On Howe Trusses Work [yet again]

The benefit of u-frame action

In previous posts we have looked at different forms of truss and how they work. We started with an intrigue about the underlying logic of the Howe Truss and moved on to look at the intricacies of various other forms. Something that may not have been obvious, perhaps until part way through the post immediately prior to this one, is that we have thus far existed in a 2D world, we have not yet looked in the third dimension and considered what happens to trusses out of plane.

The place where this is most obviously important is the top chord. We have discovered already that the top chord is in compression and that failure by compressive buckling is proportional to the square of a member’s length. In the plane of a truss its effective length is relatively short due to the position of the internal chords connected along its length.

Out of plane there is of course no restraint from and therefore the effective length of the top chord is the full length of the truss. The top chord is almost certainly going to buckle. This is a pretty big issue to have overlooked. Fortunately there are several solutions available to solve this particular problem.

In almost all cases trusses come in pairs, for example one either side of a bridge deck. Our first option is therefore to take benefit from the bridge deck, which can be attached to the top chord of both. Since the deck is relatively stiff in plane, sometimes it may even be braced, it will have the capacity to prevent the top truss chords from displacing laterally. They are therefore unable to buckle.

The trouble with this solution is that it is rarely viable in practise. If, for example, our bridge were to span over a motorway, and the deck were located on the truss top chords, one of two scenarios would occur. Either the bottom of the truss would project down into the road below or the whole truss would need to be lifted up into the air meaning that much larger ramps would be needed to get traffic up unto the bridge.

For this reason a better solution is to place the bridge deck on the bottom chords. This, however, reintroduces the problem of top chord buckling. If the trusses are tall enough the possibility exists to introduce a horizontal truss between the two top chords. The purpose of this truss is to resist the out-of-plane loading that results from top chord buckling. It does so by the same means that the vertical trusses resist the loads to which they are subject.The two vertical and one horizontal truss exist in a symbiotic relationship, because the vertical ones can equally resist buckling action in the horizontal truss.

It is also worth noting that if our hypothetical bridge structure is outside it will also be subjected to the full force of the wind and our horizontal truss can, in combination with the bridge deck, be used to resist the wind.

A problem arises when the vertical trusses are not tall enough to permit passage below the horizontal truss. You would not wish to duck as you passed along the bridge. It would of course be possible to increase their height, but this seems an inelegant solution and a waste of material if the trusses need not be so tall. A different approach is needed.

This is the perfect opportunity for u-frame action. U-frame action requires continuity between the vertical chords in the two bridge trusses. This is achieved with horizontal members that connect them beneath the bridge deck. If the connections are stiff enough to resist bending forces then a series of rigid u-shaped structures are formed along the length of the bridge. If the top truss chords then try to move laterally and buckle their vertical chords are able to provide resistance by cantilevering from the horizontal member below the bridge deck. This is known as u-frame action. In this way the bridge trusses remain stable without the need for a horizontal truss. It is a neat solution, which is not obvious to the casual observer, and makes for an elegant bridge with unobstructed views.

 

On Howe Trusses Work [again]

Consideration of some additional forms

In my last post I described the ‘deeper magic’ that underlies Howe Trusses by describing the counter intuitive logic, at least to modern engineers, which underlies them. We also referred to a textbook diagram, which included various different families of truss. On that occasion we only looked at those trusses that were necessary to explain the Howe truss. 

In this post I thought it would be useful to revisit the text book diagram, which is reproduced below, and explain some of the remaining trusses.

 

When a truss is particularly deep, perhaps because the span is large, the gap between bays can become large and the top chord of the truss, which is more heavily loaded than the internal chords, becomes vulnerable to vertical buckling [1]. To prevent this from happening the top chord needs to become thick and heavy. In these circumstances it can make sense to shorten its effective length instead by introducing secondary triangulation [g]. 

Today such trusses are not used much because the additional joints make the structure statically indeterminate, which can result in unwanted secondary stresses, if there are minor defects or errors in the jointing.

The k-truss [h] can be a good alternative when a truss needs to be deep and the verticals are prone to buckle. The jointing remains relatively simple, but the k-shape shortens the effective length of the vertical members allowing them to take up a more slender form.

A further efficiency can be made to the design of long span trusses by taking up the form of the overall bending forces to which they are subject [i]. If the truss bridges a single span it is self evident that the maximum force is in the centre of the span reducing to zero at the supports located at either end. A truss which is deep in the middle and shallow at the ends will therefore use materials more efficiently.

