On Stegosaurus

The original idea for this post was to show an animal’s skeletal structure and explain how it works. I have chosen to do this using a stegosaurus skeleton for two reasons. 

Firstly, this blog is going to be a bit speculative, because I don’t know anything about animals or their skeletons. In the event that a biologist or anatomist finds him or herself reading this blog there is a high probability I shall be politely informed that I don’t know anything. Based on this I figured that my best form of defence would be to choose an animal for which there are no living examples. This way I can argue that since there are no living examples what said animalist thinks they know is mere speculation. No-one can prove otherwise.

Secondly, stegosaurus is a dinosaur and dinosaurs are cool, every kid knows that. Of course I could have chosen T-rex or triceratops, but I think those are the obvious choices and I happen to like stegosaurus so that’s what I’ve chosen.  Here goes….

It seams to me that the spine of an animal, which supports its head and tail are analogous to a thin flexible truss; in fact a special type called a vierendeel truss. To explain this I am going to need to make a few assumptions. Self-evidently all animals walk, run, jump, climb and so forth. Okay, I don’t know whether the last two are true of Stegosaurs, but neither do you. See how effective that defence is? Anyway, movement gets a bit too complex for my stated aim of explaining how skeletal structures work. With that in mind I am going to assume, for the purposes of this blog post, that we have a stationary stegosaurus standing on all four legs.

 

Based on this the first observation we shall make concerns the legs. The fore-limbs, compared to the hind-limbs, are rarely small, however they are still stocky and have the appearance of being load bearing. The larger hind-limbs and pelvis are larger and would appear designed to carry greater load. This corresponds with the head and neck being rather small when compared to the size and bulk of the hind-quarters and tail.

The stegosaurus’ spine bridges between the two sets of legs and cantilevers beyond them at both ends in order to carry the neck and tail. To understand the structural implications of this arrangement we must first learn something about bending moments.

A bending moment is a turning force whose magnitude is the product of force (or weight) multiplied by the distance (known as the lever arm) to the nearest point of support. The greater the distance to a support; the greater the bending moment.

Consider if you will the following thought experiment. Supposing the stegosaurus’ fore-limbs were located at the end of its nose. I know that’s daft, but run with it. The distance between the nose and its legs is zero therefore no matter how heavy the nose the resulting bending moment is zero.

Now suppose the fore-limbs are located at the back of its head. The bending moment has increased from zero in proportion to the length of its head. I hope you can see where I am going with this. Now suppose the fore-limbs are back where they should be attached to the shoulders. The bending moment has now increased in proportion to the length of a stegosaurus neck and head.

If we were to portray the magnitude of the bending force graphically it would look like a triangle, being zero at the nose and increasing to a maximal point where the legs meet the shoulders. We could repeat the same process starting from the tail. The bending shape would be the same, though the height of the triangle would be bigger because the tail is longer and heavier.

We are now left with the bit in the middle, we need to join the two triangles together. Of course the  body of the stegosaur is somehow attached to the spine (I told you I am not an animalist). We’ll get back to that subject later, but for now we will simply note that the weight of the body must pull down on the spine. It follows that a line depicting the bending moment will join the two triangles, by sagging in middle. It might look a bit like the diagram below. 

The interesting bit; also the bit where I might start to get into trouble; is how the spine resists these bending forces.

The first thing we notice about a stegosaur spine is, like most spines, the odd shape of the vertebrae, which is distinguished by three parts, at least to my eye. There is the oval shaped portion located at the anterior. The lower part is of bone and the upper part has a hole where the spinal cord would be. 

The second feature is the piece of bone that projects vertically from the middle of the vertebra to form the posterior.  This is where the ligaments and muscles of the back are connected to the vertebrae.

The third feature is the two transverse projections; one either side of the vertebrae. These are where muscles and ligaments attach to the spine and are also the points from which the ribs are articulated. The fossil stegosaurus vertebrae pictured below is quite tall and for reasons that will become clear is postulated to be from the lower part of the back.

