How to Calculate how much a column buckles?

In summary, 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, you will need to perform a postbuckling analysis which requires solving the large-deflection beam equation. If you have access to finite element analysis software with nonlinear analysis capability, you could try doing your own analysis for your case.
  • #1
Quietrabbit
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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.
 

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  • #2
How are the end connections to the columns made?
That will determine the type of curve and the final chord length.
 
  • #3
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.
 
  • Informative
Likes berkeman
  • #4
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:
Euler spring.jpg

That paper also has some references that look good.
 

Related to How to Calculate how much a column buckles?

What is the Euler's formula for calculating the buckling load of a column?

The Euler's formula for calculating the critical buckling load (P_cr) of a column is given by P_cr = (π² * E * I) / (K * L)², where E is the modulus of elasticity of the material, I is the moment of inertia of the column's cross-sectional area, K is the column effective length factor (which depends on the end conditions), and L is the actual length of the column.

How do end conditions affect the buckling load of a column?

End conditions significantly affect the buckling load of a column because they determine the effective length factor (K). For example, a column with both ends pinned has K = 1, a column with one end fixed and the other end free has K = 2, a column with both ends fixed has K = 0.5, and a column with one end fixed and the other end pinned has K = 0.7. The effective length (K * L) is used in Euler's formula to calculate the buckling load.

What is the moment of inertia and how is it calculated for different cross-sections?

The moment of inertia (I) is a geometric property that measures the distribution of the cross-sectional area relative to an axis. For common shapes: for a rectangular cross-section, I = (b * h³) / 12, where b is the base width and h is the height; for a circular cross-section, I = (π * d⁴) / 64, where d is the diameter. The moment of inertia is crucial in determining the column's resistance to buckling.

What role does the modulus of elasticity play in column buckling?

The modulus of elasticity (E) is a material property that measures the stiffness of a material. It is a critical factor in the Euler's formula for buckling load because it directly influences the column's ability to resist deformation under load. Higher modulus of elasticity indicates a stiffer material, which typically results in a higher buckling load for the column.

How can I determine if a column will buckle under a given load?

To determine if a column will buckle under a given load, you need to compare the applied load (P) to the critical buckling load (P_cr) calculated using Euler's formula. If P < P_cr, the column will not buckle. If P ≥ P_cr, the column is likely to buckle. Ensure that all parameters such as the column's material properties, cross-sectional dimensions, length, and end conditions are accurately accounted for in the calculation.

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