Intermediate Column Load⚠ unverified

Understanding the Parameters

• Yield strength $\sigma_y$ — the anchor of the parabola: at zero slenderness the column simply yields, so $\sigma_{cr}=\sigma_y$. Unlike Euler theory, the Johnson formula does depend on strength, because yielding is what drives the intermediate-column failure.

• Young's modulus $E$ — appears in the reduction term; it controls how fast the parabola drops and where it meets the Euler curve. Its influence is opposite to $\sigma_y$: larger $E$ makes the reduction term smaller, keeping the stress near yield over a wider slenderness range.

• Slenderness inputs $L$, $r$, $K$ — combine into $\lambda = KL/r$ (see Slenderness Ratio). The reduction from yield grows with $\lambda^2$, so the parabola falls gently at first and then steeply as slenderness increases.

• The output regime — the calculator returns the Johnson parabola for $\lambda < \lambda_c$ and switches to the Euler stress for $\lambda > \lambda_c$, giving one continuous column curve.

Derivation (Approaching a Proof)

Johnson's formula is a fitted parabola subject to three physical constraints, not a first-principles result — but the constraints determine it uniquely:

1. Stub-column limit. At $\lambda = 0$ there is no buckling; the column yields, so $\sigma_{cr}(0) = \sigma_y$. This fixes the constant term.

2. Parabolic form in $\lambda^2$. Column strength is assumed to reduce smoothly and symmetrically with slenderness, so a parabola $\sigma_{cr} = \sigma_y - a\lambda^2$ is chosen (the simplest curve with zero slope at $\lambda=0$, matching the flat top of stub-column behaviour).

3. Tangency to Euler at the transition. The parabola must join the Euler curve $\sigma_{cr}=\pi^2E/\lambda^2$ smoothly (equal value and slope) at the transition slenderness $\lambda_c=\sqrt{2\pi^2E/\sigma_y}$, where by convention $\sigma_{cr}=\sigma_y/2$.

Applying the tangency condition determines the coefficient $a$. Requiring $\sigma_y - a\lambda_c^2 = \sigma_y/2$ with $\lambda_c^2 = 2\pi^2E/\sigma_y$:

$$a\,\lambda_c^2 = \frac{\sigma_y}{2} \;\Longrightarrow\; a = \frac{\sigma_y}{2\lambda_c^2} = \frac{\sigma_y}{2}\cdot\frac{\sigma_y}{2\pi^2 E} = \frac{\sigma_y^2}{4\pi^2 E}.$$

Hence

$$\sigma_{cr} = \sigma_y - \frac{\sigma_y^2}{4\pi^2 E}\,\lambda^2 = \sigma_y\left(1 - \frac{\lambda^2 \sigma_y}{4\pi^2 E}\right).$$

The parabola is thus pinned by yield at one end and glued tangentially to Euler at the other.

Dimensional check. $\dfrac{\lambda^2\sigma_y}{E}$ is dimensionless, so the bracket is dimensionless and $[\sigma_{cr}]=\text{Pa}$. ✓

History and Development

J. B. Johnson proposed the parabolic column formula in the 1890s to correlate the intermediate-column test data that Euler theory could not. Together with Rankine–Gordon and the later tangent-modulus theory, it addressed the "column problem" that had troubled engineers since the mid-1800s. The Johnson parabola remains the standard intermediate-column formula in mechanical machine design (Shigley), while structural steel codes use closely related smooth column curves.

Related Concepts: Critical Slenderness, Euler Buckling Stress, Euler Buckling Load column, Slenderness Ratio, Tangent Modulus Load, Factor of Safety

Notes: Empirical parabolic fit (Johnson), valid below the transition slenderness; above it the Euler stress governs. Multiply $\sigma_{cr}$ by area for the critical load, and apply a factor of safety for allowable design.