by J E Gordon
One of the earliest of these was the Bollman truss (Figure 14), which was very extensively used in America – perhaps more on account of Bollman’s political talents than his technical ones. He somehow managed to persuade the American government that his was the only ‘safe ‘design of truss, and at one time its use was made compulsory. This may not have been quite so difficult a legislative feat as one might suppose, since it came to be accepted for many years as a practical working principle, by professional engineers, that the technical ignorance of the American Congressman could safely be regarded as bottomless.*
Figure 14. Bollman truss.
Figure 14 shows a simplified Bollman truss with only three panels. In practice there were usually a great many more, and the whole thing tended to get complicated. Besides this the tension members were unnecessarily long. The Fink truss (Figure 15) does the same job as the Bollman truss, but does it rather better, using shorter members.
Figure 15. Fink truss.
We can, with benefit, put a continuous member along the bottom of the Fink truss and turn it into what is more or less a Pratt or Howe truss (Figure 16).
This is pretty well what is generally used in the traditional biplane. It will be seen that the Pratt or the Howe truss will work equally well upside-down – that is to say, either in hogging or in sagging – provided that we take certain common-sense precautions. Furthermore, if we arrange that all the members can take both tension and compression, we can simplify the structure by turning it into a Warren girder (Figure 17). It is this form, or something like it, which is most commonly used for trusses made from ordinary steelwork.
Figure 16. Pratt or Howe truss.
So far, we have considered all these bridges as being simply supported beams, and so, of course, a great many of them were and are. However, a number of beam bridges are cantilever bridges. For some reason cantilever bridges were never very popular in wooden construction, but they are widely used nowadays when built from steel and concrete. A good proportion of the bridges over the motorways are reinforced concrete cantilever bridges. Such bridges generally have a centre-section which is a simply supported beam, resting on the extremities of two cantilevers (Figure 18). This is partly because it is easier to accommodate the deflections with this arrangement. However, there are a few bridges where the two cantilevers just stick out from each side and meet in the middle.
Figure 17. Warren girder.
In the days when very long railway bridges were being built it became fashionable to construct large steel cantilever bridges. The most famous example is the Forth railway bridge, which was completed in 1890. It was the first important bridge to be built from open-hearth steel,* and, in fact, contains 51,000 tons of it. However, road bridges generally do not need so much rigidity as railway bridges (the Forth bridge is said to be the only large bridge in the world over which trains are allowed to pass at full speed), and so most long modern bridges are suspension bridges, which are usually cheaper to build. The Forth road bridge, which has a similar total span to the railway bridge next door to it, and which was finished in 1965, contains only 22,000 tons of steel.
Figure 18. Cantilever bridge with simply supported beam for centre section.
Stress systems in trusses and beams
From all this it is clear that beams and trusses of various sorts and kinds play an immensely important part in sustaining the burdens of the world. What is rather less clear is just how they do it. How do the stresses work in a beam and what is it that really keeps the thing up? As we have said, lattice trusses and solid beams can nearly always be used interchangeably, and so, as one might suppose, the stress system within a truss is not very different in principle from that in a solid beam, although it has the advantage of being rather easier to visualize. Furthermore, cantilevers are perhaps easier to think about than simply supported beams, although as we have seen from Figure 13, the two conditions are quite simply related.
Let us consider therefore a truss in the form of a cantilever which is fixed to a wall (or encastre) at one end and which sticks out and supports a load W, for instance, from the other end. Let us begin, in fact, with the embryonic or nascent cantilever which is the simple triangular arrangement shown in Figure 19. In this affair the weight, W, is directly kept from falling down by the action of the upward component of the tension in the slanting member 1. The compressive force in the horizontal member 2 can only act horizontally, and so it can play no direct part in sustaining the weight. However, they also serve who only push horizontally, and member No. 2 is performing an indirect but very necessary function in keeping the truss extended, that is to say, sticking out in the way it does.
Figure 19.
Figure 20.
Let us now add an extra panel to the truss, as in Figure 20. It is clear that the weight is now sustained directly by the combined upward action of the tension in No. 1 and the compression in No. 3. No. 4 is necessarily in tension but, like No. 2 (which is still in compression), it does not contribute directly to sustaining the weight, although the truss cannot hold up without it.
If we build the truss up into several panels, as in Figure 21, the general situation remains very much the same. The diagonal members 1 and 5 are in tension and 3 and 7 are in compression. It is still these members which directly sustain the load. Taken together, these members are resisting what is called ‘shear’. We shall have a good deal more to say about shear in the next chapter. In the meantime we may observe that the force which is acting in all of these diagonal members is numerically similar. This remains true however long the cantilever may be and however many panels it has.
This is not true, however, of the horizontal forces. The compression in 2 is greater than in 6 and, in the same way, the tension in 4 is greater than the tension in 8. The longer we make the cantilever, the higher the compression will be in member No. 2 and the greater the tension in No. 4. If we make the cantilever very long, then the horizontal or longitudinal tension and compression forces and stresses close to the fixed end may be very high indeed. In other words, such a cantilever will probably break near its root, which after all is only common sense. However, we do have the apparent paradox that the forces are highest in members which do not contribute directly to supporting the load.
