by J E Gordon
But what has all this to do with stresses? In one sense a great deal; in another very little. As long ago as the seventeenth century, Fenelon observed that classical architecture owes its effects to the fact that it appears to be heavier than it really is, Gothic to the fact that it appears to be lighter than is really the case. In this respect there appears to be no aesthetic pay-off from honest functionalism – from appearing to be just as heavy as you really are.
The classical orders, especially Doric, appear almost to stagger under the burden of their own weight. In fact there is really very little load in most of the columns, but the swelling or ‘entasis* which is given to them provides a sort of Poisson’s ratio effect to convince us that they are bulging under the compressive stress. This bulging effect is carried still further by the swelling, cushion-like capitals or ‘echinoi’ which transmit the compressive load from the lintels to the heads of the columns. The effect of weight is enhanced still further by the excessive depth of the architraves.
Although classical architecture operates on the emotions, at least in part through a subjective sense of stress, its beauty has little or nothing to do with modern ideas of structural efficiency, in the sense of the one-hoss shay. All these buildings were, in fact, thoroughly inefficient. The compressive stresses were absurdly low, while the tensile stresses in the lintels were far too high, often dangerously so (Chapter 9). The roofs of classical buildings, as we have seen, can only be described as a structural mess. But there is nothing wrong with most of these buildings aesthetically.
When we come to consider Gothic architecture, the compressive stresses in the masonry are, as a rule, a good deal higher than they are in classical buildings, and the structure as a whole is generally more stable, in spite of its airy-fairy appearance. The effect of lightness is, however, achieved, in part, by the use of pointed arches, which, again, are ‘inefficient’. These Gothic structures are, to the modern functional mind, excessively complicated. The real heroes of Gothic cathedrals seem to be the statues, whose weight, perched on pinnacles and flying buttresses, keeps the thrust lines stable (Chapter 9).
Structurally ‘inefficient’ as ancient buildings may have been, it does seem that the eye requires some subjective sense of stress if it is to find satisfaction in looking at a structure. In many modern buildings the load-bearing structure, which is often of reinforced concrete, is hidden away inside the building. All that the external observer can see is a curtain wall or ‘cladding’ of thin brick or glass which is obviously inadequate to carry any load at all. I do not think I am alone in finding these buildings unsatisfactory to look at and often downright ugly.
But supposing that we had some kind of structure whose means of support were clearly visible, and which was also highly ‘efficient ‘ in the modern manner, what might we expect it to look like? Clearly this is a subject about which one could argue for a long time. However, if we may judge from the structures which are employed for landing on the Moon – in which weight has been saved regardless of cost, the ultimate in one-hoss shays – the answer seems likely to be ‘Hideously ugly’.
On skiamorphs, fakes and ornament
The earliest surviving buildings of consequence in Greece are Mycenaean and date perhaps from some time before 1,500 b.c. These buildings were made of stone and seem to have been deliberately and intelligently designed as structures suited to the characteristics of that material. The Mycenaeans were well aware, for instance, of the danger of excessive tensile stresses in stone lintels, and they made adequate provision to relieve the bending loads on their stone beams, as one can see in the Lion Gate at Mycenae (Plate 24). To this extent, at least, Mycenaean architecture can be described as ‘structurally functional’.
When the Mycenaean civilization collapsed, around 1,400 b.c, Greece seems to have reverted to a dark and illiterate age, from which no buildings of any importance survive. No doubt people lived and worshipped in wooden huts of one kind or another. When formal architecture began to revive in early Archaic times, perhaps about 800 B.C., the early temples were built of wood, like the New England churches.
Naturally, none of the original wooden temples has survived. However, the transition from wood to stone construction seems to have been a piecemeal process; as timber became scarce, decayed wooden members were replaced by stone copies. Pausanias speaks of a temple still existing at Olympia in the second century a.d. in which some of the wooden columns still remained, mixed with more recent stone ones.
Doric architecture is thus ‘trabeate’ or beam architecture, based on wooden construction; and even when temples came to be built, de novo, entirely of stone, architects still stuck to the forms and proportions which were suited to timber. Not only did classical architects of the sophisticated fifth century use weak stone beams in the place of wooden lintels; they went to the trouble of copying in marble all sorts of irrelevant constructional details, such as the ends of the wooden pegs which had once held the wooden buildings together.
The result ‘ought’ to have been ridiculous, but it was not; it was gloriously and triumphantly successful and has served as a model for the civilized world, on and off, for two thousand years. Survivals of this sort are known as ‘skiamorphs’ (shadow shapes), and in one form or another they are very common in technology, A modern instance is the survival of timber graining on the surfaces of plastic mouldings and furniture.
Contrary to the whole ethic of the functionalist school of engineering aesthetic thought, skiamorphs are not necessarily shoddy or vulgar. Nowadays, of course, they very often are; but surely this is because of our own faulty execution, not because there is something inherently wrong with the idea.