When the vertebrae are aligned in a row, as they would be to form a spine, they start to resemble a truss with a lower chord of bone and an upper chord of ligament and muscle. You have to imagine the muscle and ligament joining the vertical projections together, as self-evidently they haven’t been fossilised. The internal members of the truss are formed by the bone surrounding the spinal cord and the projection that extends from it. There are of course no diagonal members and that is why the resemblance is to a vierendeel truss.

 

For this information to be of use we now need to describe how a truss works. We shall begin by considering a beam, which is altogether a simpler and humbler form than the truss. 

If a beam spans between two supports, one at either side, it will deflect in the direction of the bending force to form a curve. As the beam deflects the outer edge of the curve is stretched and is evidently in tension. Conversely the inside edge is squashed and must be in tension. The centre of the beam, known as the neutral axis is neither in compression nor tension; it is at rest.

If we apply the same logic to a truss the outer chord of the truss carries a tensile force and the inner chord carries compression. The internal members of the truss transfer load between the outer and inner chords. 

In the case of our stegosaur based truss the ligaments and muscle at the posterior are ideal for carrying tension and the boney part at the anterior is ideal for resisting compression. This is remarkably good luck, because as nature would have it, the bending moments imparted by the tail and neck of the stegosaur, as shown in our earlier diagram, impart tension on the posterior side and compression on the anterior.

If that is not remarkable enough the depth of the vertebrae increase towards the hind limbs to match the increasing magnitude of the bending forces. Nature has actually fine tuned the size and shape of the vertebrae so that the resulting truss matches the shape of the bending forces to which they are subject.

Having made this observation it is tempting to end this post satisfied that our work is complete, but I am not quite ready to do that. There is one more thing that is just too interesting not to pass comment. I also noted earlier in the post that I would return to this subject.

It is not lost on me that the stegosaur spine has a very distinct curve, which given the apparent fine tuning of the vertebrae, cannot be without reason. Based on that it is, I think, worth a further speculation. 

It strikes me when looking at the bulk of a stegosaur’s body, how is that great big, heavy fleshy part of the animal supported from the spine. Presumably it somehow hangs? I imagine the animal’s flank muscles, perhaps reinforced by the ribcage, are responsible for transferring the load.

It then strikes me that an arch is a rather efficient way of supporting the load and transferring it to the legs. There is of course a potential issue with this load path. Arches generally have large abutments whose purpose is to resist the lateral thrusts generated at the base of an arch and thus prevent it from spreading. It is these same thrusts that make it so difficult to build a house of cards. Of course stegosaurus have no abutments.

There is a potential solution, which is to be found in the form of a bow string arch. Sometimes when there is no opportunity to provide bridge abutments the engineer will instead join the arch supports together with a tie member. This works by causing the two sides of the arch to pull against each other. Since the pull is equal and opposite equilibrium is maintained and the arch is prevented from spreading.

My final speculation is therefore that the stegosaurus’ sternum and chest muscles provide a tie, which form’s a bow string truss made of meat and bone.

Now I can finish the post and wait to be shot down by people that know what they’re talking about. I shall be lying in wait with my pre-prepared speculation defence.

On Bell Towers & Skipping Ropes

For many years there has been considerable academic interest in the structural behaviour of masonry bell towers during earthquakes and whether their bells help or hinder performance. The observed level of interest is for the most part driven by the fact that masonry belfries are often amongst the structures most severely damaged when an earthquake strikes.

One such structure is the belfry at San Silvestro in L’Aquila Italy, which is pictured below as it was in 1906. It was damaged by an earthquake in 2009, however repair works continued until 2019. One of the difficulties of designing such repairs, or indeed pre-emptive strengthening, is the inherent difficulty of predicting how the structure will respond to the dynamic forces that seismic events generate. No two structures behave in the same way and therefore each one presents a fresh puzzle. 

When a friend, and director of an MSc course in building conservation, recently passed me an academic paper with the somewhat catchy title ‘Identification and Model Update of the Dynamic Properties of the San Silvestro Belfry in L’Aquila and Estimation of Bell’s Dynamic Actions’, I read it with interest.

