defect. Figure 7 depicts the post-buckling equilibrium path of PMMA spherical cabin with the imperfection amplitude of 0.5%\(R\). The load percentage coefficient, which is produced by standardizing the original applied pressure, is displayed on the vertical axis. The end of the curve corresponds to the post-buckling mode of the spherical shell, and the extreme point represents the critical buckling mode of the spherical shell, which is multiplied by the external pressure to give the critical buckling pressure. The findings indicate that the critical pressure for nonlinear buckling when adding the initial geometrical imperfection of 0.5%\(R\) is 5.02 MPa, and the linear buckling critical pressure calculated by eigenvalue buckling is reduced by 2.5% from 5.15 MPa, which is 5.8% lower than the theoretical solution of 5.33 MPa. This suggests that the critical pressure value of PMMA spherical cabins is significantly impacted by initial defects.  
As the initial geometrical imperfection increases, the critical pressure of the PMMA spherical cabin gradually decreases. Table 2 displays the precise values. When the ultimate strength of the PMMA spherical cabin with different size of initial geometrical imperfection is normalized by its corresponding value with imperfection of 0.1%\(R\), Figure 8 can be drawn based on the proportion of strength value reduction, which represents the ultimate strength  
Figure 6: FE model of an intact spherical cabin and boundary conditions.  
Figure 7: Load Proportional Factor versus Arc Length curve of an intact PMMA spherical cabin.  
reduction trend of the intact PMMA spherical cabin with the increase of geometrical imperfection.