On Lies, Damned Lies & Statistics

Why statistics are actually quite useful

“Figures often beguile me,” wrote the American Author Mark Twain, “particularly when I have the arranging of them myself; in which case the remark attributed to Disraeli would often apply with justice and force: ‘There are three kinds of lies: lies, damned lies, and statistics’”.

It is somewhat ironic that a sentence about degrees of falsehood appears to contain one, though most likely unintended. The earliest known appearance of the phrase attributed to British Prime Minister Benjamin Disraeli was in fact after his death.

Never-the-less it is a pithy phrase, which has remained in common parlance, most likely due to the habit politicians have for selectively marshalling statistics in support of their policy decisions. The idea being to give them a veneer of mathematical rigour.

This habit led, over time, to an acute public distrust of Government Statistics; so acute that in 2007 Parliament passed an act which made the Office for National Statistics independent of Government.

This is why I fear that this, my attempt to show why statistics matter and why they are in fact very reliable, could backfire spectacularly.

In order to make that case we must turn to the subject of how we know that a structure designed by an engineer will not fall down. It is of course self evident that some structures do fail, however before explaining why that is we must first explain why most of the time they don’t. We are not here talking about situations where a design has been created in error; that may not be fairly attributed to statistics.

It would of course be possible to build a structure and then test it for the desired load, however this is not entirely practical. If the structure fails to pass then there will have been a rather significant waste of time and materials to build it. Something more reliable is therefore required. 

A second option would be to proof test each of the components that are used to fabricate a given structure. This was in fact the method that was sometimes used, particularly for large or novel civil engineering structures. This has the advantage of not destroying the whole structure if a test specimen fails.

That said, it is still not a terribly efficient method manufacture, due to the time and expense of the testing regime, which is bespoke to the structure being constructed and cannot be re-used for different arrangements. 

It might be possible to test a model of the whole structure, however this approach has many difficulties. Firstly, most structural effects do not scale linearly i.e. behaviour at small scale is different to that at large scale, therefore in all but the simplest cases the model structure will not represent real behaviour at full scale[1]. Secondly, if we solved the scaling problem we still could not be sure that the actual model would behave as the real structure, because we have not tested the actual structure. This brings us back to the point we started.

What is needed is a reliable method of predicting the point of failure irrespective of whether a structure is a test model or the real thing.

Supposing that we required iron chain links, which are to be used in a suspension bridge, to carry 4 tonnes of load. If we tested a link and found that it could resist 1 tonne then we might conclude that 4 links are required. 

The difficulty is that if we did another test we are likely to get a different answer; perhaps less than 1 tonne; perhaps more. We could conduct 10 tests and each time we would very likely get a different answer. The reason for this is the presence of small, perhaps imperceptible, differences in the manufacturing of each one, which are impossible to eliminate completely, even with modern technology. 

The question therefore becomes, which of the test results should be used for the design?

Some would say the average figure should be used, however it does not take long to work out why this cannot be correct. It is self-evident that half of the links will have a capacity that is less than assumed in the design. That cannot possibly be a safe outcome.

Another idea would be to adopt the lowest value. Surely by definition this option must produce a safe, if cautious, result. If, however, we think a bit further we might consider how we know that any of the10 tests we have conducted thus far represent the lowest test result that we could get. After all, what if we conduct test number 11 and the result turns out to be lower than the lowest we had obtained thus far?

This turns out to be a bigger problem than initially seemed to be the case, for how do we know that test 12 won’t be worse still? The question appears to recur infinitely. We need a better answer and that is where statistics comes in.

All materials, or components for that matter, are imbued with a curious property that is wholly reliable, completely unchangeable, and is it would seem, an intrinsic property of nature itself. 

Despite its universal applicability, it is a deceptively simple concept to demonstrate. Supposing we started by plotting a bar chart [histogram] showing the distribution of our results. Along the bottom axis we plot the capacity of the links we have tested and on the vertical axis we plot the number of links that achieve a given capacity.

As we plot the test results we will find that they start to cluster around the average with fewer results corresponding to capacities either much greater or much less than the average.

With still more tests the cluster will start to develop into a recognisable pattern resembling the shape of a bell, where the top of the bell is located at the average value. It will soon become clear that no matter how many new results are added the overall pattern of the results does not change; the bell is there to stay.  