I confess that the mathematics contained in the paper are likely now beyond me. I am too far removed from doing analytical work in that depth.

That said, this is not always a bad thing as distance can sometimes bring perspective. In my view the most important diagnostic information in the paper is contained in figures 7a and 7b, because we can postulate from the depicted crack patterns what the complex mathematics mean.

That said everything that follows is complete speculation on my part. I have never visited the church, though I would like to, and I have no intimate knowledge of the way in which it was built. My entire speculation is based upon the reported pattern of cracking and some experience of designing buildings to resist earthquakes. 

The base of the tower is locked to the ground and therefore travels back and forth at the same rate as the quake shakes the ground. Since the tower is not infinitely stiff it begins to flex when the ground and tower base start to move. The time taken to flex means that the lateral movement at the top of the tower lags behind the base of the tower. At some point the base of the tower, which was moving left starts to move right, but due to the effect of lag the top of the tower is still moving left. The top and bottom of the tower are now out of sync. 

Thus, a snaking motion is set up in the tower, resulting in the crack pattern shown. It’s a bit like the wave you can generate in a skipping rope when you move the end up and down rapidly.  

In mathematical terms we might say the period of the tower is greater than the period of the earthquake. 

The maths in the paper essentially goes on to explain that the bell in the tower has no material effect on the period of the tower. This is because the bell’s mass, and hence momentum, is insufficient to overcome the stiffness and mass of the tower, even although the tower has flexed and cracked and movement at the top has lagged the base. 

Unfortunately structures like this are not at all like modern ones in the sense that they don’t have clearly discernible load paths and consistent properties. It follows that academics will always find it difficult to determine the stiffness, and hence period, of a given tower with accuracy i.e. the analytical answer can only be as accurate as the estimation of stiffness. All the maths in the world doesn’t overcome that problem. It’s painstaking detective work that is required. In fairness I think the authors of the catchily titled paper acknowledge this difficulty. 

I may well have over simplified the problem. For example, I haven’t mentioned the effect the rest of the church has, or what effect the windows in the bell tower have, or whether the tower’s stiffness changes over its height. In fact there are lots of things I haven’t mentioned. I am simply making an observation that the reported pattern of cracking is consistent with the explanation I have set out…… at least that's what I think. Feel free to differ.

On Garden Fences

The photograph below is a picture of a garden fence at my parents home. It divides a portion of their garden from their neighbours’. The fence is supported from a series of four square timber posts embedded in the ground. Horizontal timber battens are fixed to the timber posts both at the head and base of each post. Timber boarding is then fixed alternately either side of the battens. 

In one sense the fence looks unremarkable, but looks can be deceiving. On closer inspection it is evident that something is wrong. A vertical split has begun to appear at the head of the two middle posts. For the curious mind the question arises as to why this has happened and why only to the middle posts?

For those who have read my prior post titled ‘On Cladding Garden Sheds’ you may have an idea what the cause may be. For those who haven’t its worth re-capping. That said, while part of the mechanism is similar, ultimately the load-path is actually different.

‘When a tree is felled its moisture content could well be 100%. As it starts to dry out free water will evaporate from its cells until it reaches somewhere between 25 and 35%. Beyond this point water is lost from the cell walls of the timber fibres themselves. This causes the timber to shrink’.

It seems self evident that on warm days the timbers forming the fence begin to dry out causing all of the timbers to shrink. The battens are fixed rigidly to the posts supporting the fence; in the case  of the two middle post there is one either side. If both of the battens shrink at the same rate then they will pull on the middle posts in equal, but opposite directions causing a split to form in each.

Conversely the two outside posts have a timber batten fixed to one side only. They are therefore pulled from one side only. Since there is no counter pull these posts are free to flex. It follows that there are no splits in the two outside posts.

If the mechanism described above is correct the next obvious question is; why there is much less evidence of splits at the base of the fence; there are perhaps hairline splits at most? If all the battens are fixed in the same way why does the base not match the head of the fence?