Figure 21.
In Figure 21 the downward load, or ‘shearing force’, is directly supported, as we said, by the zig-zag of the diagonal members 1, 3, 5 and 7. However, there is nothing to prevent us from complicating this diagonal trellis by introducing more slanting members, which will all perform the same function. In fact this is often done for various reasons (Figure 22). This is just what Nature quite frequently does. The trunk and rib-cage of most vertebrates can be considered as a sort of simply supported beam. This is obvious in the case of a horse. The bones of the vertebrae and the ribs form the compression members of a rather elaborate Fink truss (Figures 15 and 23). The space between the ribs is criss-crossed by a web or network or trellis of muscular tissue which runs roughly at ±45° to the ribs.
Figure 22. The shear can equally well be taken by a multiple lattice or indeed by a continuous plate.
The next step in an engineering structure is to fill in the space in the middle of a truss, not with some kind of lattice, but with a continuous plate or ‘web’ of some material like steel or plywood. This sort of beam can take many forms but probably the most familiar is the ordinary H or I beam (Figure 24). The function of the plate or web in the middle of the beam is just the same as that of the zig-zag trellis in a truss, and so the loads and stresses in the web run in much the same way.
Figure 23. Many vertebrate animals form a sort of Fink truss with muscles and tendons making a rather complicated diagonal shear bracing between the ribs.
Figure 24, In many engineering beams the shear is taken by a continuous plate web. But the tension and compression stresses due to shearing are still at ±45°.
Thus, in an H beam of this type, the ‘booms5 or ‘spars’ or ‘flanges’ at the
top and bottom are there to resist horizontal or longitudinal tension and compression, while the *web in the middle, is chiefly there to resist the vertical or shearing forces.
Longitudinal bending stresses
As we have said, the longitudinal tension and compression stresses which act along the length of a beam are frequently higher and more dangerous than the shearing stresses, even though these longitudinal stresses do not themselves contribute directly towards supporting the load. In the ordinary beams which we are likely to meet in practice, it is very commonly the longitudinal stresses which are liable to cause failure, and so they are frequently the first stresses to be calculated by an engineer.
Although beams of H section (Figure 24) are common, a beam may be of any cross-sectional shape, and ordinary beam-theory calculations apply to beams of most simple shapes. In fact, the distribution of longitudinal stresses across the thickness of a beam is essentially similar to the distribution of stresses across the thickness of a masonry wall (Chapter 9), with the important difference that, whereas the masonry cannot take tensile stresses, the beam can.
Every beam must deflect under the load which is applied to it and it will therefore be distorted into a curved or bent shape. Material on the concave or compression face of a bent beam will be shortened or strained in compression. Material on the convex or tension face will be lengthened or strained in tension (Figure 25). If the material of the beam obeys Hooke’s law the distribution of stress and strain across any section of the beam will be a straight line, and there will be some point ‘0’ at which the longitudinal stress and strain is neither tensile nor compressive, but is zero. This point lies on what is called the ‘neutral axis’ (N.A.) of the beam.
Figure 25. Distribution of stress through the thickness of a beam.
Since it is important to know the position of the neutral axis in a beam it is fortunate that this is easy to determine. It is quite simple to show, algebraically, that the neutral axis must always pass through the centroid or ‘centre of gravity’ of the cross-section of the beam. For simple symmetrical sections, such as rectangles and circles and tubes and H beams, the neutral axis lies in the middle, half-way between the top and bottom of the beam. For non-symmetrical sections, such as railway lines and ships and aircraft wings, its position will have to be calculated -but this is not very difficult.
It is clear from Figure 25 that the longitudinal stress increases directly with the distance away from the neutral axis. This distance is generally called y when discussing beam theory.* Now if we are seeking structural ‘efficiency’, whether in terms of weight of material, or cost, or metabolic energy, then we do not want to keep any cats that don’t catch mice. In other words we do not want to have to provide material which carries little or no stress. This means that we want, as far as possible, to discard material which lies close to the neutral axis in favour of material as far away from it as possible. Of course, we shall need to leave some material near the neutral axis so as to carry the shearing stresses, but in practice we may not need much material for this purpose and quite a thin web may suffice (Figure 26).
Figure 26. Tension or compression stress due to bending at a point distant y from the neutral axis is s where
and M = bending moment
I = second moment of area of cross-section.
For how to arrive at M and I, see Appendix 2.
This is why, in engineering, steel beams usually have a cross-section of H or ‘channel’ or Z form (Figure 24). These sections have the advantage of being relatively easy to make from mild steel in a rolling-mill. They are often known as ‘rolled steel joists’ (R.S.J.s), and nowadays they can be bought in very large sizes. Z sections have the advantage over channels and Hs that it is easier to rivet the flanges to a plate. This is why Zs are often used for ships’ribs.
