# The Ultimate Strength of Fully Transparent Pressure-Resistant Spherical Cabins
## 3 Ultimate Strength of PMMA Spherical Cabin
### Spherical Cabin with a Single Access Opening  
#### 3.2.1 Numerical Analysis of the Spherical Cabin with a Single Access Opening  
When designing submersibles, multiple openings are usually reserved as for personnel entry and exit, electronic equipment installation, etc., and the existence of openings can affect the strength of pressure-resistant structures. The present study conducts corresponding nonlinear buckling numerical analysis considering the design of single and double openings, in order to explore the effect of openings on the ultimate strength of the PMMA cabin. The hatch cover for the opening in the present study is made of aluminum alloy 6061-T6. The thickness is determined based on the principle of equivalence in which it is assumed that the entire spherical shell is made of aluminum 6061-T6 and can withstand pressure greater than the critical pressure of the intact PMMA spherical shell. Based on Section 3.1, As can be observed, the PMMA shell's critical pressure is 5.02 MPa. According to the formula specified in CCS rule, the stress of the entire shell can be calculated as follows:  
\[\sigma=\frac{P_{j}R}{2t} \tag{4}\]  
\[\sigma\leq 0.85R_{cH} \tag{5}\]  
where \(P_{j}\) is the calculation pressure (MPa), \(\sigma\) is the Shell plate stress of spherical shell and \(R_{cH}\) is the yield stress of the material (MPa). Based on this principle, it can be calculated  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(t\)/\(R\) & & & & & \\ \hline Initial geometrical imperfection & 0.1\% & 0.2\% & 0.3\% & 0.4\% & 0.5\% \\ Theoretical value of the buckling pressure (MPa) & 5.33 & 5.33 & 5.33 & 5.33 & 5.33 \\ Numerical results of buckling pressure (MPa) & 5.31 & 5.21 & 5.13 & 5.08 & 5.02 \\ Failure mode & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Effect of initial geometrical imperfection on ultimate strength of the intact spherical cabin.  
Figure 8: Strength reduction curve with initial geometrical imperfection.  
that if a complete aluminum alloy shell needs to bear a pressure of 6 MPa, the value of thickness \(t\) should be 9.6 mm. Usually, a safety factor of 1.5 is applied for design, which leads to the thickness of 14.4 mm. The specific dimensions are shown in Figure 9.  
The procedure of finite element modeling and analysis steps has been described in Section 3.1.1. The element type will be C3D8R, and the shell part will be divided into hexahedral structural meshes. The tetrahedral free-meshing is applied for hatch cover. Then, the critical pressure of the spherical cabin with a single access opening and initial geometrical imperfection of 0.5%R is added to the model, which results in the critical pressure of 3.11 MPa, and reduces the strength of the intact spherical cabin by 38.05%. Its crushing morphology is also significantly different from that of the intact spherical cabin. Figure 10 shows the LPF curve of the PMMA spherical cabin with a single access opening. The stress concentration occurs around the opening, and the final failure also occurs near the connection between the metal hatch cover and the PMMA shell.