Columns and Buckling

A short block in compression crushes when the stress reaches yield. A slender strut never gets there: it bows sideways and collapses at a stress the material could carry comfortably.

The question changes form too. Not "is the stress acceptable" but "what happens after a small disturbance" — does the strut return to straight, or does the bow grow? That is a stability question, and it needs its own analysis.

1. 1Below the critical load, a sideways disturbance is resisted and the strut springs back. Stable.
2. 2At the critical load, straight and slightly bent require the same load. The strut is indifferent.
3. 3Above it, any disturbance grows rather than decaying, and collapse follows.
4. 4Solving the resulting differential equation gives P_cr = π²EI/Lₑ².

Look at what the formula does not contain: the yield stress. All structural steels have E ≈ 200 GPa regardless of grade, so specifying high-strength steel for a slender column buys nothing at all. The gain has to come from I or from a shorter effective length.

Since P_cr ∝ 1/Lₑ², the range from fixed-fixed to fixed-free spans a factor of sixteen for the same strut, the same material and the same section.

The practical caution is that real connections are semi-rigid. A bolted end plate is neither pinned nor fixed, and claiming full fixity from a joint that cannot deliver it is unconservative in a failure mode that gives no warning.

A column buckles about the axis with the smallest I — never the one you were designing for. An I-section is excellent in bending about its strong axis and a poor column unless its weak axis is braced.

A circular hollow section has the same I in every direction, which is why it is the natural choice for a free-standing column. Bracing exists precisely to shorten the effective length about the weak axis, and it is why a slender column with mid-height restraint carries four times as much.

Extrapolated to a short column, Euler's formula predicts a buckling stress above the yield stress — which the material cannot reach, so it crushes first. The formula is valid only above a critical slenderness ratio.

• Rankine's formula blends the crushing and buckling regimes into one expression.
• Design codes use curves fitted to test data, with separate curves for different section types and manufacturing routes.
• Slenderness ratio Lₑ/k is the parameter that decides which regime a column is in, where k is the radius of gyration.

All of them push the same way, so measured capacities sit consistently below the theoretical curve. Buckling also gives no warning — no yielding, no cracking, no visible sag — which is why column safety factors are the largest in the subject.