How to Calculate how much a column buckles?

[Quietrabbit]

TL;DR Summary

How to Calculate how much a column buckles. Ie horizontal measurement and new vertical length based on force applied.

I’m am looking for how to calculate the new horizontal and vertical “size” of a column based on the material and how much force is applied from the top/ends.
I have 3D printed TPU pillars that are made to compress and mold to a shape, but I am trying to figure out how much they will compress since that effects the shape I CAD. (I don’t want to trial and error the prints).
I have already looked into eulers equation but that seems to only give me the critical force which I know I am well past.

 

[Baluncore]

How are the end connections to the columns made?
That will determine the type of curve and the final chord length.

 

[T1m0]

You will need to perform a postbuckling analysis which requires solving the large-deflection beam equation. An example of this can be found in Section 2.7 in "Theory of Elastic Stability" by Timoshenko and Gere. If their example does not match yours, you can try a search on postbuckling of beams.

If you have access to finite element analysis software with nonlinear analysis capability (e.g., ANSYS or ABAQUS), you could try doing your own analysis for your case.

 

[jrmichler]

The Euler equation predicts the force to buckle the column. That force, as alluded to by @Baluncore, is very dependent on the end support conditions. Your sketch shows simple supports, AKA pinned ends. Note that the Euler equation only applies to elastic buckling. Elastic buckling is when the column pops back to its starting shape after the load is removed.

After buckling, the force on an Euler column is roughly constant as the displacement increases. Two different search criteria that found good information are euler column constant force spring and post buckling euler columns. A good hit is this one: https://www.researchgate.net/figure...-spring-The-example-shown-here_fig1_349575713. One figure from that paper that shows this is:

That paper also has some references that look good.