Structures- Or Why Things Don't Fall Down

by J E Gordon

  For many years professional engineers, who knew all about beam theory and neutral axes and second moments of area, despised this as so much traditional nonsense. In fact the first thing that a modern engineer does with a tree is to cut it up into small pieces, which he then glues together again – preferably into some kind of hollow section. It is only recently that we have realized that, after all, the tree does know a thing or two. Among other subtleties, the wood in various parts of the trunk grows in such a way that it is ‘pre-stressed ‘.

  Figure 6. (a) Tree bent by the wind with no pre-stress in wood. Stress distribution across the trunk is linear and maximum tension and compression stresses are equal.

  (b) Pre-stressed tree in a calm. The outside of the trunk is in tension all round; the inside is in compression.

  (c) Pre-stressed tree in a strong wind. Compression stress is halved and this tree can bend twice as far as the one in (a).

  Now in a beam such as a glider wing-spar, where the biggest bending load is practically always in one direction, it is possible, though not very efficient, to make the compression boom of the spar thicker than the tension boom to allow for the fact that wood is much weaker in compression than it is in tension. Things like trees and masts, however, may have to resist bending forces coming from many different directions – according to the caprices of the wind – and so this solution is not open to them. Trees, at any rate, have to have a symmetrical cross-section, usually a round one. For an un-prestressed section the distribution of stress under bending loads will be linear, as in Figure 6a. For such an arrangement, when the compressive stress reaches about 4,000 p.s.i. (27 MN/m2) the beam, that is the tree, will start to break.

  This is where the pre-stressing comes in. Somehow or other the tree manages to grow in such a way that the outer wood is normally in tension (to the extent of something over 2,000 p.s.i. or 14 MN/m2), while the middle of the tree, by way of compensation, is in compression. Thus the distribution of stress across the trunk, under normal conditions, is something like Figure 6b. (One of the important consequences of Hookean elasticity is that we can safely and truthfully superpose one stress system upon another.) Thus, when we add Figures 6a to 6b we get Figure 6c.

  By this method the tree roughly halves the maximum compressive stress (4,000 p.s.i.—2,000 p.s.i. = 2,000 p.s.i.) and so doubles its effective bending strength. It is true that the maximum tensile stress has been raised, but the wood has plenty in hand in this respect. What the tree does in the way of protecting itself by pre-stressing is exactly the opposite of what we do when we make a pre-stressed concrete beam. In the latter case the concrete is weak in tension and relatively strong in compression; the danger is that, when the beam is bent, failure may occur in the concrete on the tension face. To avoid this we put the steel reinforcing rods, which are inside the beam, permanently into tension, so that the concrete is permanently in compression. Thus the beam has to be bent considerably before the compressive stress in the concrete near the surface is relieved and replaced by a tension stress. Thus the cracking of the cement is postponed, since the beam has to be bent further before the critical tensile strain is reached.*

  As we have said, both timber and the fibrous composite materials generally fail in compression by the formation of bands or creases of bent and buckled fibres. My colleague Dr Richard Chaplin points out that these compression creases have a good deal in common with cracks which occur in tension. In particular they are often started by stress-concentrations at holes or other defects in the material. In general, fastenings like nails and screws do not much weaken timber, always provided that they are in place and fit tightly. Once they are removed, however, the resulting hole has a much more serious effect; and no doubt the same is true of knots in timber. In a highly stressed wooden structure, such as a glider or a yacht’s mast, it is therefore wise to leave unwanted nails and screws alone and not try to pull them out. If needs be, they can be cut off flush with the surface of the wood.

  Furthermore, as Richard Chaplin says, the formation of compression creases in a fibrous material requires energy. In fact the amount of energy required is rather larger than the work of fracture of the material in tension. It follows that the propagation of compression creases needs a supply of strain energy and that their behaviour is something like that of a Griffith crack. There are, however, some important differences.

