Structures- Or Why Things Don't Fall Down

by J E Gordon

  In an attempt to keep the wheels more or less in contact with the ground the springs and shock-absorbers of the vintage sports cars were stiffened up until they were virtually solid. As a result, of course, the ride became almost unbearably rough and jerky. Like the noisy exhaust, this kind of thing was no doubt impressive to the girl passenger, but it did not really do very much to keep the car on the road. The solution adopted by most modern car designers is to scrap the rather flimsy chassis and to take the torsion and bending loads through the pressed-steel ‘saloon’ body shell. This forms, with its roof, a big torsion box not wholly unlike the old biplanes. With so much stiffness at his disposal the designer can concentrate on providing a scientifically designed suspension which is both safe and comfortable.

  As we have said, the strength and stiffness of a structure in torsion vary as the square of the area of its cross-section. This is more or less all right with bulky things like aircraft wings and ships’ hulls and saloon cars; but when we come to shafts in engines and machinery the diameter – and therefore the area of the cross-section – is usually very limited, and so, as a rule, such members need to be made from solid steel. Even then, although they are often very massive, they are not always sufficiently strong. This is one of the reasons why engines and machinery are usually so heavy, As most experienced designers will tell you, any major requirement for torsional strength and stiffness in a structure is apt to be a curse and a blight. It puts up the weight and the expense and altogether provides a quite disproportionate amount of trouble and anxiety to the engineer.

  Nature does not seem to mind taking a lot of time and trouble, and she has no sense at all of the value of money; but she is intensely sensitive to ‘metabolic cost’ – that is to say, to the price of a structure in terms of food and energy – and she is also generally pretty weight-conscious. It is not surprising, therefore, that she seems to avoid torsion like poison. In fact she nearly always manages to dodge out of any serious requirement for the provision of torsional strength or stiffness. As long as they are not subjected to ‘unnatural’ loads, most animals can afford to be weak in torsion. None of us likes having our arm twisted, and in normal life the torsional loads on our legs are small. However, when we attach long levers called skis to our feet and then proceed to ski rather badly, it is only too easy to apply large twisting forces to our legs. Because this is the commonest cause of broken legs in ski-ing, it has led to the development of the modern safety binding, which releases automatically in torsion.

  Not only our legs, but virtually all bones, are surprisingly weak in torsion. Should you wish to kill a chicken – or any other bird -much the easiest way is to wring its neck. This is well known; what is less well known is how very weak are the vertebrae in torsion, as the beginner is apt to find out to his disgust and embarrassment when the head comes off in his hand. But then neck-wringing, like ski-ing, is an entirely artificial hazard and quite out of the ordinary course of nature. Unlike engineers, Nature has little interest in rotary motion and (like the Africans) she has never bothered to invent the wheel.

  * * *

  * Note that there is a relationship between G and E. For isotropic materials like metals

  where q = Poisson’s ratio.

  * Warp threads or yarns are those which run parallel to the length of a roll of cloth; weft threads are those which run across the cloth, at right angles to its length.

  * An understanding of this principle is very important when making things like balloons and pneumatic dinghies from rubberized fabric. If shear distortions are incurred the rubber coating is strained in such a way that the fabric will leak.

  * The cuticles of many worms and other soft animals are strengthened by systems of helically disposed collagen fibres (Chapter 8). The worm has much the same problems as the dressmaker, though it is often more successful in solving them. It is difficult to put a crease into a worm.

  * Note that, for an initially flat membrane to conform easily to a surface with pronounced two-dimensional curvature, it is necessary to have both a low Young’s modulus and a low shear modulus. This is essentially the problem of map-projection which was encountered by Mercator about 1560.

  * Nor had many of the academic engineers. Even as late as 1936, the basic Lanchester-Prandtl (or vortex) theory of fluid dynamics was neither taught nor permitted to be used in the Department of Naval Architecture in the University of Glasgow. To those of a younger generation who may not be disposed to believe this story, I would point out that (a) I was myself a student in the department at the time, and (b) much the same sort of thing happens with ‘modern’ theories of fracture mechanics (Chapter 5) in present-day engineering departments.

  * This is why a dead leaf or a sheet of cardboard falls in the way it does.

  Chapter 13 The various ways of failing in compression

  -or sandwiches; skulls and Dr Euler

  By reason of the frailty of our nature we cannot always stand upright.

