by J E Gordon
The vocabulary of shear
The elasticity of shear is very much like the elasticity of tension and compression, and concepts like shear stress, shear strain and shear modulus are pretty closely analogous to their tensile equivalents and certainly no harder to understand.
SHEAR STRESS – N
As we have said, a shear stress is a measure of the tendency for one part of a solid to slide past the neighbouring part, very much as in Figure 1. Hence, if a cross-section of material, having an area A, is acted upon by a shearing force P, then the shear stress in the material at that point will be
Figure 1.
-just like a tensile stress. The units are also the same as those of a tensile stress, that is to say, p.s.i., MN/m2 or what you fancy.
SHEAR STRAIN – g
All solids yield or strain under the action of a shear stress, in the same sort of way as they do under a tensile stress. In the case of shear, however, the strain is an angular one, and it is therefore measured, like any other angle, in degrees or in radians – usually in radians (Figure 2). Radians, of course, have no dimensions, being really a number or a fraction or a ratio. We shall call the shear strain g in this book: like the tensile strain, e, therefore, g is a dimensionless number or fraction and has no units.
Figure 2. Shear strain = angle through which material is distorted as a result of shear stress N
= g, which is an angle - usually in radians.
In hard solids like metal or concrete or bone, the elastic shearing strain is likely to be less than 1° (1/57 radian). Beyond this shearing strain, materials of this kind will generally either break or else flow in a plastic and irrecoverable way, like butter. However, with materials like rubber or textiles or biological soft tissues, recoverable or elastic shear strains may be much higher than this – perhaps 30° to 40°. With liquids and squidgy things like treacle or custard or plasticine, the shear strain is unlimited; but then it is not recoverable.
THE SHEAR MODULUS OR MODULUS OF RIGIDITY – G
Figure 3. The stress-strain diagram in shear is very like that in tension. The slope of the straight part is equivalent to the shear modulus
At small and moderate stresses most solids obey Hooke’s law in shear, much as they do in tension. Thus, if we plot the shear stress, N, against the shear strain, g, we shall get a stress-strain curve which is, at least initially, a straight line (Figure 3). The slope or gradient of the straight part represents the stiffness of the material in shear and is called the ‘shear modulus’, or sometimes the ‘modulus of rigidity’, or ‘G’. Thus
So G is the exact analogue of the Young’s modulus, E, and, like E9 it has the dimensions and units of a stress: that is to say, p.s.i., MN/m2 or whatever.
Shear webs- isotropic andaniso tropic materials
As we said in the last chapter, although there may be large horizontal tension and compression forces in the top and bottom flanges of a beam or a truss, the actual upward thrust which really enables the structure to do its job of sustaining a downward load has to be produced by the web – that is to say, by the part in the middle which joins the top and bottom booms together. In a continuous beam the web will be of solid material, perhaps a metal plate; in a truss the same function will be served by some sort of lattice or trellis.
Since the distinction between a material and a structure is never very clearly defined, it does not matter very much whether the shearing loads in a beam are carried by a continuous plate web or whether they are carried by a lattice which might be made up of rods and wires, strips of wood or whatever. There is, however, an important difference. If the web is made from, say, a metal plate, then it is of no consequence in which direction the plate is put on. That is to say, if we cut the plate for the web out of some larger sheet of metal, it does not matter at what angle we cut it, since the metal has the same properties in every direction within itself. Such materials, which include the metals, brick, concrete, glass and most kinds of stone, are called ‘isotropic’, which is Greek for ‘the same in all directions’. The fact that metals are isotropic (or nearly so) and have the same properties in all directions makes life somewhat easier for engineers and is one of the reasons why they like metal.
