Euler Buckling is a failure mode for structures that suddenly fail with huge deformations due to compressive force. It mainly applies to long and slender members (columns), but also to plates and shells.
As a practical example take a plastic ruler (steel is also acceptable, but it has to be flat - not the triangular draftsman's ruler or the roller ruler). If you pull both ends, not much happens - you cannot (by yourself) create enough stress in the ruler for it to yield. Now try the opposite. Push on both ends of the ruler (ie. try to compress it). With much less force, the ruler will now bend out of shape and lose all capacity - if you don't restrain yourself you can easily break the ruler at this point. This is the behavior that is called Euler buckling.
Because the failure is without warning, and quite dramatic, care must be taken when designing columns or beams taking compressive forces.
The theory behind buckling is strictly non-linear with respect to deformation - if you double the force applied you are not guaranteed double the deformation. Linear theory will explain a small compression of the column, but not its lateral deflection.
Leonhard Euler deducted the theory behind this problem in 1757. The result of his work was the "Euler load" - the critical load where buckling will start. For a column that is pinned at both ends, the load becomes:
Pcr = (π2 E I) / (L2)
For columns that are supported differently the effective length of the column will vary.
It should be noted that buckling is not dependant on material strength but on dimensions alone - namely the length L and the moment of inertia I. It is dependant on material though ( E = modulus of elasticity).
As a practical example we consider our plastic ruler again:
- E = 2500 N/mm2
- L = 30 cm
- I = (3 mm)3(25 mm) / 12
Inserting these values into the formula above we get Pcr
= 1.258 kg - which is more or less correct for my ruler at least. By pulling the ruler we need to apply 270 kg for it to break.
When buckling the column will always fail in the direction with the weakest moment of inertia. Therefore many columns are made up of square hollow sections, which have the same property in both directions.