Limits on the spans of beams
How long can we make a beam? There must be a limit, because we see many other types of structure, often more complicated than beams.
One thing that engineers and scientists often do is to start with a very simple case and see what happens. Sometimes we can get away with a very crude calculation, often referred to as "back of an envelope".
Suppose we want to find out whether something is possible. If a simple calculation shows that something is impossible, or easily possible, by a factor of hundred, then we don't need to do a very accurate calculation to make our decision. On the other hand, we might need to check our calculation, and of course the assumptions on which it was based. It's a good idea to calculate by two different methods as a safeguard. If our simple calculation shows that something is marginally possible, we must do a more accurate calculation to make a decision.
Even if we find that our project is "impossible", we can look for another method of achieving our goal. And if it is very easily possible, we can also look for another method that is nearer the limit, in order to save materials and money.
For a beam calculation, we will start with a solid cuboid, with a span S, a depth (top to bottom) D, and a width W. If the density is R, and the acceleration by gravity is g, then the weight is DSWGR. The bending moment B of this is found by using the centre of gravity of one half of the beam, and we find that B = (W / 2)(S / 2) = 0.25SW. We will need that result later.
The tensile/compressive stress in a beam varies linearly from top to bottom, so if we use y to denote the vertical distance from any point to the neutral axis of the beam, we can say that the force on a thin horizontal slice (depth dy) of the beam is given by F = kyWdy, where k is a constant. The element of moment about the neutral axis for this element of area is dM = kWy2dy. (We are being sloppy here about dy.)
Integrating this shows that the total moment M is kWD3/12.
We can equate this with the value of B already found, giving 0.25SW = kWD3/12. If we use T to represent the tensile stress at the top of the beam, we find the following formula - T = 3RGS2/D.
If we allow T to be as big as the ultimate tensile strength of the material, we can find the values of S and D that will break the beam. Just by looking at the formula, we can see that S and D are not proportional, which implies that there is a limit, since having D as big as S is not a useful way of building.
As an example, we will use granite. Being a natural material with variable composition, its properties are different for different samples. We will use a representative density of 2750 kg/m3. The tensile strength varies with the size of the specimen, because of the distribution of cracks. We will use 1 MN/m3.
For a span of 10 metres, we find that the depth of the beam is about 0.8 metres. This looks very reasonable, but we must consider the fact that we have used the ultimate strength of a very variable material. We should assume that some of our samples might be rather weaker than than the average, and we should even then use a safety factor. Furthermore, most rocks are not ductile in the short term, and any failure is likely to be sudden and catastrophic. We might consider reducing our limit of 10 metres to 5 metres, or even 3 metres. No wonder then, that when we look at stone structures based on beams, such as ancient Egyptian and Greek temples, and structures such as Stonehenge in Britain, we don't see very long spans. It wasn't until the arch was discovered that people could create large open spaces in stone.
There is another route, besides the arch. We no longer have to use stone, as we have materials such as iron and steel. Because they can take tension reliably, we can use them in structures that are far more efficient than solid beams. Even so, there is an economic limit. Looking at the Forth railway bridge and the Quebec cantilever bridge, we see that the cantilever arms are very deep, and that making them much longer would be impractical. Indeed, no longer ones have ever been made. They are not beams, but they are not unlike beams that are supported in the middle instead of at the ends.
The diagram below shows the maximum lengths of some types of bridges, in about 2002. The beam, represented here by the box girder, (which uses the material far more efficiently than a solid beam), does not do well. On the other hand, for short spans, the beam, because of its simplicity, is very widely used.
For a comparison of a solid beam with a truss, click here.