However, if we now consider the lattice web, it is clear that it must be constructed so that the rods and tie-bars lie nearly at ±45° to the length of the beam. If this is not done, then the web will have little or no stiffness in shear (Figures 4-5). Under load the lattice will fold up and the beam will probably collapse. Materials of this kind are called ‘anisotropic’, or sometimes ‘aelotropic’ – both of which are Greek for ‘ different in different directions’. In their different ways wood and cloth and nearly all biological materials are anisotropic and they tend to make life complicated, not only for engineers, but for a great many other people as well.
Figure 4. Shear will produce tension and compression stresses in directions at 45° to the plane of shearing.
Figure 5. Thus a system like the one on the right is ‘rigid’ in shear, and systems like the one on the left are floppy.
Cloth is one of the commonest of all artificial materials and it is highly anisotropic. As we have said repeatedly, the distinction between a material and a structure is a vague one, and cloth, though called ‘material’ by dressmakers, is really a structure, made up of separate yams or threads crossing each other at right angles; and its behaviour under load is much the same as that of the trellis web of a beam or a truss.
If you take a square of ordinary cloth in your hands – a handkerchief might do – it is easy to see that the way in which it deforms under a tensile load depends markedly upon the direction in which you pull it. If you pull, fairly precisely, along either the warp or the weft threads,* the cloth will extend very little; in other words, it is stiff in tension. Furthermore, in this case, if one looks carefully, one can see that there is not much lateral contraction as a result of the pull (Figure 6). Thus the Poisson’s ratio (which we discussed in Chapter 8 in connection with arteries) is low.
Figure 6. When cloth is pulled parallel to the warp or the weft threads, the ‘material’ is ‘stiff’ and the lateral contraction is quite small.
However, if you now pull the cloth at 45° to the direction of the threads – as a dressmaker would say, ‘in the bias direction’ – it is much more extensible; that is to say, Young’s modulus in tension is low. This time, though, there is a large lateral contraction, so that, in this direction, the Poisson’s ratio is high; in fact it may have a value of about 1-0 (Figure 7). On the whole, the more loosely the cloth is woven, the greater is likely to be the difference between its behaviour in the bias and in the warp and weft or ‘square’ direction.
Figure 7. If cloth is pulled ‘on the bias’ or at ±45° to the warp and weft, the ‘material’ is extensible, and the Poisson’s ratio – and hence the lateral contraction – is large. This is the basis of the ‘bias cut’ in dressmaking.
Although I suppose that not very many people have ever heard of the word ‘anisotropy’, the fact that cloth behaves in this sort of way must have been familiar to nearly everybody for centuries. Rather surprisingly, however, the technical and social consequences of the anisotropy of woven cloth do not seem to have been properly realized or exploited until quite recent times.
When we stop to think about the matter, it is clear that when we make anything from cloth or canvas, we can minimize the distortions by arranging for the important stresses to run, as far as possible, along the directions of the warp and weft threads. This usually involves cutting the material ‘on the square’. If the circumstances are such that the cloth is pulled at 45°, that is to say ‘on the bias’, then we shall get much larger distortions, which will, however, be symmetrical. But, should we be so inept that the cloth ends up by being pulled in some intermediate direction, which is neither one thing nor the other, then we shall not only get large distortions, but these will be highly asymmetrical. Thus the cloth will pull into some weird and almost cert
ainly unwelcome shape.*
Although sailmaking has been an important industry ever since the beginning of history, these elementary facts about canvas never fully dawned upon European sailmakers. They continued from age to age to construct sails in such a way that the pull came obliquely upon both the warp and weft threads. As a consequence, their sails quickly became baggy and could seldom be made to set properly when the wind was ahead. The situation was worsened by the European predilection for making sails from flax canvas, which distorts particularly easily because of its loose weave.
