Submarine Design 
General note regarding models: A particular model is predicating a particular failure mode. If the predicted depth is greater than another model, it doesn't mean the value is wrong, it just means that the lower depth failure mode would occur first and therefore the greater depth failure mode would not occur. Essentially the lowest predicated value is the expected failure depth.
Simple Hoop Stress  Shell Yield using the Simple Hoop Stress Formula  Marks, Lionel S."Mechanical Engineering Handbook, Fifth Edition, 1951, Page 421
This provides the theoretical maximum depth a perfect circular shell can withstand until it yields and collapses. A thicker shell will increase the depth. A larger OD will decrease the depth.
Shell Thickness = 0.5" OD = 42", Failure Depth = 2036 ft. sea water
Shell Collapse by General Instability  Shell Collapse by General Instability using Kendricks Equation as modified by Bryant  Comstock, John Paul, "Principles of Naval Architecture, 1967 p. 213, Equation[33]
This simulates the number of lobes for failure. The minimum value should be used to determine the collapse depth and will show the number of lobes for failure. The shell thickness and outside diameter are varied to show the affect on the depth of failure. The predicted depth of failure is much greater than predicted by the simple hoop stress which indicates that failure by General Instability is not the predicted failure mode.
Shell Thickness = 0.5" OD = 42", Failure Depth = 7133, 7105 ft. sea water
Elastic Shell Buckling  Elastic Shell Buckling using NonDimensional Analysis  Comstock, John Paul,"Principles of Naval Architecture", 1967, Page 208
A thicker shell will increase the depth. A larger OD will decrease the depth.
Shell Thickness = 0.5" OD = 42", Failure Depth = 2892 ft. sea water
Bodily Collapse  Bodily Collapse using Tokugawa's Equation  Trilling, Charles"The Influence of Stiffening Rings on the Strength of Thin Cylindrical Shells Under External Pressure", U.S. Experimental Model Basin Report No. 396 February 1935, p. 7, Equation [10]
A thicker shell will increase the depth. A larger OD will decrease the depth. The predicted depth of failure is much greater than predicted by the simple hoop stress which indicates that failure by Shell Buckling is not the predicted failure mode.
Shell Thickness = 0.5" OD = 42", Failure Depth = 4616 ft. sea water
Shell Buckling Von Mises  Shell Buckling using Von Mises Equation  Widenberg, D.F and Trilling, C."Collapse by Instability of a Thin Cylindrical Shells Under External Pressure" ASME Trans, Volume 56, 1934, P. 820, Equation [6]
This simulates the number of lobes for failure. The minimum value should be used to determine the collapse depth and will show the number of lobes for failure. The shell thickness and outside diameter are varied to show the affect on the depth of failure. A thicker shell will increase the depth. A larger OD will decrease the depth. The predicted depth of failure is much greater than predicted by the simple hoop stress which indicates that failure by Shell Buckling is not the predicted failure mode.
Shell Thickness = 0.5" OD = 42", Failure Depth = 3051 ft. sea water
Shell Buckling Widenberg  Shell Buckling using Widenberg's Formula  Comstock, John Paul,"Principles of Naval Architecture", 1967, Page 209, Equation [19]
A thicker shell will increase the depth. A larger OD will decrease the depth.
Shell Thickness = 0.5" OD = 42", Failure Depth = 3102 ft. sea water
Shell Yield Von Sanden and Gunther  Shell Yield using Von Sanden and Gunther's Equation  Comstock, John Paul "Principles of Naval Architecture", 1967, p. 210, Equation [21]
This predicts the depth to failure at the Frame (the Ring) and MidBay. The lower of the two values is the predicted failure depth and failure mode. A thicker shell will increase the depth. A larger OD will decrease the depth.
Shell Thickness = 0.5" OD = 42", Failure Depth = 1846 ft. sea water, Failure at the Frame.
Ring Collapse  Ring Collapse using Formula (88)  Trilling, Charles"The Influence of Stiffening Rings on the Strength of Thin Cylindrical Shells Under External Pressure", U.S. Experimental Model Basin Report No. 396
February 1935, p. 6, Equation [8]
Ring Hoop Stress  Ring Yield using Frame Hoop Stress  Comstock, John Paul"Principles of Naval Architecture", 1967, p. 211, Equation [27]
Ring/Shell Yield  Ring Study of Shell Yield using Von Sanden and Gunther's Equation  Comstock, John Paul"Principles of Naval Architecture", 1967, p. 210, Equation [21]
The Ring width is varied to see the affect. A thicker ring decreases the depth as expected. Having a stiffer ring caused the shell to bend at the ring decreasing the depth.
Weight and Displacement  Weight and Displacement of a Ring Stiffened Cylinder: This does not include the Volume and Weight of the Conning Tower or Battery Box
The predicted payload is 3205 lbs. More than enough for personnel and equipment, but added lead ballast will be needed.
Several scale models were made and tested to destruction to verify the calculated design depth. It was somewhat with a heavy heart when we put them into the pressure chamber to crush them, since one had taken us over two months to make, within 0.001” tolerance on the dimensions (as close as we could make/measure it).
Crush depth of each model (respectively):
2793 ft. 2838 ft.
2763 ft. 3086 ft.
2882 ft. 2725 ft.
3041 ft. 3625 ft.
The internal rings were made as a “T” to increase the stiffness and decrease the ring weight.
When first tested the model failure occurred at 2763 ft and 2793 ft. (fairly close). The failure occurred on the main cylinder where the conning tower connects. This seemed to be a one lobe failure.
Since the front and back portions of the model were still intact, the front and back semielliptical heads were sawn off and tested to see what their failure depth would be. The front head section with all the ports failed at 2838 ft. only 100 ft. deeper than the hull (again fairly close). The rear head containing the motor compartment failed at 3086 ft. and 2725 ft. (again fairly close) but greater than the main head. The strongest model was front head section with no ports failing at 3625 ft. These seemed to be a general instability failure.
The calculated failure depth using the Simple Hoop Stress formula is 2036 ft. This should be the theoretical maximum of a basic shell, but additional stiffening (braces and rings) could increase this.
The calculated failure depth using the Von Sanden and Gunther formula is 2061 ft. at MidBay
Both of these predictions are fairly close to the actual model failure depths, however the actual failure was 35% and 34% respectively higher than predicted. This could have been due to using material with an increased yield strength in the construction the model or increased thickness. We used certified material for the models which showed it’s tested yield strength, but I used the minimum value for the A515 Pressure Vessel Grade Steel for the calculations, any additional strength would only increase the margin of safety. Plus, we had tried to hold everything to within 0.001” (the closest we could measure and make).



