Buckling of columns (Euler)

By the end of this topic you will be able to

Explain buckling as a stability failure rather than a strength failure.
Calculate the Euler critical load for different end conditions using the effective length.
Use the slenderness ratio to decide whether Euler's formula or yielding governs.

A different kind of failure

A short, stocky column fails by crushing: the material yields. A long, slender column fails much earlier, by buckling: it suddenly bows sideways, often with a stress far below the yield stress. Buckling is a failure of stability, not of material strength. Think of pressing on the ends of a plastic ruler.

Euler's formula

For an ideal, straight, pin-ended column loaded exactly along its axis, the load at which it buckles is

P_{\text{cr}} = \frac{\pi^2 EI}{L^2}

where $I$ is the smaller second moment of area of the section (the column buckles about its weakest axis).

End conditions: effective length

Other supports change the buckled shape. We account for them with an effective-length factor $K$ :

P_{\text{cr}} = \frac{\pi^2 EI}{(KL)^2}

Ends	$K$
Pinned – pinned	1.0
Fixed – free (flagpole)	2.0
Fixed – fixed	0.5
Fixed – pinned	0.7

A column fixed at both ends carries four times the load of a pinned one; a flagpole carries only a quarter.

Slenderness

With the radius of gyration $r = \sqrt{I/A}$ , the slenderness ratio $\lambda = KL/r$ and the critical stress are

\lambda = \frac{KL}{r}, \qquad \sigma_{\text{cr}} = \frac{\pi^2 E}{\lambda^2}

Euler's curve is valid only while $\sigma_{\text{cr}}$ is below the yield stress. For stockier columns (small $\lambda$ ) the column yields before it buckles, and the load is limited by $\sigma_y A$ .

Interactive modelVary length, section and end conditions. The curve shows critical stress against slenderness.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Worked example

A 3 m steel column ( $E = 200$ GPa, $\sigma_y = 250$ MPa), pinned at both ends, has a 50 mm × 80 mm rectangular section. Find the critical load.

Weak axis. $I_{\min} = 80\times50^3/12 = 0.833\times10^6$ mm⁴, $A = 4000$ mm².

Critical load.

P_{\text{cr}} = \frac{\pi^2\times200\,000\times0.833\times10^6}{3000^2} = 182.8 \text{ kN}

Checks. $r = \sqrt{I/A} = 14.4$ mm, so $\lambda = 3000/14.4 = 208$ . The critical stress is $182\,800/4000 = 45.7$ MPa, far below 250 MPa, so buckling governs. A yield (squash) load would have been $250\times4000 = 1000$ kN: the column is limited to under a fifth of that.

Practice problems

The column in the worked example is fixed at both ends. What is its critical load?

Show answer

$K = 0.5$ , so $P_{\text{cr}}$ is four times as large: $4\times182.8 = 731$ kN (the stress, 183 MPa, is still below yield).
The same column is used as a flagpole (fixed at the bottom, free at the top). Find $P_{\text{cr}}$ .

Show answer

$K = 2$ , so $P_{\text{cr}} = 182.8/4 = 45.7$ kN.
A pinned column's length is increased from 3 m to 4 m. How does the buckling load change?

Show answer

$P_{\text{cr}} \propto 1/L^2$ : $(3/4)^2 = 0.5625$ , so it falls to 56% of the previous value.
Why is a circular tube a better column than a solid bar of the same area?

Show answer

For the same area, a tube has a larger $I$ (the material is further from the centroid), so a bigger $r$ and smaller $\lambda$ , hence a higher $P_{\text{cr}}$ .

Key formulae

Euler critical load: $P_{\text{cr}} = \frac{\pi^2 EI}{(KL)^2}$
Radius of gyration: $r = \sqrt{\frac{I}{A}}$
Slenderness ratio: $\lambda = \frac{KL}{r}$
Critical stress: $\sigma_{\text{cr}} = \frac{\pi^2 E}{\lambda^2}$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (45 min). 5 min: students push on a plastic ruler to feel buckling; 15 min: Euler's idea and $K$ ; 10 min: worked example; 15 min: model, slenderness curve and Euler validity.

Activity. Find the shortest length at which the modelled steel column switches from yielding to buckling.

Discussion. Why do scaffolding and bracing add bars at mid-height?

Assessment idea. Compare three end conditions for one section and rank the results.