On Significant Digits & Rounding

From time to time something comes along that bothers you. You know that in the big scheme of things it’s a trivial matter and that it shouldn’t bother you quite as much as it does. When it happens again it really bothers you. If it becomes a habitual occurrence then sooner or later you are going to feel the need to get it off your chest. I have arrived at that point and feel compelled to say so. I think that’s ok every now and then, right? 

My primary complaint is this: ’accuracy’ and ‘precision’ are not synonyms; they are not the same thing.

‘Accuracy’ means how close something is to being correct, whereas ‘precision’ means how specifically something is described. It is therefore entirely possible to be very precise, perhaps to several decimal places, but at the same time completely wrong i.e. inaccurate.

Similarly, it is possible to be accurate without being precise. In engineering we almost always prefer accuracy to precision, particularly when added precision makes little or no difference to the level of accuracy.

Let me now expand a little and explain why this bothers me; I assure you it isn’t to be pedantic.

If I ask someone to calculate the amount of force at the base of a column or the amount of deflection in a concrete floor slab I am definitely looking for an accurate number not a precise one. I do not expect the answer to be reported with a precision that cannot be justified.

For example, supposing I report that the force in a concrete column is 14,976.35 kN. I am effectively making the claim that I know the magnitude of the force to the nearest 1/100 of 1 kN, even although the magnitude of the force is almost 15,000 kN. This implies an accuracy of almost 1/1,500,000 and a precision of 0.01.

This is clearly nonsense. I do not know the density of concrete to a precision of two decimal places nor can I measure the floor thickness to that precision. It follows that the reported output is more precise than the input; that cannot be right. While the answer is precise, it is no more accurate than if we had rounded the last four digits.

The concept of significant digits is supposed to help us solve this problem, so perhaps it’s worth a quick recap; just to get it off my chest.

In any number a significant digit is any digit from 1 to 9 or any zero that is not used to show the position of the decimal point. For example:

345, 8.62, 3.80 and 0.00654. 

Each of these numbers have three significant digits. So far so good, but we’re not done yet, because there are still some cases where we need to clarify how to treat the zeros.

Let us use the number 76,000 as an example. Should we read this as having two significant digits with the three zeros defining the position of the decimal or is one or more of the zeros accurate giving us  three, four or five significant digits?

There is of course a convention for dealing with this, which is unfortunately seldom applied. If we write the same number 76x10³ then we immediately know there are only two significant digits. If it had been written as 760x10² then we could assume three significant digits and so on.

For easier manipulations during a calculation it is convenient to work with exponents that are multiples of three, at least it is in the SI system [our American cousins may not find this as useful], however the final answer should be converted back into the requisite number of significant digits in order to clarify the accuracy of the output.

Now, if you thought this post was trivial thus far its about to get worse. We need to talk about rounding and why most calculators and computer software does it wrong. I am talking to you Microsoft.

The way in which we set aside insignificant digits is called rounding. The rules for rounding are straightforward. If the number being discarded is less than 5 we round down and if it is greater than 5 then we round up. For example:

456.33 rounded to 4 significant digits becomes 456.3. Rounding 674.68 to 4 significant digits we get 674.7

The tricky decision is what to do when the insignificant digit is 5. “Round it up”, I hear you say. That is what Microsoft Excel would do, however I say there is a better way.

An improved rule would be to round off to the even digit. For example 43.25 rounds to 43.2, whereas 43.35 rounds to 43.4[1].

The reason for this rule, as apposed to always rounding up, is to stop rounding errors from accumulating, particularly in long calculations. Since odd and even digits occur in a more or less random sequence the rounding up cancels out the rounding down. It is therefore a better way of calculating an answer.

We are now full circle and back to where we started. My request is this: Please select a number of significant digits [and round the insignificant ones] so that the answer isn’t more precise than the input. Choose accuracy over precision. 

The primary advantages to this approach are:

1. You are far less likely to make a mistake if you're not carrying all those digits. 
2. You’ll do the calculation quicker if you’re not carrying all those digits.
3. Most importantly, you won’t look daft on a construction site when you ask the contractor to measure the floor thickness to two decimal places.

