The core idea
Compress a short block and it fails by crushing when the stress passes the material's strength. Compress a long slender member the same way and it fails very differently: sideways, suddenly, at a fraction of the crush load. A tiny sideways bow reduces the load's ability to stay straight, which deepens the bow — a self-reinforcing runaway called elastic instability, or buckling.
Euler's critical load:
P_cr = π² · E · I / L_eff²
- E — Young's modulus of the material (Pa); steel ≈ 210 GPa
- I — second moment of area (m⁴), resistance to bending from the cross-section's shape
- L_eff — effective length (m): pinned ends count the full length; fixed ends cut it roughly in half
Note what is absent: there is no strength term. Buckling is a stiffness problem, so it is decided by E and I, not by how strong the material is.
The L_eff² in the denominator is the headline. Double a column's effective length and its safe load drops to a quarter; quadrupling the length cuts capacity to one sixteenth. Length is punishing; shape and end restraint are where the wins are.
Real-world example
A galvanised scaffolding tube — 48.3 mm outside diameter, 4 mm wall — has I ≈ 1.38 × 10⁻⁷ m⁴. In steel (E = 210 GPa) with pinned ends, Euler predicts P_cr ≈ 71 kN at 2 m — about 7 tonnes. Stretch the same tube to 4 m and capacity collapses to ≈ 18 kN: a quarter, for double the length. This is why scaffold standards braced by ledgers every 2 m can stack ten storeys high, and why the unbraced length between connections — not the total height — governs the design. It is also why replacing those tubes with the same size in aluminium (E ≈ 70 GPa) would cut capacity to one third, and why real slender columns are designed with a generous safety factor: crookedness, off-centre loads and rust all shave the ideal figure.
Common pitfall
Designing a compression member for crush strength and assuming the safety factor carries over to the real thing. For a slender column, buckling may govern at a small fraction of the crush load, and the classic Euler formula assumes a perfectly straight member loaded perfectly axially — neither exists in reality. Slender compression members need stability checks with imperfection allowances, not just a stress check. Corollary: making a column "stronger" (higher-strength steel) buys almost nothing if buckling governs — spend the material on I instead, by moving it away from the axis: a tube beats a solid rod of the same weight.