Structures: Or Why Things Don't Fall Down

by J.E. Gordon

A structure has been defined as ‘any assemblage of materials which is intended to sustain loads’, and the study of structures is one of the traditional branches of science.

To make strong structures without the benefit of metals requires an instinct for the distribution and direction of stresses which is by no means always possessed by modern engineers; for the use of metals, which are so conveniently tough and uniform, has taken some of the intuition and also some of the thinking out of engineering.

Yet most of the artefacts of the eighteenth century, even quite humble and trivial ones, seem to many of us to be at least pleasing and sometimes incomparably beautiful. To that extent people – all people – in the eighteenth century lived richer lives than most of us do today. This is reflected in the prices we pay nowadays for period houses and antiques.

Most engineers have had no aesthetic training at all, and the tendency in the schools of engineering is to despise such matters as frivolous.

If you go and look at a cathedral you may well wonder whether you are impressed more deeply by the skill or by the faith of the people who built it. These buildings are not only of very great size and height; some of them seem to transcend the dull and heavy nature of their constructional materials and to soar upwards into art and poetry.

Naturally, the buildings we see and admire are those which have survived: in spite of their ‘mysteries’ and their skill and experience, the medieval masons were by no means always successful.

A fair proportion of their more ambitious efforts fell down soon after they were built, or sometimes during construction.

Mathematics is to the scientist and the engineer a tool, to the professional mathematician a religion, but to the ordinary person a stumbling-block.

In the first place, Hooke realized that, if a material or a structure is to resist a load, it can only do so by pushing back at it with an equal and opposite force. If your feet push down on the floor, the floor must push up on your feet. If a cathedral pushes down on its foundations, the foundations must push up on the cathedral. This is implicit in Newton’s third law of motion, which, it will be remembered, is about action and reaction being equal and opposite.

In other words, a force cannot just get lost. Always and whatever happens every force must be balanced and reacted by another equal and opposite force at every point throughout a structure.

By about 1676 Hooke saw clearly that, not only must solids resist weights or other mechanical loads by pushing back at them, but also that 1. Every kind of solid changes its shape – by stretching or contracting itself- when a mechanical force is applied to it.

2. It is this change of shape which enables the solid to do the pushing back.

All materials and structures deflect, although to greatly varying extents, when they are loaded

It is important to realize that it is perfectly normal for any and every structure to deflect in response to a load. Unless this deflection is too large for the purposes of the structure, it is not in any way a ‘fault’ but rather an essential characteristic without which no structure would be able to work.

The science of elasticity is about the interactions between forces and deflections in materials and structures.

With a thing like a plant or a piece of rubber the deflections are often very large and are easily seen, but when we put ordinary loads on hard substances like metal or concrete or bone the deflections are sometimes very small indeed.

Although such movements are often far too small to see with the naked eye, they always exist and are perfectly real, even though we may need special appliances in order to measure them.

When you climb the tower of a cathedral it becomes shorter, as a result of your added weight, by a very, very tiny amount,...

Hooke made a further important step in his reasoning which, even nowadays, some people find difficult to follow. He realized that, when any structure deflects under load in the way we have been talking about, the material from which it is made is itself also stretched or contracted, internally, throughout all its parts and in due proportion, down to a very fine scale – as we know nowadays, down to a molecular scale. Thus, when we deform a stick or a steel spring – say by bending it – the atoms and molecules of which the material is made have to move further apart, or else squash closer together, when the material as a whole is stretched or compressed.

Having hung a succession of weights upon them and measured the resulting deflections, he showed that the deflection in any given structure was usually proportional to the load. That is to say, a load of 200 pounds would cause twice as much deflection as a load of 100 pounds ‘and so forward’.

In fact he could usually go on loading and unloading structures of this kind indefinitely without causing any permanent change of shape. Such behaviour is called ‘elastic’ and is common. The word is often associated with rubber bands and underclothes, but it is just as applicable to steel and stone and brick and to biological substances like wood and bone and tendon.

However, a certain number of solids and near-solids, like putty and plasticine, do not recover completely but remain distorted when the load is taken off. This kind of behaviour is called ‘plastic’. The word is by no means confined to the materials from which ashtrays are usually made but is also applied to clay and to soft metals.

However, as a broad generalization, Hooke’s observations remain true and still provide the basis of the modern science of elasticity.

If we think for one moment, it is obvious that the deflection of a structure is affected both by its size and geometrical shape and also by the sort of material from which it is made.

A great deal of Newton’s time, however, was spent in a curious world of his own in which he speculated about such perplexing theological problems as the Number of the Beast. I don’t think he had much time or inclination to indulge in the sins of the flesh.

The concept of the elastic conditions at a specified point inside a material is the concept of stress and strain.

In the Two New Sciences, the book he wrote in his old age at Arcetri, he states very clearly that, other things being equal, a rod which is pulled in tension has a strength which is proportional to its cross-sectional area. Thus, if a rod of two square centimetres cross-section breaks at a pull of 1,000 kilograms, then one of four square centimetres cross-section will need a pull of 2,000 kilograms force in order to break it, and so on.

