by J E Gordon
† Recent finds at Kouklia in Cyprus indicate the existence of military catapults during the fifth century, though nothing is known about them. In any case Dionysius’s seems to have been the first ‘scientific’ approach to the problem.
‡ These were probably derived from the ‘Spanish windlass’ used in ancient ships. See Chapter 11, p. 224.
* During the 1940 invasion scare two versions of the Roman ballista were made for use by the Home Guard in England. These weapons were intended to project petrol bombs against German tanks. However, since the range of both of these catapults was only about a quarter of that of the classical prototypes, it seems likely that their designers had omitted to read Vitruvius sufficiently carefully.
*Actually, much of the resilience of anchor cables and tow ropes comes from their own weight, which causes them to sag. This is one of the reasons for preferring heavy wire or chain to organic ropes, which are much lighter.
*The ‘true’ or theoretical maximum tensile stress required actually to pull the atoms apart is very high indeed, far higher than the ‘practical’ strength determined by means of ordinary tensile tests. See The New Science of Strong Materials, Chapter 3.
*This is often the same thing as the ‘free surface energy’, which is closely related to the surface tension of both liquids and solids, and which is frequently bandied about in discussions on materials science. See for instance, The New Science of Strong Materials, Chapter 3.
*See The New Science of Strong Materials, Chapters 3 and 9, for an elementary account of the dislocation mechanism; for a fuller description see, for instance, The Mechanical Properties of Matter by Sir Alan Cottrell (John Wiley, 1964 etc.).
† Again, see The New Science of Strong Materials (second edition), Chapter 8.
* It might perhaps be supposed that Lg would correspond to O Y on the diagram, but a little thought will show that this is not the case. The negative amount of energy, ZX, which we have to feed into the system to get the crack going represents the margin of safety or threshold energy. (This is, in fact, the true ‘factor of safety’.)
† Because strain energy = ½ es, which can be written s2/2E, since E = s/e.
* There are medical conditions where the bones of quite young people become very brittle, but this state of affairs is rare. An orthopaedic surgeon tells me that the causes are by no means understood.
Part Two
Tension structures
Chapter 6 Tension structures and pressure vessels
-with some remarks on boilers, bats and Chinese junks
That the ship went faster through the water, and held a better wind, was certain; but just before we arrived at the point, the gale increased in force. ‘If anything starts, we are lost, sir,’ observed the first lieutenant again.
‘I am perfectly aware of it,’ replied the captain, in a calm tone; ‘but, as I said before, and you must now be aware, it is our only chance. The consequences of any carelessness or neglect in the fitting and securing of the rigging will be felt now; and this danger, if we escape it, ought to remind us how much we have to answer for, if we neglect our duty.’
Captain Marryat, Peter Simple
The easiest structures to think about are generally those which have to resist only tensile forces – forces which pull rather than push – and, of these, the simplest of all are those which have to resist only a single pull: in other words unidirectional tension, the basic case of a rope or a rod. Although simple unidirectional tension is sometimes to be seen in plants – especially in their roots – the muscles and tendons of animals provide better biological examples and so do vocal cords and spiders’ webs.
Muscle is a soft tissue which, when it receives an appropriate nerve signal, is able to shorten itself and so produce tensile forces by pulling in an active way.* However, although muscle is a more efficient device than any artificial engine for converting chemical energy into mechanical work, it is not very strong. So, to produce and sustain any considerable mechanical pull, muscles have to be thick and bulky. Partly for this reason muscles are often attached to the bones which they have to manipulate by means of an intervening cord-like tension member made of tendon. Although tendon is unable to contract itself, it is very many times as strong as muscle and therefore needs only a small fraction of the cross-section to take a given pull. Thus the function of tendon is partly that of a rope or wire, although it can also act as a spring, as we saw in the last chapter.
Although some tendons are quite short, many of those in our arms and legs are very long indeed, and they run through the body in almost as complicated a way as the wires of an old-fashioned Victorian bell system. As far as our legs are concerned, muscle is not only bulky but heavy, and the object seems to be to arrange for the centre of gravity of our legs to be as high up in the body as possible. The reason for this is that, in normal walking, the leg operates as a pendulum swinging freely in its own natural period and therefore consuming as little energy as may be. It is because we have to force our legs to oscillate faster than their natural frequency that running is so tiring. The natural period of swing of our legs will be faster the nearer the centre of gravity of the limb is to the thigh-joint. This is why we have thick calves and thighs and, hopefully, small feet and ankles.
However, large feet are not generally so severe a handicap in life’s struggle as large hands, whatever people may say about policemen. Our arms, of course, have evolved from front legs, and they seem to have taken the process of remote control even further. Thus, by means of even longer and thinner tendons than exist in our legs, our hands and fingers are operated by muscles located quite a long way away, high up in our arms. So the hand is enabled to have much more slender proportions than would be the case if it had to contain all its own muscles. The advantage of this arrangement mechanically – and perhaps aesthetically – is obvious.