  We have said that, in materials of the kind we have been discussing, compression creases can occur both at 45° and also at 90° to the direction of loading. (They can also occur at other angles between 45° and 90°.) The 45° crease is effectively a shear crack, and, if the conditions are right, it will spread right across the material, much like a Griffith crack in shear. However, the 90° crease is shorter – and therefore consumes less energy – for a given depth of penetration below the surface of the material.

  For this reason the 90° crease is, on the whole, more likely to occur. However, although the 90° crease seems to be easier to start off, it is more likely to come to a halt after travelling for a short distance. This is because, as the crease advances, its two sides tend to get pinched together (or ‘come up solid’) and so cease to release much strain energy. Thus complete failure is unlikely to take place, at any rate immediately.

  What may happen in these circumstances is that many little creases will form, one behind the other, all along the compression surface of a beam. This can be seen on the compression face of a wooden bow, and sometimes with oars (Figure 7). Although engineers often advocate Efficient’ H sections or box sections for beams, this can be a mistake. For reasons which are easily demonstrated,* the strain energy release conditions are often less favourable to the propagation of both cracks or compression creases when the beam section is rounded – like a tree – and this is probably the rationale behind the rounded cross-sections of most wooden bows. No doubt something of the kind is also relevant to the rounded cross-sections of the bones of animals.

  Figure 7. Multiple compression creases on the compression face of a round piece of timber such as a tree, a mast, an oar or a bow. These creases may not be able to spread, and so complete fracture does not occur.

  So long as the material is stressed consistently in compression there are many hindrances to the spread of compression creases. This is one of the reasons why wood is generally such a safe material. However, under conditions of reversed loading, it can be very dangerous indeed. This is because the buckled fibres which constitute a compression crease have little or no tensile strength, and so, under tension forces, the crease acts like an ordinary crack. It is especially dangerous because, in tension, there is now no restriction on the release of strain energy since the two sides of the crack are free to spring apart.

  One of the best ways to arrange for a wing to come off a wooden glider in flight is to make a heavy landing with it. If one puts the aircraft down with a really bad bump, the wings will, momentarily, be bent downwards towards the ground. This may cause compression creases in the wood of what is normally the tension part of the main spar. If this happens, the creases are most unlikely to be spotted during routine inspections. The next time the glider is flown the spar may break in tension at this point, after which, of course, the wing will fall off.

  Leonhard Euler and the buckling of thin struts and panels

  What we have said so far applies to struts and other compression members which are fairly short and thick. As we have seen, these usually fail in compression by a diagonal shearing mechanism, or sometimes by the formation of local creases in the fibres. However, a large number of compression structures of one sort or another involve members which are long and thin and which fail in a totally different way. A long rod, or a membrane such as a thin sheet of metal or a page of this book, fails in compression by buckling, as can very easily be seen by doing the simplest experiment. (Take a sheet of paper and try to compress it lengthwise.) This mode of failure – which has important technical and economic consequences – is called ‘Euler* buckling’
since it was originally analysed by Leonhard Euler (1707-83).

  Euler came from a German-Swiss family well-known for its mathematical ability, and he very soon acquired fame as a mathematician: so much so that, while still quite young, he was invited to Russia by the Empress Elizabeth. He spent most of his life at the Court of St Petersburg, taking refuge for a time with Frederick the Great at Potsdam when the political situation in Russia got too exciting. Life at the courts of the Enlightened Despots in the middle of the eighteenth century must have been both interesting and colourful, but little of this is reflected in Euler’s voluminous writings. As far as I can trace, there appears to be no incident of any noticeable human interest recorded of him in any of his biographies.* He simply went on for a very long time doing mathematics and writing it all down in an enormous number of learned papers, the last of which were still being published forty years after his death.

  As a matter of fact, Euler did not really mean to do anything about columns at all. What happened was that, among a great many other mathematical discoveries, he had invented something called the ‘calculus of variations’, and he was looking for a problem to try it out on. A friend suggested that he might use this method to calculate the height of a thin vertical pole which would just buckle under its own weight. It was necessary to make use of the calculus of variations to tackle this rather hypothetical problem because, as we mentioned in Chapter 3, the concepts of stress and strain were not invented until much later.

