Squeeze a plastic ruler between your palms, end to end. For a while nothing happens. Then, with no warning, it snaps into a bow and gives way sideways. The ruler never got crushed. It buckled, and the same thing decides how thin a steel column in a building or a scaffold pole can be.
A straight column under load has two options. It can stay straight and shorten a tiny bit, storing energy as compression. Or it can bend, which costs bending energy but lets the ends move closer together. At small loads, staying straight is cheaper. At one particular load the two options cost the same, and past it a bowed column is the cheaper shape. A perfectly straight column is balanced on a knife edge; any tiny flaw, wobble or off-centre push tips it over.
Leonhard Euler worked out that threshold in 1757. For a column with both ends free to pivot:
E is how stiff the material is. L is the length. I is the "second moment of area", a number that describes how the material is spread across the cross-section. Notice the square: double the length and the load a column can carry falls to a quarter. Notice also that strength of the material never appears. Buckling is about stiffness and shape, not about how hard the steel is.
This is a steel rod with the same amount of metal in every case (400 mm², about the cross-section of a 22.6 mm solid bar). You can make it longer, spread the same metal into a wider tube, change how the ends are held, and push. Watch which failure comes first.
With the default 1 m rod, buckling arrives at about 25 kN, roughly the weight of two and a half tonnes. Crushing the steel itself would take around 100 kN. The rod fails at a quarter of what the metal could take, because it runs out of stiffness long before it runs out of strength. Shorten it and the two limits meet; below that length the rod simply crushes, and Euler's formula no longer matters.
Now keep the length and drag the diameter up. The amount of steel does not change, but pushing it away from the centre line makes the cross-section far harder to bend. Going from a solid bar to a 40 mm tube with a wall of about 3.5 mm makes it more than five times stiffer, so it carries five times the buckling load for the same weight. That is why bike frames, scaffold poles, tent poles and even your long bones are hollow tubes.
The tube trick has a limit. Make the wall too thin and it stops buckling as a whole and instead wrinkles locally, like a drinking straw bent too far. Engineers have to watch both.
Euler's formula assumes both ends can pivot. Clamp them rigidly and the column bows in a shorter, tighter wave: its "effective length" is half the real one, so it carries four times the load. Clamp one end and leave the other free, like a flagpole, and the effective length doubles, so it carries only a quarter. Swap the end buttons above and watch the buckling load jump around while the rod itself stays exactly the same.
Buckling is also why designers brace tall, thin members. A cross-brace halfway up a column forces it to bend in two shorter waves, and that cuts the effective length in half. Cutting the unsupported length in half is worth four times the strength, which is far cheaper than a thicker column.
Next time you see a thin pole holding up something heavy, check how it is held and braced. Long and slender members fail by bending, not by crushing, and the cure is usually geometry, not more steel: make it shorter, wider, hollow or braced.