Structures- Or Why Things Don't Fall Down Read online

  The New Science of Strong Materials – or Why you don’t fall through the floor (Chapter 2)

  We might start by asking how it is that any inanimate solid, such as steel or stone or timber or plastic, is able to resist a mechanical force at all – or even to sustain its own weight? This is, essentially, the problem of ‘Why we don’t fall through the floor’ and the answer is by no means obvious. It lies at the root of the whole study of structures and is intellectually difficult. In the event, it proved too difficult for Galileo, and the credit for the achievement of any real understanding of the problem is due to that very cantankerous man Robert Hooke (1635–1702).

  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. This is true for any kind of structure, however small and simple or however large and complicated it may be. It is true, not only for floors and cathedrals, but also for bridges and aeroplanes and balloons and furniture and lions and tigers and cabbages and earthworms.

  If this condition is not fulfilled, that is to say if all the forces are not in equilibrium or balance with each other, then either the structure will break or else the whole affair must take off, like a rocket, and end up somewhere in outer space. This latter result is frequently implicit in the examination answers of engineering students.

  Let us consider for a moment the simplest possible sort of structure. Suppose that we hang a weight, such as an ordinary brick, from some support – which might be the branch of a tree -by means of a piece of string (Figure 1). The weight of the brick, like the weight of Newton’s apple, is due to the effect of the earth’s gravitational field upon its mass and it acts continually downwards. If the brick is not to fall, then it must be sustained in its position in mid-air by a continuing equal and opposite upwards force or pull in the string. If the string is too weak, so that it cannot produce an upward force equal to the weight of the brick, then the string will break and the brick will fall to the ground -again like Newton’s apple.

  Figure 1. The weight of the brick, acting downwards, must be supported by an equal and opposite upward pull or tension in the string.

  However, if our string is a strong one, so that we are able to hang not one, but two, bricks from it, then the string will now have to produce twice as much upward force; that is, enough to support both bricks. And so on, of course, for any other variations of the load. Moreover, the load does not have to be a ‘dead’ weight such as a brick; forces arising from any other cause, such as the pressure of the wind, must be resisted by the same sort of reaction.

  In the case of the brick which hangs from a tree the load is supported by the tension in the string, in other words by a pull. In many structures, such as buildings, the load is carried in compression, that is by pushing. In both cases the general principles are the same. Thus if any structural system is to do its job – that is to say, if the load is supported in a satisfactory way so that nothing very much happens – then it must somehow manage to produce a push or a pull which is exactly equal and opposite to the force which is being applied to it. That is, it has to resist all the pushes and pulls which may happen to arrive upon its doorstep by pushing and pulling back at them by just the right amount.

  This is all very well and it is generally fairly easy to see why a load pushes or pulls on a structure. The difficulty is to see why the structure should push or pull back at the load. As it happens, quite young children have had some inkling of the problem from time to time.

  ‘Do stop pulling the cat’s tail, darling. ‘

  'I'm not pulling, Mummy, Pussy’s pulling.’

  In the case of the cat’s tail the reaction is provided by the living biological activity of the cat’s muscles pulling against the child’s muscles, but of course this kind of active muscular reaction is not very often available, nor is it necessary.

  If the cat’s tail had happened to be attached, not to the cat, but to something inert, like a wall, then the wall would have to be doing the ‘pulling’; whether the resistance to the child’s pull is generated actively by the cat or passively by the wall makes no difference to the child or to the tail (Figures 2 and 3).

  How then can an inert or passive thing like a wall or a string -or, come to that, a bone or a steel girder or a cathedral – produce the large reactive forces which are needed?

  Figure 2. ‘Do stop pulling the cat’s tail, darling.’

  ‘I’m not pulling, Mummy, Pussy’s pulling.’

  Figure 3. It doesn’t make any difference whether Pussy pulls or not.

  Hooke’s law – or the springiness of solids

  The power of any Spring is in the same proportion with the Tension* thereof: That is, if one power stretch or bend it one space, two will bend it two, three will bend it three, and so forward. And this is the Rule or Law of Nature, upon which all manner of Restituent or Springing motion doth proceed.

