### Buckling and Elasticity

Buckling is a form of instability which leads to failure of a structural element. It is one of the various modes of material failure, and is caused by high compressive stresses along the axis of a structural member. Buckling can be manifested both in vertical columns and horizontal members – such as railroad tracks, also known as sun kink.

Buckling can be described as the more or less sudden “popping out”, or bending outward of a slender structural member due to high compressive stress. These compressive stresses at the point of buckling are lower than what is required to cause failure in compression. This does not mean that a combination of both cannot occur simultaneously, as explained further below.

Slenderness Ratio

The slenderness ratio is a ratio that is used to classify various column dimensions based on the length and cross sectional area. To be more specific, it is the ratio of the effective length of a column to its least radius of gyration, expressed as the Greek letter lambda, λ. The radius of gyration is equal to the square root of the quotient of the second moment of area and the area.

Columns are often classified as either short, intermediate, or long, and are based on both material-type and slenderness ratio, not just its length. Obviously, materials like steel can afford to have a higher slenderness ratio than something like wood or concrete due to its higher overall strength and elastic stability.

A short column will most likely fail in compression before failing by buckling, whereas the tendency of intermediate and long columns will be to fail by a combination of both and pure buckling, respectively. Buckling may also be more of a concern when the load is eccentric – loads applied outside of the member’s center of gravity – as opposed to an axial load.

Euler’s Formula

This is a formula that was derived by mathematician Leonhard Euler way back in 1757, and is used to determine the maximum load a structural member can take in compression before failure by buckling, and sheds light on several interesting points. Remember that this assumes the sample member is “ideal”, which means it is homogeneous, perfectly straight, and free from any initial stress.

where

F = maximum or critical force (vertical load on column)
E = modulus of elasticity
I = second moment of area
L = unsupported length of column
K = column effective length factor, whose value depends on the conditions of end support of the column, as follows:

• If both ends are pinned (hinged, free to rotate), K = 1.0.
• If both ends are fixed, K = 0.50.
• If one end is fixed and the other end is pinned, K = 0.699….
• If one end is fixed and the other end is free to move laterally, K = 2.0.

KL is the effective length of the column.

A point of particular interest is that a column’s elasticity – and not its compressive strength – determines the critical load. Of course, high compressive strength may prevent buckling if the given material has a high slenderness ratio, but this is not always feasible. Ideally, second moment of area is maximized while cross sectional area is minimized.

And as you can see, whether or not the ends are fixed or are free to rotate make a very crucial difference as well. What might seem like a small difference between both ends fixed and both ends pinned but free to rotate, ends up being four times. – Yes, having your ends fixed solid as a rock makes it capable of withstanding four times more force without buckling!

Buckling in Surface Materials

This mode of mechanical failure is not reserved only for structural members like columns and beams, but can be manifested in railroad tracks and even pavement surfaces. As the sun beats down on a pavement surface, absorbed heat causes expansion in the material. If the material expands enough, it will cause buckling due to the high compressive stress exerted.

Railroad tracks are in most cases a pair of steel “I-beams” secured to concrete or wood sleepers. Being that the second moment of area for the steel tracks are higher in the vertical direction to support the weight of the train, sun kink more often occurs in the lateral direction – dragging the sleepers with them.