Structures- Or Why Things Don't Fall Down

by J E Gordon

  * * *

  *The mechanical problem is often much complicated by the association of muscle tissue and other active devices for contraction, but we shall ignore this for the present.

  * The theory of surface tension was originally worked out, independently by Young and by Laplace, about 1805.

  * Since in all such cases, e2 is always of opposite sign to eu q or Poisson’s ratio ought always to be negative, and it should therefore carry a minus sign about with it. However, we choose to forget about this and omit the minus sign; this is compensated for by putting a minus sign in the sums, such as those we are now doing.

  * To save indignant elasticians from the trouble of unnecessary correspondence, I do know about the energy changes involved. These anomalies have a rational explanation.

  *Note for bio-elasticians. This Hookean analysis is simplistic. For a non-Hookean system, where the tangent moduli are E1 and E2, then, approximately, the change of longitudinal strain is zero when

  Although most soft tissues preserve approximately constant volume – that is, they seem to have a true Poisson’s ratio around 0.5 – most membranes choose to deform in plane strain, that is to say, they do not get thinner when they are stretched, and so they show an apparent Poisson’s ratio of about 1-0 – like my tummy. This fits with a value of E1/E2 of around 2-0, which is likely enough. But why does the membrane not get thinner when it is strained? See, for instance, E. A. Evans, Proc. Int. Conf. on Comparative Physiology (1974; North Holland Publishing Company).

  * The shape of the stress-strain curve for most animal tissues – such as skin – is very much like that of a knitted fabric, which it is almost impossible to tear.

  * Since this was written, Dr J. M. Gosline has put forward an alternative hypothesis to account for the behaviour of elastin.

  Part Three

  Compression and bending structures

  Chapter 9 Walls, arches and dams

  -or cloud-capp’d towers and the stability of masonry

  What are you able to build with your blocks?

  Castles and palaces, temples and docks.

  R. L. Stevenson, A Child’s Garden of Verses

  As we have seen, unless one is as clever as Nature is, the whole business of making tension structures is set about with difficulties, complications and treacherous traps for the unwary. This is especially the case when we want to make a structure from more than one piece of material, so that we are faced with the problem of preventing it from coming apart at the joints. For these reasons our ancestors generally avoided tension structures as far as they could and tried to use constructions in which everything was in compression.

  Much the oldest and the most satisfactory way of doing this is to use masonry. As a matter of fact the immense success of masonry buildings has really been due to two factors. The first is the obvious one about avoiding tension stresses, especially in the joints; the second reason may be less obvious. It is that the nature of the design problem in large masonry buildings is peculiarly adapted to the limitations of the pre-scientific mind.

  Out of all the different kinds of structures which might be made, the masonry building is, as we shall see, the only one in which a blind reliance on traditional proportions will not automatically lead to disaster. This is why, historically, masonry buildings were by far the largest and most imposing of the works of man. The desire to build cloud-capp’d towers and solemn temples goes far back into history and indeed into prehistory. There is a quotation from Genesis about the Tower of Babel at the head of Chapter 1. It may be remembered that this was a project to build ‘a tower with its top in the heavens’. However, I do not think any theologian has ever inquired to what height such a tower could really have been built.

  Nearly all the load upon the walls would have been due to the effect of their own weight, and one way of looking at the problem is to calculate the direct compressive stress which would be caused near the bottom of the tower by the vertical dead weight of the masonry. A limit will be set to the height of the structure when the bricks begin to be crushed by the superincumbent weight.

  Now brick* and stone weigh about 120 lb. per cubic foot (2,000 kg/m3), and the crushing strength of these materials is generally rather better than 6,000 p.s.i. or 40 MN/m2. Elementary arithmetic shows that a tower with parallel walls could have been built to a height of 7,000 feet or 2 kilometres before the bricks at the bottom would be crushed. However, by making the walls taper towards the top, it could have been made much higher still; this is more or less how mountains work. Mount Everest is 29,028 feet or about 8 kilometres high and shows no signs of collapsing. Thus a simple tower, preferably with a broad base and tapered towards the top, could well have been built to such a height that the men of Shinar would have run short of oxygen and had difficulty in breathing before the brick walls were crushed beneath their own dead weight.

  Although there is nothing very much wrong with this sum, in fact even the most ambitious towers have never been built to anything remotely approaching that kind of height. The tallest ‘building’ which actually exists today is probably the New York Trade Center, which is only about 1,350 feet or 400 metres high; and this, like other skyscrapers, could be said to be cheating, since its structure is made of steel. The Great Pyramid and the highest cathedral spires reach a little more than 500 feet or 150 metres, but very few other masonry buildings are more than half so tall; the great majority are much lower still.