For obvious reasons a curved line which bounds all of the plotted results is called a ‘bell curve’. Johann Carl Friedrich Gauss was the first person to describe such curves mathematically, which is why they are also known as Gaussian distributions.

As soon as it is possible to describe the curve mathematically we need only a limited number of test results to do so. Once we have done this mathematics also gives us the ability to predict the probability of a given capacity being exceeded, or in other words we can predict how likely the material or component is to fail. All we then need to do is decide what probability of failure is acceptable.

This is an interesting question for several reasons. Firstly, it is not entirely an engineering question. It requires a decision to be made about what level of failure is tolerable. This is unquestionably influenced by public perception of what ought to be. Public opinion can be a fickle thing, which can often be contradictory, especially when emotion is involved. 

For example, if something has been done a particular way for many years the public are loath to have it changed or restricted in any way. You are bound to hear someone say “it has always been done this way and it has performed been perfectly well. Why do you feel the need to change it.”

In reality this is often not the case. What someone thinks represents the way something has always been done often turns out to be nothing of the sort. It is in fact an imitation of what has been done previously. In reality it has been changed over the years in many small imperceptible ways, but the cumulative effect of those changes has not always been appreciated and may one day be the cause of failure.

Conversely if there has already been a major failure, particularly one that has caused the loss of life, then the public demand will be for immediate change and for someone to be held to account. It will be said “how did nobody foresee that this could happen? There must be either negligence of incompetence. This cannot be allowed to happen again; something must be done.”

In this way public opinion is largely based on momentum. We do not wish things to change until there has been a rude awakening that forces us to re-think and then we must have change and the status quo will not do.

One of the ways that engineers think about this problem is to consider both the likelihood and the consequences of failure i.e. how probable is it for something to go wrong and what would happen if it did? 

It is obvious that the failure of a nuclear power plant or a hydroelectric dam would have consequences far greater than the failure of a roof purlin in an agricultural barn. Quite reasonably one might wish the former examples to be less likely than the latter.

It is also obvious that it is not economically viable to design barns like power stations. Nor is it aesthetically desirable to design barns that way. It may seem curious to discuss aesthetics in this context, however humans do place a value in art and culture. We do not wish to live in homes that look like nuclear bunkers and we rarely purchase a car based wholly on the results of safety testing. Aesthetics are important to us.

Perhaps a more subtle example of likelihood and consequence is to re-consider our bridge links. Since we have determined that more than one link is required. It seems improbable that all the links would be sub-standard, which opens the intriguing possibility that the systemic probability of failure may be quite different to the probability of an individual component failing. This would of course rely on the remaining links being capable of sustaining extra load due to the failure of their neighbour. This introduces the concepts of redundancy and margins of safety, but these are for a different blog post.

For now we are simply concerned with the bell curve and its ability to predict the probability of failure in a reliable way, based on a finite set of test results. It gives engineers the ability and the freedom to make reasoned judgements about cause and effect, however it does not provide engineers with an answer to the question of what an acceptable failure is.

This is of course the reason why some failures do happen; it is simply because they have been deemed to be acceptable, perhaps on the basis of a more robust structure being too expensive when compared to the perceived benefits. However, as we have learned, what is acceptable is not a fixed thing and may change with public perception.

A good example might be the failure of a flood prevention barrier shortly after it is completed. Presumably people chose to live close to the water, because they like the view and they do not wish to live behind a large wall that obscures said view. 

That being said, those same residents, when flooded, are not particularly interested in hearing that given the magnitude of flood event that has occurred failure of the barrier had been judged tolerable.

Statistics are therefore an eminently useful thing, which can inform our decisions, but cannot make tough choices on our behalf. We must decide what our appetite for risk is based on many subjective factors. Whether they know it or not this is the dilemma faced by all politicians and is probably what tempts some of them to misuse statistics. Like anyone else they do not wish to be held responsible for a decision that might be criticised in retrospect. They would rather believe that statistics can absolve them of that responsibility.

Unfortunately statistics can only ever tell us the probability of an event and not whether said event is good or bad. Perhaps Mark Twain should have said:

"There are three kinds of lies: lies, damned lies, and self-deception”.

[1]I am aware that there was a period in time where testing of models was adopted, but this is a rather complicated subject involving some interesting mathematics and is perhaps a subject for another time

 

On Hambly's Paradox

Some thoughts about the complexity of design

Hambly’s Paradox is an example which illustrates the difficulty of assessing so called hyperstatic structures i.e. those with redundant members. It was first posited by the eminent structural engineer Edmund Hambly, whose name it bears. 

