Finds the load at which a slender column buckles sideways, from Euler's formula P = π²EI ÷ (kL)². It falls with the square of the length, so a column twice as long carries a quarter of the load. How the ends are held matters just as much.
A slender column fails by bowing sideways long before the material itself is crushed. That is buckling, and this finds the load at which it happens.
is Young's modulus, the second moment of area, the length and the effective length factor.
The denominator carries the square of the length, so a column twice as long carries a quarter of the load. Length matters far more than thickness here. It is why a thin rod bends easily between your hands but becomes stubbornly stiff once cut short.
The effective length factor depends on the restraint at each end: 1.0 for pinned at both ends, 0.7 for one fixed and one pinned, 0.5 for both fixed and 2.0 for a cantilever. Fixing both ends halves and quadruples the load; a cantilever cuts it to a quarter. The restraint alone spans a factor of sixteen.
A column with a Young's modulus of 8 GPa, a second moment of area of 1013 cm⁴ and a length of 3 m, pinned at both ends, buckles at 88.9 kN. With a cross-section of 110.25 cm² the slenderness ratio is 99 and the buckling stress 8.06 N/mm².
Euler's formula applies to slender columns. A short, stout column with a low slenderness ratio crushes before it buckles, and its capacity is set by the compressive strength of the material instead.