That said there is a disadvantage to this form of truss. As the profile of the truss changes along its length the angle of the internal chords must also change. Near the ends of the truss their angle of inclination becomes quite acute and it is there difficult to form the joint and the member is rather inefficient.

A sensible compromise is therefore to add a web-post at either end. This brings the benefit of matching the overall bending forces while making the internal chords at either end of the truss more practical [j].

There are of course many other permutations of truss design, but between this and my prior post we have outlined some of the key principles that underly many of the most common forms.

[1] we have not yet talked about out of plane buckling. This will be the subject of a further post on the subject of trusses. 

 

On Howe Trusses Work

A search for deeper magic

‘"It means," said Aslan, "that though the Witch knew the Deep Magic, there is a magic deeper still which she did not know. Her knowledge goes back only to the dawn of time. But if she could have looked a little further back, into the stillness and the darkness before Time dawned, she would have read there a different incantation”‘ [1]

Trusses are a great way of spanning a long distance; without a web plate they are much lighter than an equivalent beam or girder. Nonetheless not all trusses are equal. There are many different ways that truss chords can be arranged making some more efficient than others. Most arrangements will fall within a family of trusses that share a standard arrangement of chords. Normally structural analysis text books will contain a diagram showing each family stating their names underneath. An example is shown below.

 

For many years there was something that bugged me about this type of diagram. It wasn’t the diagram per se that bugged me, but rather a particular family known as the Howe truss. This truss bugged me, because it didn’t make sense. I couldn’t work out why on earth Mr Howe, whoever he was, had come up with the design that he did. It is just so inefficient, or so it seemed. It turns out, however, that Mr Howe was right and I was wrong. It turns out that he knew deeper magic than I did. In this metaphor that made me the Witch, which I didn’t like very much!

This post is therefore about truss design and about Howe I learned some ‘deeper magic’.

Before we get to my mistake we need to learn something about how trusses work, which will help you understand why I made a mistake in the first place. It will also help you appreciate the existence of deeper magic and how clever Mr Howe was.

In the beginning iron and steel bridges were quite complex, but progressed to simpler arrangements over time. The progression has an intuitive feel to it. At the outset engineers were perhaps thinking about how to lighten the web of a beam or girder and decided to do so by creating a trellis arrangement [a]. For this type of truss there would have been great expense making all the joints and it would have no doubt taken a long time to fabricate. It would also have been clear that being highly redundant it could not be designed by normal static methods.

Engineers would soon have been realised that more economic trusses could be made if the number of internal chords could be reduced. This would make them lighter; they would require fewer joints; and they would be simpler to analyse.

The Bollman truss [b] was designed with pairs of diagonal members extending from the supports at either end to intermediate points on the span. It has an interesting aesthetic, but has several drawbacks. Firstly, if load is applied at a node point only the pair of diagonals connected to that node is mobilised to carry the load; all the others become redundant. A more economic structure would utilise all the chords simultaneously. A second disadvantage is that, except for the middle set, pairs of diagonals are necessarily inclined at different angles. This means that they must carry different magnitudes of load and consequently rather unhelpful secondary stresses are induced. It turns out that diagonals are better suited to being set at a regular angle of inclination, perhaps somewhere between 45 and 60 degrees.

The Warren truss [c] is a good example of this. It has a simple arrangement of diagonals arranged in the shape of equilateral triangles. It minimises the number of members and joints, and what is more, all members are the same length. It is an efficient truss that is simple to fabricate and is statically determinate [easy to analyse]. What is not to like?

Nevertheless, for all its advantages the Warren Truss does have a disadvantage and we can therefore improve on its design. That disadvantage comes from a consideration of the type and distribution of forces its chords must carry. 

If we imagine for a moment that we are dealing with a beam and not a truss a useful analogy can be made. As the truss takes up load and starts to bend the top chord shortens and the bottom one lengthens. There is therefore compression at the top and tension at the bottom.

The internal chords, which must transfer load between the upper and lower chords behave differently. Those which point towards the supports shorten and are in compression; those which point towards the centre of the span lengthen and are in tension.

There is of course a difference between tension and compression forces. Members that are in compression are prone to buckle in the middle while tension members are not. Since the buckling capacity of compression members is proportional to the square of their length there is a distinct advantage to being shorter. 

This brings us nicely to the Pratt truss [d], whose compression members are arranged vertically. Since the verticals are necessarily shorter than the tensioned diagonals this is a highly efficient form of truss, which recognises the type of loads each member carries.