A milkmaid weighing 600 N sits on a three-legged stool. For what basic force should each leg of the stool be designed?

The answer to the question is of course 200 N. We are, of course, assuming the stool is symmetrical and that the milkmaid is sitting in the centre of the stool.

The milkmaid now sits on a square stool with four legs, one at each corner, and again the stool and the loading are symmetrical. For what basic force should each leg of the stool now be designed?

The obvious answer would be 150 N, however as Hambly points out that would not necessarily be the right answer.

If we assume that the milking stool is stiff and so is the floor on which it sits then inevitably the stool will rock and only three of the legs will be in contact with the floor. Even if raised by only a fraction of a millimetre the load in the fourth leg is unquestionably zero. 

Since, the stool rocks there must be a point where both the fourth leg and the one diagonally opposite are both free of the ground. In such circumstances both legs carry zero load and the remaining two legs must therefore carry 300 N each.

If the stool is instead placed on rough ground the we have no way of knowing which of the legs will or will not be in contact with the ground and therefore every leg must be designed for 300 N.

This is, of course, the paradox. By increasing the number of legs supporting the stool the design load for each has effectively doubled.

The conclusion that we have reached is based on static analysis i.e. our only concern has been to ensure that the stool remains in equilibrium. Such analysis contains a prior assumption the consequences of which have gone unnoticed until now. That is the assumption that we are dealing with rigid bodies. 

This is not how the real world works. All materials deform when subjected to load. The amount of deformation depends on their stiffness. We could consider the stiffness of the milking stool seat and its legs in our assessment, but it is probably sufficient, and somewhat simpler, to consider only the legs.

When the legs are loaded they will shorten a little. The amount they will shorten is proportional to the stiffness of the material they are made of. If the gap between the so called fourth leg and the ground is small it may, as a result of shortening, become in contact with the ground causing the weight of the milkmaid to be redistributed according to the stiffness of each leg. Therefore, depending on the size of the gap the fourth leg may experience a load anywhere between 0 and 300 N.

It follows that thus far geometric imperfections make it almost impossible to predict the actual load in a 4-legged milking stool, even after the elastic properties of each leg has been taken into account.

There is of course one further step we can take, which will restore common sense to milking stool design, but first we must make a further assumption. If we make the milking stool legs relatively stocky then they can reach a state of plasticity [1] without their legs buckling.

In this scenario let us suppose that each leg of the milking stool has been designed to yield at a load P. Let us then suppose that the weight of our milkmaid is 4P. At the outset of our problem the stool is potentially supported on only two legs this means that those two legs are momentarily exposed to a load of 2P each and 4P in total. Each leg shortens elastically until it reaches its yield point whereupon it continues to squash until load is shed into the two adjacent legs. Since the yield point is P load will continue to shed until it is shared equally between all 4 legs. Thus, each leg of the milking stool may be designed for 150 N. 

And just like that our paradox vanishes!

[1] plasticity is a state that exists in a material beyond the point at which it yields. In this state the material will continue to deform without being exposed to any additional load. I may write more about this in another post

[2] My thinking for this post borrows from Jacque Heyman

On Waterloo

One of the questions sometimes asked of scientists is whether learning how the universe works somehow diminishes their sense of wonder and mystery? I am not a scientist, but I have been interested in science ever since I was a child. I cannot speak for anyone else, but my answer would unequivocally be no It does not diminish my sense of wonder; it actually increases it.

The image below is of the Eurostar terminal at Waterloo station in London. The terminal was closed in 2007 when it was replaced by St Pancras, which is why it looks a little tired. Nevertheless, I think that it remains an elegant structure, however the question arises does it provide more or less wonder when you know how it works? I will leave it to the reader to decide.

The first thing that anyone looking at the structure should understand is that its form is deliberate. In common with many significant structures it is not an architectural whim. The respective roles of the architect and the engineer in relation to aesthetics is not the subject of this post, however I do hope that it will leave the reader with an increased appreciation for the engineer’s contribution.

The next thing that we should note is blindingly obvious.  The structure is formed of a series of steel arches, however the arches are certainly not conventional. They are formed from two curved trusses joined with a pin at the crown and further pins at each of the two bases; though only one of them can be seen in the photo.

