On Jenga

Why the middle-third rule doesn’t cause instability

Jenga is a family game, which is much enjoyed in our house. The game starts by constructing a wooden tower using layers of three timber bricks. The orientation of the bricks alternates by 90 degrees between layers.

Each player takes a turn to remove one brick from the tower and then to place it on top. This simultaneously adds to the height of the tower while reducing the cross section of the lower portion. This process continues until eventually the tower topples, either during withdrawal of a brick or as it is placed on top. The last person to touch the tower is the loser. 

In my experience collapse happens far more frequently while bricks are removed than when they are added. I think this is because the bricks are not exactly the same size and therefore, while some can be removed with considerable ease, others are held tightly by friction. I do not know whether this is an intended jeopardy, but it definitely adds to the game, as you are never quite sure, which bricks will stick until after you are committed.

That said, all things being equal, it is possible to take a brick from either the middle position in any given layer, or from one of the two outer positions, without toppling the tower.

This is interesting, because it undermines a common usage of the so called ‘middle-third’ rule. The middle-third rule essentially states that a structure will avoid tension providing its centre of gravity lies within the middle third of its thickness. Since masonry structures cannot resist tension this is often taken to mean that they will become unstable if the middle third rule is breached, however, as we are about to see, this is not so.

The centre of gravity of a structure is the axis through which its weight acts. For example, the centre of gravity of a uniformly thick wall would be a vertical axis through the middle of the wall. Self-evidently the center of gravity for such a structure falls within the wall’s middle third.

If we apply this logic to a Jenga tower we can remove the middle brick from any layer and neither the centre of gravity nor the middle third will change i.e. the centre of gravity remains in the middle of the tower and therefore the tower remains stable. If, however, we remove a brick from one of the two outer positions something interesting happens.

 

The centre of gravity above this level remains in the middle of the tower, however at the level with a missing brick the middle third has shifted into the two remaining bricks i.e. it is the middle third of two rather than three bricks.

At this level the middle third therefore extends from 2/9 [1] to 4/9 [2] of the full tower thickness. The centre of gravity of the wall above remains 1/2 of the tower thickness. For ease of comparison we can convert these fractions so that they have the same denominator. The middle third is from 4/18 to 8/18 of the tower thickness, while the centre of gravity is 9/18 i.e. 9/18 sits outside the middle third, which is limited to 8/18. 

Since the tower does not collapse the middle third rule, while a cautious limitation, cannot represent the point at which the tower becomes unstable. 

The next step would be to modify the rule from the middle third to the middle half. In this scenario the middle half, at the level with two bricks, would extend from 1/6 [3] to 3/6 [4] of the full tower thickness. Of course 3/6 is equivalent to 1/2, which corresponds to the exact position of the tower’s centre of gravity at the level above. This means that the tower is in theory just stable, but ought to topple if it moves even a tiny fraction. Obviously, this can’t be right either or the game would be impossible to play.

Perhaps a good way to think about why this can’t be the correct limit is to imagine what would happen if we removed the remaining outside brick leaving only the centre brick. In this scenario clearly the structure remains balanced and will not topple. 

If we now apply the middle third rule to the remaining middle brick the edge of the no-tension zone would extend to 5/9 [5] of the overall tower thickness. This provides a margin of safety of 1/18 relative to the centre of gravity of the tower above. This margin is probably slightly inaccurate, because the tower has not been built under laboratory conditions with perfect alignment between layers. We also know from our earlier discussion that the blocks are not all precisely the same size. This means that in practical terms the tipping point probably occurs if the centre of gravity moves either a little less or a little more than 5/9 of the tower thickness.

If we take the traditional Jenga tower to be 45 mm wide then the margin of safety is circa 2.5 mm. This is perhaps just big enough to tolerate a small disturbance while removing and placing a brick. It is also just small enough to provide a level of jeopardy that makes the game interesting.

It is also worth noting that 5/9 of the full tower width is 25 mm, which equates to 5/6 of the two bricks we started with after removing one of the outer bricks. Thus the 1/3 rule is exceeded by a significant margin.

Of course it becomes easier to breach the margin of safety as the structure becomes taller. This is because the structure becomes top heavy and therefore has greater momentum if disturbed.

[1] 1/3 x 2/3
[2] 2x 1/3 x 2/3 
[3] 1/4 x 2/3
[4] 1/4 x 2/3 + 1/2 x 2/3
[5] 1/3 + 2/3 x 1/3

 

On Hennebique

Understanding an early patent system

In 1892 Francois Hennebique patented his eponymous ferro-cement system, which is today recognised as one of the earliest forms of reinforced concrete. It was used under licence in many countries, including the UK, where Mouchel was the local partner. The Hennebique system was conceived before the era of codified design, so its worth trying to understand its structural load-paths. 