Grade 

60 [415] 
65 [450] 
70 [485] 

Tensile strength, ksi [MPa] 
6080 [415550] 
6585 [450585] 
7090 [485620] 
Yield strength, min, ksi [MPa] 
32 [220] 
35 [240] 
38 [260] 
Elongation in 8 in. [200 mm], min, % * 
21 
19 
17 
Elongation in 2 in. [50 mm], min, % * 
25 
23 
21 
* See Specification A20/A20M for elongation adjustment.
This gave us good confidence that the “Delta” would not fail under normal operation. We designed it to 2640 ft. which provided a 2 : 1 safety factor and tested the actual sub to 1920 ft which provided 1.5 : 1 safety factor. So by only diving to a maximum of 1320 ft., we would never get close to yield in actual use.
The designed maximum depth is 1320 ft, but the American Bureau of Shipping (ABS) certified “Delta” to 1200 ft. and reduced the rated depth due to the low temperature characteristics of A515 and restricted the water temperature that “Delta” could dive in. It turned out we should have used A516 Pressure Vessel Grade Steel, which has better low temperature properties and we would not have had a low water temperature restriction.
During the certification process we came to an impasse with the ABS. Their rules mandated that an ungrounded electrical system be used. We analyzed this and determined that their regulation was actually not as safe or reliable as a grounded system with the voltage that was being used in "Delta". This went on for some time and resulted in the writing of: Grounded
vs. Ungrounded Electrical Systems for Use in Manned Submersibles  Marine Technology Society Journal, Fourth Quarter 1981 Vol. 15  No.
4 . ABS finally relented and did certify Delta and "grandfathered" the grounded electrical system since it was previously used in the ABS certified "Nekton" submersibles. “Delta” was marked with a A1 on the right side ring to indicate it was not built under ABS survey. If it had been built under ABS Survey it would have been marked with ✠ A1 .
ABS: Rules for Building and Classing Underwater Vehicles, Systems and Hyperbaric Facilities 2021