  Put in modern terms, what Euler came up with was what we now call ‘the Euler formula for the buckling load of a strut’, which is

  (see Figure 9)

  where P = load at which the column or panel will buckle

  E = Young’s modulus of the material

  I = second moment of area (the so-called ‘moment of inertia’) of the cross-section of the strut or panel (Chapter 11)

  L = length of strut.

  Naturally, all these quantities must be in mutually consistent units.

  (It is curious, but convenient, that so many of these important structural formulae should be, algebraically, so very simple.*)

  Euler’s formula applies to all sorts and kinds of long, thin columns and struts – both solid and hollow – and, perhaps even more importantly, to thin panels and plates and membranes such as occur in aircraft and ships and motor cars.

  Thus, if we plot the failing load of a strut or a panel against its length we get a diagram something like Figure 8, which shows two regimes of failure. For a short strut, failure will be by crushing. When the ratio of length to thickness increases to a value between about five and ten, then this line will be crossed by the curve which represents Euler buckling failure. Buckling now becomes the weaker mode, and so long struts will fail in this way. In practice the change-over from crushing failure to Euler buckling is not a sharp one and there will be a transitional region, something like the dotted line in the diagram.

  Figure 8. Variation in the compressive strength of a column with its length.

  The form of the Euler formula which has just been given assumes that the strut or panel is’ pin-jointed’, or free to hinge, at both ends (Figure 9). Usually, anything which tends to prevent a strut or panel from hinging at the ends will increase the buckling load. For the extreme case, where both ends are rigidly restrained, the buckling load, P, is multiplied by as much as 4. Very frequently, however, the achievement of any considerable degree of end restraint involves extra weight and complication and cost and may not be worth doing. Furthermore ‘rigid’ end-connections will transmit any misalignment of the end-attachments to the strut. If this happens, the strut may be prematurely bent, and so, in practice, made weaker. For this reason the ‘rigid’ stepping of masts, by attaching them both to the deck and the keel, is no longer usual (Figure 10).

  Figure 9. Various Euler conditions.

  (a) Both ends pin-jointed.

  (b) Both ends fixed in direction and position.

  (c) One end encastre, the other pin-jointed and free to move sideways.

  It will be noticed that, in the Euler formula which we have just written down, there is no term which represents a breaking stress. The buckling load of a strut or a panel of a given length depends solely upon the ‘/’ (or second moment of area) of the cross-section and upon the Young’s modulus or stiffness of its material. A long strut does not ‘break’ when it buckles. It just bends elastically in such a manner as to get out of the way of the load. If the ‘elastic limit’ of the material has not been exceeded during buckling, then, when the load is removed, the strut will simply spring straight again and recover its original shape, quite undismayed by its experience. This characteristic can often be a good thing, for it is possible to design ‘unbreakable’ structures in this way. Broadly speaking, this is how carpets and doormats work. Predictably, Nature uses the principle very widely, especially for small plants like grasses which inevitably get trodden on. This is why it is possible to walk on a lawn without doing it any harm. It is the ingenious combination of spiky thorns with Dr Euler’s principle which makes a quickset hedge practically indestructible and impenetrable to both men and cattle. On the other hand, mosquitoes and other insects which make use of long slender stabbing weapons have to employ an indecent amount of low structural cunning to prevent these thin struts from buckling when they sting you.

  Figure 10. If a column is clamped at the ends in such a way as to force it out of alignment, its buckling load may be reduced. Since rigging is liable to stretch, it is no longer customary to fix masts at both deck and keel.

