Structures- Or Why Things Don't Fall Down

by J E Gordon

  For a semi-circular notch or a round hole (when r — L) the stress will thus have the value of 3s; but for openings like doors and hatchways, which often have sharp corners, r will be small and L large, and so the stress at the comers may be very high -quite high enough to account for ships breaking in two.

  In the Wolf experiments, extensometers, or strain-gauges, were clamped to the ship’s plating in various positions. By this means the extension or elastic movement of the steel plates could be read off. From this the strain – and thus the stress – in the steel was easily calculated. As it happened none of the extensometers was placed close to the corners of hatchways or other openings. If this had been done some very frightening readings would almost certainly have been obtained when the ship was plunging into a head sea in Portland Race.

  When we turn from hatchways to cracks the situation is even worse, because, while cracks are often centimetres or even metres long, the radius of the tip of the crack may be of molecular dimensions – less than a millionth of a centimetre – so that is very large; thus the stress at the tip of the crack may well be a hundred or even a thousand times higher than the stress elsewhere in the material.

  If Inglis’s results had to be taken entirely at their face value it would scarcely be possible to make a safe tension structure at all. In fact the materials which are actually used in tension, metals, wood, rope, Fibreglass, textiles and most biological materials, are ‘ tough’, which means, as we shall see in the next chapter, that they contain more or less elaborate defences against the effects of stress concentrations. However, even in the best and toughest of materials, this protection is only relative, and every tension structure is susceptible to some extent.

  The ‘brittle solids’, however, which are used in technology, like glass and stone and concrete, do not have these defences. In other words they correspond pretty closely to the assumptions which were made in Inglis’s calculations. Moreover we do not need to put in stress-raising notches artificially in order to weaken these materials. Nature has already done this liberally, and real solids are nearly always full of all kinds of small holes and cracks and scratches, even before we begin to make a structure out of them.

  For these reasons it is rash to use any of the brittle solids in situations where they may be subject to appreciable tension stresses. They are, of course, very widely used in masonry and for roads and so on where they are, at least officially, in compression. Where we cannot avoid a certain amount of tension, for instance in glass windows, we have to take care to keep the tensile stresses very small indeed and to use a large factor of safety.

  In talking of stress concentrations we must note that weakening effects are not exclusively caused by holes and cracks and other deficiencies of material. One can also cause stress concentrations by adding material, if this induces a sudden local increase of stiffness. Thus if we put a new patch on an old garment or a thick plate of armour on the thin side of a warship, no good will come of it.*

  The reason for this is that the stress trajectories are diverted just as much by an area which strains too little, such as a stiff patch, as they are by an area which strains too much, such as a hole. Anything which is, so to speak, elastically out of step with the rest of the structure will cause a stress concentration and may therefore be dangerous.

  When we seek to ‘strengthen’ something by adding extra material we have to be careful we do not in fact make it weaker. The inspectors employed by insurance companies and government departments who insist on pressure vessels and other structures being’ strengthened’ by the addition of extra gussets and webs are sometimes responsible, in my experience, for the very accidents which they have tried to prevent.

  Nature is generally rather good at avoiding stress concentrations of this and other kinds. However, one would think that stress concentrations must be of significance in orthopaedic surgery, especially when the surgeon fits a stiff metal prosthesis to a relatively flexible bone.

  NOTE. In Inglis’s formula (p. 67) L is the length of a crack proceeding inwards from the surface, i.e. half the length of an internal crack.

  * * *

  *See, for instance, The Double Helix, by James D. Watson, Weidenfeld & Nicolson, 1968.

  †The process also works in reverse; the bones of astronauts lose calcium and become weaker after a period of weightlessness in space.

  * Almost the only woman to have gained distinction in elasticity, Mademoiselle Sophie Germain (1776-1831), was French. It may be relevant that two of our most highly educated and theoretically-minded engineers during this period, Sir Marc Brunei (1769-1849) and his son, Isambard Kingdom Brunei (1806-59), were of French origin.

  †The British tradition of totally ignoring mathematics has been splendidly continued in the present century by a number of distinguished engineers, notably Sir Henry Royce, who did, after all, create the ‘best car in the world’.

  * Factors of safety as high as eighteen were used in the design of connecting-rods for steam locomotives at least as late as 1910.

  * As a matter of fact the effect of a round hole in a plate under tension had been calculated by Kirsch in Germany in 1898 and that of an elliptical hole by Kolosoff in Russia in 1910, but, as far as I know, little notice was taken of these results in English shipbuilding circles.

  *’Partial strength produces general weakness’ (Sir Robert Seppings (1767-1840), Surveyor of the Navy 1813-32).