In other words the ‘stress’ in a solid is rather like the ‘pressure’ in a liquid or a gas. It is a measure of how hard the atoms and molecules which make up the material are being pushed together or pulled apart as a result of external forces.

If the brick weighs 5 kilograms and the string has a cross-section of 2 square millimetres, then the brick pulls on the string with a force of 5 kilograms, and the stress in the string will be: or, if we prefer it, 250 kilograms force per square centimetre or kgf/cm2.

MEGANEWTONSPER SQUARE METRE: MN/m2. This is the SI unit. As most people know, the SI (System International) habit is to make the unit of force the Newton. 1-0 Newton = 0-102 kilograms force = 0-225 pounds force (roughly the weight of one apple). 1 Meganewton = one million Newtons, which is almost exactly 100 tons force.

POUNDS (FORCE) PER SQUARE INCH: p.s.i. This is the traditional unit in English-speaking countries, and it is still very widely used by engineers, especially in America.

KILOGRAMS (FORCE) PER SQUARE CENTIMETRE: kgf/cm2 (sometimes kg/cm2). This is the unit in common use in Continental countries, including Communist ones.

FOR CONVERSION

Since the calculation of stresses is not usually a very accurate business, there is no sense in fussing too much about very exact conversion factors.

It is worth repeating that it is important to realize that the stress in a material, like the pressure in a fluid, is a condition which exists at a point and it is not especially associated with any particular cross-sectional area, such as a square inch or a square centimetre or a square metre.

Just as stress tells us how hard – that is, with how much force – the atoms at any point in a solid are being pulled apart, so strain tells us how far they are being pulled apart – that is, by what proportion the bonds between the atoms are stretched.

Thus, if a rod which has an original length L is caused to stretch by an amount l by the action of a force on it, then the strain, or proportionate change of length, in the rod will be e, let us say, such that: To return to our string, if the original length of the string was, say, 2 metres (or 200 cm), and the weight of the brick causes it to stretch by 1 centimetre, then the strain in the string is:

Like stress, strain is not associated with any particular length or cross-section or shape of material. It is also a condition at a point. Again, since we calculate strain by dividing one length by another length – i.e. the extension by the original length – strain is a ratio, which is to say a number, and it has no units, SI, British or anything else.

The strength of a structure is simply the load (in pounds force or Newtons or kilograms force) which will just break the structure. This figure is known as the ‘breaking load’, and it naturally applies only to some individual, specific structure.

The strength of a material is the stress (in p.s.i. or MN/m2 or kgf/cm2) required to break a piece of the material itself.

Stress is not the same thing as strain. Strength. By the strength of a material we usually mean that stress which is needed to break it.

To quote from The New Science of Strong Materials: ‘A biscuit is stiff but weak, steel is stiff and strong, nylon is flexible (low E) and strong, raspberry jelly is flexible (low E) and weak. The two properties together describe a solid about as well as you can reasonably expect two figures to do.’

All the same, after about 1850 even British and American engineers did begin to do calculations about the strength of important structures, such as large bridges. They calculated the highest probable tensile stresses in the structure by the methods of the day, and they saw to it that these stresses were less than the official ‘tensile strength’ of the material. To make quite sure, they made the highest calculated working stress much less – three or four or even seven or eight times less – than the strength of the material as determined by breaking a simple, smooth, parallel-stemmed test-piece.* This was called ‘applying a factor of safety’. Any attempt to save weight and cost by reducing the factor of safety was only too likely to lead to disaster.

Geometrical irregularities, such as holes and cracks and sharp corners, which had previously been ignored, may raise the local stress – often only over a very small area – very dramatically indeed. Thus holes and notches may cause the stress in their immediate vicinity to be much higher than the breaking stress of the material, even when the general level of stress in the surrounding neighbourhood is low and, from general calculations, the structure might appear to be perfectly safe.

When we seek to ‘strengthen’ something by adding extra material we have to be careful we do not in fact make it weaker.

Using the elasticians’ own algebraical methods, he pointed out that the existence of even a tiny unexpected defect or irregularity in an apparently safe structure would be able to cause an increase of local stress which might be greater than the accepted breaking stress of the material and so might be expected to cause the structure to break prematurely.

The scientific kind of energy with which we are dealing is officially defined as ‘capacity for doing work’, and it has the dimensions of ‘force-multiplied-by-distance’.

In our material world, every single happening or event of whatever kind involves a conversion of energy from one into another of its many forms.

Energy can neither be created nor destroyed, and so the total amount of energy which is present before and after any physical transaction will not be changed. This principle is called ‘the conservation of energy’.

Thus energy may be regarded as the universal currency of the sciences, and we can often follow it through its various transformations by means of a sort of accounting procedure which can be highly informative.