Structures- Or Why Things Don't Fall Down

by J E Gordon

  This is best determined by means of what is called the ‘thrust line’, that is to say, a line passing down the wall of a building from the top to the bottom which defines the position at which the vertical thrust can be considered as acting in each successive joint. The thrust line is a French invention and seems to have been first thought of by Coulomb (1736-1806).

  Figure 7. For the simplest symmetrical case, the ‘thrust line’ passes down th e centre of a wall.

  For a very simple symmetrical wall or column or pillar, such as Figure 7, the thrust line will clearly pass down the middle of the wall and so there is really no problem. However, in a building with any pretension to sophistication there is most likely to be at least one oblique force arising from the sideways thrust of the roof members, from archways or vaultings or from various other forms of asymmetrical construction. In such a case the thrust line will no longer pass neatly down the middle of the wall but will be displaced to one side, frequently into a curved path such as Figure 8.*

  If, on plotting the thrust line, we find that it is in danger of reaching the surface of the wall at any point, then we shall clearly have to think again, and think hard, because there is a good chance that a building designed like that will fall down.

  Figure 8. The effect of oblique loading is to deflect the thrust line in this kind of way.

  One of the things we can do, and it may well be one of the most effective, is to add weight to the top of the wall. What then happens can be represented diagrammatically by Figure 9. Contrary to what one might suppose, weight at the top is likely to make a wall more, and not less, stable and will bring an erring thrust line back, more or less, to where it ought to be.

  One way to do this is simply to build the wall to a greater height than appears to be really necessary, and, in addition, anything like heavy balustrades and copings are a good thing. If it is that sort of building and you can afford it, a line of statues will always help (Figure 10). This is the structural justification for the pinnacles and statuary on Gothic churches and cathedrals. They are really up there to say ‘boo’ to the functionalists and to all the dreary people who bleat too much about ‘efficiency’.

  Figure 9. The effect of an additional load at the top of the wall is to reduce the eccentricity of the thrust line.

  Figure 10. This can be done by adding top weight in the form of pinnacle, statues etc.

  It used to be supposed that it was absolutely essential that the thrust line* should be kept within the ‘middle third’ of a wall because, if cracks appeared, the wall might fall down. This is a sound conservative principle which makes for safety and ought to be observed, but, in this permissive age, I am afraid that it seldom is. Anyone who looks at a modern housing estate or a new university cannot help seeing that the walls are full of cracks, and, where there is a crack, there must once have been a tension stress. However, although these cracks do a good deal of damage to the plaster-work and interior decoration,† they seldom constitute any danger to the stability of the main structure.

  The basic condition for the safety of masonry is that the thrust line should always be kept well inside the surface of a wall or column.

  Dams

  Like walls, masonry dams usually fail not from lack of strength but from lack of stability; again they are liable to tip up. The sideways thrust on a dam due to the pressure of the impounded water is generally comparable to the weight of the masonry used in its construction. For this reason there are apt to be very large variations in the position of the active thrust line between the ‘full’ and the ‘empty’ conditions. With dams, unlike ordinary buildings, one cannot take any liberties at all with the ‘middle third’ rule. It is quite essential that there should be no cracks of any kind in the masonry, especially on the upstream side. If there are cracks, water under pressure is likely to get inside the structure of the dam and to have two effects, both bad.

  The first effect is that the flow of water will damage the masonry; to counteract any seepage, arrangements are generally made to drain the interior of large dams. The second effect is more dramatic. It is that the water pressure within the crack will exert a vertical lifting force (about 5 tons per square foot at a depth of 100 feet, i.e. 0-5 MN/m2 at a depth of 30 metres) which, acting upon an already rather critical situation, will overturn the dam.

  It is probable that the destruction of the Mohne and Eder dams by the R.A.F. in 1943 was accomplished in two stages, separated by a short space of time. In the first stage Barnes Wallis’s bombs were dropped against the upstream faces of the dams, where they sank before exploding. When they did explode, the structure of the dams would have been cracked deep down, and after a short delay the actual overturning of the dams themselves was caused by the penetration of high-pressure water into the cracks. Those who have read accounts of the operation will remember that there was an appreciable pause between the explosion of the bombs and the visible failure of the dams. The breaching of these dams, of course, did an immense amount of damage in the Ruhr.

  The failure of a dam in peace-time is an engineer’s nightmare. Even if the dam is made, not from stone, but from unreinforced concrete, it will be unwise to count upon any reliable tensile strength. Thus in all unreinforced dams the thrust line must not move upstream beyond the ‘middle third’ when the dam is empty nor downstream of it when it is full, and it is just as well to leave something in hand. These requirements usually result in the tapering, asymmetrical shape with which most of us are familiar (Figure 11).

  However, dams are expensive in relation to the value of the water which they impound, and engineers are continually looking for cheaper ways of making them. A considerable saving in the weight and cost of cement can generally be achieved by reinforcing the concrete with steel rods, especially if the reinforcement is under tension. However, unless the reinforcing rods are anchored to solid rock beneath the foundations of the dam, there is a real danger that the whole dam, reinforcements and all, will be uprooted and overturned.

