Propellant tanks are a key part of any propulsion system; usually they are the heaviest single parts of a launch vehicle or sounding rocket. They are also one of the more dangerous as the somewhat recent 2007 Scaled Composite accident has shown. So let’s spend a couple of blog posts discussing the design.
First off, today is the basic stress analysis. We will just focus on the general sizing analysis for tanks today and move to materials and joints in the next post.
In a perfect world, you would just have spherical tanks that were not part of the primary structure load path. In that case, the analysis is very easy: Stress = (Pressure x Radius) / (2 x Thickness). So if we had a 6 inch diameter sphere at 200 psi out of Al-6061 with a FS of 2, that is a (42 ksi / 2) = (220 psi x 3 in) / (2 x thick) which mean we can have a 0.029″ wall thickness. Now, a sphere is pretty hard to package so if we use a cylinder with spherical endcaps, the cylinder max stress is Stress = (Pressure x Radius) / (Thickness). Thus, a cylinder would have to be twice as thick at 0.057″. Of course, if you don’t use a spherical endcap, things get trickier and you need to either do FEA analysis or SP-125 has some rules of thumb for ellipsoidal endcaps on page 338.
And for a flat constant thickness endcap assuming simply supported there is a simple solutions given to us by the trusty Roark’s
Max Stress = 3/4 * Pressure * radius^2 / (4 * thickness ^2 )
Now that we have general thickness sizing of the tanks, we can check for buckling loads and for bending loads. For buckling loads, we can just assume that if the tank is pressurized it will not buckle, using the same rational as a pressurized soda can cannot be smashed but a empty can is easy to smash. This is called pressure stabilizing and it is really convenient. If you want more information on buckling, you should check out SP-8007. For most tanks, you can do a first pass for beam bending by assuming Sigma = My / I, or Stress = Moment / ( pi * Radius ^2 * Thickness) . For a gimballed engine, Moment can be assumed to be the Distance from the Cg to the gimbal plane * sin (gimbal angle ) * Thrust. For an ungimbaled engine, assume 5 degrees for the gimbal angle for a first pass. So with 200 lbf, set 40 inches for the Cg and a 5 degree angle this gives us a moment of 700 lb*in and a stress of 450 psi for our 6″ diameter 0.057″ thick tank.
As you can see, pressure loads dominate for initial sizing and other flight loads are relatively trivial. In a more in depth analysis, other flight loads can have a large impact, especially at joints.
Since pressure loads are the driving factor, a term called Tank Factor is fairly common for initial tank sizing. Tank Factor (m) = Tank Volume (m^3) * Pressure ( Pa) / ( Tank Mass (kg) * g (m/s^2)). This equation is used to initially size tanks and compare tanks made with a common fabrication technique as they should all share Tank Factors. A good first pass for an Aluminum tank with high end amateur construction is 2000-2500 meters.
Stay tuned next time for some more tank analysis.