[1] I realise someone smarter than me is probably about to point out that there is a menu option in excel that can fix my gripe. In which case my argument is softened, but not defeated. Why isn’t it the default? 

p.s. Microsoft, please don’t get upset and crush my blog.

On Breaking Trains

Or why systems need to be robust

On 22 October 1895 a steam locomotive approached its Paris terminus slightly faster than normal hoping to make up lost time. Except that rather than stopping at Gare Montparnasse, as planned, it crashed through the end of the platform, over the concourse, through the station facade and down onto Place de Renne a full storey below. Apparently, the only casualty was a woman who had the misfortune to be standing in the street and was struck by falling masonry.

 

The question arises, why did the train fail to stop? 

Following the subsequent accident inquiry the hapless locomotive driver is reported to have been fined 50 francs and sentenced to two months in gaol, because he approached the station too fast. One of the guards was fined 25 francs, because he was apparently too pre-occupied with paperwork to apply the hand brake.

One might assume that this was all there was to it, however as it turns out the driver and the guard were not solely responsible. 

It is also believed that the train’s Westinghouse air brakes had rather tragically failed, which seems to me a rather more significant event, but not for the reason that you might think.

Trains were of course a wonderful invention, which had transformed the world by making mass transit possible over long distances. The trouble with steam locomotives, at least in their infancy, was that nobody was quite sure how to stop them.

They travelled faster than anyone had travelled before, but were also big and heavy. The locomotive had a hard enough time stopping itself let alone the passenger cars and goods wagons that followed behind. It was not unusual for the following carriages to catch up with the locomotive when the brakes were applied causing them to collide, first with each other, and then with the back of the locomotive.

In order to make the train stop within a reasonable distance it was realised that brakes had to be added to the carriages too. The obvious difficulty was how to apply the brakes on the locomotive, and all the carriages, at the same time.

Initial solutions were somewhat rudimentary. In the United States a brake man sat on top of the first carriage. When the driver blew the train’s whistle he was responsible for applying the brake on the first carriage. He was then required to run down the roof before leaping onto the next carriage whereupon he applied its brake. This process was repeated until he reached the back of the train. This was, as one could imagine, a rather precarious job and not surprisingly there were many casualties.

The American entrepreneur and engineer George Westinghouse, like everyone else, saw the problem. Unlike everyone else, Westinghouse came up with a solution. He joined the carriages together with airtight hoses and used compressed air to apply the carriage brakes almost simultaneously. The system worked brilliantly, bringing trains to a halt with great effect. For many people the idea of stopping a large heavy object travelling at high speed with nothing more than air had initially seemed a little crazy. When it worked Westinghouse was rightly seen as a genius.

Except that there was a problem that no one at the time had foreseen. If there was a loss of air pressure, due to leak in the system, the breaks wouldn’t work. It is believed that this is exactly what happened at Gare Montparnasse. Understandably, fail safe systems where added to subsequent designs.

Knowing this story I was rather intrigued by the heritage steam train that I happened across while on a recent camping trip in Cumbria. In the images below you can see a red  pipe on the back of the locomotive, which passes between all the carriages and can also be seen at the tail of the last carriage. There is also a small pressure vessel beneath one of the seats in each carriage, but you can’t see that in these pictures.

   

In case you haven’t guessed I rather suspect that what we have here is a rather old fashioned air-break system not unlike those used on early locomotives. I didn’t get chance to investigate further, but I am going to assume its the mark two version.

The next question is what this has to do with a structural engineering blog? The reason I decided to write, other than the fact that I found it interesting, is the principle of robustness. An otherwise brilliant idea, which made a big difference to the safety of trains, was, in its earliest conception, flawed. It wasn’t flawed because it didn’t work. It was flawed because it was vulnerable to miss-use or accidental damage. In short it was not a robust system.