There are several possible answers to this question. The true answer may be a combination of each. 

Firstly, the base of the fence is closer to the ground and therefore when it rains timbers in this location will receive splash back. Secondly, when it rains water will tend to run down the fence under gravity and will collect near the base of the fence. These factors would make the timber at the bottom likely to be more wet than those at the top.

A third factor would be that sunlight would reach the top of the fence for a longer period of the day and the base would tend to be in shadow for longer. Fourthly, there is dwarf wall built on the neighbours side of the fence located close to the base of the fence. This side of the fence would definitely be in shade for longer. When the sun shines from the neighbours side of the fence this effect would be exaggerated. Together these two effects mean that timber at the base of the fence would dry out more slowly.

Taken together it is self-evident that the timbers at the base of the fence are on average wetter than those at the top thus providing less potential for shrinkage. It follows that the forces experienced by the timber posts are less at the base.

It is also possible that since the bottom of the fence is close to the point at which the posts are embedded in the ground, the posts are harder to split due to their confinement by the ground.

The question now arises how could the fence builder have prevented the posts from splitting. The obvious answer would have been to double up the internal fence posts to create a movement joint, but of course that would cost more money. The moral of the story is that being cheaper carries its own price.

Postscript

For the most observant it is also interesting to note that the nails fixing the battens are corroded and stains are starting to appear in the timber. Conversely the nails connecting the timber boarding to the battens are not corroding.

One can see that the nail heads vary in size suggesting that they are of a different type. It would appear that the smaller nails used to connect the vertical boards are of the stainless steel variety while those connecting the battens to the posts are not. 

On Snow & Ice

Engineers are supposed to be numbers people who thrive on logic, objectivity and data. Engineers are not supposed to appreciate subjectivity; and we’re definitely not supposed to like art. At least that’s the theory. The truth is that art isn’t always as subjective as you might think and engineers can and do like art.

Fillippo Brunelleschi is most famous for designing the dome of Florence Cathedral. It is a spectacular structure designed and built using an exceedingly novel method of construction. Both architects and engineers claim Brunelleschi as one of their own, yet his apprenticeship was served in the Arte Delle Seta, where he became a master goldsmith and sculptor. That is how it was in the Renaissance, master builders and designers learned about material and form in the artist’s studio.

One of my favourite modern artists is Andy Goldsworthy. I wouldn’t remotely consider myself an art critic nor would I claim to know what Mr Goldsworthy was thinking when he conceived a particular piece, but I am going to speculate that he has developed a keen sense of material and form through hours of trial and error in the artist’s studio.

I have made this speculation, because he succeeds in combining natural materials with shapes and forms that make complete sense from a structural perspective. He appears to understand exactly what he is doing.

Perhaps my favourite examples of Goldsworthy’s work are those which he creates from snow and ice. I like them for several reasons. The first is because snow and ice illustrate particularly well that material properties can and do vary. For example ice remains solid when cold, but melts when warm.

Just as important, but perhaps more subtle, snow can be squashed and moulded into different shapes while ice is hard and brittle. It would rather fracture than bend.

Both materials have a dislike for tension; though they express their dislike in different ways. Snow will disintegrate and crumble, while ice will crack and fracture. Conversely both snow and ice will quite happily resist compression without difficulty.

It turns out that materials, like people, have temperaments that must be understood to get the most out of them. This Goldsworthy achieves exceedingly well.

If we consider, for example, the ice sculpture shown below. It consists of eight storeys each resembling the columns and entablature of a greek temple or if you prefer the sarsens at stone henge.

 

Just like the designers of those ancient structures, Goldsworthy has realised that tension is the enemy he must subdue. By placing the ice columns close together he prevents the lintels from developing excess tension on their soffits. For a similar reason the columns have been carefully aligned so that load can travel from top to bottom in direct bearing.

Further examples of matching form to material are shown in the sculptures below. One shows an arch constructed from thin wedges of ice and the other from stone and snow. Both resemble the classical form of a traditional masonry arch. 