Observant readers will no doubt have foreseen what comes next. The Howe truss [e] is a complete reversal of the Pratt truss. The tension members are now arranged vertically and the compression members diagonally i.e. the longest members are in compression. If you think this doesn’t make sense you are just like I was and don’t appreciate the deeper magic involved. 

There are of course some other truss families in our diagram, but we are going to leave those for another time so that we can concentrate on the mysterious Howe Truss. In order to get to the bottom of the conundrum we need, like Aslan, to go back in time.

Perhaps the first trusses were timber roof trusses. Unlike trusses of iron and steel they would have started simple and became more complex over time. A pitched roof would have been an advantage to early builders, as it is today, because it encourages water to flow away from the building; it is less likely to develop a leak. 

It is self-evident that inclined rafters are required to create the pitch. This in turn implies rafters, which lean against each other and therefore tend to spread at their bases. The associated spreading force is rather unhelpful to the supporting walls. A horizontal thrust applied at their head will of course make them unstable and push them over. To stop this from happening a horizontal tie is added forming the most basic form of triangular truss.

With the success of this form it stands to reason that the designer will soon want to bridge a bigger span. This would have brought a new effect to the designers attention. With increased span the tie beam would begin to sag under its own self weight. To prevent this from happening the solution would be to suspend the middle of the tie beam from the rafters using a new tie member called a King Post. If the spans increased again the rafters would surely begin to sag too. It therefore becomes necessary to prop them with struts supported on the tie beam. To prevent the tie beam from being bent the struts would be joined at the same point the king post is connected, thereby transferring the propping load back into the rafters in tension.

The next development would be to minimise the length of our new struts to stop them from buckling. This can be done by making them perpendicular to the rafters. The trouble with this arrangement is that they now bear on the bottom ties at a distance from the King Post and thus bending is reintroduced to the system. For obvious reasons this is undesirable. The solution is to replace the King post with two tension ties, which join the struts to the apex of the rafters, thus eliminating tension. 

We have nearly reached the end of our detour into timber roof trusses and are almost ready to return to the Howe Truss. Before we get there we must learn one more thing about timber roof trusses. When timber members are in compression they are squashed together and load is conveyed at the joint in bearing. This allows relatively high loads to be transferred. Conversely, when two members are in tension they require timber pegs inserted between them to hold them together. In this case all of the load is transferred through the pegs alone, which are much smaller than the overall member size and are not terribly strong. For this reason tension joints are the weak link in the system.

To overcome this problem engineers began to use wrought iron straps to transfer loads at tension joints. It could not have been long before it was realised that the tension members themselves could take the form of wrought iron rods. The advantage being that they could be inserted through the timber chords and clamped tight with large washer plates.

This results in a hybrid structure, which is very efficient, because in a time when iron was expensive and difficult to produce, it uses relatively cheap timber to carry compressive loads and iron to form the tension members and joints.

This is the key to unlocking the mysterious Howe Truss. The thing which structural analysis, text books never tell you is that the vertical members are wrought iron or steel and the diagonals are in timber. They also don’t tell you that the verticals are tightened at the fixings until they carry a pre-stress, which keeps them permanently in tension. By doing so the compression members are clamped tightly together. The timber members are stocky compared to the slender ties and are therefore not vulnerable to buckling.

This is where that deeper magic begins. The logic which governs the design of modern steel trusses does not apply to older trusses made of timber, due to the limited capacity of timber joints in tension and the cost of making iron.

That said there is something else that you need to know. William Howe was an American engineer who invented his eponymous truss for use in the construction of railroad bridges. In the vast spaces of the United States it stands to reason that it would have been more efficient to cut timbers from trees near to the site of a planned bridge rather than having to transport all the iron members from a fabrication yard.

It all makes sense now, Mr Howe was in fact a rather clever man.

Before I finish this post there is one further thing that I need to explain. Most archive images of Howe trusses have diagonal members in two directions rather than one, they don’t actually look like those found in structural analysis text books. This tells me that their authors are only familiar with the theoretical form of the truss and not how they were made or what they were used for.

The reason for the additional chords is also related to the construction of railroad bridges. Howe realised that as a heavy steam locomotive passed over one of his bridges it would experience uneven loading and this could potentially cause load reversal in some of the chords. The additional members were used to ensure that there were always members acting in compression. 

[1] chapter 15, ‘The Lion the Witch and the Wardrobe’ by C S Lewis.

