2.4 Structural Stability

In structural stability of a rocket there are many different forces that are acting upon the fuselage causing bending and compression/buckling which if your rocket is not strong enough would cause it to snap and blow up. These forces are the following

Axial Stress

Bending

Shear Stress

Hoop

2.4.1 Axial Stress:

Axial stress is essentially the resistance of a material to a load that's applied on its longitudinal axis. This means that it's not a point load, it's a force applied onto a cross sectional area. This is the most basic type of stress, its given by the formula

$\sigma = \frac{F}{A}$ where F is the force and A is the cross sectional area. This formula makes sense intuitively because your force is essentially acting upon that entire surface which creates a stress upon it. There are two different types of axial stress, compressive and tensile, compressive is when it's pushing into the object while tensile is pulling it apart.

How does this apply to rockets?

When you think about this it makes a lot of sense, each engine is applying axial stress to the body of the rocket, this means that at all times during boost the force of your engines is the same force that's acting upon your fuselage, which if too large can compound with other forces to cause buckling which we will talk about next.

2.4.2 Bending Stress

Take a straw or a piece of pasta and slowly bend it with your fingers at the edges, what's happening? Well first of all you probably snapped your pasta, when you bend a cylinder you're applying a force away from the center of mass that's taking one side and compressing it and taking the other and stretching it. This essentially causes two distinct forces on your cylinder and if it can withstand both then it snaps. Now imagine this but with 7600KN being applied at the same time from axial stress at the bottom, that's what a rocket has to go through and that's why bending moments are so dangerous.

The formula for bending stress is given by $\sigma = \frac{M*y}{I}$ where M is the bending moment inside, y is the distance from the neutral axis and I is the moment of Inertia of the cross section. This makes sense because at distance r from the center line you get your maximum bending moment.

When $\sigma_{current} > \sigma_{\max}$ Then the beam breaks. Keep in mind that the bending moment is proportional to the distance from the center of gravity, the maximum bending moment will be at the center because it has to hold onto all the fuselage above and below it.

How does this apply to rockets?
Well quite obviously if your going through air really really fast and you're not perfectly aligned with your thrust vector the aerodynamic lift at certain points will cause bending stress, and as mentioned earlier even the slightest bit of being stressed under that strong of forces it can cause your whole rocket to pass the critical stress and cause structural failure.

2.4.3 Shear stress

Shear stress is incredibly similar to bending stress; they are both caused by the same type of load. Bending stress is a stress that's perpendicular to the cross sectional area while shear stress is a stress that's parallel to the cross sectional area.{width="2.6666666666666665in" height="2.0520833333333335in"}

Here's the best way to imagine shear stress: stack a whole bunch of super thin objects on top of each other, when you get high enough push on the top one, if it's not connected to the bottom one it will slide. Now if you connect them they will still be trying to slide. That's sheer stress.

2.4.4 Hoop stress

This one is probably the most straightforward to imagine, if your tank is pressurized in a rocket obviously it will be pushing out against the material uniformly in every cross sectional area. This causes the material to essentially pull against each other and try to expand

There are two types of hoop stress, thin walled and thick walled. If your tube is thick walled then the stress on the inner wall vs outer wall will be different because of the pressure gradient on an object, essentially it dissipates because there's more area for it to act upon. An object is considered thin walled if its wall width is 10x smaller than its inner radius. A thin hoop stress is given by $\sigma = \frac{pr}{t}$ where r is the radius p is the internal pressure and t is the wall thickness

2.4.5 The combined loading problem

Look at this image, this is a very simplified look at what a rocket is going through every moment. You have your axial load coming from your thrusters and bending stress coming from aerodynamic forces, and hoop stress coming from your pressurized tank. All of these are in the thousands of KN and are insane forces. And at the same time you need to minimize the width of your walls to reduce weight for the rocket. Oh My God.

How can this be visualized mathematically?

${\sigma_{axial} \pm \sigma}_{bending} = \sigma_{total}$ The plus minus depends on what side you're analyzing for the bend, the side that's compressing gets added and the size that's under tensile subtracts. To account for hoop stress and shear stress you need specific equations to find stress in certain dimensions and accommodate them . In this case you can use von mises for two dimensions.

$\sigma =$ where $\tau$ is shear stress and if that stress exceeds the material yield strength then the load fails and it causes structural failure.