Most web-sites and books about bridges provide a short list of bridge types, such as beam, arch and cantilever. A bridge may be regarded as a simple case of a more general class of spanning structures, such as ceilings and roofs. These more general structures may be regarded as fully three dimensional, while bridges are extended structures mainly in two dimensions, vertically and spanwise with a smaller extension in width. The width is of course necessary to accommodate a road, railway or canal, but a bridge intended to carry a pipeline might be wider than the load. In the latter case, the extra width is needed to provide stiffness.

Why are there so many different kinds of bridge? Why isn’t there an optimal type that will serve in all situations? This would avoid a lot of design work. One answer can be found by looking at other groups of objects that include a large range in size. A small one-seat aircraft may have the same functional parts as a huge airliner – wings, tailplane, fin, fuselage, power unit, but the appearance is obviously different. The sizes of the parts are not simply scaled up and down uniformly. The same goes for mammals, reptiles, birds and plants. For example, if we compare the legs and neck of an elephant, hippopotamus, or rhinoceros with the legs and neck of a small gazelle, we will see a clear difference.

Galileo explained this phenomenon very clearly. Simply doubling the size of a bird, for example, would quadruple the wing area, and multiply the weight by eight. For a given speed, the ratio of weight to area would double, and so would the ratio of weight to lift. The bird could, in principle, flap and glide quicker to overcome this problem, which would produce problems in landing and take-off, because the eight times weight would be loading legs of only four times the cross-sectional area.

Therefore, when scaling structures up or down, changes in the proportions are obviously required.

This is all very well, but there’s no suggestion here of radically different structures in the manner of beam, arch, cantilever, etc.  All we are doing is changing shapes.

Now let us consider an aircraft such as an A330 airliner, with tens of thousands of rivets holding the parts together. If we scaled it down literally to a tenth of the length, the rivets would be ridiculously small and weak. Futhermore, making the rivets and their holes, and inserting the rivets, might be almost as expensive as in the large aircraft, and would be uneconomic for such a small plane. A real smaller aircraft would have fewer rivets, not smaller ones. This reminds that economics, as well as physics, provides reasons for crude scaling being impractical. For similar reasons, using large components made of composite materials, as in the A787 dreamliner, saving tens of thousands of rivets, would not at present be economical for a very small aircraft.

The picture below shows an example from the natural world, a part of the skull of an elephant. The cellular structure provides stiffness with far less weight than a solid mass of bone would achieve. The skull of a small mammal, such as a fox or a mouse, would be much simpler.

Thus changing scale results in changing proportions, not only in outline, but in small details as well. Similarly, the Tour Eiffel and the Forth rail bridge are made of giant trusses. Many of the members of these trusses are smaller trusses. This subdivision does not go further, and is not used at all in smaller structures. On other other hand, for packaging and building, many cellular materials exist. These, however, are made without having to create and assemble the cells individually. Examples are plastic foam, corrugated cardboard, and honeycomb sandwich slabs.

How much does something deflect when a force is applied? It will deflect until the deformation generates exactly enough force to cancel the applied force, so that the net force at any point in the beam is zero. The sum of the upward forces at the supports will be equal to the sum of the weight of the beam and any loads placed on it. And for any part of the bridge, the sum of the forces on that part will also be zero. The whole thing is then said to be in equilibrium.

Although a rectangular beam seems to be the simplest shape we can make, it is not necessarily the best. The bending stresses in the beam will distributed roughly as in the diagram below, in which we use red for compression and blue for tension. The stresses are greatest near the middle of the span, and at the top and bottom of the beam. Thus the material of the uniform beam is not distributed very efficiently.  (Please note that in all the coloured diagrams, the indication of stress is only qualitative and not quantitative.)

Why are all beams not built with the material distributed according to the stresses? Economics gives us a reason. To make a footbridge across small stream with a width of two metres, the simplest solution is to cut down a tree and make a rectangular beam from it. It simply isn’t worth spending time on something more efficient. As with the elephant and mouse, smaller is often simpler. We will return to the efficiency question later.