Another observation that we might make is that the depth of the truss on the near side projects above the roof while the situation is reversed on the far side. Similarly, while both trusses have three longitudinal chords, the two outer chords are more slender on the near side and the single inner chord is more slender on the far side.

A more subtle observation would be that the thicker chords get thicker towards the middle of their span. This is seen most clearly on the near side. 

Less obvious, due to the angle of the photograph, is that the overall span of the arch is not symmetrical. The span of the near side truss is shorter and more steeply inclined. 

If these forms are deliberate the question must be asked; what on earth is going on? To answer this question we cannot avoid learning some basic structural principles. The first of these is the principle of strut buckling.

If we were to apply load to a short squat object, for example a rock, we would effectively have to apply sufficient force to crush the rock before it would fail. The same cannot be said of, for example, a thin branch taken from the edge of a tree canopy. If we apply a compressive force to the branch it will bow in the middle and fail in bending long before the compressive strength of wood has been reached. This effect is known as strut buckling. There is clearly a relationship between the length of a member and its predilection to buckling in compression. As it happens buckling capacity is proportional to the square of the length.

Conversely the load which causes a member to buckle in compression can by applied to the same member in tension without effect. We all know this in principle, but perhaps without realising it. Consider the absurdity of trying to apply a compressive load to a piece of string and compare that with an application of tensile load to the same piece of string.

We may now apply this logic to the structure in the photograph. Those members with a thick cross section can be inferred to be in compression to avoid buckling while those that are slender must be in tension.

This inference is interesting in so far as it goes, but if we are to follow the logic to its conclusion then that would mean compressive and tensile forces must reverse either side of the central pin. How or perhaps why should that be so? Surely, everyone knows that an arch is a compression structure so what’s going on?

I think the answer lies in the asymmetry of the arch. Supposing there happens to be a day of heavy snowfall. It seems reasonable to suppose that the snow would slide off the steep side of the arch while settling on the flat side. Perhaps there is also a light wind that exacerbates the effect by causing the snow to drift further onto the flatter far side. There is now an uneven load on the arch due to both the self-weight of the asymmetric arch and the snow that has gathered on top.

This would of course cause the far side of the arch to dip while the near side would respond by rising. At some intermediate point the arch would be neither up nor down. We call this the point of contraflexure and amazingly its position will almost certainly correspond with the position of the pin at the crown of the arch; almost like someone intended it to be so.

Once we have grasped this idea the next mental step is not too far away. If the near side has risen the outside chords of the truss will be stretched while the inside chord will be squashed. On the far side the relationship is reversed. We now have an explanation for exactly why the members are fat and slender on each side of the arch. 

We can also make the quite reasonable assumption that the compression chords get thicker towards the middle, because that is the point at which buckling wants to occur. 

The only question left is why the arch is asymmetrical in the first place. It could be because of the route the railway takes into Waterloo station and the implications this has on the shape of the structure……or it could be an architectural whim. It’s a bit of a mystery, perhaps I’ll let you decide.

On Camping

Some observations about tents

A recent camping trip got me thinking about how tents work; specifically my own. I think it seems obvious to most casual observers that tents are tension structures, but perhaps there is also more to it than that.

I like camping, but I am not obsessive about it. I am therefore not sure if there is a specific name for the type of tent that I own. I shall simply be referring to it as a horseshoe tent, because its cross-section is roughly that shape.

The horseshoe shape is created by an exoskeleton consisting of three arches, two larger ones at the front end and a slightly smaller one at the rear. The smaller one is inclined a little towards the rear of the tent. The rear of the tent is also the location of the sleeping compartment, which is hung between the two arches closest to the back.

The arches themselves are formed of thin plastic tubes, which slot together to form long flexible rods. These are held together by an elastic chord which passes through their center and allows them to be folded away without coming apart. Each rod is inserted into a long pocket stitched to the tent’s waterproof flysheet. Each end of the rod is then inserted over a brass pin attached to the base of the flysheet. Each pin is connected to its neighbour via a nylon strap, which sits beneath the tent’s ground sheet. In the photo below you can see them on the grass before the ground sheet has been laid. 

This is the first structural feature of the tent. The flexible arches have a natural tendency to spring apart, but the tension strap prevents them from doing so. It locks the springiness of the tent rods into place and keeps the rather thin structure rigid.

