Disclaimer – this includes a back-of-the-envelope calculation. Take it with a pinch of salt.

If a steel beam is exposed to an increased temperature, it will tend to expand. If there is nothing to oppose that expansion, then the beam increases in length by ΔL.

If, however, the beam’s ends don’t allow this expansion then the beam will tend to buckle.

But how big the temperature variation needs to be for this beam to noticeably buckle?

Let’s simplify the things a bit.

That sinusoidal shape of the buckle can be simplified to a triangle. The beam expands from the initial length L to 2H and the maximum lateral movement of the buckled beam is v.

Pythagoras theorem now…

For the following:

L = 40m

v = 200mm (0.2m) maximum movement

H= 20.001m

ΔL = 2 * 20.001 – 40 = 0.002m = 2mm

Surprising, huh?

The beam needs to expand by 2mm to get a 200mm buckle wave!

(Am I wrong?)

Good! Now let’s see how much temperature variation we need to get this 2mm expansion and 200mm buckle.

ΔL=αLΔt

α is the expansion coefficient of beam’s material. For steel this is 1.15 10^-5 mm/mm °C

Δt = 4.35°C

That’s quite a simplification, Constantin!

Yes, indeed.

Imagine that that beam is a rail. The rail buckling has three sections:

We have longitudinal movements on a significant length outside of the area where we can notice the buckle. In fact the length involved in the second equation is different from the one we used in the first. Bigger.

But the railway tracks don’t buckle from just 5°C variation!

Yes, they don’t.

Let’s get real.

The tracks are formed of two rails fixed on sleepers – a frame, a truss.

That truss doesn’t just hang in space. It is fixed in ballast and that opposes friction forces to any movement, lateral or longitudinal.

This truss develops a significant and complex set of lateral resistances.

Isn’t it then easier for the buckle to happen in the vertical plane?

Well, the rail has a funny shape. And that shape has a property called area moment of inertia – defined by the distribution of the points in a section relative to an axis. This parameter defines the resistance to the bending relative to the axis for which is calculated.

You know how easy is to bend a ruler on one axis but impossible to do it on the other? This area moment of inertia is to blame for that. Further away from the axis the section points are, the better.

The rail moment of inertia for the XX plane (opposing vertical bending/buckling) is 5 to 6 times higher than the one for the YY plane (opposing lateral bending/buckling).

So a single rail will buckle laterally rather than vertically.

And, yes, the rail foot of the Vignole rail gives it a significant advantage over the bullhead rail. About 20% better for the same weight.

For a railway track (rails+sleepers) this XX/ YY factor is not that high – quite the opposite – although the moment of inertia of the truss is smaller vertically (it’s easier to bend the track vertically rather than horizontally), the weight of the frame alone keeps the track in place in the vertical plane. Not quite alone, but it carries almost all the weight.

When the hot weather comes however, some minute vertical bends appear, either just due to temperature alone or due to the passage of a train – in front of it or between bogies. When these bends develop the friction forces that retain the truss decrease and the likelihood of lateral buckling increases. The rails+sleepers truss starts to move through the ballast – a central zone will move laterally developing the well known wave. At both extremities of that wave the track will move longitudinally towards the buckled section, feeding the system with the energy required to move.