The development of the Watson steam yacht is a splendid example of a successful skiamorph. The classical form for large steam yachts was evolved in latish Victorian times by the greatest of all yacht designers, G. L. Watson (who had for his epitaph ‘Justice to the line and equity to the plummet’). For his fully powered vessels Watson retained, not only the graceful ‘clipper’ bow of the sailing ship, but also the now functionless bowsprit. The result is one of the most beautiful ship conventions which has ever been developed (Plate 22).
If all this be so, what are we to think about ‘honesty’ in design? Honesty compels me to say ‘Not much’. If skiamorphs are permissible in Greek temples and steam yachts, what are we to think of the total ‘fake’? Is there any reason why we should not dress up suspension bridges as medieval castles, motor cars to look like stage coaches, or yew-trees to look like peacocks?
Personally I am rather in favour of it. After all, the results could hardly look worse or more depressing than the results of modern functionalism, and they might be a lot more fun. What is wrong with eighteenth-century ‘Gothick’ buildings? The best of them are tremendous fun and perfectly lovely. Horace Walpole was no fool, and the Pavilion at Brighton is a delight.
There are those who moan about ‘meaningless ornament’; but the phrase is surely an oxymoron, for no ornament can be ‘meaningless’ – even if it means something pretty frightful. If the critic wants to imply ‘ornament which is unsuitable or unrelated to its substrate’, that is fair enough; but all ornament must have some effect. It seems to me that what we want is more, not less ornament. The truth seems to be that we are frightened to express ourselves in ornament. We don’t know how to handle it, and fear that we may expose the nakedness of our mean little souls. Medieval masons did not have that kind of inhibition, and they were probably psychologically healthier in consequence.
Is it not fair to ask the technologist, not only to provide artefacts which work, but also to provide beauty, even in the common street, and, above all, to provide fun! Otherwise technology will die of boredom. Let us have lots of ornament. Let there be figure-heads on ships, gilded rosettes on the spandrels of bridges, statues on buildings, crinolines on women, and, everywhere, lots and lots of flags. Since we have created a whole menagerie full of new artefacts, motor cars, refrigerators, wireless sets and th
e Lord knows what, let us sit down and think what fun we can have in devising new kinds of decorations for them.
* * *
* W. M. Dixon, The Human Situation (Penguin, 1958).
* Vide the recent vogue for collecting chamber-pots. Aristophanes regarded Greek oil-bottles as essentially ridiculous but he never implied that they were ugly: indeed the ones in museums are much admired.
Appendix 1 Handbooks and formulae
Over the last 150 years the theoretical elasticians have analysed the stresses and deflections in structures of almost every conceivable shape when subjected to all sorts and conditions of loads. This is all very well, but usually the results, in the raw form as published by these people, are too mathematical and too complicated to be of much direct use to ordinary human beings who are in a hurry to design something fairly simple.
Fortunately a great deal of this information has been reduced to a set of standard cases or examples the answers to which can be expressed in the form of quite simple formulae. Formulae of this sort, covering almost any possible structural contingency, are to be found in handbooks, notably R. J. Roark’s Formulas for Stress and Strain (McGraw-Hill). These formulae can be used by people like you and me equipped with little more than common sense, a knowledge of elementary algebra and the contents of Chapter 3. A few of these formulae are given in Appendixes 2 and 3 which follow.
Used with caution, such formulae really are very useful indeed, and indeed they form the professional stock-in-trade of most engineering designers and draughtsmen. There is not the slightest need to be ashamed of using them; in fact we all do. But they must be used with caution.
Make sure that you really understand what the formula is about.
Make sure that it really does apply to your particular case.
Remember, remember, remember, that these formulae take no account of stress concentrations or other special local conditions.
After this, plug the appropriate loads and dimensions into the formula – making sure that the units are consistent and that the noughts are right. Then do a little elementary arithmetic and out will drop a figure representing a stress or a deflection.
Now look at this figure with a nasty suspicious eye and think if it looks and feels right. In any case you had better check your arithmetic; are you sure that you haven’t dropped a two?
Naturally neither mathematics nor handbook formulae will ‘design* a structure for us. We have to do the designing ourselves in the light of such experience and wisdom and intuition as we may possess; when we have done this the calculations will analyse the design for us and tell us, at least approximately, what stresses and deflections to expect.
In practice therefore design procedure often runs something like this. First, one determines the greatest loads to which the structure may be subjected and the deflections which can be allowed. Both of these are sometimes laid down by existing rules and regulations, but, where this is not the case, they may not be particularly easy to determine. This sort of thing calls for judgement, and in case of doubt it is clearly better to err on the conservative side, although, as we have seen, it is quite possible to go too far and incur danger from too much weight in the wrong places.