This ought to be a concept familiar to all structural engineers. The archetypal accident, at least in the UK, was the partial collapse of a 22 storey tower block in 1968. The tower stood quite happily until a gas explosion blew out one of its walls, causing the walls and floors above to collapse like a pack of cards. Unfortunately the component parts were not adequately tied together and were therefore unable to bridge over the damaged section of the structure. Four people died and 17 were injured.

The concept of robustness is not necessarily aimed at particular events or circumstances, rather it is intended to provide a degree of resilience against the unforeseen and the unknown. It now seems obvious that structures should not fail the moment the design load case has been exceeded, but it was not always so. 

Of course a supplementary question one might ask is how robust does a structure need to be? How robust is enough? That’s a difficult question to answer, but an ingenuous formulation has been devised, which has come to be known as the principal of ‘disproportionate collapse’. Put simply this means that any damage suffered by a building should not be disproportionate to the event that caused it.

So what is considered proportionate? That’s a rather big question, which perhaps needs its own post at some future point. 

On Carbuncles

Why its worth preserving ugly structures? 

One of the questions I am frequently asked is why buildings and structures that seem to have no aesthetic merit have been Listed for preservation by the conservation authorities. 

I confess that, despite being an enthusiastic exponent of conservation engineering, I too struggle with the requirement to preserve some structures. 

For example, I am quite sure that I share the majority view that brutalist architecture is ugly and the genre has not delivered the utopia that was promised. In saying this I recognise that I am wholly out of step with many, possibly the majority, of architects.

Conversely, I have found that when reading Le Corbusier’s philosophy of brutalist design I am wholly enthused and compelled by his logic. It jars that there is a complete disconnect between the eloquence of his written intent and how it has transferred into the real world. Sadly, it has always been this way with utopian ideas.

So while the public declares that ‘the emperor has no clothes’ the architectural profession continues to be enthralled by Le Corbusier’s philosophy and principles, seemly blind to the simple fact that brutalism has created a legacy of truly miserable buildings that do not work in practise. Lest my architectural friends chide me for suggesting brutalist buildings do not work let me clarify what I mean. 

Whatever their perceived architectural merits, direct experience has taught me that their fabric has not stood the test of time. In many cases the distress and deterioration are intrinsic to their design and detailing. 

Another interesting case would be the preservation of industrial buildings. In this instance I am possibly further away from the majority public opinion, though I don’t think there is an equivalence with brutalist architecture. Redundant industrial architecture generally served a useful purpose in its day and often had a cleverness about its design. For example, the structural efficiency of some cooling towers and gas holders is remarkable.

I would also adopt the view that some industrial design does actually have an aesthetic quality, though perhaps, as an engineer, that is my blind spot. I rather suspect that at some subliminal level understanding why something works converts into an enhanced aesthetic appreciation. 

I suppose if this hunch is true, and I am to apply it consistently, then I fear I must grant my architectural friends some latitude too.

All that being said, while I cannot bring myself to appreciate some Listing decisions, I think that I can shed some light on why the system is so.

The first step is to understand that the point of the Listing system is not to preserve aesthetically beautiful buildings, although many Listed buildings do fulfil that criteria. Nor does it mean that the decision to preserve a particular architect or engineer’s work is because their work is beautiful.

In such cases the important question is why the work of said designer is considered important not whether their work is beautiful. More often than not it is because they changed the way something was done and caused us to think differently about design.

As I have previously noted I can accept that there is an elegance to the writings of Le Corbusier; though I would argue that the theory he expounded has been proven hollow. I would also accept that his designs definitely represent a change in how things were done, though I would argue not necessarily for the better. I imagine that contention will not meet with universal approval and is an argument that is unlikely to be resolved any time soon, but clearly that is not the point.

So what is the point?

In a prior post ‘On Conservation Principles’ I noted that buildings reflect changes in the way we live and work, and also the cultural values that were prevalent at the time. An industrial chimney may be worth preserving, because in its day it was a community’s raison d’etre. The community exists because the factory exists. Preserving the factory preserves evidence of a way of life and a way of working.

