Can a bridge collapse if a group walks over it in cadence?

Answer

Dear Elly,

Yes, that’s possible. That is why soldiers are instructed to step over a bridge out of cadence.

How come? This has to do with a concept in physics called natural frequency. Every object that exists has a specific vibration frequency with which it vibrates easily. This ranges from atoms to glasses to bridges to stars. If you tap a wine glass or brandy glass, it will produce a very specific tone (the chime you hear). The glass is then vibrating at its natural frequency. Now to understand why soldiers marching in cadence across a bridge can be a problem, let’s start with another example of natural frequency: Someone on the swing.

If you swing someone on a swing (you are the only one pushing), the person on the swing will move back and forth. So far nothing new. What you may not have noticed is that the time it takes to go back and forth does not depend on how high the person goes on the swing. The moving back and forth happens at the natural frequency of the object (swing + person). To make someone go higher, give a nudge at set times, and you do this (if you do it right) with the same frequency as the rocking (ie every time the person comes back to you). If, on the other hand, you were to give a nudge at random times, you would slow down the swing one moment and speed it up the next, with no net effect. This is a property of the natural frequency of an object: if you push an object (=give extra energy) with its natural frequency, it will vibrate harder and harder. And eventually it will vibrate to pieces.

Now that we can imagine this with a swing, we can go back to the wine glass. It’s a well-known spectacular trick: breaking a wine glass by singing to it. This is possible by singing the note with the natural frequency of the glass. Because of this, the sound waves you produce while singing will be the ones that give the glass that extra “swing” push over and over until it finally breaks. (The gentlemen of Mythbusters had a lot of fun with this: video in English.)

Now we can go one step back: the bridge. Like all other objects, a bridge has a natural frequency. If a bridge starts to vibrate at this frequency, it will rock harder and harder and eventually break, just like the glass. The soldiers who step over the bridge in Kadans can give the bridge a nudge with every footstep. If the natural frequency of the bridge is accidentally the same as the step frequency of the soldiers, this will have disastrous consequences. By letting the soldiers step out of cadence you get the same as with the swing where you give a push at random moments, all pushes cancel each other out so that the bridge cannot swing too hard. However, it is not only soldiers who are dangerous for bridges, wind can also make a bridge vibrate with its natural frequency with disastrous consequences. Today this is very carefully taken into account when designing new bridges, but there was a time when this was not the case. This resulted in the Tacoma bridge collapsing (with spectacular images as a result: see here and here (and yes there you see people and cars on the bridge)). This bridge was also called a Galloping Gertie because of the swinging behavior it could exhibit.

Regards,

Danny

PS: There is some debate whether the Tacoma bridge actually collapsed due to resonance or not (e.g. here and here, where the 1.5x accelerated playback from the first link is a little relevant argument regarding resonance or not if the resonance frequency is not given is). Several aspects need to be considered separately.

1. Tocama bridge showed oscillations as you can imagine after the example of the swing (eg first 10-15 seconds of this movie)
2. Although the example of the swing or of a string will give you 1 natural frequency that is not the case for a 3D object, it can have multiple natural frequencies, each linked to a different vibration mode. The twisting of the Tacoma bridge is then a different vibration mode than the up and down movement from the first second of point 1). The question you can now ask yourself in connection with resonance is whether the oscillation can be seen before the collapse occurs at the corresponding natural frequency or not (the cause of the swishing, being the source that provides the power to keep the movement going is not important for this, as long as it pushes it with the natural frequency).

PS2: Another example of a bridge that vibrates when too many people walk across it is the Millennium Bridge. It also began to move briskly when tens of thousands of pedestrians crossed it together when it opened. In the meantime, additional shock absorbers have been installed to prevent or limit this in the future. (links: here and here)