When simple sections of this sort are unsuitable it is quite common to use built-up ‘box’ sections. The first and most important use of these was in Stephenson’s Britannia bridge over the Menai Straits (1850; Plate 16 and Chapter 13, Figure 11, p. 291). Since the introduction of waterproof glues and reliable plywood, box beams are often used in wooden construction, particularly in the wing-spars of wooden gliders (Chapter 13, Figure 5, p. 279).
The same sort of arguments apply, of course, when we come to consider sheet materials. Thin sheet metal is weak and flexible in bending, and, to save weight, we want, if possible, to achieve a deeper section. This is often done by rolling corrugations into the metal sheet – with corrugated iron as the unfortunate result.* Corrugated metal sheet has been used in the past for the outside skins of both ships and aircraft, notably with the old Junkers monoplanes. However, the objections are obvious, and it is much more usual nowadays to stiffen and strengthen metal skins in shipbuilding and in aerospace by riveting or welding metal angles, called stringers, to the inside surfaces of the skin.
In all these situations the load commonly comes upon the beam from one direction only, and the shape of the cross-section is optimized with regard to this condition. In some engineering structures and in very many biological ones, however, the load may come from any direction. This is roughly true for lamp-posts, chair legs, bamboos and leg-bones. For such purposes it is better to use a round, hollow tube, and of course this is what is very often done. An intermediate case occurs with bermuda masts. These are generally made from tubes of oval or pear-shaped section. This is not primarily so as to reduce wind-drag by ‘streamlining’, as is often supposed, but rather to cater for the fact that it is much easier to stay a modem mast laterally than it is in the fore and aft plane, and so the mast section has to take account of this by providing more strength and stiffness fore and aft.
* * *
* Of course, a great many small Norman churches have simple wooden roofs, but the design of these roofs is often such that they thrust outwards upon the walls nearly as badly as a stone vault.
† In Pompeii, where the windows are inadequate and the artificial light must have been bad, the walls of nearly all the rooms are painted either dark red or black. One wonders why.
*’I am not a Pillar, but a Buttress, of the Established Church, since I support it from without’ (Lord Melbourne).
* 1 Kings 5 (where there is a strong hint that Solomon had to pay a stiff price).
*The Nine Tailors (Gollancz, 1934). But the roof-trusses of the little church of St Swithin at Wickham in Berkshire are decorated with large Victorian papier-mache elephants.
* For the benefit of any unfortunate airmen who may have had involuntary experience of these devices, I would explain that I would go about the job quite differently nowadays.
*The cost per mile of American railways was one fifth of that of English lines, although American wages were much higher.
* As late as 1912, during the American governmental inquiry into the loss of the liner Titanic, the following exchange was recorded:
Senator X.: You have told us that the ship was fitted with watertight compartments?
Expert witness: Yes.
Senator X.: Then will you explain how it was that the passengers were not able to get inside the watertight compartments when the ship sank?
* The New Science of Strong Materials, Chapter 10.
*See Appendix 2.
* Notice also the corrugations in clam-shells and in many kinds of leaves, such as hornbeam.
Chapter 12 The mysteries of shear and torsion
-or Polaris and the bias-cut nightie
Twist ye, twine ye! even so
Mingle shades of joy and woe, Hope and fear,
and peace and strife,
In the thread of human life.
Sir Walter Scott, Guy Mannering
There is supposed to have been a book review by Dorothy Parker which started off ‘This book tells me more than I care to know about the Principles of Accountancy’. And indeed I dare say that many of us are apt to come to the conclusion that the way in which things behave in shear might, after all, be left to the exper
ts. Tension and compression we feel we can cope with, but when it comes to shear we think we can detect a tendency for the mind to boggle.
It is unfortunate, therefore, that the shear stresses to which we are introduced in the elasticity text-books are assumed to spend their time inhabiting things like crankshafts or the more boring sorts of beams. Though undeniably worthy, this approach somehow lacks human appeal, and it also diverts attention from the fact that shearing stresses and shearing strains are by no means confined to beams and crankshafts but keep intruding into practically everything we do – sometimes with unexpected results. This is why boats leak, tables wobble and clothes bulge in the wrong places. Not only engineers, but also biologists and surgeons and dressmakers and amateur carpenters and the people who make loose covers for chairs would live better and more fruitful lives if they could only look a shear stress between the eyes without flinching.
If tension is about pulling and compression is about pushing, then shear is about sliding. In other words, a shear stress measures the tendency for one part of a solid to slide past the next bit: the sort of thing which happens when you throw a pack of cards on the table or jerk the rug from under someone’s feet. It also nearly always occurs when anything is twisted, such as one’s ankle or the driving shaft of a car or any other piece of machinery. Materials which are being sheared or twisted usually behave in quite straightforward and rational ways, but, rather naturally, when we come to discuss this behaviour it helps a good deal to make use of the appropriate vocabulary. So we might begin with a few definitions.