  Figure 11. Unreinforced masonry dam.

  One way of dealing with the situation is shown in Figure 12. Here simple vertical steel tie-rods are anchored to the rock beneath and carried up through the concrete to the top of the dam, where they are tensioned by means of a jacking arrangement. It will be seen that these rods are really doing the same job as the angels and pinnacles in a cathedral. Of course, all traditional heavy masonry may be regarded as a structure which is ‘pre-stressed’ by its own weight. No doubt a line of heavy statues along the top of a dam would be effective and might look rather nice, but I am afraid they would turn out to be more expensive than the steel rods.

  Figure 12. Reinforced dam. A thinner, cheaper dam can sometimes be achieved by using pre-tensioned steel rods anchored in the rock beneath. This is equivalent to extra weight on top of the dam and so restricts the movement of the thrust line.

  Arches

  Although the arch is not quite as old as masonry itself, it is certainly very old. There is evidence of fully developed brick arches, both in Egypt and in Mesopotamia, going back to about 3,600 B.C. The stone arch seems to have evolved separately, and possibly independently, from the idea of ‘corbelling’, that is to say, building out the masonry step-wise from each side until the stones met in the middle. The vaulted chambers (Plate 5) deep under the walls of the Mycenaean city of Tiryns – which were old when Homer marvelled at them – are roofed in this way. The postern gate in these immense walls (Plate 6) might be regarded as a development of corbelling. It was probably built before 1,800 B.C.

  However, the corbelled* or the semi-corbelled arch, like the gate at Tiryns, is rather a crude affair. Arches soon developed a construction in which the bricks or stones of the arch-ring are made slightly wedge-shaped and are called ‘voussoirs’. The various parts of a traditional arch are shown in Figure 13.

  Figure 13. Various parts of an arch.

  The voussoir at the top or crown of the arch is called the ‘keystone’ and is sometimes made larg
er than the rest. Although poets, politicians and other non-technical people have attributed special qualities to real and figurative keystones, in fact the keystone is functionally no different from all the other voussoirs, and its distinction, if it has any, is purely decorative.

  The structural function of an arch is to support the downward loads which come upon it by turning them into a lateral thrust which runs round the ring of the arch and pushes the voussoirs against each other. The voussoirs, naturally, push in their turn against the abutments or springings of the arch. The manner in which this process works is pretty clear from common sense (Figure 14).

  Figure 14. An arch collects the vertical loads and turns them into lateral ones. These lateral loads run round the arch ring and are reacted by the abutments.

  The arch ring, with its voussoirs, is very much like a curved wall, and the position of the compressive loads at each joint can be indicated by a thrust line in the same sort of way. In this case the thrust line has to curve round and follow, more or less, the shape of the arch. We shall talk about thrust lines in arches in the next chapter; for the moment let us accept that there is a thrust line. Also, as with the wall, we can assume that the voussoirs cannot slide over each other and that the joints cannot take tension.

  The joints between the voussoirs will behave in much the same way as the joints between the stones in a wall. If the thrust line strays beyond the ‘middle third’ then a crack will appear; also, if the thrust line moves to the edge of a joint, that is to say, to the boundary of the arch ring, then a ‘hinge’ will develop. What makes the arch dramatically different from a mere plebeian wall, however, is that, whereas the wall now falls down, the arch does not. From Figure 15 it can be seen that no fewer than three hinge-points can develop in an arch without anything very dramatic happening. In fact a good many modern arch bridges are deliberately built with three hinged joints so as to allow for thermal expansion.

  Figure 15. An arch can put up with three hinge-points without collapsing; in fact many modern arches are deliberately built in this way.

  If we really want the bridge to fall down then we shall need four hinge-points so that the arch can become in effect a three-linked chain or ‘mechanism’ which is now at liberty to fold itself up and collapse (Figure 16). Incidentally, this is why, if you want to demolish a bridge – for good or bad reasons – it is best to put the explosive charge somewhere near the ‘thirds point’ of the arch. This generally involves digging down through the roadway so as to reach the top of the arch ring. Since this takes time, the demolition of bridges behind a retreating army is often ineffectual.

  All this means that arches are extraordinarily stable and are not unduly sensitive to the movements of their foundations. If there is any appreciable movement in the foundation a wall will probably collapse*; arches do not much mind, and some sort of distortion is quite common. Clare bridge, for instance, in the Backs of Cambridge (Plate 7) is very noticeably bent in the middle because of the movement of the abutments. It has been like that for a long time and is quite safe. In the same way arches stand up remarkably well to earthquakes and to other kinds of abuse, such as modern traffic.

  Figure 16. An arch needs to develop four hinge-points before it can collapse.

  Altogether it is not surprising that many of our ancestors were so addicted to arches, for they will probably go on standing up even if you have got all your sums wrong (or not done any sums at all) and, in addition, placed the foundations of the whole thing iij a bog – as indeed is the case with several of the English cathedrals.