  Robert Hooke

  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.

  Figures 4 and 5. All materials and structures deflect, to greatly varying extents, when they are loaded. The science of elasticity is about the interactions between forces and deflections. The material of the bough is stretched near its upper surface and compressed or contracted near its lower surface by the weight of the monkey.

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

  Thus, when we hang a brick from the end of a piece of string, the string gets longer, and it is just this stretching which enables the string to pull upwards on the brick and so prevent it from falling. All materials and structures deflect, although to greatly varying extents, when they are loaded (Figures 4 and 5).

  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.

  Although every kind of solid changes its shape to some extent when a weight or other mechanical force is applied to it, the deflections which occur in practice vary enormously. 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, but it really does become shorter. As a matter of fact, masonry is really more flexible than you might think, as one can see by looking at the four principal columns which support the tower of Salisbury Cathedral: they are all quite noticeably bent (Plate 1).

  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.

  As we also know nowadays, the chemical bonds which join the atoms to each other, and so hold the solid together, are very strong and stiff indeed. So when the material as a whole is stretched or compressed this can only be done by stretching of compressing many millions of strong chemical bonds which vigorously resist being deformed, even to a very small extent. Thus these bonds produce the required large forces of reaction (Figure 6).

  Figure 6. Simplified model of distortion of interatomic bonds under mechanical strain.

  (a) Neutral, relaxed or strain-free position.

  (b) Material strained in tension, atoms further apart, material gets longer.

  (c) Material strained in compression, atoms closer together, material gets shorter.

  Although Hooke knew nothing in detail about chemical bonds and not very much about atoms and molecules, he understood perfectly well that something of this kind was happening within the fine structure of the material, and he set out to determine what might be the nature of the macroscopic relationship between forces and deflections in solids.

  He tested a variety of objects made from various materials and having various geometrical forms, such as springs and wires and beams. 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’.

  Furthermore, within the accuracy of Hooke’s measurements -which was not very good – most of these solids recovered their original shape when the load which was causing the deflection was removed. 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. It is in this wider sense that engineers generally use it. Incidentally, the ‘ping’ of the mosquito, for instance, is due to the highly elastic behaviour of the resilin springs which operate its wings.

  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. Such plastic substances shade off into things like butter and porridge and treacle. Furthermore, many of the materials which Hooke considered to be ‘elastic’ turn out to be imperfectly so when tested by more accurate modern methods.

  However, as a broad generalization, Hooke’s observations remain true and still provide the basis of the modern science of elasticity. Nowadays, and with hindsight, the idea that most materials and structures, not only machinery and bridges and buildings but also trees and animals and rocks and mountains and the round world itself, behave very much like springs may seem simple enough – perhaps blindingly obvious – but, from his diary, it is clear that to get thus far cost Hooke great mental effort and many doubts. It is perhaps one of the great intellectual achievements of history.

  After he had tried out his ideas on Sir Christopher Wren in a series of private arguments, Hooke published his experiments in 1679 in a paper called ‘De potentia restitutiva or of a spring’. This paper contained the famous statement ‘ut tensio sic vis’ (‘as the extension, so the force’). This principle has been known for three hundred years as ‘Hooke’s law’

  How elasticity got bogged down

  But to make an enemy of Newton was fatal. For Newton, right or wrong, was implacable.

  Margaret ’Espinasse, Robert Hooke (Heinemann, 1956)

  Although in modern times Hooke’s law has been of the very greatest service to engineers, in the form in which Hooke originally propounded it its practical usefulness was rather limited. Hooke was really talking about the deflections of a complete structure – a spring, a bridge or a tree – when a load is applied to it.

  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. Materials vary very greatly in their intrinsic stiffness. Things like rubber or flesh are easily distorted by small forces which we can apply with our fingers. Other substances such as wood and bone and stone and most metals are very considerably stiffer, and, although no material can be absolutely ‘rigid’, a few solids like sapphire and diamond are very stiff indeed.

