One disadvantage of the arch is that it needs firm support from the sides. If there are no abutments or banks to build against, the arch will spread and collapse. This is why large cathedrals need 'buttresses' to provide the horizontal support force necessary to counteract the horizontal dissipation of the load from the roof and walls.
The wider one makes an arch – a wide arch is known as an 'elliptical arch' – the more it resembles a beam, and the higher the stresses are at the midpoint.
For hundreds of years, bridge-builders have tried to balance the aesthetic advantages of a wide and shallow elliptical arch against the strength advantages of a steeper arch. Many of their efforts didn't stand the test of time, but we can only see those that did!
So why not keep the strong, semicircular arch shape, but just make the whole thing bigger? It seems reasonable to expect arches of the same shape to be as strong as each other, whether they are one metre or 100 metres wide. It's the shape that's important, after all.
Well, actually, it’s not.
An arch ten times as big in every way as another perfectly strong arch will behave completely differently to the smaller arch. The larger arch is, in fact, ten times weaker - or ten times more prone to stress - than the smaller one. This is because the weight of the larger arch goes up by a factor of ten cubed, since it depends on the number of cubic metres of material in the bridge.
Hence it weighs 1000 times as much as its smaller counterpart. Unfortunately, the strength of the arch depends on its cross-section, which only increases by a factor of ten squared. Hence the stress on each part of the bridge increases by a factor of 1000/100, or by a factor of ten. In other words, the larger bridge is ten times weaker, or ten times more prone to stress, than the smaller one.
As with the beam bridge, the limits of scale will eventually overtake the natural strength of the arch. So bridge-builders need to find a way of keeping the strength of an arch but reducing its weight, or of keeping the arch's weight but increasing its strength.
The first man-made materials that had strength comparable to stone were metals like iron, bronze and brass - all used in ancient times for tools and weapons. But the breakthrough in bridge design had to wait for large-scale iron-smelting processes to be developed, such as those undertaken by the Darby family in Coalbrookdale, Shropshire.
The chief advantage of an arch built from iron (and later of those built from steel) is that it doesn’t have to be solid; instead, it can be built from struts and girders. So the arch can be much less heavy, without losing any of its strength. Another advantage, of course, is that an iron or steel structure won't rot as quickly as a wooden structure.
There it looked like the story of the arch had come to an end, until the ancient Roman art of using 'pozzalana', or 'concrete' as we know it today, was rediscovered and improved upon between 1830 and 1930.
Here was a material that behaved like stone – it was very strong under compression - but could be shaped like clay, and it continued to set harder with the passage of time, even under water.
The modern innovation, however, was the use of iron or steel bars within the concrete – known as reinforcing. This enabled the concrete to take tensile stresses, as well as compressive ones, thus making it suitable for slabs and beams, as well as for piers and towers.
Modern concrete bridges are usually built from pre-stressed slabs. In these slabs the steel rods are stretched before the concrete sets, thereby placing the entire slab under compression. This makes the material far stronger and opens the way to building shallower arches and longer spans than could ever be coped with by stone.
This new wonder material of the construction industry has been largely responsible for the worldwide proliferation of roads and large buildings since the Second World War.
Robert Maillart, from Switzerland, is perhaps the twentieth century's most inspirational bridge engineer, his name ranking alongside that of Isambard Kingdom Brunel in architectural and engineering circles. His use of concrete to enable both roads and railways to leap across the chasms of the Alps typifies modern construction techniques.
At last . . . the perfect arch?
So can the perfect arch finally be built? In a sense, yes. We can now play with the shape, the interplay of tension and compression, the combination of materials and even the colour of a bridge.
But remember that the effects of scale still put a top limit on how long a beam or an arch can be, and any bridge longer than 300 metres for concrete or 500 metres for steel is pushing that limit. For bigger spans – those needed to cross the Straits of Gibraltar, perhaps – a completely different type of bridge would be needed.