On Frocks and Bias

Or why fashionable dresses cling

I like watching movies, but I don’t much care for the process of promoting them or the somewhat cringeworthy awards season that culminates with the Oscars. Writing this in that season, albeit you will be reading this later in the year, I set myself the challenge of finding something Oscars related that involves structural engineering. I assumed that it would likely involve something to do with the staging or set, but eventually I decided to go in a different direction.

Just how do they make those frocks seen on the red carpet hug the body so tightly?

It may not be immediately obvious what figure hugging dresses, of the type worn to award ceremonies, have to do with structures, so your going to need to bear with me. I hope I am not biting off more than I can chew, because I am no fashionista and I don’t know anything about clothes. As always I shall be relying on engineering principles.....

It seems to me that there has never been a time when the human body has not been viewed as art and its form put on display. In this sense not much has changed between the marble sculptures of antiquity to modern day frocks.

That said, one thing that does seem to have changed is that extensive rigging no longer seems necessary to create a tight fit. I suspect that corsets are not terribly comfortable and therefore I expect most ladies welcome their demise.

To understand why we need to understand cloth and how it behaves. We also need to revisit phenomena we have met in a prior post; namely Young’s modulus and Poisson’s ratio. When a material stretches it narrows and when it is squashed it bulges. The amount of narrowing and bulging is directly related to the stiffness of the material; the stiffer the material the less there is. This is why we rarely notice the effect in steel or concrete. Stiffness is the ratio of stress to strain and is known as Young’s modulus, while the ratio of axial stretching or squashing to lateral narrowing or bulging is known as Poisson’s ratio.

Cloth is unlike steel, because its stiffness varies in different orientations. This is due to the way in which it is made. There are two sets of fibres, which are arranged in perpendicular directions. The warp threads are aligned vertically and the weft are weaved above and below. If the cloth is pulled in the direction of the warp there is little stretch in the direction of the tensile force. Similarly the weft threads prevent narrowing in the perpendicular direction. Alternatively, pull in the direction of the weft and again there is little movement in either direction. For this reason cloth is quite stiff in tension, except that is, if it is stretched at 45 degrees to the warp and weft. In this orientation there is considerable stretching and narrowing. Thus, at 45 degrees cloth has a low Young’s Modulus and a high Poisson’s ratio.

A useful analogy would be to compare the stiffness of a structural grid formed of perpendicular members with a lattice formed of diagonal members. Due to rotation at the joints the lattice is poor in tension, but good in shear, while the perpendicular grid is good in tension, but poor in shear.

Now supposing a dressmaker were to arrange and cut the cloth for a dress such that the warp and weft were aligned at 45 degrees to the vertical. I gather this is known as a bias cut. In such circumstances the self weight of the dress would place the cloth in tension, which would cause stretching in the vertical axis and a corresponding contraction in the perpendicular direction. 

Self-evidently the noted contraction would cause the dress to hug more tightly around its wearer. 

I am quite sure that dress makers must have lots of crafty tricks to enhance this effect. Perhaps they use thicker material, folds or stitching to increase the weight of the dress in certain locations or perhaps they vary the composition of the cloth by using threads of different stiffness or looser weave.

I have no idea if these are actual things or not, but they do seem to be reasonable suppositions based on the need to minimise Young’s modulus and maximise Poisson’s ratio. 

I don’t suppose dressmakers communicate in terms of Young’s modulus & Poisson’s ratio, but to make their dresses work as they do I suspect they do have a rather good empirical understanding of cloth. 

All this means that while I have little time for red carpets and awards ceremonies I can at least admire the skill of the dressmaker and their intimate knowledge of fabric. Whether they know it or not the figure hugging designs that have replaced laces and corsets rely on sound engineering principals.

On Grass & Buckling

Why you shouldn’t walk on a frosty lawn

An interesting thought that hadn’t struck me before is why it is possible to play sport on grass without it being ruined. I am not suggesting that sports pitches don’t suffer from wear and tear; self evidently they do. After two weeks of tennis in the summer the courts at Wimbledon are not in the same condition they were at the start of the tournament. Similarly, after a season of football or rugby pitches around the country need time to recover, albeit pitches today fair better than they did in the past, but that’s not really what I mean. How is it that sport can be played on grass at all? 

It takes no effort at all to pluck a blade of grass and almost none to tear or cut it, how then can we walk on grass without damaging it let alone run and jump? I think the answer is to be found in a structural principle known as Euler[1] or Strut buckling. Euler buckling is a special form of compression failure, which applies to slender structures and is named after Leonhard Euler who sorted out the mathematics. Slender structures are those, which are thin relative to their height.