  During Euler’s lifetime the actual technological uses for his formula were very few. Practically the only important application would have been in the design of ships’ masts and other spars. However, contemporary shipwrights had already got this problem under control in a pragmatic way. The magnificent eighteenth-century text-books on shipbuilding, such as Steele’s Elements of Mastmaking, Sailmaking and Rigging, contain extensive tables of the dimensions of every kind of spar, based on experience, and it is doubtful if these recommendations could have been much improved upon by calculation.

  Serious interest in buckling phenomena only began about a century after Euler’s time and was largely due to the increasing use of wrought-iron plates in constructional work. These plates were naturally much thinner than the masonry and woodwork to which engineers had been accustomed. The problem was first tackled seriously in the case of the Menai railway bridge, about 1848. The design of this bridge was the joint responsibility of three great men, Robert Stephenson (1803-59), Eaton Hodgkinson (1789-1861), a mathematician and one of the first professors of engineering, and Sir William Fairbairn (1789-1874), a pioneer in the structural use of wrought-iron plates.

  Stephenson’s railway suspension bridges had been a failure because they were too flexible. Furthermore, the Admiralty, not unreasonably, insisted upon a clear 100 feet (30 metres) headroom beneath the bridge for shipping. The only way of combining the necessary stiffness with the headroom which was demanded seemed to be to design a beam bridge far longer than had ever been built before. For various reasons it seemed best to make the beams, each of which had to be 460 feet (140 metres) long, in the form of tubes built up from wrought-iron plates, with the trains running inside the tubes.

  It fairly soon became evident that one of the most serious design problems lay in the buckling of the iron plates which formed the upper or compression side of the beams. Although Euler’s formula is accurate enough for simple panels and struts, the shape of the bridge tubes was necessarily complicated, and no adequate mathematical theory existed at that time. The three designers had thus no option but to experiment with models. As might have been expected, these proved to be confusing and unreliable – so much so that the three men quarrelled among themselves and at one time it looked as if the partnership would break up with no really safe design for the tubes in sight. Eventually, however, a cellular box beam was decided upon (Figure 11). To everybody’s immense relief this
proved satisfactory, and it is there to this day.

  Figure 11. Britannia bridge: tubular box beam.

  Since Stephenson’s time, a very great amount of mathematical research has been carried out on the buckling of thin shells; but the design of such structures is still accompanied by even more than the usual degree of uncertainty. So the development of critical structures of this kind is likely to be expensive, because of the full-scale strength tests which may be needed before the design can be finalized.

  Tubes, ships and bamboos – and something about Brazier buckling

  Since, according to Euler, the buckling load of a strut varies as EI/L2, the compressive strength of a long column is liable to be very low indeed. The only thing we can do about this is to increase EI- if possible in proportion to L2. For most materials E, the Young’s modulus of elasticity, is pretty well constant, so what we have to do in practice is to increase I, the second moment of area of the cross-section. This means that we have got to make the column fatter. That, of course, is exactly what we do in masonry, for instance in the sturdy columns of a Doric temple. The result, however, is excessively heavy, and if we want to make a light structure then we shall have to design some sort of expanded section. This sometimes takes the form of an ‘H’ or star shape, or sometimes a square box. On the whole, however, round tubes are usually better and more efficient.

  Figure 12. ‘Brazier’ or local buckling of a thin-walled tube under axial compression.

  The use of tubes is extremely popular both with engineers and with Nature, and tubular struts are very widely used for all sorts of purposes. However, a tube under compression has a choice of two modes of buckling. It may buckle in the way we have been describing: that is to say, in a long-wave mode, over its whole length, Euler-fashion. Alternatively, it may buckle in a short-wave mode, that is to say, locally, by putting a sort of crease or crumple into the wall of the tube. If the radius of the tube is large and if the wall is thin, then the strut may well be safe against Euler, or longwave, buckling; but it will fail by the local crumpling of the skin. This is easily demonstrated with a thin-walled paper tube. One form of this local buckling or crumpling is called ‘ Brazier buckling’ (Figure 12). It is this effect which sets a limit to the use of simple tubes and thin-walled cylinders in compression.*