  Therefore the compressive stresses in everyday masonry due to its own vertical dead weight are very small indeed. In general they are seldom more than a hundredth part of the crushing strength of the stone, and so this factor is not, in practice, a limitation upon the height or the strength of buildings. However – to be biblical again – the Tower of Siloam, which was probably not particularly high, fell and killed eighteen people, and it is notorious that in spite of the confidence of builders and architects, walls and buildings do fall down unexpectedly. They have been doing so for a very long time and they still sometimes do so today. Since masonry is heavy, people often get killed.

  If walls do not collapse because of the direct crushing stress upon the material, why do they fall down? Once again, we can learn from what children do. When we were very young, most of us played with ‘bricks’, and about the first thing we did was to build a tower by piling one brick upon another rather erratically. Usually, when the tower had reached a modest height, it fell down. Even the child knew perfectly well, although he could not have expressed the idea in scientific words, that there was no question of the bricks being crushed under a compressive stress. The actual stress in the bricks was negligible; what happened was that the pile of bricks tipped up and fell over because the tower was not straight and vertical. In other words the failure was due to lack of stability and not to lack of strength. Although this distinction soon becomes evident to young children, it is not always clear to builders and architects. For the same reason the reflections of art historians who write about cathedrals and other buildings are apt to make rather distressing reading.

  Thrust lines and the stability of walls

  How reverend is the face of this tall pile,

  Whose ancient pillars rear their marble heads,

  To bear aloft its arched and ponderous roof,

  By its own weight made steadfast and immoveable,

  Looking tranquillity. It strikes an awe

  And terror on my aching sight.

  William Congreve, The Mourning Bride

  There was only one culture in Queen Anne’s time, and there is very little doubt that Congreve (1670-1729) talked and drank with Vanbrugh, who wrote plays and designed Blenheim Palace, and also with Sir Christopher Wren himself. To all these people it was perfectly clear – in a general kind of way – that what kept a building from tipping up and collapsing was not so much the strength of the stones and mortar as the weight of the material, acting in the right places.

  However, it is one thing to be aware of this in a general ki
nd of way and another to understand what is happening in detail and to be able to predict just when a building is safe and when it is not. In order to get a proper scientific understanding of the behaviour of masonry it is necessary to treat it as an elastic material; that is to say, one must take into account the fact that the stones deflect when they are loaded and that they obey Hooke’s law. It is also a considerable help, though perhaps not absolutely essential, to make use of the concepts of stress and strain.

  At first sight it does, of course, seem improbable that solid brick and stone should deflect to any significant extent under the loads which occur in a building. In fact for at least a century after Hooke’s time the common-sense view prevailed, and builders and architects and engineers persisted in ignoring Hooke’s law and treating masonry as if it were perfectly rigid. In consequence, their buildings sometimes fell down because they got their sums wrong.

  As a matter of fact the Young’s moduli of brick and stone are not particularly high, and, as one can see from the bent pillars in Salisbury Cathedral (Plate 1), the elastic movements in masonry are by no means so tiny as one might suppose. Even in an ordinary small house the walls are likely to be shortened or compressed elastically, in the vertical direction, by something like a millimetre under their own weight. In a large building the movements are naturally much greater. Incidentally, when the house is shaken by the wind during a gale, you are not imagining the effect; the house is being shaken by the wind. The top of the Empire State building sways something like two feet during a storm.*

  The modern analysis of masonry structures is based upon simple Hookean elasticity and also upon four assumptions, all of which turn out to be justified by practical experience. These are:

  That the compressive stresses are so small that the material will not be broken by crushing. We have already discussed why this is so.

  That, owing to the use of mortar or cement, the fit between the joints is so good that the compressive forces will be transmitted over the whole area of the joint and not just at a few high spots.

  That the friction in the joints is so high that failure will not happen because of bricks or stones sliding over each other. In fact no sliding movements at all will take place before the structure collapses.

  That the joints have no useful tensile strength. Even if, by chance, the mortar does have some strength in tension, this cannot be relied upon and must be neglected.

  Thus the function of the mortar is not to ‘glue’ the bricks or stones together but simply to transmit the compressive load more evenly.

  As far as I know, the first person to take the elastic deformations of masonry into account was Thomas Young. Young considered what would happen in a rectangular block of masonry, such as a piece of a wall, when it had to carry a vertical compressive load, P, let us say. In what follows I have simplified Young’s arguments by translating them into the language of stress and strain, which of course was not available in his time.

  As long as P acts symmetrically along the centre-line, that is, down the middle of the wall, then the masonry will be compressed uniformly, and, because of Mr Hooke, the corresponding distribution of compressive stress across the thickness of the wall will also be uniform (Figure 1).

  Suppose, now, that the vertical load, P, becomes a little eccentric, that is to say, it no longer acts exactly along the centreline; then the compressive stress can no longer be spread evenly but must be higher on one side than on the other so as to react properly against the load and keep it in balance. If the material obeys Hooke’s law, then Young showed that the stress will be distributed linearly and the stress-distribution diagram will look like Figure 2.