When a squat structure is subjected to compression it will fracture and split if it is made of a brittle material. Alternatively, if it is made from a ductile material it will bulge and deform, however a slender structure will buckle before any of these states are reached.

Buckling is essentially the point at which a structure subjected to compression gives way due to a rapid increase in lateral deflection. The reason deflection increases rapidly is because the onset of buckling instigates a feedback loop.

When buckling begins the structure is displaced laterally, which causes the compressive load it carries to be applied eccentric to the line of support. As we have learned in earlier posts an eccentric force generates a bending moment, which causes increased lateral displacement. Thus the feedback loop is set in motion.

Euler’s work teaches us that the load at which buckling commences is directly proportional to the stiffness of the structure and inversely proportional to the square of its length. This tells us two important things:

Firstly, doubling the stiffness of a structure will double the buckling load, whereas doubling its length will reduce the buckling load by three quarters. The length of the structure is therefore the most significant factor.

Secondly, providing the material is ductile and will therefore remain elastic, its strength plays no role in Euler buckling. This means that once the compressive load has been removed it will recover and return to its original shape. For those who are interested elasticity is covered in a prior post, On Ductility.

These are the properties which make walking on grass possible. A blade of grass has a low material stiffness and is tall relative to its thickness i.e. it is slender. For this reason when you step on grass the blades simply buckle elastically and then recover when the foot is lifted. It is also worth noting that grass grows from the root rather than the tip so any damage that does occur can be repaired from below the damaged part.

It is often said that you should not walk on a frosty lawn. In light of our buckling logic the reason for this becomes clear. Grass, being a plant, is made or cells which contain water. I am sure biologists would put it better than this, but I think that’s pretty much the case. When a frost sets in water contained in the cells will freeze and make the normally flexible blades of grass brittle. It follows that if you were to step on a lawn in this condition the frozen blades would no longer be elastic and instead of buckling they will fracture. Thus permanent damage is done.

So that’s my answer; sport can be played on grass, because each blade is a small slender structure that behaves according to the rules of Euler Buckling. I suspect that the normal wear and tear that we see on sports pitches is primarily due the lateral movement of feet, which would presumably apply horizontal shears that tear and rip at the buckled blades. That said, this cannot be the dominant behaviour or pitches would become rather more damaged than they actually are.

[1] Pronounced Oiler

On San Petronio

Gravitational equilibrium & the square cube law

‘Gulliver’s Travels’ is a classic of English Literature written by Jonathan Swift in 1726. It is intended to be a satire of human nature and ‘travellers’ tales’. In the book its protagonist visits the fictional countries of Lilliput and Brobdingnag. The citizens of the former are 12 times smaller than Gulliver and in the latter they are 12 times bigger.

Of course every reader of the book knows that Lilliput and Bribdingnag are fictional, but perhaps fewer might realise that they are necessarily fiction in any possible world that follows the same physical laws as our own. The reason for this is described by the square cube law, which was first attributed to Galileo.

According to this law the size of things cannot be indefinitely scaled up, because the physical properties of objects change as they increase in size. Consider a cube with sides one unit long. If we were to double the length of each side then the surface area of each face will increase from 1 to 4 units. The volume of the cube, and by extension the quantity of stuff from which it is made, increases from 1 to 8 units i.e. doubling the linear dimensions causes the surface area to be squared and the volume of stuff to be cubed.

In other words the rate at which the volume and weight of an object increase with size is greater than the rate at which its linear dimensions and surface area increase.

Conversely, the strength of things, normally expressed as a limiting stress, is independent of size [1]. It follows that as things become larger, while retaining their original proportions, they will eventually reach a point where they can no longer support their own weight. This places an upper limit on how big things, including people, can get.

This is a vexing problem for modern day architectural students, who are surprised to learn that the model they have spent hours building does not prove that the structural gymnastics their design requires are viable in the real world.

There is of course an exception to the square cube law, though not a true exception. The square cube law does not cease to work, rather it is not discernible within the normal range of scaling for certain objects.

An example of such an object from the natural world would be a mountain. An equivalent structure form the man-made world would be a pyramid. Both structures come from a class of things that share three key ingredients. Firstly, they are both made of stone, which is a natural material with a very high compressive strength and low tensile strength. Secondly, they are both compression structures that assiduously avoid tension. Thirdly, they are stocky and solid structures. Not solid in the sense that they are strong, though that is indeed true, but solid in the sense that they are not hollow. The implication of being stocky and solid is that they are not prone to buckle, as a slender or thin walled structure is. A second implication of stockiness is a large cross-section, which implies low stress.