This is the point. Some structures are preserved not because of what they look like, but rather because of what they represent. This is the reason I don’t mind working on projects where I don’t necessarily like the aesthetics. I can nevertheless appreciate the history and culture that is involved. If I get really stuck I can still fall back on finding enjoyment in whatever puzzles the project throws up; there are always engineering puzzles to solve[1].

In this sense a nation’s infrastructure and building stock is much like the rest of its history. There are parts to be proud of and there are parts you rather wish hadn’t happened. Unfortunately only the future is still to be written. For better or worse we own our past both good and bad. We should try to understand our Carbuncles**.

[1] that’s not the only reason. I am also not sure that many people would like to live in a world where I got to decide what qualifies as being aesthetic and only what pleased me could be built!

[2] for those too young to remember, in 1984 Prince Charles described a proposed extension to the National Gallery in the following way. “What is proposed seems to me a monstrous carbuncle on the face of a much-loved and elegant friend”. His view got him into bother with Architects at the time, but I just think its a funny phrase.

On Pile Caps [again]

The single pile cap

At the conclusion of my prior post ‘On Pile Caps’ I posed the question; would a single column supported by a single pile result in a straightforward pile cap design? It has to, right? The load-path is so obviously simple. Load passes from the column directly into the cap and from there directly into the pile. It’s surely a clear example of load transfer by direct bearing? There isn’t any bridging to be done.

We are only one paragraph into this post so patently there’s something else going on. The question is what? To answer that question we need to take a step back and start thinking about the mechanics of materials in a more nuanced way.

We are going to do this by putting concrete aside for moment and considering some other materials that make it easier to visualise what might be going on.

When a bar or rod is loaded in tension it will start to stretch in the direction of the load. The amount it stretches is proportional to the material’s stiffness. What is perhaps less obvious is that while the rod stretches it is also experiencing a lateral contraction perpendicular to the action of the load.

You can actually see this happen if you stretch an elastic band, however in stiffer materials the phenomenon is harder to detect without instrumentation.

We can also reverse this process. If we take a short rod [to ensure it does not buckle] and apply a compressive load it will start to squash in the direction of the load. As before the amount it squashes is proportional to the materials stiffness. Since we are reversing the load’s direction we can reasonably expect the rod to experience a lateral expansion perpendicular to the direction of the applied load.

This phenomenon can also be seen visually if, for example, you were to squash a bathroom sponge.

In engineering terminology the amount a material shortens or lengthens is known as strain. The ratio of lateral strain to axial strain is known as Poisson’s ratio after the French Mathematician and Physicist Simeon Poisson.

Now, since lateral strain results from the application of a vertical strain [tensile or compressive] we can logically deduce that there must also be a lateral force, which is proportional to the applied vertical load. The relationship between vertical and lateral load is of course Poisson’s ratio. Posssion’s ratio for concrete is approximately 0.2, which means that the magnitude of lateral load is roughly 20% of the vertical load.

If we now return to our pile cap we can deduce that there is in fact a limit to the bearing pressure that may be applied to the pile cap by the column. As the applied vertical load increases the lateral load will also increase causing the sides of the cap to push apart. This is of course a tensile action, which we know concrete does not like. It follows that the point is quickly reached where the sides of the pile cap will literally burst apart. To prevent this from happening containment reinforcement is required to resist the tensile forces.

Having established the effect of Poisson’s ratio we are still not quite done.

One of the practicalities of pile design, particularly friction piles, [see prior post] is that their circumference has to be large or there will be insufficient friction to prevent it from sinking into the ground. This means that the pile diameter is likely to be much bigger than the column diameter. 

The implication is that load must spread from beneath the column into the pile. If the difference in diameter is sufficiently large then a ‘strut and tie’ arrangement is set up in the pile cap, which is akin to that described in my prior post about two-pile caps.

It turns out that passing load through a single pile cap has a more complicated load-path than you might think, because the effect of Posison’s ratio and load spreading must both considered.