  Chapter 5 Strain energy and modern fracture mechanics

  - with a digression on bows, catapults and kangaroos

  An unwise man doth not well consider this: and a fool doth not understand it.

  Psalm 92

  As we said in the last chapter, it was the considerable achievement of the nineteenth-century mathematicians to find ways of calculating the distribution and the magnitude of the stresses in most kinds of structures in a rather broad, generalized or academic way. However, many practical engineers had not long come to terms with calculations of this kind before Inglis planted the seeds of doubt at the back of their minds. 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.

  In fact, using Inglis’s formula (p. 67), it is easy to calculate that, if you were to scratch a girder of the Forth railway bridge, moderately hard, with an ordinary sharp pin, the resulting stress concentration should be sufficient to cause the bridge to break and fall into the sea. Not only do bridges seldom fall down when they are scratched with pins, but all practical structures such as machinery and ships and aeroplanes are infested with stress concentrations caused by holes and cracks and notches which, in real life, are only rarely dangerous. In fact they generally do no harm at all. Every now and then, however, the structure does break; in which case there may be a very serious accident.

  When the implications of Inglis’s sums began to dawn upon engineers some fifty or sixty years ago, they were apt to dismiss the whole problem by invoking the ‘ductility’ of the metals which they were accustomed to use. Most ductile metals have a stress-strain curve which is shaped something like Figure 9, and it was commonly said that the overstressed metal at the tip of a crack simply flowed in a plastic sort of way and so relieved itself of any serious excess of stress. Thus, in effect, the sharp tip of the crack could be considered as ‘rounded off’ so that the stress concentration was reduced and safety was restored.

  Like many official explanations, this one has the merit of being at least partly true, though in reality it is very far from being the whole story. In many cases the stress concentration is by no means fully relieved by the ductility of the metal, and the local stress does, in fact, quite often remain much higher than the commonly accepted ‘breaking stress’ of the material as determined from small specimens in the labor
atory and incorporated in printed tables and reference books.

  For many years, however, embarrassing speculations which were likely to undermine peoples’ faith in the established methods of calculating the strength of structures were not encouraged. When I was a student Inglis’s name was hardly ever mentioned and these doubts and difficulties were not much spoken about in polite engineering society. Pragmatically, this attitude could be partially justified, since, given a judiciously chosen factor of safety, the traditional approach to strength calculations – which virtually ignores stress concentrations – could generally be relied upon to predict the strength of most conventional metal structures. In fact it forms the basis of nearly all of the safety regulations which are imposed by governments and insurance companies today.

  However, even in the best engineering circles, scandals occurred from time to time. In 1928, for instance, the White Star liner Majestic of 56,551 tons, which was then the largest and finest ship in the world, had an additional passenger lift installed. In the process rectangular holes, with sharp corners, were cut through several of the ship’s strength decks. Somewhere between New York and Southampton, when the ship was carrying nearly 3,000 people, a crack started from one of these lift openings, ran to the rail, and proceeded down the side of the ship for many feet before it was stopped, fortuitously, by running into a port-hole. The liner reached Southampton safely and neither the passengers nor the press were told. By an extraordinary coincidence, very much the same thing happened to the second largest ship in the world, the American transatlantic liner Leviathan, at about the same time. Again the ship got safely into port and publicity was avoided. If the cracks had run a little further, so that these ships had actually broken in two at sea, the loss of life might have been severe.

  Really spectacular accidents of this kind to large structures such as ships and bridges and oil-rigs became common only during and after the last war, and latterly they have been growing more, and not less, frequent. What has emerged rather painfully over a number of years – at a vast cost in life and property – is that, although the traditional view of elasticity as hammered out by Hooke and Young and Navier and by scores of nineteenth-century mathematicians is extremely useful and certainly ought not to be neglected or spurned, yet it is not really enough, by itself, to predict the failure of structures – especially large ones -with sufficient certainty.

  The approach to structures through the concept of energy

  I saw the different things you did,

  But always you yourself you hid.

  I felt you push, I heard you call,

  I could not see yourself at alL.

  R. L. Stevenson, A Child*s Garden of Verses

  Until fairly recently elasticity was studied and taught in terms of stresses and strains and strength and stiffness, that is to say, essentially in terms of forces and distances. This is the way in which we have been considering it so far, and indeed I suppose that most of us find it easiest to think about the subject in this manner. However, the more one sees of Nature and technology, the more one comes to look at things in terms of energy. Such a way of thinking can be very revealing, and it is the basis of the modern approaches to the strength of materials and the behaviour of structures; that is, to the rather fashionable science of ‘ fracture mechanics’. This way of looking at things tells us a great deal, not only about why engineering structures break, but also about all sorts of other goings-on – in history and in biology, for instance.