In artificial structures there are a number of simple examples of unidirectional tension, such as fishing lines and loads hanging from cranes. These differ very little from the problem of the brick and the string which we discussed in Chapter 3. However, many of the more interesting cases, such as the rigging of a ship or the design of aerial cableways, are apt to be beset by uncertainties and complications.
In the rigging of a ship there would, of course, be no difficulty about determining a safe thickness for each rope, provided only that one knew what loads they would have to carry. The difficulty lies in predicting the magnitudes of the many different forces which operate in so complicated an affair as a sailing ship. Although there are several ways in which one might set about this, I strongly suspect that most yacht designers prefer to rely on what might be described as experienced guesswork. However, it is just as well to get one’s guesses right, since the failure of a vital piece of rigging is likely to result in the loss of a mast. If this happens when the ship is caught on a dangerous lee shore, like Marryat’s frigate, then the consequences will be serious.
Nowadays ski-ing is a vast international industry which is dependent upon the reliability of many thousands of cable-cars and ski-lifts. I suppose that most of us have worried, in our more vertiginous moments, about the strength of the wire ropes which support chair-lifts and cable-cars above what seem to be rather frightening chasms. Actually, accidents very seldom occur directly from the failure of one of these cables in tension. This is because in this case the static loads are known pretty accurately, and it is not difficult to do the sums and ensure an ample factor of safety. More serious risks arise from such matters as the excessive swaying of the cables in the wind, so that the cars are likely to strike each other as they pass or perhaps to hit the supporting pylons. Here again, designers seem to rely mainly on precedent and guesswork.
A very different application of unidirectional tension theory is concerned with the strings of musical instruments. The frequency* of the note given out by a stretched string depends, not only on its length, but also upon the tensile stress in it. In stringed instruments the appropriate stres
ses are produced by stretching the strings – which are made of stiff material, such as steel wire or catgut – across a suitable framework, which may be the wooden body of a violin or the cast-iron frame of a piano. Since both the strings and the framework are stiff, very small extensions greatly affect the stress in a string and therefore the frequency of its note. This is why such instruments are so sensitive to ‘tuning’. It is also why one can use the note emitted by a rope when it is ‘twanged’ as an indication of the stress in the material. The Roman army used to require that the officers in charge of military catapults should have a good musical ear, so that they could assess the tensions in the tendon ropes of these weapons when they were set up and tuned for action.
Although the human voice differs in many ways from a stringed instrument, somewhat similar considerations apply to it. The mechanisms of voice production are rather complicated, but our larynx plays an important part in both singing and talking. It may be interesting to note that the various tissues of the larynx are among the few soft tissues in the body which conform approximately to Hooke’s law; most of the other body tissues obey quite different and rather weird laws of their own when they are stretched, as we shall see in Chapter 8.
The larynx contains the ‘vocal cords’, which are strips or folds of tissue whose tensile stress can be varied by muscular tension so as to control the frequency with which they vibrate. Because the Young’s modulus of the vocal folds is rather low, large strains sometimes have to be applied to them in order to cause the necessary stresses; they are, in fact, stretched by a good 50 per cent when we want to achieve the top notes.
Incidentally, the higher frequencies of the voices of women and children are caused, not by higher tensions in their vocal cords, but simply by the fact that the larynx is smaller and the vocal cords therefore shorter. There is a surprising difference in this respect between grown-up men and women, the relevant larynx measurements being about 36 millimetres for men against about 26 for women. However, the larynxes of both boys and girls are of very similar size up to the age of puberty. The ‘breaking’ of boys’ voices is due, not to any change of tension in the cords, but to a rather sudden enlargement of the larynx around the age of fourteen.
Pipes and pressure vessels
Plants and animals might be regarded to a considerable extent as so many systems of tubes and bladders whose function is to contain and to distribute various liquids and gases. Although the pressures in biological systems are not usually very high, they are by no means negligible, and living vessels and membranes do burst from time to time, often with fatal results.
In technology the provision of reliable pressure vessels is a fairly modern achievement and we seldom stop to think how we should get on without using pipes. For the lack of pipes capable of conveying liquids under pressure the Romans incurred enormous expenses in building masonry aqueducts upon tall arches in order to carry water in open channels across miles of undulating country. The earliest approximations to pressure-tight containers were the barrels of guns, and, historically, these were never very satisfactory and quite frequently failed. A list of the people who have been killed by the accidental bursting of guns, from King James II of Scotland downwards, would be long and impressive. Nevertheless, when gas lighting began to be installed in London, soon after 1800, the pipes had to be made by Birmingham gunsmiths, and in fact the earliest gas-pipes were actually made by welding musket-barrels end to end.
Although there are innumerable accounts of the history of the steam-engine, relatively little has been written about the development of the pipes and boilers on which it depended and which, in reality, presented more difficult problems than the actual mechanism. The earliest engines were very heavy and bulky and consumed vast amounts of fuel, chiefly because they worked at very low steam pressures – which was perhaps just as well in view of the nature of contemporary boilers.
