Euler Column Buckling Calculator

A long, slender member in compression does not crush. At a certain load it bows out sideways and keeps going, well before the material yields. Euler's formula gives that load for a straight column with the load on its axis:

P_cr = π²EI ÷ (KL)²

E is the modulus of elasticity, I the least moment of inertia of the section (it buckles about its weakest axis), L the unsupported length and K the effective length factor, which says how the ends are held. For a long column, where Euler governs, stiffness sets the answer and strength does not enter: a high-strength steel buckles at the same load as a mild one. A short column is another matter, below.

Where Euler stops. Divide the load by the area and Euler gives a stress of π²E ÷ (KL/r)², where r = √(I/A) is the radius of gyration. For a short column that stress climbs toward and past the yield strength, which cannot happen. Below the transition slenderness (KL/r)_t = √(2π²E ÷ S_y) the calculator uses the J. B. Johnson parabola instead, P_cr = AS_y [1 − S_y (KL/r)² ÷ 4π²E], and says by how much Euler overpredicts. The two meet at the transition, both at half the yield strength. Most compression members in machines are short enough for Johnson, which is why the check matters.

Worked example, inch. The handbook's own: a 1/2 × 1 in steel bar, 24 in long, pinned ends, yield 40 ksi, E 30 Msi, carrying 4,000 lb. The least r is 0.1443 in, so KL/r = 166, above the transition of 122: Euler applies. P_cr = 5,355 lb, a factor of safety of 1.34 (the handbook rounds r to 0.145 and prints 5,400 lb and 1.35).

Worked example, metric. A 20 mm round 6061-T6 bar, 400 mm long, fixed at both ends (design K 0.65), carrying 30 kN. KL/r = 52, below the transition of 70.9, so Euler's 80.6 kN is 27% high. Johnson gives 63.4 kN, a factor of safety of 2.11. Load either example with the button under the inputs, in its units.

Not covered: the load must act along the column's axis and the column must start straight and be of one section along its length. Short columns, which yield as they buckle, are covered only by the Johnson parabola, an empirical fit. An offset load or a crooked column bends as well as buckles and fails sooner; side loads, heat and fatigue are not included. Thin walls can wrinkle locally (crippling) before the whole tube buckles, and open sections can twist; neither is checked. The material values are typical; for a design, use the minimum from the material's own specification, and remember the factor of safety covers what this idealisation leaves out.

References: Machinery's Handbook, 27th ed., pp. 286 to 288 (Euler formula, J. B. Johnson formula, the rule for choosing between them, factors of safety and the worked example); p. 291 (effective length factors, theoretical and recommended design values, from the AISC Manual of Steel Construction, 8th ed.); pp. 239 and 241 (moments of inertia and radii of gyration). Typical material values: Tables 7b, 11a, 12 and 13, and p. 420.

Reviewer: pending.