by J E Gordon
Good ceramics 5,000-50,000 35-350
Silk 50,000 350
Cotton fibre 50,000 350
Catgut 50,000 350
Flax 100,000 700
Fibreglass plastics 50,000-150,000 350-1,050
Carbon-fibre plastics 50,000-150,000 350-1,050
Nylon thread 150,000 1,050
Metals
STEELS
Steel piano wire (very brittle) 450,000 3,100
High tensile engineering steel 225,000 1,550
Commercial mild steel 60,000 400
WROUGHT IRON
Traditional 15,000-40,000 100-300
CAST IRON
Traditional (very brittle) 10,000-20,000 70-140
Modern 20,000-40,000 140-300
OTHER METALS
Aluminium: cast
wrought alloys 10,000
20,000-80,000 70
140-600
Copper 20,000 140
Brasses 18,000-60,000 120-400
Bronzes 15,000-80,000 100-600
Magnesium alloys 30,000-40,000 200-300
Titanium alloys 100,000-200,000 700-1,400
The tensile strengths of a good many materials are given in Table 2. As with stiffness, it will be seen that the range of strengths in both biological and engineering solids is very wide indeed. For instance, the contrast between the weakness of muscle and the strength of tendon is striking, and this accounts for the very different cross-sections of muscles and their equivalent tendons. Thus the thick and sometimes bulging muscle in our calves transmits its tension to the bone of our heel, so that we can walk and jump, by means of the Achilles or calcaneal tendon, which, although it is pencil-thin, is generally quite adequate for the job. Again, we can see why engineers are unwise to put tensile forces on concrete unless that weak material is sufficiently reinforced with strong steel rods.
The strong metals are rather stronger, on the whole, than the strong non-metals. However, nearly all metals are considerably denser than most biological materials (steel has a specific gravity of 7-8, most zoological tissues about 1-1). Thus, strength for weight, metals are not too impressive when compared with plants and animals.
We might now sum up what has been said in this chapter: load
It expresses how hard (i.e. with how much force) the atoms at a point within a solid are being pulled apart or pushed together by a load.
It expresses how ./or the atoms at a point within a solid are being dragged apart or pushed together.
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.
It expresses how stiff or how floppy a material is.
Strength is not the same thing as stiffness.
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.’
In case you should ever have felt any trace of doubt or confusion on these points, it might be of some comfort to know that, not so long ago, I spent a whole evening in Cambridge trying to explain to two scientists of really shattering eminence and worldwide fame the basic difference between stress and strain and strength and stiffness in connection with a very expensive project about which they were proposing to advise the government. I am still uncertain how far I was successful.
* * *
*Except, apparently, the Oxford Dictionary, The words are used, of course, in casual conversation to describe the mental state of people and as if they meant the same thing. In physical science the meanings of the two words are quite clear and distinct.
*How can a ‘point’ have an ‘area’? Consider the analogy of speed: we express speed as the distance covered in a certain length of time, e.g. miles per hour, although we are concerned usually with the speed at any given -infinitely brief- moment.
* ‘Though science is much respected by their Lordships and your paper is much esteemed, it is too learned... in short it is not understood’ (Admiralty letter to Young).
Chapter 4 Designing for safety
-or can you really trust strength calculations?
That with music loud and long,
I would build that dome in air,
That sunny dome! Those caves of ice!
And all who heard should see them there,
And all should cry, Beware! Beware!
S. T. Coleridge, Kubla Khan
Naturally all this business about stresses and strains is only a means to an end; that is, to enable us to design safer and more effective structures and devices of one kind or another and to understand better how such things work.
Apparently Nature does not have to bother. The lilies of the field toil not, neither do they calculate, but they are probably excellent structures, and indeed Nature is generally a better engineer than man. For one thing she has more patience and, for another, her way of going about the design process is quite different.
In living creatures the broad general arrangement or lay-out of the parts is controlled during growth by the R N A-DNA mechanism – the famous ‘double helix’ of Wilkins, Crick and Watson.* However, in each individual plant or animal, once the general arrangement has been achieved, there is a good deal of latitude about the structural details. Not only the thickness, but also the composition of each load-carrying component is determined, to a considerable extent, by the use which is actually made of it and by the forces which it has to resist during life.† Thus the proportions of a living structure tend to become optimized with regard to its strength. Nature seems to be a pragmatic rather than a mathematical designer; and, after all, bad designs can always be eaten by good ones.