The production of engines which were light, compact and altogether more economical was wholly dependent on the use of much higher working pressures. In the steamships of the 1820s, with steam pressures of about 10 p.s.i. – provided by square ‘haystack* boilers – the coal consumption was around 15 lb. weight per horsepower hour. In the 1850s engineers were still talking in terms of 20 p.s.i. and about 9 lb. per horse-power hour. By 1900, pressures had gone up to well over 200 p.s.i., and coal consumption had fallen to 1-5 lb. per horse-power hour – a tenfold reduction in eighty years. It was not the steamship, as such, which drove sailing ships from the high seas, but the high-pressure steamship with triple-expansion engines, ‘Scotch’ boilers, low fuel costs and long range.
The high-pressure boiler was not developed without incident. Throughout most of the nineteenth century boiler explosions were relatively frequent and the consequences were sometimes very terrible. The American river steamers, in particular, were pioneers of high-pressure working. During the middle years of the century the Mississippi steamboats used regularly to indulge in dramatic races over thousands of miles of river. The designers of these vessels were prepared to sacrifice almost everything to speed and lightness, and they took what might charitably be called an optimistic approach to boiler design. As a result, during the years 1859-60 alone, twenty-seven of these ships were lost as a result of boiler explosions.*
Although some of these accidents were due to criminal practices such as the tying down of safety valves, most of them were basically caused by lack of proper calculations. This was a pity because in fact the basic calculations needed to determine the stresses in simple pressure vessels are very easy – so easy indeed that, as far as I can find out, nobody has ever bothered to claim the credit for originating them, and only the most elementary kind of algebra is required.†
Spherical pressure vessels
As soon as we come to consider any kind of pressure vessel or container – which includes such things as balloons and bladders and stomachs and pipes and boilers and arteries – we have to deal with tensile stresses which operate in more than one direction at the same time. This may possibly sound complicated but presents, in fact, no cause for alarm. The skin of any pressure vessel really performs two functions. It has to contain the fluid by being watertight or gastight, and it has also to carry the stresses set up by the internal pressure. Nearly always this skin or shell is subjected to tension stresses acting in both directions in its own plane, that is to say, parallel to its surface. The stress in the third direction, perpendicular to its surface, is usually negligibly low and can be forgotten about.
It is convenient to look first at pressure vessels of spherical shape. The skin or shell of the bladder-like object in Figure 1 is supposed to be reasonably thin, say less than about a tenth of the diameter. The radius of the shell, taken at the middle of the wall thickness, is r. The thickness of the wall or shell is l and the whole thing is subject to an internal fluid pressure of p (all these being in whatever units we happen to patronize).
If we imagine that we slice the thing in two, like a grapefruit, then from Figures 1, 2 and 3 it is pretty clear that the stress in the shell – in all directions parallel to its own surface – will be
This is a useful practical result, and it is in fact a standard engineering formula.
Cylindrical pressure vessels
Spherical containers have their uses, but clearly cylindrical vessels have wider applications, especially to things like pipes and tubes. The surface of a cylinder has no longer the same sort of symmetry as that of a sphere and so we cannot assume that the stress along a cylinder is the same as that around its circumference; and in fact it isn’t. Let us call the stress in the shell along the length of the cylinder sx and that around the circumference of the shell s2.
Figure 1. A spherical vessel with internal pressure p, mean radius r, and wall thickness t.
Figure 2. Imagine the vessel sliced in two across any diameter. The resultant of all the pressure forces acting on the inside of each half of the shell must equal the sum of all the stresses which would have acted
on the cut surface, whose area is 2πrt.
Figure 3. The resultant of all the pressure forces acting on the inside curved surface of a hemisphere will be equal to the same pressure acting on a flat disk of the same diameter, which must be πr2p. Hence
From Figure 4 we can see that the stress along the shell, s1 must be the same as that in the sphere which we have just been considering, that is to say
To get at s2, the circumferential stress in the shell of the cylinder, we now slice, in our imaginations, in the other plane, after the fashion of Figure 5; from which we see that
Thus the circumferential stress in the wall of a cylindrical pressure vessel is twice the longitudinal stress, i.e. s2 = 2s1 (Figure 6). One consequence of this must have been observed by everyone who has ever fried a sausage. When the filling inside the sausage swells and the skin bursts, the slit is almost always longitudinal. In other words, the skin has broken as a consequence of the circumferential, not the longitudinal, stress.
Figure 4. The longitudinal stress, s1 in the shell of a cylindrical pressure vessel is the same as that in the equivalent spherical vessel.
Figure 5. Circumferential stress in a cylinder, s2
Figure 6. Stress in the wall of a cylindrical pressure vessel.
These sums are continually cropping up in engineering and in biology. They are used to calculate the strength of pipes and boilers and balloons and air-supported roofs and rockets and space-ships. As we shall see in Chapter 8, the same simple piece of theory applies to the whole question of development from amoeba-like forms of life towards more elongated and mobile primitive creatures.