On Waterloo

One of the questions sometimes asked of scientists is whether learning how the universe works somehow diminishes their sense of wonder and mystery? I am not a scientist, but I have been interested in science ever since I was a child. I cannot speak for anyone else, but my answer would unequivocally be no It does not diminish my sense of wonder; it actually increases it.

The image below is of the Eurostar terminal at Waterloo station in London. The terminal was closed in 2007 when it was replaced by St Pancras, which is why it looks a little tired. Nevertheless, I think that it remains an elegant structure, however the question arises does it provide more or less wonder when you know how it works? I will leave it to the reader to decide.

The first thing that anyone looking at the structure should understand is that its form is deliberate. In common with many significant structures it is not an architectural whim. The respective roles of the architect and the engineer in relation to aesthetics is not the subject of this post, however I do hope that it will leave the reader with an increased appreciation for the engineer’s contribution.

The next thing that we should note is blindingly obvious.  The structure is formed of a series of steel arches, however the arches are certainly not conventional. They are formed from two curved trusses joined with a pin at the crown and further pins at each of the two bases; though only one of them can be seen in the photo.

Another observation that we might make is that the depth of the truss on the near side projects above the roof while the situation is reversed on the far side. Similarly, while both trusses have three longitudinal chords, the two outer chords are more slender on the near side and the single inner chord is more slender on the far side.

A more subtle observation would be that the thicker chords get thicker towards the middle of their span. This is seen most clearly on the near side. 

Less obvious, due to the angle of the photograph, is that the overall span of the arch is not symmetrical. The span of the near side truss is shorter and more steeply inclined. 

If these forms are deliberate the question must be asked; what on earth is going on? To answer this question we cannot avoid learning some basic structural principles. The first of these is the principle of strut buckling.

If we were to apply load to a short squat object, for example a rock, we would effectively have to apply sufficient force to crush the rock before it would fail. The same cannot be said of, for example, a thin branch taken from the edge of a tree canopy. If we apply a compressive force to the branch it will bow in the middle and fail in bending long before the compressive strength of wood has been reached. This effect is known as strut buckling. There is clearly a relationship between the length of a member and its predilection to buckling in compression. As it happens buckling capacity is proportional to the square of the length.

Conversely the load which causes a member to buckle in compression can by applied to the same member in tension without effect. We all know this in principle, but perhaps without realising it. Consider the absurdity of trying to apply a compressive load to a piece of string and compare that with an application of tensile load to the same piece of string.

We may now apply this logic to the structure in the photograph. Those members with a thick cross section can be inferred to be in compression to avoid buckling while those that are slender must be in tension.

This inference is interesting in so far as it goes, but if we are to follow the logic to its conclusion then that would mean compressive and tensile forces must reverse either side of the central pin. How or perhaps why should that be so? Surely, everyone knows that an arch is a compression structure so what’s going on?

I think the answer lies in the asymmetry of the arch. Supposing there happens to be a day of heavy snowfall. It seems reasonable to suppose that the snow would slide off the steep side of the arch while settling on the flat side. Perhaps there is also a light wind that exacerbates the effect by causing the snow to drift further onto the flatter far side. There is now an uneven load on the arch due to both the self-weight of the asymmetric arch and the snow that has gathered on top.

This would of course cause the far side of the arch to dip while the near side would respond by rising. At some intermediate point the arch would be neither up nor down. We call this the point of contraflexure and amazingly its position will almost certainly correspond with the position of the pin at the crown of the arch; almost like someone intended it to be so.

Once we have grasped this idea the next mental step is not too far away. If the near side has risen the outside chords of the truss will be stretched while the inside chord will be squashed. On the far side the relationship is reversed. We now have an explanation for exactly why the members are fat and slender on each side of the arch. 

We can also make the quite reasonable assumption that the compression chords get thicker towards the middle, because that is the point at which buckling wants to occur. 

The only question left is why the arch is asymmetrical in the first place. It could be because of the route the railway takes into Waterloo station and the implications this has on the shape of the structure……or it could be an architectural whim. It’s a bit of a mystery, perhaps I’ll let you decide.