  We can make objects of the same size and shape, such as ordinary plumber’s washers, out of steel and also out of rubber. It is clear that the steel washer is very much more rigid (in fact about 30,000 times more rigid) than the rubber one. Again, if we make a thin spiral spring and also a thick and massive girder from the same material – such as steel – then the spring will naturally be very much more flexible than the girder. We need to be able to separate and to quantify these effects, for in engineering, as in biology, we are ringing the changes of these variables all the time and we need some reliable way of sorting the whole thing out.

  After such a promising start it is rather surprising that no scientific way of coping with this difficulty emerged until 120 years after Hooke’s death. In fact, throughout the eighteenth century remarkably little real progress was made in the study of elasticity. The reasons for this lack of progress were no doubt complex, but in general it can be said that, while the scientists of the seventeenth century saw their science as interwoven with the progress of technology – a vision of the purpose of science which was then almost new in history – many of the scientists of the eighteenth century thought of themselves as philosophers working on a plane which was altogether superior to the sordid problems of manufacturing and commerce. This was, of course, a reversion to the Greek view of science. Hooke’s law provided a broad philosophical explanation of some rather commonplace phenomena which was quite adequate for the gentleman-philosopher who was not very interested in the technical details.

  With all this, however, we cannot leave out the personal influence of Newton (1642–1727) himself or the after-effects of the bitter enmity which existed between Newton and Hooke. Intellectually, Hooke was probably nearly as able as Newton, and he was certainly even more touchy and vain; but in other respects they were men of totally different temperaments and interests. Basically, although they both came from fairly modest backgrounds, Newton was a snob whereas Hooke, though a personal friend of Charles II, was not.

  Unlike Newton, Hooke was an earthy sort of person who was occupied with an enormous number of very practical problems about elasticity and springs and clocks and buildings and microscopes and the anatomy of the common flea. Among Hooke’s inventions which are still in use today are the universal joint, used in car transmissions, and the iris diaphragm, which is used in most cameras. Hooke’s carriage lamp, in which, as the candle burnt down, its flame was kept in the centre of the optical system by means of a spring feed, went out of use only in the 1920s. Such lamps are still to be seen outside people’s front doors. Furthermore, Hooke’s private life out-sinned that of his friend Samuel Pepys: not only was every servant girl fair game to him, but he lived for many years ‘perfecte intime omne’* with his attractive niece.

  Newton’s vision of the Universe may have been wider than Hooke’s, but his interest in science was much less practical. In fact, like that of many lesser dons, it could often be described as anti-practical. It is true that Newton became Master of the Mint and did the job well, but it seems that his acceptance of the post had little to do with any desire to a
pply science and a lot to do with the fact that this was a ‘place under Government’ which, in those days, conferred a much higher social position than his fellowship of Trinity, not to mention a higher salary. 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.

  In short, Newton was well constituted to detest Hooke as a man and to loathe everything he stood for, down to and including elasticity. It so happened that Newton had the good fortune to live on for twenty-five years after Hooke died, and he devoted a good deal of this time to denigrating Hooke’s memory and the importance of applied science. Since Newton had, by then, an almost God-like position in the scientific world, and since all this tended to reinforce the social and intellectual tendencies of the age, subjects like structures suffered heavily in popularity, even for many years after Newton’s death.

  Thus the situation throughout the eighteenth century was that, while the manner in which structures worked had been explained in a broad general way by Hooke, his work was not much followed up or exploited, and so the subject remained in such a condition that detailed practical calculations were scarcely possible.

  So long as this state of affairs continued the usefulness of theoretical elasticity in engineering was limited. French eighteenth century engineers were aware of this but regretted it and tried to build structures (which quite often fell down) making use of such theory as was available to them. English engineers, who were also aware of it, were usually indifferent to ‘theory’ and they built the structures of the Industrial Revolution by rule-of-thumb ‘practical’ methods. These structures probably fell down nearly, but not quite, as often.