Why do thin columns suddenly fail?

01

The idea

Thin posts don't crush. They bow out.

Press down on a ruler standing on end. It doesn't get shorter, it suddenly bends sideways. That's buckling.

02

Play with it

Tall, short, held, free

Change the height, the load and how the ends are held. Watch the red curve on the right.

Try this · 0/3Make it buckleMake it crush insteadMake a 4 m post carry 120 kN

03

The rule

Twice as tall, a quarter of the strength

Buckling load (Euler)

\force{P_{cr}} = \dfrac{\pi^2 E I}{(K\,\len{L})^2}

K: how the ends are held

Fixed both ends	0.5
Fixed + pinned	0.7
Pinned both ends	1
Flagpole (fixed + free)	2

Height is squared, so it matters most. A stiffer shape (bigger $I$ ) or ends held firmly helps too. Real design adds safety factors and the fact that no post is perfectly straight.

04

Your turn

Prove it

▲ Build1/3

The tallest post that carries 150 kN

To do

Both ends pinned. Find the tallest post (on the slider) that does not buckle.

= Solve2/3

When does this post buckle?

To do

A post pinned at both ends is $\len{3}\,\mathrm{m}$ tall with $EI = 90\,\mathrm{kN{\cdot}m^2}$ . At what load does it buckle?

A post pinned at both ends is $\len{10}\,\mathrm{ft}$ tall with $EI = 140\,\mathrm{kip{\cdot}ft^2}$ . At what load does it buckle?

◆ Pick3/3

Make the post twice as tall. The load it can carry before buckling becomes…

To do

05 · Takeaway

Short posts crush. Tall posts buckle.

\force{P_{cr}} = \dfrac{\pi^2 E I}{(K\,\len{L})^2}

Related: why an I-beam is shaped like an I (a bigger I fights buckling too).