Suppose we now have to make a beam with the span twice as big.  We can keep the width the same, as people and vehicles don’t get bigger when they cross longer bridges. We can also keep the depth the same. What will happen to the deflection, or sag? The beam is twice as long, and therefore twice as heavy. We can therefore expect the sag to be twice as much. But there is another consideration. Every part of the bridge is twice as far from a support as the corresponding part in the smaller bridge. This is like using a spanner or wrench twice as long as another one. Therefore the forces will be twice as effective as in the smaller example. Thus the beam sags four times as much as the smaller one. This is very unfortunate. Going to four times the span will make the deflection sixteen times as big.

What can we do? We must make the beam stiffer. We can make the beam deeper. That will make it stiffer, but it will also increase the weight, so we will not gain as much as we might have hoped. All this can be calculated simply if we know the properties of the material.

If we want to keep the same ratio of sag to span, we find that we have to increase the depth rapidly compared with the increase in span. Eventually the beam will become ridiculously deep, and we might as well fill the valley with earth, with a tunnel through it.

To find that better way we need to think about what happens when we bend a beam. We do it by exposing it to external forces. The greater these forces, the greater the bend.

As the beam gradually bends, it absorbs energy from the sources of the forces, but it does not do this uniformly. It is obvious that the top of the beam is compressed more than any other part, and that the bottom is stretched more than any other part. These parts therefore store more energy. This shows clearly in the diagram, repeated below.

The next diagram shows the direction of the stresses as well as the magnitude, represented by the length of the lines, and it indicates shear stress, concentrated near the ends, in green. Only the right hand end of the beam is shown.

The bending will be reduced if we choose a stiffer material, that is, one that needs more energy input for a given amount of bend. If we have already chosen the stiffest economical material, how can we arrange that more energy is needed for a given bend? We can do this by placing more material where the deflections are greatest, and less material where they are least. Thus we want more material at the top and bottom of the beam.

One way to achieve this is the I-beam, which has flat plates along the top and the bottom, connected by a vertical plate that holds them together.  The stresses are shown in the diagram below. The diagram ignores the stresses in the vertical plate.

Other shapes, such as tubes with various cross-sections, have been used. Having decided to use an I-beam or a tube, can we do better? Yes, we can. Instead of solid vertical plates, it turns out that we can use narrow strips of metal, arranged an angle to the horizontal, and thus we have a truss. The gaps between represent a saving in weight. Using these ideas, we can now make a longer span than we could before. Between the I-beam and the truss, there is an intermediate type called a castellated beam, which is described elsewhere in the site.

There is still at least one more idea that we can use. We have seen that an important function of a beam is to resist bending. To improve the efficiency of our use of materials, we could ask where the bending is greatest. The answer is that it is around the middle of the span. We can see that a beam resting freely on supports at each will not bend there, because in order to bend something, several forces are needed, and at the ends, any forces will tend to turn it about the support, rather than bend it. Given this fact, a long beam can be less deep at the ends than in the middle, leading to a shape rather like a cross section through a convex lens. Such a beam is called a lenticular (lens shaped) beam, shown in the diagram below. The distribution of stress is not uniform, but it is much more evenly distributed than before.

This isn’t the only way of achieving the same effect: more commonly, the beam would have a straight top or a straight bottom, which is very convenient for attaching a roadway. As before, the grey area can represent a vertical plate or a truss.

Once we have used up all the weight-saving ideas, there must be a maximum length that can be spanned by a beam.

To summarize what we have looked at so far, we can say that larger structures are likely to be more complicated than smaller ones. We can, for example, make a list of beams, roughly in order of length, as follows –

Simple solid beam, I-beam or tube, castellated beam, trussed beam, lenticular or tapered trussed beam.

These details do not matter very much for the moment. The important point is that there is a maximum practicable length.