These then are the ingredients for subverting Galileo’s square cube law and within the class of things which have these ingredients there is a group of structures that does so with a style and panache that is difficult to surpass. They laugh in the face of the square cube law.

I am of course talking about gothic structures; those great stone cathedrals of the late medieval period with their quadripartite vaults and flying buttresses. I am often asked, mainly by my dad, how it was that medieval masons created such structures in the absence of modern structural theory. The stock answer is that they used rules of thumb, but for me there has to be more to it than that.

How on earth did they manage to design structures, using rules of thumb, that modern analysis shows us to have near perfect proportions. Rules of thumb are intended to apply generally, but must necessarily be derived from the particular. The further a design departs from the particular the less useful a rule of thumb is. 

Gothic structures do have similarities, but there is also great variety in their design, which ought to make rules of thumb less helpful.

Perhaps the answer to this dilemma is to be found in a process of trial and error. While there was undoubtedly a role for trial and error I am not convinced that it played a central role, at least not in the commonly understood sense. I hold this view for several reasons:

Firstly, most cathedrals took hundreds of years to build and though master masons may have worked on several, they would not, in their own lifespan, have time to learn all the necessary forms by trial and error. This brings us back to rules of thumb. Secondly, while there is evidence of design evolution over time there is relatively little evidence of major failures, which is odd given the innovative, and often spectacular, designs adopted. Furthermore, when known failures occurred they were often, though not always, associated with abnormal events like earthquakes or phenomena external to the structure like differential settlement.

We therefore have much evidence of trying daring new things, but scant recorded evidence of failure [2]. This is not entirely a surprise, because cathedrals are expensive and their proprietors were not stupid. Master masons would not have been in charge for long if their structures kept collapsing.

To find a satisfactory explanation I think we need to return to the square cube law. If a structure could be designed to subvert the square cube law then a successful pattern would be successful at any scale. It follows that if you could demonstrate a particular form of structure would stand using a small model made of wood then it would also stand if it was scaled up to full size. 

I do not know whether masons understood that they were subverting the square cube law. I suspect that they didn’t, but I do think that they understood perfectly well the load-paths and principles that were necessary to keep a stone structure in equilibrium. I think they fully understood that tension was the enemy and that gravity must be harnessed to maintain compression in all parts of the structure. 

They were specifically designing structures to achieve gravitational equilibrium and they were doing it by experimentation with scale models. They then used rules of thumb, derived from these models, to scale up their findings to full size structures.

 

Happily for the masons this methodology was just perfect for avoiding the square cube law, whether they knew it or not, and in this way spectacular and innovative designs could be realised without needing to know anything about materials science, stresses or strains.

While this theory involves a heavy dose of speculation it is not without evidential support. For example, we know that Antonio Vicenzo the designer of San Petronio church in Bologna commissioned a model of brick and plaster at circa one eighth scale. It was around 19m long and 6m high. I expect that it was used to convey the design to its proprietors, but I don’t think it is too great a leap to posit that it might also have played a role in the church’s structural design.

[1] I accept that all bets are off at the atomic scale, but we don’t normally consider atomic forces when designing building structures, bridges and the like.

[2] I know that absence of evidence is not evidence of absence, nevertheless evidence is sometimes notable by its absence, particularly when there might be a reasonable expectation to find some.

On Britannia Bridge

An empirical masterpiece

When I studied engineering at university one of the structures that was frequently put forward by Professors, as an excellent example of structural design was Britannia Bridge. Throughout my time at university and probably for a time afterwards I didn’t really understand why. It didn’t seem a particularly great design to me. I don’t suppose for one moment that was my Professors’ fault; I had good teachers. I suspect it was more likely that I wasn’t paying sufficient attention.

Though I was unquestionably wrong, if I give my younger self the benefit of the doubt, my view was not entirely without merit. Firstly, I had not seen lots of other bridges old or modern that resembled Britannia; suggesting that despite its supposed brilliance few saw fit to copy it[1]. Secondly, passing steam trains through a tubular structure, as Britannia did, didn’t seem like a good idea to me. The smoke and soot must have been pretty unbearable. Furthermore, the integrity of the bridge was compromised by fire in 1970, which brought about its eventual demise. It therefore failed in its primary purpose of carrying trains from Wales to Anglesey. 170 years of service is by most accounts rather good, but I reasoned this was luck. What would have happened had the fire occurred earlier in its life? 