The next structural feature of the tent is its guylines, which we shall divide into two. The transverse guys i.e. those in the same plane as the arches have only one role, at least so far as I can tell. They anchor the tent to the ground, via the arches, and prevent uplift in stormy weather. I believe this to be the case, because the tent will stand happily without them in calm weather. 

The second set of guylines we shall call longitudinal guys. They are located to the front and rear of the tent. I am sure that they help anchor the tent in stormy weather too, but they also have another, perhaps more important role, as I suspect the transverse guys could do most of the wind resisting work themselves.

It seems to me that the primary role of the longitudinal guys is to stretch out the shape of the tent and to provide it with longitudinal stiffness. The more taught they are made, by adjusting the tensioners, the more rigid the tent becomes. 

It is self evident that to ensure equilibrium i.e. to stop the tent being pulled either forward or back the front and rear guylines must balance each other. What is perhaps less obvious, and is therefore more interesting, is what happens to the tension force after it leaves the guylines. Put another way, what happens between the front and rear guylines to ensure equilibrium. This is what caught my eye and is the underlying reason for this post.

While erecting my tent I had become a little frustrated that the flysheet appeared to be wrinkled and was not sitting flat and smooth as I thought that it should. Then I noticed that the wrinkles had a pattern and that pattern was structurally significant. 

I realised that in any other context an engineer would describe a structure like a tent’s flysheet as a stressed skin. Stressed skin structures resist out of plane loads, for example the wind, by being stretched tight. 

Normally a stressed skin will be fixed along the edges either with a continuous seam or with fixings at close centres. This is because you want the load to be imparted into the thin stressed skin in an even manner. This is not, however, the case with tents.

The tensile loads, which convert the flexible flysheet into a taught stressed skin structure, are imparted at discreet points by the guylines. In the case of my tent via the thin rods that form the horsehoe arches.

The slender rods have little stiffness perpendicular to the arches they form and are of little assistance in distributing the tensile load from the guylines. It is therefore no coincidence that the observed creases extend diagonally from the guyline connections on one arch to the base of the adjacent arch, where it is pegged to the ground.

On closer inspection there is a second set of creases, which start midway between the guyline connection and ground level. This happens to be the point where the rods are connected to the flysheet with plastic clips. 

The question arises, why does the fabric crease and what does it indicate? 

As the flysheet has little thickness it is not good a distributing concentrated load across its own surface. For this reason load applied at discreet points by the guylines causes localised stress in the fabric between points of restraint. Since these parts of the fabric are subject to more stress they stretch more than neighbouring fabric causing the observed creases to appear.

What this phenomena indicates is none-other than the arrangement of internal forces, one might say the load path, in the flysheet. We can visibly see that it has taken up the form of a truss with node points at the parts of the tent that are held stiff. Structures do not often reveal their load path in this way, which makes it all the more interesting.

It is also interesting to note that there are diagonal creases near the top of the tent too. These appear to be the result of the additional section of flysheet that helps ventilates the roof and provides additional stiffness.

One can also detect some creases in a rectangular section of the flysheet at mid height of the first bay. It is difficult to see in the photo, but at this location there is a clear plastic window, covered on the inside by a canvas flap that is used to blank the window for privacy. The clear plastic is a stiffer material than the flysheet around it.

Something else that interested me was why the direction of the creases was in one direction rather than the other. The answer I believe lies in the different geometries, which exist at the front and rear of the tent, and thus vary the angle at which the guylines apply load to the structure. That said, I am also curious to know whether the order in which the guylines were tightened also plays a role. Next time I pitch the tent I shall reverse the sequence and see what happens.

The final observation that ought to be made about my tent, or any tent for that matter concerns the effectiveness of tent pegs. These are short lengths of wire, which are pushed into the ground in order to provide resistance to the the tensile loads in the guylines, including those generated by the full force of the weather. This seems remarkable given their small stature. 

While the angle at which the pegs are pushed into the soil is surely important, friction between the soil and the circumference of the pegs is the primary guard against the pegs being pulled free of the ground. Thus, we can see the ability of friction to resist load is not to be underestimated. It is this same force by which many buildings are supported in soft ground on concrete piles, though in this case the piles are in compression rather than tension like tent pegs.

So it turns out that I needn't have been concerned about the creases in my tent; there doesn’t seem to be anything I could have done about them. This of course wasn’t the main reason I enjoyed my camping trip. Good humour and good company had something to do with that too, however it was quite interesting.