Critical Slenderness⚠ unverified
Mechanical / Columns · Compute the transition slenderness between short and long columns
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 1.0 | Young's modulus of the material |
| sigma_y | σy | Pa | 1.0 | Yield strength of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | srcrit | — | Dimensionless transition slenderness ratio separating Johnson (short) and Euler (long) column behaviour |
The science & history
Understanding the Parameters
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Young's modulus $E$ — sets the Euler branch $\sigma_{cr}=\pi^2 E/\lambda^2$. A stiffer material pushes the Euler curve up, so the two curves cross at a higher slenderness — $\lambda_c$ grows with $\sqrt{E}$.
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Yield strength $\sigma_y$ — sets the ceiling that the Johnson parabola descends from. A stronger material lowers $\lambda_c$ (the crossover happens sooner), because the Euler curve has to fall further before it drops below the higher yield line. So high-strength steels reach the Euler regime at lower slenderness than mild steels.
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The result $\lambda_c$ — the dividing line. It embodies the design rule: material strength helps only for columns below $\lambda_c$; above it, all that matters is $E$ and geometry.
Derivation (Approaching a Proof)
The transition is defined as the slenderness where the Johnson parabola and the Euler curve join smoothly. The J. B. Johnson parabola is constructed to be tangent to the Euler hyperbola while passing through $\sigma_{cr}=\sigma_y$ at $\lambda=0$:
$$\sigma_{cr} = \sigma_y\left(1 - \frac{\sigma_y\,\lambda^2}{4\pi^2 E}\right).$$
Convention places the join at the slenderness where the Euler stress has fallen to half the yield stress, $\sigma_{cr}=\sigma_y/2$. Setting the Euler stress equal to $\sigma_y/2$:
$$\frac{\pi^2 E}{\lambda_c^2} = \frac{\sigma_y}{2} \;\Longrightarrow\; \lambda_c^2 = \frac{2\pi^2 E}{\sigma_y} \;\Longrightarrow\; \lambda_c = \sqrt{\frac{2\pi^2 E}{\sigma_y}}.$$
At exactly this slenderness the Johnson parabola and Euler curve share both value and slope (tangency), so the composite column curve is smooth. Substituting $\lambda_c$ back into the parabola confirms $\sigma_{cr}(\lambda_c)=\sigma_y/2$ — the parabola has descended to half yield, meeting Euler there.
Dimensional check. $\dfrac{E}{\sigma_y}$ is dimensionless (Pa/Pa), so $\lambda_c$ is dimensionless. ✓
History and Development
The parabolic intermediate-column formula and its transition slenderness are due to J. B. Johnson (late 19th century), one of several empirical fixes (alongside Rankine–Gordon and the tangent-modulus theory) to Euler's overprediction for stocky columns. The "half-yield" tangency construction became standard in American machine-design practice (Shigley) and survives as the boundary between column formulas in mechanical design, echoed by the $C_c$ transition in older AISC allowable-stress design.
Related Concepts: Intermediate Column Load, Euler Buckling Stress, Slenderness Ratio, Euler Buckling Load column, Tangent Modulus Load
Notes: This uses the common half-yield tangency ($\sigma_{cr}=\sigma_y/2$ at the join). Some codes define the transition differently; confirm which convention applies. Below $\lambda_c$ use the Johnson parabola, above it use Euler.