Rational modern sailmaking began in the United States early in the nineteenth century. American sailmakers used tightly woven cotton canvas, and they arranged their seams in such a way that the direction of the threads corresponded more nearly to the direction of the applied stresses. Although the consequence was that American ships could frequently sail faster and also closer to the wind than British ones, it required something like an earthquake to bring the facts home to English sailmakers. This was provided by the publicity associated with the schooner yacht America, which came over from New York to Cowes in 1851 to compete with the fastest English yachts. She was entered for a race round the Isle of Wight which was to be sailed for a rather ugly piece of silverware presented by Queen Victoria. This jug-like object has since acquired a certain fame as the ‘America’s Cup’. When the Queen was told that the America was the first yacht to have crossed the finishing line, she asked ‘And who is second?’
Figure 8. In modern sailmaking it is usual to arrange the weft threads of the canvas so that they are parallel to the free edges of the sail.
‘There is no second in sight yet, your majesty.’
After this, the English sailmakers mended their ways – so much so that, within a few years, American yachtsmen would be buying their sails from Mr Ratsey of Cowes. The lessons taught by the American sailmakers have stuck, and, although the majority of modern sails are made from Terylene, not cotton, if you look at any modern sail (Figure 8) you can see that it is cut in such a way that the weft threads are, as far as possible, parallel to the free edges of the sail, which is usually the direction of greatest stress.
In many respects the problems of persuading cloth to conform to a desired three-dimensional shape are not very different in sailmaking and in dressmaking. However, tailors and dressmakers seem to have been more intelligent about the matter than sailmakers. As far as was practicable they cut their cloth on the square, so that most of the circumferential or hoop stresses came directly along the line of the yarns. When a close fit was wanted it was achieved by what might be described as a system of Applied Tension: in other words, by lacing. At times the Victorian young lady seems to have had nearly as much rigging as a sailing ship.
With the virtual abandonment of systems of lacing in post-Edwardian times – possibly on account of a shortage of ladies’ maids – women might well have had to face a shapeless future. However, in 1922 a dressmaker called Mile Vionnet set up shop in Paris and proceeded to invent the ‘bias cut’. Mile Vionnet had probably never heard of her distinguished compatriot S. D. Pois-son – still less of his ratio – but she realized intuitively that there are more ways of getting a fit than by pulling on strings or straining at hooks and eyes. The cloth of a dress is subject to vertical tensile stresses both from its own weight and from the movements of the wearer; and if the cloth is disposed at 45° to this vertical stress one can exploit the resulting large lateral contraction so as to get a clinging effect. The result was no doubt cheaper and more comfortable than the Edwardian solutions to the problem and, in selected instances, probably more devastating (Plates 17 and 18).
An analogous problem arises with the design of large rockets. Some rockets are driven by combinations of liquid fuels such as kerosene and liquid oxygen, but these systems involve elaborate plumbing which is liable to go wrong. Thus it may be better to use a ‘solid’ fuel such as that known as ‘plastic propellant This stuff burns vigorously but relatively slowly, producing a great volume of hot gas which escapes through the rocket nozzle with a most impressive noise, driving the thing along as it does so. Both the propellant and the gas which it produces are contained within a strong cylindrical case or pressure vessel, whose walls must not be unduly exposed to flames or to high temperatures. For this reason the rather massive propellant charge is shaped in the form of a thick tube which fits tightly into the rocket casing. When the rocket is fired, combustion takes place at the inner surface of the plastic propellant, so that the tubular charge burns from the inside outwards. In this way the material of the case is protected from the flames up to the last possible moment by the presence of the remaining unburnt fuel.
Plastic propellant looks and feels rather like plasticine, and, like plasticine, it is apt to break in a brittle way, especially when it is cold. When a rocket is firing, the case naturally tends to expand under the gas pressure, rather as an artery expands under blood-pressure; if it does so, then the propellant has to expand with4t. If the interior of the charge is still cold, it is likely to crack when the circumferential strain in the case reaches about 1-0 per cent. If this happens, then the flames will penetrate down the crack and destroy the case. This naturally results in a sensational explosion as another Polaris bites the dust.
