## 2 Experimental Studies of Spherical Shells
### Experimental Studies of Steel Spherical Shells Performed by [PERSON] _et al._  
Another study on real spherical shell models that deserves attention was described in 2020 by [PERSON] et al. in [8] \"Ultimate Strength Assessment of Steel-Welded Hemispheres under External Hydrostatic Pressure\", published in the Journal of Marine Science and Application. In this study, four models of steel spherical shells HS-1 through HS-4 were tested. They were fabricated using petal welding procedures. This is the same method used to fabricate actual full-size submarine end closures. Residual stresses and imperfections appeared in the tested shells because the fabrication method uses sheet pressing and welding processes. The manufacturing process consists of several steps. First, each piece of spherical shell was individually cold-pressed to the desired sphericity. In this case, cold forming was repeatedly performed until the required bending curvature was achieved. Once this process was completed, each of these bent plates was welded together. In order to maintain sphericity, the welding process begins by welding one of the plates to the crown of the shell with a 60-degree opening angle. The remaining plates are then welded. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Test** & **Numerical** & **Numerical** & **B\&M Procedure** \\ \hline _Sample_ & \(p_{test}\) [MPa] & \(p^{\text{-}PP}\) [MPa] & \(p^{\text{-}PP}\) [MPa] & \(p^{\text{-}PP}\) [MPa] & \(p^{\text{B\&M}}\) [MPa] \\ \hline
1 & 5.280 & 5.333 (1.010) & 5.468 (1.036) & 5.307 (0.995) \\ \hline
2 & 5.553 & 5.185 (0.934) & 5.310 (0.956) & 5.397 (1.029) \\ \hline
3 & 5.255 & 5.348 (1.018) & 5.506 (1.048) & Q = 40 5.410 (0.971) \\  & & & & (0.25 5.246 (1.002) \\ \hline
4 & 5.580 & 5.116 (0.917) & 5.257 (0.942) & 5.369 (1.039) \\ \hline
5 & 5.356 & 5.357 (1.000) & 5.500 (1.027) & Q = 40 5.431 (0.986) \\  & & & & Q = 25 5.246 (1.017) \\ \hline
6 & 5.647 & 5.282 (0.935) & 5.429 (0.961) & 5.389 (1.047) \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental and numerical buckling resistance [MPa]. Figure 13: Comparison of the test results of [PERSON] et al. [7] with proposed B&M buckling capacity curves. A detailed assessment of the geometry, including its imperfections and a measurement of the shell thickness, is conducted after the welding work is completed. The exact process of manufacturing test specimens is described in [8], and photos of the specimens before testing are shown below in Figure 14.  
The welding diagram and photos of the spherical shells prepared for testing are shown in Figure 14. Models HS-1 and HS-4 consist of four segments and one crown. Model HS-2 has three segments and one crown. Finally, a single shell segment was used in model HS-3. All the test models are composed of SS41 mild steel. Details of the determination of the material parameters of the tested shells are provided in [8]. For the purpose of the analytical estimation of the buckling capacity of spherical shells, [PERSON]'s modulus \(E\) = 206 GPa, [PERSON]'s ratio, \(\
u\) = 0.3, and yield strength of \(fyk\) = 290 MPa and \(fyk\) = 332 MPa were used for the calculations (hence the two different graphs for comparison. The experimental tests were conducted at the Ultimate Limit State Analysis Laboratory (ULSA Lab) at Ulsan University. Each model was measured for initial imperfections and thickness. Measurements were completed with a proprietary instrument on a rotary table with a dial gauge. For thickness measurement, an ultrasonic meter was used. All the spherical shell models except HS-2 were tested in a pressure chamber with an inner length of 2.8 m and a diameter of 1.3 m. The maximum pressure in this chamber can be 80 bar. The HS-2 model, for which the highest buckling capacity was expected, was tested using a chamber with an inner length of 1.8 m and an inner diameter of 0.65 m, in which a pressure of 120 bar is possible. Figure 15 shows a schematic of the test stand and a view of the two pressure chambers.