Unfortunately, these design methods are not, as yet, available to human engineers, who are therefore driven to use either guesswork or calculation or, more often, some combination of the two. Both for safety and for economy it is clearly desirable to be able to predict how the various parts of an engineering structure will share the load between them and so to determine how thick or how thin they ought to be. Again, we generally want to know what deflections to expect when a structure is loaded, because it may be just as bad a thing for a structure to be too flexible as for it to be too weak,
French theory versus British pragmatism
Once the basic concepts of strength and stiffness had been stated and understood, a considerable number of mathematicians set themselves to devise techniques for analysing elastic systems operating in two and three dimensions, and they began to use these methods to examine the behaviour of many different shapes of structures under loads. It happened that, during the first half of the nineteenth century, most of these theoretical elasticians were Frenchmen. Although very possibly there is something about elasticity Which is peculiarly suited to the French temperament,* the practical encouragement for this research seems to have come, directly or indirectly, from Napoleon I and from the ficole Poly-technique, which was founded in 1794.
Because much of this work was abstract and mathematical it was not understood or generally accepted by most practising engineers until about 1850. This was especially the case in England and America, where practical men were regarded as greatly superior to ‘mere theoreticians’. And besides, one Englishman had always beaten three Frenchmen. Of the Scottish engineer, Thomas Telford (1757-1834), whose magnificent bridges we can still admire, it is related that:
He had a singular distaste for mathematical studies, and never even made himself acquainted with the elements of geometry; so remarkable indeed was this peculiarity that when we had occasion to recommend to him a young friend as a neophyte in his office, and founded our recommendation on his having distinguished himself in mathematics, he did not hesitate to say that he considered such acquirements as rather disqualifying than fitting him for the situation.
Telford, however, really was
a great man, and, like Nelson, he tempered his confidence with an attractive humility. When the heavy chains for the Menai suspension bridge (Plate 11) had been hoisted successfully in the presence of a large crowd, Telford was discovered, away from the cheering spectators, giving thanks on his knees.†
Not all engineers were as inwardly humble as Telford, and Anglo-Saxon attitudes at this time were often tinged, not only with intellectual idleness, but also with arrogance. Even so, scepticism about the trustworthiness of strength calculations was not unjustified. We must be clear that what Telford and his colleagues were objecting to was not a numerate approach as such -they were at least as anxious as anybody else to know what forces were acting on their materials – but rather the means of arriving at these figures. They felt that theoreticians were too often blinded by the elegance of their methods to the neglect of their assumptions, so that they produced the right answer to the wrong sum. In other words, they feared that the arrogance of mathematicians might be more dangerous than the arrogance of pragmatists, who, after all, were more likely to have been chastened by practical experience.
Shrewd North-Country consulting engineers realized, as all successful engineers must, that when we analyse a situation mathematically, we are really making for ourselves an artificial working model of the thing we want to examine. We hope that this algebraical analogue or model will perform in a way which resembles the real thing sufficiently closely to widen our understanding and to enable us to make useful predictions.
With fashionable subjects like physics or astronomy the correspondence between model and reality is so exact that some people tend to regard Nature as a sort of Divine Mathematician. However attractive this doctrine may be to earthly mathematicians, there are some phenomena where it is wise to use mathematical analogies with great caution. The way of an eagle in the air; the way of a serpent upon a rock; the way of a ship in the midst of the sea and the way of a man with a maid are difficult to predict analytically. One does sometimes wonder how mathematicians ever manage to get married. After King Solomon had built his temple, he would probably have added that the way of a structure with a load has a good deal in common at least with ships and eagles.
The trouble with things like these is that many of the real situations which are apt to arise are so complicated that they cannot be fully represented by one mathematical model. With structures there are often several alternative possible modes of failure. Naturally the structure breaks in whichever of these ways turns out to be the weakest – which is too often the one which nobody had happened to think of, let alone do sums about.
A deep, intuitive appreciation of the inherent cussedness of materials and structures is one of the most valuable accomplishments an engineer can have. No purely intellectual quality is really a substitute for this. Bridges designed upon the best ‘modern’ theories by Polytechniciens like Navier sometimes fell down. As far as I know, none of the hundreds of bridges and other engineering works which Telford built in the course of his long professional life ever gave serious trouble. Thus, during the period when French structural theory was outstanding, a great proportion of the railways and bridges on the Continent were being built by gritty and taciturn English and Scottish engineers who had little respect for the calculus.
Factors of safety and factors of ignorance
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.