  It is noticeable that, in ruins, it is generally the arches which have survived best. This is partly due to the inherent stability of arches, though it may well have more to do with the fact that the wedge-shaped stones of the voussoirs are less attractive to the local peasantry than the rectangular ones in the walls. The preservation of the round columns of many Greek temples, long after the ashlar of the walls has been stolen, is no doubt due to similar causes.

  It is generally easier to keep the thrust line well inside a wall or an arch if the masonry is thick; but of course solid brick and stone-work are expensive. To get extra thickness at a low price the Romans introduced mass concrete. This was usually made by mixing pozzolana (pulvis puteolanis) – a natural earth which is fairly common in Italy – with lime and adding sand and gravel.

  If walls and arches are made thicker they are generally more stable and may not need to be made so heavy. If less weight of material has to be transported and handled, then the cost of construction is likely to come down. Vitruvius (fl. 20 B.C.), who was a very distinguished architectural writer as well as an artillery officer, tells us that in his day low-density concretes were often made by incorporating pumice powder. The great dome of the HagiaSophiain Constantinople (a.d. 528) is made in the same way.

  Reduction of weight and cost can be taken still further by incorporating empty containers of one sort or another in the concrete. In the ancient world the very extensive and prosperous wine trade was carried on by means of amphorae. These large earthenware containers were strictly non-returnable and they tended to accumulate in embarrassing quantities. The obvious solution was to cast them into concrete, and in fact many late Roman buildings are made in this way. In particular, the beautiful early Byzantine churches at Ravenna are said to be composed largely of disposable empties.*

  Scale·, proportion and safety

  Although some structures are alleged to be sustained by the Power of Faith and others to be held together entirely by paint or rust, unless a designer is totally irresponsible, he likes to have some kind of objective assurance about the strength and stability of whatever he is proposing to make. If one is unable to do the right sort of modern calculations then the obvious thing to do is either to make a model or else to scale up from some previous smaller version of the structure which has proved to be successful.

  This, of course, is pretty well what people used to do down to quite recent times. Perhaps they still do. The difficulty is that models are all very well if one just wants to see what the thing will look like, but they can be dangerously misleading if they are used to predict strength. This is because, as we scale up, the weight of the structure will increase as the cube of the dimensions; that is, if we double the size, the weight will increase eightfold. The cross-sectional areas of the various parts which have to carry this load will, however, increase only as the square of the dimensions, so that, in a structure of twice the size, such parts will have only four times the area. Thus the stress will go up linearly with the dimensions, and, if we double the size, we double the stress and we shall soon be in serious trouble.

  The strength of any structure which is liable to fail because the material breaks cannot be predicted from models or by scaling up from previous experience.

  This principle, which was discovered by Galileo, is known as the ‘square-cube law’ and it is one good reason why vehicles and ships and aircraft and machinery need to be designed by proper modern analytical methods. This is probably why such things were so late in developing, at least in their modern forms. However, we can neglect the square-cube law with most masonry buildings because, as we have said, buildings do not normally fail by reason of the material breaking in compression. The stresses in masonry are so low that we can afford to go on scaling them up almost indefinitely. Unlike most other structures, buildings fail because they become unstable and tip up; and for any size of building this can be predicted from a model.

  Plate 1

  Chapter 2

  Each of the four columns which support the 400 foot (120 metre) tower of Salisbury Cathedral is very noticeably bent. Masonry is much more elastic than is generally supposed.

  Plate 2

  Chapter 4

  Stress concentration at a crack tip. The shear stress in a transparent material is revealed by polarized light. The bands in this photograph are, in effect, contours of equal shear stress.

  Plate 3

  Chapter 8
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br />   Rubber has a ‘sigmoid’ stress-strain curve like Figure 4, Chapter 8. A tube made from such a material will not distend evenly under pressure but will bulge into an ‘aneurism’. This is why artery walls do not have rubbery elasticity.

  Plate 4

  Chapter 8

  Artery walls and other living soft tissues have a special kind of elasticity like that in Figure 5, Chapter 8. The artery wall is constructed partly of elastin reinforced by kinked collagen fibres. This helps to produce the required ‘safe’ type of elasticity. (The artery tends to flatten when it is emptied of blood after death.)

  Plate 5

  Chapter 9

  Corbelled vault at Tiryns (c. 1,800 B.C.). Corbelled arches and vaults preceded the true arch.

  Plate 6

  Chapter 9

  Semi-corbelled postern gate at Tiryns. These walls were old when Homer marvelled at them.

  Plate 7

  Chapter 9

  It is very difficult to get a true arch to fall down. The foundations of Clare bridge, Cambridge, moved a long time ago, but the bridge is perfectly safe though the arch has distorted.

  Plate 8

  Chapter 9

  Part of the enormous temple of the Olympian Zeus at Athens. It was built in the Corinthian style by the Emperor Hadrian about a.d. 138. One of the architraves can be seen to be cracked. Note the walls of the Acropolis, which tower above Hadrian’s temple.