## 2 Experimental Studies of Spherical Shells
### Experimental Studies of Stainless Steel Alloy Spherical Shells Performed by [PERSON] et al.  
In 2018, [PERSON] et al. published a paper [6] entitled \"Buckling of stainless steel spherical caps subjected to uniform external pressure\" in the journal _Ships and offshore structures_. The paper includes experimental tests conducted on six nominally identical specimens, designated SC1, SC2, SC3, SC4, SC5, and SC6. The material from which they were composed was 304 stainless steel alloy. To ensure the geometric repeatability of the individual shells, advanced laser machining techniques with a numerical control system were used to produce them. The hemispherical shells were cut and cold-stamped from sheets of 304 stainless steel. The exact manufacturing process of the samples is described in [8]. This material has advantages such as high ductility, which can prevent the caps from disintegrating into small pieces, and it enables the post-buckling modes to be determined conveniently.  
The spherical shells produced have a nominal base diameter of \(d=146\) mm, a nominal wall thickness of \(t=1\) mm, and a nominal height of \(h=37\) mm, and the shell radius \(R\) was determined to be 90.51 mm Figure 11.  
During the entire fabrication process, the shells were not heat-treated because of their relatively small thickness-to-radius ratios. Before the hydrostatic test, the wall thicknesses and geometrical shapes of all the shells were measured carefully. The results show that the actual wall thickness values of the manufactured shells varied from 0.978 to 1.077 mm due to the stamping process.  
Based on the laboratory tests, the material properties were determined for each individual sample. Then, the averaged values of [PERSON]'s modulus \(E\), Poisson's coefficient \(v\), and yield stress _fyk_ served as the starting parameters in the numerical analyses and were used to create buckling capacity curves. The values are yield stress _fyk_ = 335.408 MPa, [PERSON]'s modulus \(E=159.208\) GP, and [PERSON]'s coefficient \(v=0.291\).  
Figure 10: Comparison of the test results of [PERSON] and [PERSON] [13] with proposed B&M buckling capacity curves.  
After the initial measurements, the samples were slowly pressurized to destruction in a cylindrical pressure chamber at Jiangsu University of Science and Technology. The chamber has an inner diameter of 200 mm, an overall length of 400 mm, and a maximum pressure of 20 MPa, with water as the pressure medium. The pressure inside the chamber was recorded with a pressure transducer; the pressure was applied slowly in \(\sim\)0.1 MPa increments. The exact process of loading and conducting the test is described in [6; 7]. The effect of the experimental tests regarding the form of spherical shells that buckled is shown in Figure 12.  
The comparison of the experimental results with the proposed B&M procedure for estimating the buckling capacities of spherical shells is shown in Figure 13. This comparison very well demonstrates how accurate analytical calculations can be. The differences between the experiment and analytical estimation are within 0.2-4.7%. This is also confirmed in Table 1, which compares the results of the experiment with the results from two types of numerical analysis (\(p^{\text{e-P}}\)--numerical buckling loads obtained from elastic-plastic modeling and \(p^{\text{e-PP}}\)--numerical buckling loads obtained from elastic-perfectly plastic modeling) and analytical calculations. It can be seen that the analytical estimation, in this particular case, shows greater convergence with the experimental results than the numerical calculations. Figure 13 shows the buckling capacity curves with marked points from the experimental measurements.  
Figure 11: Idealized geometry diagram of the tested spherical shell. According to [6].  
Figure 12: The form of buckling of the tested spherical caps SC1–SC6. According to [7].