 

On Spider Webs

The influence of structural form.

The last time I tackled a topic from the animal kingdom I looked at Stegosaurus, which is a rather large animal, so on this occasion I have decided to look at something much smaller. As before I still know nothing about animals and have no expertise whatsoever in the fields of biology, zoology and so forth. I shall be looking at the topic from the perspective of a structural engineer and will likely be making all sorts of terminological and other obvious errors. 

My prepared defence against basic dinosaur errors was based around stegosaurus being extinct and therefore nobody really knowing for sure. This time around I don’t have that luxury. Instead I shall base my rebuttals on that most modern phenomena of ‘getting my message out there’. In this mode of thinking errors are acceptable so long as the direction of travel is correct and ones motivation is honourable. 

Now that my new excuse has been set out, lets jump right in.

There is a cliche about spider silk being stronger than steel, which is obligatory to every discussion of Spider’s webs. I thought that I would get it out of the way early. It is perhaps less well known that spider silk is a non-linear material who’s stiffness varies depending on the applied load; it has both a slack and a stiff phase. This unusual property helps webs absorb impact from captured prey.

Although interesting the properties of spider silk are not actually the topic of this post. I shall be taking them as a given. Rather, I would like to take a brief look at the influence of structural form on spider’s webs.

In the hypothetical scenario of Sir Attenborurgh stumbling across this post and deciding to read it he would no doubt want to point out that there are many species of spider and consequently there are many kinds of web to contend with. For simplicity I shall be sticking with the common variety that most people, including me, are familiar with.

It seems to me that webs consist of several different types of member, which exist within a distinct structural hierarchy. In the first instance there are a series of threads which anchor the web to its surroundings, we shall call these the moorings.

The moorings are connected to the corners of an outer primary frame, which encloses the web. At each corner a secondary frame joins both sides of the outer frame together, but without touching the corner.

A series of radial threads extends from the centre of the web onto the primary and secondary frames. Together these members give the web its overall shape.

A spiral thread winds from the centre of the web towards the outer frames. Unlike those discussed thus far the spiral thread is made of a sticky silk, which is thinner than the other members, and is intended to catch the spider’s prey.

Before we consider how this rather spindly arrangement of threads manages to resist the impact of spider prey, and the force of strong winds, it is worth explaining a key structural principle.

The theory of elasticity dictates that when a load has several different load paths to choose from it will always prefer the one which has the greatest stiffness. In simple terms load is distributed between members according to their stiffness with the stiffest parts attracting the most load.

The importance of this principle may be illustrated by considering what would happen if spider’s designed their webs a little differently. Let us suppose that the hypothetical Institution of Web Safety, were to decree that secondary framing was no longer permitted and therefore radial threads must connect directly to the outer primary framing on all sides. Connecting directly to the primary frame, sounds like a simplification of the structural load-path. That must be good, right?

A further implication of the IWS’s directive would be manifest at the corners of the frame; the radials would now be connected directly to the moorings. Again, that must be a worthwhile safety improvement, because load is directed straight to the point of support.

We can test our theory by imagining a hypothetical fly careering into the web. We want the web to absorb the impact without breaking; that would be bad for the spider’s prospect of lunch.

This means that we want the web to spread the impact force across as many structural members as possible. The more members mobilised to resist the applied load the smaller the load each will carry.

Immediately after the fly strikes the sticky spiral the web’s load-path swings into action. The spiral is connected to lots of radials and begins to share its load. But now something has gone wrong, the load has reached a radial which is connected directly to the moorings. Being connected to the point of support this radial is much stiffer than adjacent radials, which are attached to the outer frame, which has started to flex. Load is immediately attracted out of the radials connected to the outer frame in favour of the stiffer pathway. Soon the radial connected to the mooring is carrying nearly all of the load and is stressed to breaking. Our spider’s lunch is about to escape.

Surprised by the evidence of systemic failure in radial web members the IWS takes the decision to withdraw its directive and reinstates secondary framing, which once again must be connected to the outer frame either side of corners.

Soon an unfortunate fly finds itself bumping into a web with newly reinstated secondary framing. Once again the spiral members spring into action and begin to transfer load into the radials, but this time something different happens. Instead of load being directed straight to the moorings it finds itself being directed into the secondary framing which begins to flex before re-directing load back towards the middle of the outer framing and away from the stiff corners.