To do this we must think of Hennebique’s creation, not as a beam, but as a truss made of composite materials. This may seem like an odd thing to do, but it is necessary to explain how stresses are distributed throughout the section. This approach also, as we shall see, highlights several weaknesses in Hennebique beams.

To make sense of this analogy we need to remember that concrete is strong in compression, but weak in tension. Conversely, wrought iron is equally strong in both. It follows that the key to visualising the load-path is see tension where there is iron and compression where it is absent. 

That said, before we look at the Hennebique system itself it is useful to remind ourselves of the alternatives that were available at the time.

The picture below shows a brick jack arch floor, which was conceived as a fire proof system, although given the exposure of the iron flange on the soffit it is more correctly described as non-combustible. The load paths for a jack arch floor are straightforward. The brick arch spans laterally and is supported on iron beams spanning into the page. There are tie rods evident so that arch spreading is contained and only vertical load is transferred by the brickwork.

 An improvement on this design was to replace the brickwork by fully encasing the iron beams with concrete. In principle this would certainly improve the fire resistance of the floor and make it more durable. At least it would have if the concrete didn’t include breeze, which sometimes contained unburnt coke, and sulphates, which could form a mild acid in the presence of water.

This form of construction is known as filler joist construction. It was very common, particularly in the early twentieth century, and many examples remain today. The load path is essentially the same as for the jack arch, except the arch form must be imagined within the body of the concrete, because it would have been much simpler to create a flat soffit than a curve.

The filler joist floor therefore remains a rudimentary structure; there is no composite behaviour between the iron and concrete.

This relationship was changed when it was realised that an inverted T-shape would be more efficient because the concrete fill could resist the compressive stress to which the top iron flange had been subject. The bottom flange was still required to resist tension and the web transferred load between the two by resisting shear.

 

This was an important breakthrough, because the floor was now a composite system, which shared load between concrete and iron. At that time wrought iron was exceedingly expensive and therefore this was much more than an analytical curiosity.

Hennebique’s genius was to make two further steps; or perhaps two and half. Firstly, he replaced the bottom flange of the beam with round iron bars. These were easier to make than a T-beam and had a greater surface area than a single flange with which to bond with the concrete.

 

Secondly, he did away with the iron web, which transferred load between the top and bottom of the beam. To understand the way in which he did this it is helpful to look at other systems that were common at the time.

The next image shows a trussed girders taken from a carpentry manual written in the 1860’s. I have previously written a longer post about this topic, however the key issue in this instance is the way in which timber at the top of the section is used in compression and iron rods are used in tension. Short compression stools are used to transfer load between the two.

  

As can be seen in the following image Hennebique uses exactly the same load path for his concrete system. It is reasonably straightforward to see the tension elements, highlighted in red, however the compression parts, highlighted in blue, must be imagined within the body of the concrete, much as the compression arch is imagined within the filler joist system. I do not mean imagined in the sense that the load path does not really exist. Imagined only in the sense that not all of the concrete in the beam is contributing to the load path.

When Hennebique tested his system he found that though it was successful it did not work as well as it ought; diagonal cracks formed with increasing frequency towards the end of the beam. Such cracks are related to the interaction of shear and bending forces. From Hennebique’s writing I am not entirely sure that he fully understood this mechanism, though nevertheless he found an effective solution. That may be because he found it difficult to describe or I have found it difficult to follow his writing.

 

I think Hennibque believed that there were longitudinal shear forces parallel to the length of the beam, which were causing the failure he had observed in testing. He thought that by intercepting these longitudinal stresses with vertical stirrups of bent iron he could improve the strength of his beam. While such stresses do exist there are also vertical shear stresses, which means that Hennebique’s thinking, if this was what he thought, is incomplete.

Nevertheless, Hennebique’s solution did work and cracking was avoided, though perhaps not wholly for the reasons he thought. We can understand why by referring to the next image, which shows a modern understanding of shear transfer; again red represents tension and blue compression.

We can see in this example that the load path is a fully formed truss with the stirrups and iron bars resisting tension and the top chord and diagonals resisting compression. This modern understanding highlights one reason, beyond a lack of clarity in Hennebique’s writing, that I think Hennebique did not wholly understand the load path.

The case we have thus far examined has a single span with tension at the bottom and compression at the top, however if we were to add additional spans the relationship we have established reverses at the support, with tension at the top and compression at the bottom. In such circumstances we encounter a problem with Hennebique’s stirrups. At mid span they are hooked around the tension bars, but at the support they are open at the top and can be pulled clear. Thus, for multi-span beams the Hennebique system is less efficient than single spans. Had he fully grasped the load-path I am sure Hennebique would have corrected this. Perhaps, as shown above, his tests were all conducted on single spans.