  Collect for the 4th Sunday after Epiphany

  As one would expect, the ways in which structures fail under compressive loads are rather different in their nature from the ways in which they break in tension. When we stress a solid in tension we are, of course, pulling its atoms and molecules further apart. As we do so, the interatomic bonds which hold the material together are stretched, but they can be safely stretched only to a limited extent. Beyond about 20 per cent tensile strain, all chemical bonds become weaker and will eventually come unstuck. Although the actual details of the tensile fracture process are complicated, it is broadly true to say that, when a sufficient number of interatomic bonds have been stretched beyond their breaking point, the material itself will break. The same sort of thing is also true when a material is broken by shearing. Strictly speaking, however, there is normally no analogous case of interatomic bond failure which is simply and directly due to compression. When a solid is compressed, its atoms and molecules are being pressed closer to each other, and under any ordinary conditions the repulsion between the atoms goes on increasing indefinitely as the compressive stress is raised. It is only when subjected to the enormous gravitational forces which exist in those stars which astronomers call ‘dwarfs’ that the compressive resistance between atoms collapses – with nightmarish consequences.*

  Nevertheless, lots of very ordinary earthly structures do break by what is commonly described as ‘compression’. What is really happening in failures of this sort is that the material or the structure finds some way of evading an unduly high compressive stress, usually by moving ‘out from under’ the load: that is to say, by running away in a sideways direction, using one of the escape routes which are practically always available. Looked at from the energy point of view, the structure ‘wants’ to get rid of an excess of compressive strain energy, and it will do so by means of whatever energy-exchange mechanism happens to be practicable in the circumstances.

  Compression structures are thus apt to be rather shifty characters, and the study of compressive failure is more or less the study of ways of getting out of a tight place. As one might suppose, there are a number of different means of doing this. The escape method which the structure will use naturally depends upon its shape and proportions and upon the material from which it is made.

  We have already discussed masonry at some length. Although buildings are essentially compression structures – and masonry must be kept in compression all the time – yet they cannot be said to fail by compression at all. Paradoxically, they can only fail by getting into tension. When this happens walls have a bad habit of developing hinge-points, as a consequence of which they tip up and fall down. Although arches are rather more stable and responsible structures than walls, they are capable at times of producing four hinge-points, after which they diminish both their strain energy and their potential energy by folding themselves up and reducing themselves to a heap of rubble. In any case, as we calculated in Chapter 9, the actual values of the compressive stresses in masonry are usually very
low, far below the official ‘crushing strength’ of the material.

  Crushing stresses – or the failure of short struts and columns in compression

  However, if we take a brick or a block of concrete of fairly compact shape and subject it to a large compressive load – in a testing machine or by any other method – the material will eventually break in a manner which is conventionally called ‘compression failure’. Although brittle solids like stone, brick, concrete and glass are generally crushed in such a way that they are reduced to fragments, or sometimes to powder, the failure is still not, in the strict sense, a compressive one. The actual fracture nearly always takes place by shearing. As we said in the last chapter, both tensile and compressive stresses necessarily give rise to shears at 45°; it is these diagonal shears which generally cause ‘compressive failure’ in short struts.

  Figure 1. Typical ‘compression failures’ for a brittle solid such as cement or glass. Fracture is really due to shearing.

  As we also said earlier on, all practical brittle solids are full of cracks and scratches and defects of one kind or another. Even if this is not the case when they are first made, such materials very soon become abraded from all sorts of virtually unavoidable causes. Naturally these cracks and scratches point in all directions in the material. It follows that a fair number of them will always be found to lie in directions which are diagonal to an applied compressive stress, that is to say, more or less parallel to the resulting shear stress (Figure 1).

  Like tensile cracks, these shear cracks have a ‘critical Griffith length’. In other words, a crack of a given length will propagate at a certain critical shear stress. When such conditions are reached in a brittle solid, such as concrete, the shear cracks will propagate suddenly, violently and perhaps explosively. When a shear crack has run diagonally across the width of a strut or other compression member, the two parts naturally slide past each other, so that the strut is no longer capable of carrying a compression load. The resulting collapse is likely to result in a large release of energy, and this is why brittle materials like glass and stone and concrete throw out splinters, which can be dangerous, when they are crushed or hit with a hammer. In fact the release of strain energy is quite often large enough to ‘pay’ for reducing the material to a powder. This is what happens when we crush lumps of sugar with a hammer or a rolling-pin.

  The failure of a ductile metal – or, come to that, of butter or plasticine – under compressive stress is due to similar causes. What happens is that the metal ‘slips’ or slides (because of the dislocation mechanism) within itself under the shearing stress. Again this happens along planes roughly at 45° to the compressive load: thus a short metal strut bulges outwards into some barrel-like shape (Figure 2). Because of the high work of fracture of ductile metals, such materials are far less likely to throw off splinters during compression failure, and the immediate consequences of the fracture are likely to be less dramatic and a good deal less dangerous. It is this effect, the tendency to bulge under compression, which we make use of when we spread the head of a metal rivet by hammering it or by squeezing it in a hydraulic press.

  Materials like wood and the artificial fibrous composites such as Fibreglass and carbon fibre materials generally fail in compression in a rather different way. In such cases the reinforcing fibres ‘buckle’ or fold in sympathy with each other under the compres-sive load, so that what is called a ‘compression crease’ runs across the material. These compression creases may run either diagonally or at 90° to the direction of the applied compressive stress or sometimes at various angles in between (Figure 3). Unfortunately compression creases often tend to form in fibrous materials at quite low stresses. These materials are therefore sometimes ‘weak in compression ‘, and this point needs to be considered when using them.