When the loading conditions have been determined we can sketch out, to scale, a rough design – designers often use pads of squared paper for their preliminary sketches – and we can then apply the appropriate formulae to see what the stresses and deflections are going to look like. At the first shot these will probably be too high or too low, and so we go on modifying our sketches until they seem about right.
When all this has been done, ‘proper’ drawings may have to be made from which the thing can be manufactured. Formal engineering drawings are very necessary when components have to be made by the usual industrial procedures, but they are troublesome to make and may not be needed for simple jobs or amateur work. For anything of a commercial and potentially dangerous nature, however, it is my experience that a firm can look remarkably silly in a court of law if the only ‘drawing’ they can produce is a sketch on the back of an envelope.
When you have got as far as a working drawing, if the structure you propose to have made is an important one, the next thing to do, and a very right and proper thing, is to worry about it like blazes. When I was concerned with the introduction of plastic components into aircraft I used to lie awake night after night worrying about them, and I attribute the fact that none of these components ever gave trouble almost entirely to the beneficent effects of worry. It is confidence that causes accidents and worry which prevents them. So go over your sums not once or twice but again and again and again.
Appendix 2 Beam theory
The basic formula for the stress s at a point P distant y from the neutral axis of a beam is
so
Figure 1.
where s = tensile or compressive stress (p.s.i., N/m2 etc.)
y = distance from neutral axis (inches or metres)
I = second moment of area of cross-section about the
neutral axis (inches4 or metres4)
E = Young’s modulus (p.s.i., N/m2 etc.)
r = radius of curvatures of the beam at the section under consideration due to the elastic deflections set up by the bending moment M (M in. inch-pounds, Newton-metres etc.).
Position of neutral axis
The ‘neutral axis’ (N.A.) will always pass through the centroid (’centre of gravity’) of the cross-section. For symmetrical sections, such as rectangles, tubes, T sections etc., the centroid will be in the ‘middle’ or centre of symmetry. For other sections, it can be calculated by mathematical methods. For some simple asymmetrical sections (e.g. railway lines) one can determine the centroid accurately enough by balancing a cardboard model of the section on a pin. For more elaborate structures, such as ships’ hulls, the position of the neutral axis really will have to be calculated by sheer arithmetic.
‘I’, the second moment of area of a cross-section
This is often (though incorrectly) called the ‘moment of inertia’.
Thus, if an element, at the point P, distant y from the neutral axis, has a cross-sectional area a, say, then the second moment of area of this element about the neutral axis will be ay2.
Figure 2.
Thus the total I or second moment of area of the cross-section is the sum of all such elements, i.e.
For irregular sections this can be calculated by arithmetic, or there is a version of ‘Simpson’s rule ’which gives the answer.
For simple symmetrical sections:
For a rectangle about the neutral axis,
Figure 3.
For a circle about the neutral axis,
Figured 4.
Thus simple box and H sections as well as hollow tubes can be calculated by subtraction.
For a thin-walled tube of wall thickness t, however,
I = πr3t
Figure 5.
The Is of a great many standard sections can be looked up in reference books.
‘Radius of gyration’, k
For some purposes, it is useful to know the value of what is called the ‘radius of gyration’ of a beam section: that is to say, the distance from the neutral axis at which the area of the cross-section may be considered as acting,
i.e. I = Ak2
where A = total area of cross-section
k = ‘radius of gyration’
For a rectangle (see above) k = 0·289 d
For a circle (see above) k = 0·5 r
For a thin-walled annulus k = 0·707 r
Some stock beam situations
CANTILEVERS
Point load W at end
Condition at distance x from end of beam:
M= Wx Max M= WL at B Deflection at x is
Max deflection at A
Figure 6.
Uniformly distributed load W= wL
Deflection at x is
Max deflection at tip
Figure 7.
SIMPLY SUPPORTED BEAMS
3. Simply supported beam with load in centre
Bending moment Mat point x
Deflection y at x
Figure 8.
4. Simply supported beam with single point load not in centre Bending moment M at point x
Max deflection
at when a > b
Figure 9.
5. Simply supported beam with uniform load W = wL
at point x:
Max deflection at centre
Figure 10.
For further information, see Roark, R. J., Formulas for Stress and Strain (McGraw-Hill, current edition).
Appendix 3 Torsion
Torsion
For a parallel bar or prism or tube under torsion the twist or angular deflection θ (in radians) is given by
where θ = angle of twist in radians
T = torque in inch-pounds or Newton-metres
L = length of member subject to torsion (inches or metres)
G = shear modulus (Chapter 12), N/m2 or p.s.i.
K is a factor to be found from the following table.
Section K Max shear stress N
Solid cylinder radius r ½πr4 (at surface)
Hollow tube radii r1 and r2 ½πr(r14-r24) (at outer surface)
Hollow tube longitudinally slit (i.e.’C section)