# The Ultimate Strength of Fully Transparent Pressure-Resistant Spherical Cabins
## 3 Ultimate Strength of PMMA Spherical Cabin  
### Intact Spherical Cabin  
#### 3.1.1 Failure Theory for Intact Spherical Cabin  
When the deep-sea submersible is considered, the medium-thick shell is typically used with strength theory. But when the working depth of the submersible is relatively shallow, the thickness of the shell is usually designed as a thin shell, then nonlinear buckling theory of pressure-resistant shell can be applied for calculation. The ratio of thickness to inner radius (\(t/R\)) has a boundary point relative to the two sets of theories. The present study is conducted within the scope of nonlinear buckling theory, so for the PMMA cabin, the boundary point should be determined first.  
It is easier for the pressure-resistant spherical shell to establish the equilibrium equation in the spherical coordinate system. The maximum stress \(\sigma_{max}\) is generated on the inner surface of the pressure-resistant spherical shell [27].  
\[\sigma_{max}=\sigma_{i}=\frac{3P\big{(}2+\frac{t}{R}\big{)}^{3}}{2\Big{[} \big{(}2+\frac{t}{R}\big{)}^{3}-\big{(}2-\frac{t}{R}\big{)}^{3}\Big{]}} \tag{1}\]  
Here, \(R\) is the inner radius of the spherical shell and \(t\) is of the middle surface of the spherical shell and \(t\) is the thickness of the shell; \(P\) is the outer uniform pressure acting on the pressure-resistant spherical shell. The pressure gradually increases with the increase of depth, the spherical shell first enters the elastic state, eventually achieves its plastic limit condition after going through the elastic-plastic deformation [28].  
The classical definition of buckling can be attributed to the transfer of equilibrium of a structure. When a stable equilibrium state is out of equilibrium after an arbitrarily small, applied disturbance, the structure is thus in an unbalanced state, and this transfer from a stable equilibrium state to an unbalanced state is referred to as the structure buckling. The term \"buckling critical pressure\" refers to the pressure on the structure in the critical condition and may be written as follows:  
\[P=\frac{2 Et}{R(1-v^{2})}\Bigg{\{}\frac{\big{(}1-v^{2}\big{)}+t^{2}\big{[}\xi(2 +\xi)+\big{(}1+v^{2}\big{)}/12R^{2}\big{]}}{\xi+1+3v}\Bigg{\}} \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Property** & **Value** \\ \hline Elastic Modulus/MPa & 2575 \\ Yield Strength/MPa & 126 \\ Poisson’s ratio & 0.35 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material Properties of PMMA.  
Figure 4: Engineering and true stress–strain curves of PMMA.  
In the above formula, \(P\) is the pressure, \(E\) is the elastic modulus, and \(v\) is the Poisson's ratio. In the above equation, according to the stable equilibrium condition \(\mathrm{d}P/\mathrm{d}\xi=0\), substitution into the above equation yields the following:  
\[P_{\sigma}=\frac{2 Et}{R(1-v^{2})}\left(\sqrt{\frac{(1-v^{2})}{3}}\frac{t}{R}- \frac{vt^{2}}{2R^{2}}\right) \tag{3}\]  
According to above equations, the yield critical pressure and buckling critical pressure of PMMA spherical shell were plotted as a function of \(t/R\) ratio, as shown in Figure 5. According to the theoretical formula curve, when \(t/R=0.0424\), the yield critical pressure and buckling critical pressure of the PMMA spherical shell approach to the same point, both of which are 6.06 MPa. In addition, this marks a shift in the PMMA spherical shell's mechanism of failure. The buckling critical pressure is smaller than the yield critical pressure when the \(t/R\) ratio is less than 0.0424, causing buckling failure to occur first. Should this not occur, yield failure replaces buckling failure as the mechanism of failure. Therefore, in order to ensure that the intact PMMA spherical shell undergoes buckling failure first, in the subsequent numerical analysis, a tentative design of the PMMA spherical shell with the internal radius \(R=735\) mm and the thickness \(t=29.4\) mm (\(t/R=0.04\)) are adopted for analysis.