  Figure 2. Failure of a ductile material, such as a metal in compression. Failure is again due to shearing, but this time the effect is to cause the metal to bulge.

  Figure 3. Failure of a fibrous material such as wood or Fibreglass in compression. Note that the 90° crease involves a volume contraction and can therefore only take place in a material containing voids, such as wood. ‘Solid’ composites must fail by mode (b), which does not involve a change of volume.

  Breaking stresses of materials in tension and in compression

  The various text-books and reference books generally make a great parade of tabulating the ‘tensile strengths’ of common engineering materials. As a rule, however, these books are a good deal more reticent about compressive strengths. This is partly because the experimental values of the compressive failing stresses of materials vary much more with the shape of the test-piece which has been employed than do the tensile strengths. Sometimes this effect is so great that it becomes almost meaningless to quote a figure. However, although a cautious attitude to compression strengths is in some ways justified, it does have the effect of glossing over some of the facts of structural life. One of these facts is that there is really no consistent relationship at all between the tensile and the compressive strength of a material.* Some rather approximate figures for common materials are given in Table 5. The compressive strength values are those which might be obtained using test-pieces having a ratio of length to thickness of something like three or four to one. For specimens much fatter or thinner than this the breaking stresses might be quite different.

  One of the obvious lessons to be drawn from Table 5 is that, when we come to design a thing like a beam which is stressed in both tension and compression, we may need to watch our step. It may be necessary to design a beam which has a highly asymmetrical section. In Victorian cast-iron beams the tension side is usually very much thicker than the compression side – because cast iron is weaker in tension than in compression (Figure 4). Contrariwise, the wing-spar of a wooden aircraft, such as a sailplane, is always much thicker on the upper or compression side, since wood is weaker in compression than in tension (Figure 5).

  TABLE 5

  Some materials with unequal tensile and compression strengths. (These figures are approximate.)

  Tensile strength Compressive strength

  Material p.s.i. MN/m2 p.s.i. MNim2

  Wood 15,000 100 4,000 27

  Cast iron 6,000 40 50,000 340

  Cast aluminium 6,000 40 40,000 270

  Zinc die castings 5,000 35 40,000 270

  Bakelite, polystyrene and other brittle plastics 2,000 15 8,000 55

  Concrete 600 4 6,000 40

  Figure 4. Cast-iron beams are usually made thicker on the tension face than on the compression face because cast iron is weaker in tension.

  Figure 5. A wooden glider wing-spar is usually made thicker on the compression side than on the tension face because wood is weak in compression.

  The compressive strength of timber and of composite materials

  He said he had been making masts for over fifty years, and, as far as he knew, they had all been sound spars. He said I was the only man he had ever met who deliberately planned to ruin a good mast by cutting the heart right out in a most sensitive spot. He said that any man who could do a thing like that would – (and here I tone down his words a lot) – curse aloud in church, wipe his nose on the table-cloth, take soundings in a cess-pit and eat the arming.

  . . . And that was that. Both George and I thought secretly that the spar was a great deal too whippy for comfort, but in the face of those experts we decided it might be wise to keep our opinions locked up within us. Which was well. For the experts were expert. Later on, when our main shrouds did carry away in a wicked Gulf Stream squall, that mast bent – and bent -and bent, until it looked like the letter S; but it would not break.

  Weston Martyr, The Southseaman

  In real life, as soon as we start to deal with columns of any length the distinction between a column and a beam becomes a good deal confused. A longish column – such as a leg-bone of an animal – is nearly always subject to some degree of bending, and as a result the material on t
he concave side is compressed more than it is elsewhere. Contrariwise, in a beam or a truss, especially one of sophisticated design, the ‘compression boom’ must be considered as a strut. In either case, if the material itself tends to be weak in compression, whether we call the structure a ‘beam’ or a ‘column’, failure will generally begin when the total compressive stress at the worst place reaches a dangerous level. The best examples of columns which are also subject to bending are provided by trees and the masts of traditional sailing ships. Tree-trunks have to sustain the weight of all the bits and pieces of the tree in direct compression, but, in practice, the stresses set up by bending forces caused by wind pressure are likely to be larger and more important. Again, masts are nominally struts, carrying only axial compression, but, because of the stretching of the rigging, and for other causes, they are in fact subject to a good deal of bending, especially if anything in the rigging should happen to break.

  The masts of big ships like H.M.S. Victory had to be built up by joining many pieces of wood together with iron hoops, but for masts of more moderate size the traditional spar-makers preferred to use single pine or spruce trees, left as nearly as possible in their original condition. Not only did these craftsmen strongly resist any suggestion that a mast should be built up or hollowed out in such a way as to produce a more ‘efficient’ tubular section; they also took care to remove as little as possible – beyond the bark -of the outer surface of the tree. In other words they tried, as far as they could, to use the tree in its natural state.