Round about 1950, it occurred to some of us that it would be advantageous to make the rocket case, not from a metal tube, but in the form of a cylindrical vessel, wound from a double helix of strong glass fibres, bonded together with a resin adhesive. If the fibre angles are calculated correctly, it is possible so to arrange things that the change of diameter of the tube under pressure is small. It is true that, in such a situation, the tube will elongate more than it otherwise would, like Mile Vionnet’s waists, but, for various reasons, a longitudinal extension is less damaging to the propellant. As I seem to remember, this idea about rockets stemmed from the bias-cut nighties which were around at the time.
The strain requirements for rockets are generally just the opposite of what is needed in blood-vessels. As we saw in Chapter 8, one wants an artery to maintain a constant length while exposed to fluctuations in blood-pressure (but changes in artery diameter are not important). Either condition can be met by making suitably designed tubes from helically disposed fibres. Problems of this kind keep cropping up in biology, and it was most interesting to find that Professor Steve Wainwright of Duke University, who is concerned with worms, has derived, quite independently, just the same mathematics as we had worked out twenty years or so before for use in rocketry.* On inquiry, I find that in this case too the inspiration arose, via Professor Biggs, from the bias cut.
The invention of the bias cut brought fame to Mile Vionnet in the world of haute couture. She lived to a great age and died, not long ago, at ninety-eight, quite unaware of her very significant contributions to space travel, to military technology and to the biomechanics of worms.
Shear stress is only tension and compression acting at ±45° – and vice versa
A very little further thought about plate webs in beams and lattice webs in trusses and about bias-cut nighties makes it obvious that a shear stress is merely tension or compression (or both) acting at 45°, and that, furthermore, there is a shear stress acting at 45° to every tension and compression stress.
In fact solids, especially metals, very frequently break in tension by reason of the shear stress at 45°. It is this which leads to the ‘necking’ of metal rods and plates in tension and to the mechanics of ductility in metals (Figure 9 and Chapter 5).
As we shall see in the next chapter, very much the same thing can also occur in compression. That is to say, many solids break in compression by sliding away from the load in shear.
Creasing-or the Wagner tension field
A thick plate or a solid piece of metal is able to resist compression, and so, when such things are subjected to shearing loads, there will exist, at ±45°, both tension and compressive stresses. Thin panels and membranes and films and fabrics are scar
cely able to resist compression forces in their own plane, and so, when they are sheared, they are apt to crease. This creasing in shear is quite common in thin metal panels, such as occur in aircraft, and it is quite usual to see a creased or quilted effect on the surfaces of wings and fuselages due to this cause (Plate 19). This is called by engineers a ‘Wagner tension field’.
Figure 9. In ductile materials both tension and compression failure tend to occur by shear.
The same effect is even more common in clothes and loose covers and tablecloths and badly cut sails. I suppose dressmakers do not very often talk about Wagner tension fields, but they do sometimes refer to that slightly mysterious quality which is known in the textile trade as ‘drape’. The drape of a fabric depends mainly upon its shear modulus, and although, very probably, few couturiers could quote any figures – in SI or any other units – for the shear modulus, G, of their silks and cottons, on the whole, the lower the shear modulus of a ‘material’, the less its tendency to unwanted creasing. The reason why we cannot dress ourselves in paper or Cellophane without appearing ridiculous is mainly that these substances have too high a shear stiffness, so that they will not drape properly. Contrariwise, knitted and creped fabrifcs have both a low Young’s modulus and a low shear modulus, so that it is easy to get a close and flexible fit – as girls have discovered with knitted sweaters. In the same way the skin of young people has a low initial Young’s modulus and a low shear modulus and therefore conforms easily to the shape of the body.* In later life the skin becomes stiffer in shear, with obvious results. Recently Professor R. M. Kenedi of the University of Strathclyde has made an extensive study of elastic conformity in human skin. So, for the first time, the wrinkles of age are likely to be put on to a numerical or quantitative basis.