Mechanics of deformable solids Static problems and stability Fact

Johnson's parabolic formula

Builds on Euler's critical load Young's modulus

Statement

Let a column have length $L$ , effective length factor $K$ , cross-section area $A$ and second moment of area $I$ . Let $r = \sqrt{I/A}$ be its radius of gyration and $s = KL/r$ its slenderness ratio. Let its material have Young's modulus $E$ and compressive yield stress $\sigma_y$ . Set $s_t = \sqrt{2 \pi^2 E / \sigma_y}$ . Johnson's parabolic formula is the following empirical rule. For $s < s_t$ the critical compressive stress of the column is $\sigma_{cr} = \sigma_y - \frac{1}{E} \left( \frac{\sigma_y \, s}{2 \pi} \right)^2,$ while for $s \ge s_t$ the critical stress is given by Euler's critical load, $\sigma_{cr} = \pi^2 E / s^2$ .

Remarks

The formula is not derived from a model; it is a parabola fitted to tests on columns of intermediate slenderness. It is chosen so that the two curves join smoothly. At $s = s_t$ both give $\sigma_{cr} = \sigma_y / 2$ , and both have slope $-\sigma_y / s_t$ in $s$ , so the parabola is tangent to the Euler curve there. At $s = 0$ the parabola gives $\sigma_{cr} = \sigma_y$ , the crushing stress of a very short block.

Example: for a steel with $E = 200$ GPa and $\sigma_y = 250$ MPa, $s_t = \sqrt{2 \pi^2 \cdot 800} \approx 126$ . A column with $s = 60$ then has $\sigma_{cr} \approx 250 - (250 \cdot 60 / 2\pi)^2 / 200000 \approx 221.5$ MPa, with stresses in MPa.

Going deeper

For a student

The formula sits between pure yielding and pure elastic buckling and reflects that intermediate columns fail by inelastic buckling. Tangent-modulus and reduced-modulus theories give a mechanical account of this regime; Johnson's parabola is a simple design substitute. Another approach to the same regime is the Perry–Robertson formula, which instead models initial crookedness.

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Tag	`624BQC6T8E`
URI	`m2://624BQC6T8E`
Kind	statement, fact
Writing	W0 stub
MSC	74G60, 74K10
Status	This copy of the site has no build data. Run `m2 status 624BQC6T8E`.
About	Buckling
Uses	Euler's critical load, Young's modulus
Same as	Wikidata Q28405466 Wikipedia Johnson's parabolic formula
Changed	3 November 2026