  So far the mortar in the joint is quite happy because the whole width of the joint is still safely in compression. However, if the position of the load is displaced still further from the centre – in fact to the edge of what is called the ‘middle third’ of the wall -then a situation like Figure 3 will arise where the load distribution is now triangular and the compressive stress at the outside edge of the joint is zero.

  Figure 1. Load P acting in centre

  Figure 2. Load P slightly eccentric of joint A B. but within the ‘middle third’of A B.

  This, in itself, does not matter too much, but it must be becoming clear to the percipient mind that something is about to happen. In fact, if the load is now displaced a little further outwards, something will happen: that is to say, a situation like Figure 4.

  The stress at the opposite surface of the wall has now changed from compression to tension. We said, however, that mortar cannot be trusted to take tension, and this is generally only too true. What one would expect to happen usually does happen: the joint cracks. Of course it is a bad thing for walls to crack, and it should not be allowed to happen in well-regulated buildings, but it does not necessarily follow that the wall is going to fall down immediately. What is likely to occur in real life is simply that the crack will gape a bit but the wall will continue to stand up, resting on the parts which are still in contact (Figure 5).

  Figure 3. Load P acting at the edge of the ‘middle third’of AB.

  Figure 4. Load P acting outside the ‘middle third’of A B.

  All this savours somewhat of living dangerously, and one of these days the line of the thrust may stray outside the surface of the wall, when, as a little thought will show, since no tension forces are available, one or more of the joints will hinge about its outside edge and the wall will tip up and fall down (Figure 6). It really will.

  Figure 5. What really happens as a result of the condition drawn in Figure 4. The joint cracks from B to C, and the load is now carried over the area AC- effectively a narrower wall.

  At the time when he came to these conclusions, that is, about 1802, Young, a rising man of twenty-nine, was appointed to the chair of Natural Philosophy at the Royal Institution in London. His colleague, and in some sense his rival, was Humphry Davy, who was made Professor of Chemistry in the same year at the improbable age of twenty-four. It was the custom of the professors of the Royal Institution, then as now, to deliver series of lectures to popular audiences. In those days, however, these lectures had very much of a television character, and the Institution relied heavily upon them for both money and publicity.

  Figure 6. When load P acts beyond A, i.e. outside the surface boundary of the wall, the wall will hinge about A, tip up and fall.

  Young took his educational mission seriously, and, filled with the enthusiasm of discovery, he launched into a series of lectures about the elasticity of various kinds of structures, with many useful and novel observations on the behaviour of walls and arches.

  The audience at Albemarle Street in those days was a fashionable one and is said to have consisted largely of ‘silly women and dilettante philosophers’. Young by no means neglected the feminine part of his audience, and he remarked in his opening lecture:

  A considerable part of my audience, to whose information it will be my particular ambition to accommodate my lectures, consists of that sex which, by the custom of civilized society, is in some measure exempted from the more laborious duties which occupy the time and attention of the other sex. The many leisure hours which are at the command of females in the superior orders of society may surely be appropriated, with greater satisfaction, to the improvement of the mind and to the acquisition of knowledge than to such amusements as are only designed for facilitating the insipid consumption of superfluous time...

  However, fortune does not always attend those who, however earnestly, strive to communicate useful information, and one may suspect that some of the females of the superior orders of society slipped away, preferring insipidly to consume their superfluous time. In any case Davy, who exhibited in his own lectures some of the exciting phenomena associated with the new electric fluid, together with a range of colourful chemical experiments, was a pushing young particle with what we should now call a television personality. Davy was also remarkably good-looking, and young women flocke
d to his lectures for reasons which were not always strictly academic; ‘those eyes’, one of them was heard to say, ‘were made for something besides poring over crucibles.’ The result, in box-office terms, could not be in doubt, and we are told that

  Dr Young, whose profound knowledge of the subjects he taught no one will venture to question, lectured in the same theatre and to an audience similarly constituted to that which was attracted to Davy, but he found the number of his attendants diminish daily and for no other reason than that he adopted too severe and too didactic a style.

  This kind of failure might not have mattered too much if Young could have attracted the interest and support of practical engineers. However, the engineering profession at that time was led, and frequently dominated, by the great Thomas Telford (1757-1834), whose views, as we have seen, were severely pragmatic and anti-theoretical. In consequence Young resigned his chair almost immediately and returned to his medical practice.* The development of elasticity passed, for many years, to France, where, at this time, Napoleon was actively encouraging the study of structural theory.

  The theory about elastic compression, the ‘middle third’ and instability which so bored the fashionable females at Young’s lectures does really tell us practically all we need to know about the behaviour of joints in masonry, provided that we also know the position at which the weight can be considered as acting. In other words, how eccentric is the load?