It is of course well known that arches are compression structures and are therefore inherently suited to materials that dislike tension. With an ample supply of stone, and a primitive form of concrete, it is unsurprising that Roman architecture features arches so prominently.

That said, it is Goldsworthy’s decision to construct his arches in the traditional way using wedges that is interesting, particularly the one made of ice. This choice allows us to get a deeper sense of how arches work.

After finding ourselves unintentionally seated on the ground, everyone has undoubtably discovered that ice is slippery. Knowing this to be true why don’t the pictured ice wedges at the crown of the arch simply slip past each other, under the action of their own self-weight, thus causing the arch to collapse? This is not a trivial question.

In 1695 the Frenchman Philippe de la Hire was the first to compose a theory of masonry arches using mathematics. He began by assuming that the wedge shaped stones (voussoirs) from which arches were formed have infinitely slippery surfaces i.e. the joints between them are frictionless. He then set about tackling the question, how heavy [and by inference how thick] should the voussoirs be to keep an arch stable?

In this most slippery of scenarios it is, by definition, impossible for forces to develop parallel to the joints between voussoirs. The weight of the arch must therefore thrust exactly perpendicular to the joints.

It can be seen from the photos above that the wedges of ice [and thus the joints between them] are vertically aligned at the crown of the arch and therefore at this point the weight of the arch must act horizontally i.e. perpendicular to the joints.

Moving away from the centre of the arch the inclination of the ice becomes steadily flatter. Since, the weight of the arch must still act perpendicular to the joints it is bent around the curve of the arch. 

This is where the problem starts to get interesting. If the arch is built on a flat base, as shown in the image above, La Hire discovered that the weight acting at its base must be infinite or the arch will be unstable, which is obviously wrong.

La Hire rightly concluded that friction must therefore be present between the voussoirs [even if they are made of ice] though it was left for others to account for it in subsequent theories. It is this frictional force that stops the wedges of ice slipping from the arch’s crown.

In some ways it would be satisfactory to end here, but we are not quite ready to finish. There is something else that turns out to be important, which we have not yet discussed. Thus far we have been taking about wedges or voussoirs and the analogy holds reasonably well in the instance of the Goldsworthy’s snow arch.

In the case of his ice arch the analogy is a little imperfect, because the shards of ice are in fact flat and not really wedge shaped al all. We would normally think of a wedge as having a fat and a thin end.

This doesn’t at all undermine what we have said thus far; all of that still holds. What is interesting is that without a fat and thin end the shards of ice are in contact on the inside of the arch, but gaps necessarily open up at the outside edge. The thicker the arch the more pronounced would be the gaps.

The significance of this is the contact area between the shards of ice is only a fraction of their surface areas. Since the ice does not fracture we may infer that the stress in the arch must be quite low and would certainly be in no danger of crushing its component parts. 

This principle was demonstrated in 1846 by Barlow at the Institution of Civil Engineers in London. He built a model arch with six voussoirs using slender prices of wood en lieu of mortar. In progressively withdrew the slips of wood in three locations to show the stability of arch would be maintained.

Taken together the thoughts we have outlined illustrate the key principles of masonry arch design. Namely, friction must be present between the stones; the strength of the stone is of little importance; and finally the stability of an arch relies entirely on its geometry and weight. Since weight is a function of geometry and friction is a function of weight we might just as easily say the stability of an arch is a function of its geometry.

I have no idea whether or not Andy Goldsworthy’s thinking has extended this far, nevertheless the question was worth addressing, because the answer surely enhances our appreciation of his art.

Now I realise in reaching this point that some may be thinking that I have, in discussing geometry and forces, undermined the original premise of this post. You might say that I have turned art into science.

I beg to differ. 

Isn’t the point of modern art it’s subjectivity? Isn’t it supposed to make us think individually about what it represents and then decide how that makes each of us feel? Well, in my subjective view this is what I think it represents. It satisfies my curiosity and that makes me feel happy.