Column Buckling Formulas & Effective Length Factors (K)
Pick the right buckling model—Euler (elastic), Johnson/Parabolic (inelastic), or Secant (eccentric load)—and the correct effective length factor \(K\) for your end conditions.
Basic definitions
| Parameter | Formula |
|---|
| Radius of gyration \([r]\) (a.k.a. \(k\)) | $$r=\sqrt{\dfrac{I}{A}}$$ |
| Eccentricity ratio \([er]\) | $$er=\dfrac{e\,c}{r^{2}}$$ |
| Slenderness ratio \([S]\) | $$S=\dfrac{L}{r}$$ |
| Effective slenderness \([S_{\text{eff}}]\) | $$S_{\text{eff}}=\dfrac{K\,L}{r}$$ |
| Transition slenderness \([C_c]\) | $$C_c=\sqrt{\dfrac{2\pi^{2}E}{S_y}}$$ |
Which formula should I use?
Concentric axial load (\(e=0 \Rightarrow er=0\))
- If \(S_{\text{eff}} \ge C_c\): use Euler (elastic).
$$P_{cr}=\dfrac{\pi^{2} E I}{(K L)^{2}}=\dfrac{\pi^{2} E A r^{2}}{(K L)^{2}}$$
- If \(S_{\text{eff}} < C_c\): use Johnson (parabolic) (inelastic).
$$\sigma_{cr}=S_y-\dfrac{S_y^{2}}{4\pi^{2}E}\left(\dfrac{K L}{r}\right)^{2},\qquad P_{cr}=\sigma_{cr}\,A$$
Eccentric load (\(e\neq 0\)) — use the
Secant formula:
$$\sigma_{\max}=\dfrac{P}{A}\left[1+\dfrac{e\,c}{r^{2}}\sec\!\left(\dfrac{K L}{2 r}\sqrt{\dfrac{P}{A E}}\right)\right]$$
For an allowable/limit stress \(\sigma_{\max}=S_{yc}\), solve for \(P\).
Small-angle check: if \(\phi=\dfrac{K L}{2 r}\sqrt{\dfrac{P}{A E}}\lesssim 0.3\), then \(\sec\phi\approx1+\phi^{2}/2\) and \(P\approx \dfrac{S_{yc}A}{1+e\,c/r^{2}}\).
Try it with numbers in the
Column Buckling Calculator.
Use the
minimum radius of gyration \(r_{\min}\) and a realistic \(K\) for your connection.
Effective length factors \(K\)
End conditions are idealized; the “Suggested” values include practical allowances for joint flexibility and alignment.
“Guided” means rotation fixed and translation free. Note symmetry: interchanging left/right ends gives the same \(K\).
| Boundary condition |
Theoretical \(K\) |
Suggested (engineering) |
| Free–Free | — | — (unstable) |
| Pinned–Free | — | — (unstable) |
| Pinned–Pinned | 1.0 | 1.0 |
| Fixed–Pinned | 0.7 | 0.8 |
| Fixed–Fixed | 0.5 | 0.65–0.85 |
| Fixed–Free (cantilever) | 2.0 | 2.1 |
| Fixed–Guided (rotation fixed, translation free) | 1.0 | 1.2 |
| Pinned–Guided (rotation free, translation fixed) | 2.0 | 2.0 |
| Guided–Guided (rotation free, translation free) | 2.0 | 2.0–2.4 |
FAQs
When do I use Euler vs. Johnson?
If \(S_{\text{eff}}=\dfrac{K L}{r}\ge C_c=\sqrt{\dfrac{2\pi^{2}E}{S_y}}\) the column buckles elastically → use Euler.
If \(S_{\text{eff}}<C_c\) inelastic effects govern → use Johnson (parabolic).
Which \(r\) (radius of gyration) do I use?
Use the minimum \(r\) about the two principal axes; buckling occurs about the weaker axis.
How do I pick \(K\)?
Start with theoretical \(K\) for your end conditions. If joint flexibility/misalignment is expected, adopt the “Suggested” value unless a frame analysis provides a better estimate.
What units should I use?
Any consistent set is fine (e.g., N–mm–MPa or kip–in–ksi). Ensure \(E\) and \(S_y\) use the same stress units; \(L\) and \(r\) in the same length units.
How do I handle eccentricity?
Use the Secant formula. Include initial crookedness by adding an equivalent eccentricity \(e_0\) if needed (e.g., fabrication tolerances).
References
- Shigley, Budynas, Nisbett. Mechanical Engineering Design, McGraw-Hill.
- Beer & Johnston. Mechanics of Materials, McGraw-Hill.