Accidents were very apt to be put down as due to ‘defective material’, and a few of them may have been. Metals, of course, do vary in strength between different samples, and there is always some risk of poor material being built into a structure. However, iron and steel usually vary in strength by only a few per cent and very, very rarely by anything like a factor of three or four, let alone seven or eight. Practically always discrepancies as big as this between the theoretical and the actual strengths are due to other causes; at some unknown place in the structure the real stress must be very much higher than the calculated stress, and thus the ‘factor of safety’ is sometimes referred to as the ‘factor of ignorance’.
Nineteenth-century engineers usually made things which were subject to tension stresses, such as boilers and beams and ships, out of wrought iron or mild steel, which had, with some justice, the reputation of being ‘safe’ materials. When a large factor of ignorance had been applied to the strength calculations, such structures often turned out to be quite satisfactory, although in fact accidents continued to occur fairly frequently.
Trouble became increasingly common with ships. The demand for speed and lightness led both the Admiralty and the shipbuilders into difficulties, since ships tended to break in two at sea although the highest calculated stresses seemed to be quite safe and moderate. In 1901, for instance, a brand-new turbine destroyer, H.M.S. Cobra, one of the fastest ships in the world, suddenly broke in two and sank in the North Sea in fairly ordinary weather. Thirty-six lives were lost. Neither the subsequent court martial nor the Admiralty Committee of Inquiry shed much light on the technical causes of the accident.
In 1903, therefore, the Admiralty made and published a number of experiments with a similar destroyer, H.M.S. Wolf at sea in rough weather. These showed that the stresses deduced from strain measurements made on the hull under real conditions were rather less than those calculated by the designers before the ship was built. Since both sets of stresses were far below the known ‘strength’ of the steel from which the ship was constructed – the factor of safety being between five and six – these experiments were only moderately enlightening.
Stress concentrations – or how to start a crack
The first important step towards the understanding of problems of this kind was achieved, not by very expensive practical experiments on full-scale structures, but by theoretical analysis. In 1913 C. E. Inglis, who was later Professor of Engineering at Cambridge and was the very opposite of a ‘remote and ineffectual don’, published a paper in the Transactions of the Institution of Naval Architects whose consequences and applications extend far beyond the strength of ships.
What Inglis said about elasticians was really very much what Lord Salisbury is supposed to have said about politicians, namely that it is a great mistake to use only small-scale maps. For nearly a century elasticians had been content to plot the distribution of stresses in broad, general or Napoleonic terms. Inglis showed that this approach can be relied on only when the material and the structure have smooth surfaces and no sudden changes of shape.
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.
This fact had been known, of course, in a general kind of way, to the people who put the grooves in slabs of chocolate and to those who perforate postage stamps and other kinds of paper. A dressmaker cuts a ‘nick’ in the selvedge of a piece of cloth before she tears it. Serious engineers, however, had not shown much interest in these fracture phenomena, which were not considered to belong to’ proper’ engineering.
Figure 1. Stress trajectories in a b
ar uniformly loaded in tension (a) without and (b) with a crack.
That almost any hole or crack or re-entrant in an otherwise continuous solid will cause a local increase of stress is easily explained. Figure la shows a smooth, uniform bar or plate of material, subject to a uniform tensile stress, s. The lines crossing the material represent what are called ‘stress trajectories’, that is to say, typical paths by which the stress is handed on from one molecule to the next. In this case they are, of course, straight parallel lines, uniformly spaced.
If we now interrupt a number of these stress trajectories by making a cut or a crack or a hole in the material, then the forces which the trajectories represent have to be balanced and reacted in some way. What actually happens is more or less what one would expect; the forces have to go round the gap, and as they do so the stress trajectories are crowded together to a degree which depends chiefly upon the shape of the hole (Figure lb). In the case of a long crack, for instance, the crowding around the tip of the crack is often very severe. Thus in this immediate region there is more force per unit area and so the local stress is high (Plate 2).
Inglis was able to calculate the increase of stress which occurs at the tip of an elliptical hole in a solid which obeys Hooke’s law.* Although his calculations are strictly true only for elliptical holes they apply with sufficient accuracy to openings of other shapes. Thus they apply not only to port-holes and doors and hatchways in ships and aeroplanes and similar structures but also to cracks and scratches and holes in all sorts of other materials and devices – to fillings in teeth, for example.
In terms of simple algebra what Inglis said was that, if we have a piece of material which is subject to a remotely applied stress s, and if we make a notch or a crack or a re-entrant of any kind in it having a length or depth L, and if this crack or re-entrant has a radius at the tip of r, then the stress at and very near to the tip is no longer s but is raised to: