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A.5.2.1.6.1 Pressure Vessel Buckling- Support Rings
A.5.2.1.6.1 Pressure Vessel Buckling- Support Rings
Buckling due to stress is a very important design consideration for launch vehicles2. Research on the stresses on similar launch vehicles, such as the Vanguard, confirms this5. We design the pressure vessels and solid engines not to buckle by designing support rings inside the cylindrical tanks. The ring requirements are found by comparing the applied buckling to a critical buckling based on the geometry.
The first step in the buckling analysis was determining the load. We discovered the axial load applied to a certain tank to be equal to the mass of the entire vehicle above the tank multiplied by the acceleration. The resulting equation for the applied axial load is Eq.(A.5.2.1.6.1.1)below.
(A.5.2.1.6.1.1)
where mabove is the mass of the tank being analyzed with the propellant, and mass of the launch vehicle above the tank being analyzed, l is the g loading, and g is the acceleration due to gravity on the surface of the Earth.
The launch vehicle had a large mass, and the acceleration could have a potential maximum of 30 g’s, so we were correct in our initial assessment of the importance of buckling support. Then, the dimensions and materials of the tank determine the critical buckling load for the launch vehicle. The first method we used was Euler’s buckling method1, using the equations below.
The moment of inertia is calculated using Eq.(A.5.2.1.6.1.2)
(A.5.2.1.6.1.2)
where d is the diameter of the tank and t is the thickness.
The critical buckling load is then calculated using Eq.(A.5.2.1.1.6.1.3)
(A.5.2.1.6.1.3)
where E is Young’s Modulus for the tank material and L is the tank length.
For initial analysis, a code gathers the masses and dimensions of the tanks and other components. Then the code calculated both the applied load and the critical buckling load. If the applied load was the greater value, the code advised the user to resize the tank. With more research and consideration, the code evolved to add inner support structures to the tank instead of resizing3. We determined that adding support rings inside the tank would significantly aid in buckling prevention and not require the tank functions to be run again.This would be necessary if the tank thickness were changed. We made an assumption that such support rings would not reduce the available volume in the tank. Without this assumption, the tanks would not have to be resized each time the support rings were added. Therefore the code was modified to determine the support needed to keep the tanks from buckling as opposed to just determining whether it fails or not.
After extensive research we could not find a complete, simple solution for design the tanks with support rings. From the research we did gather, the main source of support the rings provide is by essentially separating the length of the tank into parts, This increases the critical buckling load for the entire launch vehicle2. The code started with no rings and reiterated the analysis, adding a ring each time, until the tank was structurally sound. This code could only find the number of rings needed; a method to find ring dimensions was not known at the time. It also was created with the realization that this would be overestimating, as it does include the effect of internal pressure, which aids in buckling support. The code could be run for the worst cases, such as if the pressure dropped but nearly all of the mass was still in the launch vehicle, or if something happens that can not be anticipated.
This code was run many times with many varying parameters. It was first tried for values currently thought to be average, such as a first stage tank 4 meters long, 1 meter wide, with 2000 kg above it, and undergoing 2g’s acceleration. These values required no support rings. The code was run several times to determine the conditions that did require support rings. All of the conditions were found to be outrageously extreme before support was necessary, such as a thickness of 10-6 meters and diameter of 5 meters, or a mass of 100,000 kilograms, or an acceleration of 5000 g’s. The research on launch vehicles did not match these findings, nearly all vehicles required buckling support of some kind and vehicles did not have parameters similar to the extremes we found5. Due to these tests, the method of stress analysis was revived.
After reviewing the code and methods gathered from the research, we chose to perform analysis with methods other than Euler’s buckling method. Baker et. al. provided a critical buckling stress specifically for thin shells1. The equations are below.
The curvature parameter is calculated from the below Eq.(A.5.2.1.6.1.4)
(A.5.2.1.6.1.4)
where L is the tank length, ν is Poisson’s Ratio for the tank material, d is the diameter, and t is the thickness.
The buckling coefficient is calculated from the below Eq.(A.5.2.1.6.1.5)
(A.5.2.1.6.1.5)
and the critical buckling stress is calculated from the below Eq.(A.5.2.1.6.1.6)
(A.5.2.1.6.1.6)
where E is Young’s Modulus and t is the tank thickness.
Using this analysis, while the current design test cases still did not need any support, for the maximum value of 30 g’s and for only twice the mass or length, the larger tanks do require support, so logically this method seemed correct. Bruhn’s text confirmed this method4 by mentioning the very same equations.
With the correct buckling analysis, we returned to the issue of sizing the support rings. For our analysis we assumed the rings take all of the stress and used the same method as the whole tank to find the maximum load before the rings fail. A rectangular cross section was determined to be the simplest method as well as being the easiest to manufacture and weld. Starting very small, the analysis reiterated until the minimum size ring was found that could support the load. The largest size rings found for extreme test cases were only around half a millimeter square, thus our initial assumption that the rings add minimal volume was confirmed. The code was then ready to be incorporated into the costing codes and the rest of the structures codes.
References:
1. Baker, E.H., Kovalevsky, L., and Rish, F.L., Structural Analysis of Shells, Robert E. Krieger Publishing Company, Huntington, NY, 1981.
2. Bedford, A., Fowler, W., and Liechti, K., Statics and Mechanics of Materials, Prentice Hall, Englewood Cliffs, NJ, 2002.
3. Boddy, J., Mitchell, J., and Harris, L., “Systems Evaluation of Advanced Structures and Materials in Future Launch Vehicles,” AIAA Journal no. 1103-391, 1967.
4. Bruhn, E.F., “Buckling Strength of Monocoque Cylinder,” Analysis and Design of Flight Vehicle Structures, S.R. Jacobs, 1973.
5. Klemans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No.11022, April 1960.
Author: Steven Izzo
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A.5.2.1.6.1 Pressure Vessel Buckling- Support Rings
Author: Steven Izzo