Notwithstanding all of this, Britannia Bridge was, in my humble opinion, ugly. My architect friends might well quip that this is exactly what to expect from a bridge designed by an engineer. I beg to differ, but that debate is for another day.

Having got that historical baggage off my chest I am now, as you might have guessed, wholly convinced of Britannia’s brilliance and it’s aesthetic has sort of grown on me. I’m not saying it has become easy on the eye; rather an appreciation of how it works and how it came to be has resulted in a greater appreciation of its aesthetic.

So why did I change my view of Britannia’s structural design and what was it that persuaded me? Much like students today, my younger self lacked an appreciation of context and when judging the achievements of the past context is everything.

Thomas Telford, the first president of the Institution of Civil Engineers, completed the Menai Strait suspension bridge in 1826. At that time it held the record for the longest suspended span in the world and it is an elegant and beautiful bridge. That said Telford’s masterpiece [perhaps the subject of a future post] was not without problems. It suffered aerodynamically and was damaged on several occasions before being stiffened. 

When Robert Stephenson was faced with the problem of bridging the Menai Strait for the Chester & Holyhead railway a suspension bridge, like Telford’s, was not considered stiff enough to carry the weight of steam locomotives [the suspension form was still in its infancy].

The obvious answer would have been to construct intermediate piers from which arch centerings could be erected, however the currents in the strait were considered to be treacherous and therefore the Admiralty insisted that there be no obstructions placed in the channel, which would make it more difficult to navigate. The only place an intermediate support could be found was at Britannia rock, which was located almost halfway between either shore. 

An alternative idea was to construct an arch on centring built from floating pontoons, however this was also ruled out as there would have been insufficient clearance for ships near the abutments at high tide.

Since a suspension bridge was deemed too flexible and the Admiralty’s intervention had ruled out an arch, Stephenson had to invent something altogether new. 

To achieve the required strength and stiffness Stephenson had some notion that the answer would involve creating large wrought iron plate girders by riveting. He later settled on the idea of a wrought iron tube through which the trains would pass. He had originally thought of the tube as being suspended from chains, but decided instead that it should behave as a hollow beam spanning from end to end.

In 1845 there was no structural theory that could be used to design Stephenson’s hollow tube, albeit British engineers had little time for the theories of their European neighbours. This was not unreasonable as theory was unable to justify the structures that British Engineers were already producing at that time. Telford himself was on occasion disdainful of theory, his Menai Bridge was entirely justified by rigorous testing of its components.

Stephenson turned to William Fairbairn, who was known for his investigation of iron and his experience of ship hulls, which seemed analogous to his hollow tube idea. He engaged Fairbairn to investigate a hollow tube design by conducting experiments on scale models.

Fairbairn realised however, that extrapolating behaviour observed in models to full scale was not straightforward. He therefore recommended Eaton Hodgkinson to Stephenson for the purpose of deriving empirical relationships from his models that could be used to size the bridge. We have of course met this pair before in a prior blog post ‘On Fish-bellies’. 

And so it was that three great pioneers Stephenson, Fairbairn and Hodgkinson set about designing Britannia bridge.

Amongst their findings the trio discovered the mechanism of plate buckling, which overturned the idea that wrought iron was weaker in compression than tension. They also discovered that rivets work by clamping and friction, when previously it was assumed their behaviour was governed by dowel action. They looked at the effects of continuity over an intermediate structural support, which enabled a more efficient design to be realised. They also explored thermal movements and wind effects. Their collaboration was a remarkable piece of work.

That said, it wasn’t all plain sailing. They argued about whether suspension chains were in fact required so late into the process that the bridge piers had to be built full height just incase they were needed. Fairbairn took the position, based on his experiments, that the chains were not required, while Hodgkinson, based on the empirical theory he was developing, took the position that they were. In the end Fairbairn’s argument prevailed; Stephenson omitted the chains and Fairbairn was proved right. 

The resulting bridge was a mixed success. It performed as intended, however it was exceedingly expensive, owing to the cost of iron in the mid nineteenth century. This is probably the main reason its form was not adopted more widely. Nevertheless, Stephenson’s design methodology was exceedingly influential, as were the empirical insights gleaned from Fairbairn’s models.

In this sense Britannia Bridge is archetypal of the empirical approach to design that was practised by British Engineers at that time.

Ironically the fire, which caused its eventual demise in 1970 was caused by ignition of the wooden roof covering that was later added to protect the bridge from corrosion. It wasn’t even part of the original bridge, so my younger self got that completely wrong too!

[1] In a very pure sense this isn’t strictly true, because modern bridges do make use of box girders, but not in the same way as Britannia.