The outer framing begins to flex and in doing so starts to engage other radials to which it is connected, before long much of the web is flexing and load sharing is being maximised. On this occasion there is going to be no failure. It looks increasingly like the fly is doomed and the spider will be enjoying lunch.

It seems to me that the web’s structural arrangement is designed to avoid stress concentration. This key feature maximises load spread and minimises the stress in individual members. Another consequence of the redistributive process is that the web becomes less vulnerable to local damage; because load can by-pass those areas.

Thus spider webs have a highly efficient structural form optimised for absorbing impact and for ensuring spiders remain well fed.  

On Negative Skin Friction

An unusual case of piling

Many years ago I designed a building in central London, which was later occupied by a well known tech giant. There were many interesting features to the building, however perhaps the most interesting is concealed from view below ground level.

Like many large buildings in London its weight is too heavy to be borne on shallow foundations therefore concrete piles embedded deep into the soft London clay were required.

The structure was to be built on the site of a prior building which had also been piled. We soon discovered that is was going to prove difficult to match new pile positions with proposed column positions, because the existing ones were always in the way, though never quite in the right place we needed them. Grubbing out the existing piles was very expensive and would have loosened the soils somewhat. We decided to build a deep raft on top of the existing piles that would allow us to transfer new load into them from our planned column positions. Justification of the the existing pile’s load capacity was rather interesting, but that is not the topic on this occasion.

The footprint of the new building was required to extend beyond that of its predecessor and therefore a line of new piles was required in front of the existing ones. This was to prove an opportunity to use some creativity, because we knew that this part of the site fell within the influence zone of the planned Cross Rail underground tunnels. The route of the tunnels had been secured with legislation for many years.

To understand the influence this would have it is first beneficial to understand how friction piles work and what the effect of future tunnelling would be.

Friction piles, as the name would suggest, resist load by generating friction between the surface of the pile and the surrounding ground. The larger the circumference of the pile and the longer the pile is the more friction will be generated. Consider trying to push a tent peg into the ground or hammering in a fence post. It becomes progressively harder the deeper they penetrate into the ground; this is friction in action. It is a powerful force.

Tunnelling is a complex operation, but in principle it involved pushing a circular shield into the ground and digging the ground out within the safety shield, which prevents the ceiling from collapsing. A structural ring is then constructed inside the shield before it is advanced to the next section of the tunnel. When the shield moves on the ground fills the gap that is left behind by settling on to the newly constructed structural ring. As the tunnel progresses a wave of settlement follows the head of the tunnel.

An interesting thing happens when concrete piles fall within the zone of settlement. As the ground settles and drops it tugs down on the embedded piles. This action pulls the piles downwards due to friction between the ground and the pile shaft. In other words the tunnelling has caused friction to work in reverse. Instead of friction pushing against the weight of the structure it is now acting in concert with the weight of the building ‘sucking’ it into the ground. This is known as ‘negative skin friction’ and it would be bad for the building, particularly of the rest of the building, which is out with the tunnelling zone, stays where it is and does not settle.

 

The conventional way to overcome this problem is to sleeve the piles in steel tubes over the full length of the tunnel’s zone of influence. To ensure the piles still have capacity to resist the building’s weight they extend beyond the sleeves and are embedded below the depth of the tunnel. This means that when friction reverses the steel sleeve is tugged downwards, but the pile remains static inside the sleeve while load resistance continues to be provided by friction generated below the tunnel.

This is a rather expensive option, because each pile needs a long steel sleeve. We therefore decided to think about whether we could design the foundations without sleeves. If we could it would save both time and money for the client.

We ‘war gamed’ many scenarios, but eventually reasoned that, providing the piles extended below the depth of the planned tunnels, the most likely scenario was that the expected negative skin friction would cause an ‘apparent’ increase in the weight of the building. We were confident that we could design the piles to deal with that ‘apparent’ increase and so that’s what we did. It took a while to design, and being an unconventional approach, it also took a while to obtain formal approval.

It all seemed very theoretical at the time. Cross Rail had been spoken of for years, but had never been realised. Many thought it would never happen. Of course that view has since changed. Cross Rail is due to become operational in 2022. So far the building is still standing quite happily; our hard work appears to have paid off.

Something also worth mentioning; the route of the tunnels originally passed beneath the existing piles too, which would not have done them any good at all. Within the approval process for our new design we therefore proposed that they be altered to avoid damaging the existing foundations. Our proposal was eventually accepted and the layout of the tunnels was changed. In this location they run on top of each other instead of side by side.

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.