  It is a pity, therefore, that the whole idea of energy has been confused in many people’s minds by the way in which the word is often used colloquially. Like ‘stress’ and ‘strain’, ‘energy’ is used to refer to a condition in human beings: in this case one which might be described as an officious tendency to rush about doing things and pestering other people. This use of the word has really only a tenuous connection with the precise, objective, physical quantity with which we are now concerned.

  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’. So, if you raise a weight of 10 pounds through a height of 5 feet, you will have to do 50 foot-pounds of work, as a result of which 50 foot-pounds of additional energy will be stored in the weight as what is called ‘potential energy’. This potential energy is locked up, for the time being, in the system, but it can be released at will by allowing the weight to descend again. In doing so the released energy could be employed in performing 50 foot-pounds’ worth of useful tasks, such as driving the mechanism of a clock or breaking the ice on a pond.

  Energy can exist in a great variety of different forms – as potential energy, as heat energy, as chemical energy, as electrical energy and so on. 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. In a physical sense that is what ‘happenings’ or ‘events’ are about. Such transformations of energy take place only according to certain closely defined rules, the chief of which is that you can’t get something for nothing. 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. To do this, we need to use the right kind of units; and, rather predictably, the traditional units of energy are in a fine, rich state of muddle. Mechanical engineers have tended to use foot-pounds, physicists are addicted to ergs and electron-volts, chemists and dietitians like to use calories, but our gas bills come in therms and our electricity bills in kilowatt-hours. Naturally, all these are mutually convertible, but nowadays there is a good case for using the S I unit of energy, which is the Joule, that is the work done when one Newton acts through one metre.*

  Although we can measure it in quite precise ways, many people find energy a more difficult idea to grasp than, say, force or distance. Like the wind in Stevenson’s verse we can only apprehend it through its effects. Possibly for this reason the concept of energy came rather late into the scientific world, being introduced in its modern form by Thomas Young in 1807. The conservation of energy was not universally accepted until quite late in the nineteenth century, and it is really only since Einstein and the atom bomb that the enormous importance of energy as a unifying concept and as an underlying reality has been sufficiently appreciated.

  There are, of course, a great many ways, chemical, electrical, thermal and so on, of storing energy until it is wanted. If we are going to use a mechanical means then we could use the method we have just been talking about, that is to say, the potential energy of a raised weight. However, this is rather a crude way of storing energy and, in practice, strain energy, the energy of a spring, is generally more useful and it has much more widespread applications in biology and engineering.

  It is obvious that energy can be stored in a wound-up spring, but, as Hooke pointed out, official springs are only a special case of the behaviour of any solid when it is loaded. Thus every elastic material which is under stress contains strain energy, and it does not make much difference whether the stress is tensile or compres-sive.

  If Hooke’s law is obeyed, the stress in a material starts at zero and builds up to a maximum when the material is fully stretched. The strain energy per unit volume in the material will be the shaded area under the stress-strain diagram (Figure 1), which is

  Figure 1. Strain energy = area under stress-strain curve = .

  Cars, skiers and kangaroos

  We are all of us familiar with strain energy in the springs of our car. In a vehicle with no springs there must be violent interchanges of potential and kinetic energy (energy of motion) every time a wheel passes over a bump. These energy changes are bad for the passengers
and bad for the vehicle. Long ago some genius invented the spring, which is simply an energy reservoir which enables changes of potential energy to be stored temporarily as strain energy so as to smooth the ride and prevent the vehicle and its occupants from being racketed to bits.

  Latterly engineers have spent a great deal of time and effort on the improvement of car suspensions, and no doubt they have been very clever about it. However, cars and lorries run on roads whose main purpose is, after all, to provide a smooth surface. The suspension of the car has only to even out the minor or residual bumps. The problem of designing a suspension for a car which had to be driven really fast across rough country would be a very difficult one. In order to store enough energy to cope with such a situation the steel springs would have to be very large and heavy and would in themselves constitute so much ‘unsprung weight’ that the whole project might prove to be impracticable.

  Consider now the situation of a skier. In spite of the snow covering, most ski-runs are vastly more bumpy than any normal road. Even if a typical run could be covered with some effective non-skid surface, such as sand, so as to enable a car to go on it without slipping, any attempt to drive the car down the run at the speed of a fast skier – say 50 m.p.h. – would be suicidal, because the suspension would be completely inadequate to absorb the shocks. But, of course, this is exactly what the body of a skier has to do. In fact, much of this energy seems to be absorbed by the tendons in our legs, which, taken together probably weigh less than a pound.* Thus, if we are to ski fast without disaster or to perform other athletic feats, our tendons have to be able to store reliably and to give up again very large amounts of energy. This is partly what they are for.