# Acceleration effect on buckling?

The formulas for buckling under axial force only involve the applied force or pressure, such as with Euler buckling. But if you look at the case of a rocket as the fuel is burned off, the force, i.e., the thrust, stays the same but the acceleration increases as the mass decreases.
But we are told the thrust has to be limited to limit the structural stress on the vehicle during the high portions of the flight, i.e., limit the acceleration. So are there more advanced formulas that do include the acceleration?

Bob Clark

## Answers and Replies

A related question: how do you calculate the force on an upper stage when a lower stage is firing? You would think the entire force of the thrust is communicated up through the rocket so that the same force operates on the upper stages.
But this would lead to the conclusion that for a rocket with a huge amount of thrust in the first stage such as the Saturn V this huge force would be operating also on the smaller, less solidly built upper stages. This does not seem reasonable.
Another possibility is that you calculate the force operating on the upper stages by the acceleration times the mass of the stage. But in this case suppose the thrust is just canceling the weight so you have zero acceleration. The axial force on the upper stage should not be zero then but the weight due to gravity, just like when the stage is resting on the ground.
Putting these facts together the axial force on a stage for the rocket moving upwards should be the acceleration times mass of the stage plus the weight of the stage.

Bob Clark

The formulas for buckling under axial force only involve the applied force or pressure, such as with Euler buckling. But if you look at the case of a rocket as the fuel is burned off, the force, i.e., the thrust, stays the same but the acceleration increases as the mass decreases.
But we are told the thrust has to be limited to limit the structural stress on the vehicle during the high portions of the flight, i.e., limit the acceleration. So are there more advanced formulas that do include the acceleration?

Bob Clark

Hey,

If this is for a preliminary static stress analysis, I would begin with looking at the most critical buckling case e.g max g in the axial direction. Combination cases under max g as well as taking into account vibration, local I values for max stress, panel shear, bearing area around fasteners - the most applicable for the metallic parts. Composite parts require slightly more extensive analysis due to them not being isotropic - the launch trajectory will give you at least some indication of lateral g which is very important in your combination cases!

To have a model/drawing or anything to go by?

Hope my extremely generic explanation has been useful,

Kind regards,

The Jericho.

Hi,

buckling is evaluated (and avoided) by considering stress, but in an in-line launcher, a constant thrust can produce locally increasing stress as the tanks deplete and the acceleration increases.

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In a upper stage or a payload, the stronger acceleration provided by the lighter lower stage increases the stress.

At the top of the lower stage mainly, which is loaded by an upper stage or payload, the stronger acceleration increases the stress.

Against both, lower stages reduce their thrust when they go empty - sometimes last stages do it as well, but these give smaller accelerations overall.

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During the flight phase of maximum dynamic pressure, usually around 10km height where speed has increased but air is still dense, vibrations and bending moments are big. Vibrations often dimension the payload, and bending moments (from wind gusts, planned manoeuvres, steering tolerances) the launcher's tanks, including by buckling the skin.

Against that, many first stages and side boosters, including solid ones, reduce their thrust short after take-off, to pass more slowly the dense atmosphere, and resume full thrust at altitude.

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That's more than a gadget. As vibrations and loads define the mass of a spacecraft, making the launcher careful brings as much as a bigger capacity.

Hey,

If this is for a preliminary static stress analysis, I would begin with looking at the most critical buckling case e.g max g in the axial direction. Combination cases under max g as well as taking into account vibration, local I values for max stress, panel shear, bearing area around fasteners - the most applicable for the metallic parts. Composite parts require slightly more extensive analysis due to them not being isotropic - the launch trajectory will give you at least some indication of lateral g which is very important in your combination cases!
To have a model/drawing or anything to go by?
Hope my extremely generic explanation has been useful,
Kind regards,
The Jericho.

Thanks. There were a couple scenarios I'm interested in. They have to do with using an upper stage on a different rocket. I want to know how the stress on the stage would be changed.
One example is the proposed Liberty rocket that used an Ariane 5 core stage as an upper stage on top of a SRB. The SRB had 12 times the thrust of the Ariane 5 core stage but the increase in tank thickness to deal with the increased axial stress was only 50%, very much less than having to be 12 times thicker.
Another example would be for using the Ariane 5 core as the upper stage on the SLS. This would save on development cost to NASA. Here again though the lower stage thrust would be very much more than that of the Ariane 5 core. But following the idea that it's just the acceleration g's that matter, it might not take terribly much of an increase in tank thickness just as in the Liberty rocket case.

Bob Clark

Ariane 5's core is pulled by its boosters, not pushed. This avoids buckling and saves dry mass. Pushing it means a redesign.

Doubling a wall thickness:
- Doubles Euler's buckling force, for the stage buckling globally in flexion as a beam, which is rarely dimensioning;
- Quadruples the local wall buckling resistance, both when compressing or bending (often dimensioning) the stage.

Probably, much of Ariane 5's core was dimensioned by the propellants' pressure under booster acceleration more than by buckling, so the thickness didn't need a big increase.

Ariane 5's core as an SLS upper stage: its 5.4m diameter must be too narrow to resist bending moments. The SLS core has D=8.4m, the fairing is wide also and creates in the atmosphere a big bending moment.

Widening Ariane 5's core would mean to keep only the Vulcain engine, which is an ageing design: expensive, moderate performance. The J-2X benefits from being more recent and has potential uses beyond the SLS; I wish it provided a roll control.

By the way, since my Solar thermal engine looks like an enabling technology for manned Mars missions and more, I'd like the SLS to have a WIDE fairing to accommodate the concentrators. (Other launchers as well, please...)

Thanks. There were a couple scenarios I'm interested in. They have to do with using an upper stage on a different rocket. I want to know how the stress on the stage would be changed.
One example is the proposed Liberty rocket that used an Ariane 5 core stage as an upper stage on top of a SRB. The SRB had 12 times the thrust of the Ariane 5 core stage but the increase in tank thickness to deal with the increased axial stress was only 50%, very much less than having to be 12 times thicker.
Another example would be for using the Ariane 5 core as the upper stage on the SLS. This would save on development cost to NASA. Here again though the lower stage thrust would be very much more than that of the Ariane 5 core. But following the idea that it's just the acceleration g's that matter, it might not take terribly much of an increase in tank thickness just as in the Liberty rocket case.

Bob Clark

Hey Bob,

The dimensioning generally isn't based on thrust alone, but by acceleration mainly i.e. critical g, (quasi-static loading) and not forgetting vibration (dynamic loading). Therefore, the axial/longitudinal stress will not necessarily increase by the same factor as thrust, but by said loads, as well as aero-thermic loads, in compound effect.

If one was to begin a feasibility study of an alternative rocket for a given upper stage, a brief checklist may consist of the following:
a. Thrust profile
b. Mission design and trajectory
i. A change in thrust will require a different trajectory to be fuel efficient
ii. Lateral-g will change because of the different trajectory - some components, which are sensitive to changes in lateral-g (sometimes due to the bending moment of the slender shape of the launch vehicle), may be pushed near there allowable limits and thus would require design development. Adjusting the mission design would not pose as an ecological solution in overcoming the extra stresses caused by an alternative rocket with different thrust.
c. Acoustic properties - vibration and modes
d. Peak operational temperature and heat transfer/function of time

Regards

TheJez