Another shortfall of the Hennebique system is with the tension bars themselves. It will not have escaped the notice of readers with a keen eye that Hennebique’s tension bars have hooks at the end. These were, I am sure, intended to improve the transfer of load between the iron bars and the concrete. At face value this was a sensible measure, because at failure the bars were simply pulled through the concrete. This meant a premature bond failure before the iron had reached yield. This happened because, unlike modern bars, which are deformed to improve the bond, Hennebique’s bars had a smooth surface.

While seemingly a good idea the noted hooks did not really work, though we can forgive Hennebique for this, because the reason why is really rather complicated. In simple terms the bond on the bars must fail before the hooks can be mobilised. Why this happens is perhaps a subject for a different post.

Notwithstanding these shortfalls, which are made with the benefit of hindsight, Hennebique’s system was ultimately very successful and marks him out as a significant figure in the pantheon of structural engineering.

While it is true that Hennebique was not the only person to develop a patent system for reinforced concrete; his was perhaps the most successful. This was probably due to the licensing system he operated marking him out as a great businessman as well as a great engineer. 

On Canaletto & Bridges

Tangent & Radial Trussing

The eighteenth century painter Canaletto was well known for his Venetian paintings, many of which were commissioned by British Patrons. This gave him a reason to move to the UK when war broke out.

One of his commissions, painted during his stay in the UK, is ‘A View of Walton Bridge’. The painting is interesting, not because of Canaletto’s skill in depicting the British weather, nor is it because Canaletto doctored the scene by omitting the masonry viaducts leading up to the bridge. Rather it is the nature of the bridge itself. 

At first sight Walton Bridge appears to be a traditional arch structure, however it is in fact a fine example of the tangent and radial truss. The main span was reported to be 130 feet with two side arches of 44 feet. Sadly, the bridge is no longer in existence, due to timber decay.

Tangent and radial trusses are special, because they take up the form of an arch using only straight members. A simpler example, known as the “Mathematical Bridge’, still exists at the University of Cambridge. 

As the name suggests the primary structural members are arranged at tangents to an imaginary circle, thus giving the appearance of an arch. The radial members are aligned perpendicular to the imaginary circle and would, if extended, meet at its centre.

The form is also interesting because the tangential members, which spring from the abutments, are predominantly compression members, while the the radial members predominantly resist tension. Most articles written about ‘Mathematical Bridge’, or at least the ones I have seen, note that this pattern of tension and compression are an elegant visual representation of the forces within an arch. It seems to me that while oft repeated this observation is not strictly true.

A true arch is a pure compression structure where the line of thrust is contained within its depth. Many arches, including most classical and medieval examples, are not actually true arches, they are in fact spandrel arches. Spandrel arches are characterised by an infill [spandrel] structure located above the true arch. If the infill is of sufficiently depth then the line of thrust can move outside the true arch and load can be conveyed within the spandrel. At a certain point the stresses in the spandrel become more akin to those found in a beam and the arch becomes nothing more than an ornamental soffit to the spandrel. Something like this phenomenon is demonstrated in an earlier post titled ‘On Accidental Bridges’.

It follows that tangent and radial trusses are a better depiction of the forces in a spandrel arch than they are a true arch. The tangential members join with the top chord to convey compressive stress, while the radial members combine with the bottom chord to convey tension. Based on this the arrangement would be more efficient if the radial members were in fact not radial, but were instead arranged perpendicular to the tangential members i.e. if their angle of inclination were defined by the stress pattern rather than the centre of an imaginary circle.

Perhaps the reason for using radial members was to avoid a structure with more complex geometry and more awkward jointing, however there are also several other considerations. 

Firstly the radials must provide restraint to the top chord of the truss so that it does not buckle. They do this by generating u-shape action in combination with members below the bridge deck[1]. Secondly, they form a useful part of the balustrade, which would be less effective were they located at a steeper angle.

Before we finish it is also worth noting that tangent and radial trusses were commonly used as centring [temporary support] for masonry arch bridges while they were being built.

One such example was Westminster Bridge in London. We know tangent and radial trusses were used, because the original drawings have been preserved. That little detail didn’t bother Canaletto, who once again used artistic licence when framing a view of London in his painting ‘London: Seen through an Arch of Westminster Bridge’. The timber centring is clearly not formed of tangent and radial trusses. Perhaps Canaletto judged that the elegant structural form would detract from the view he wanted to paint and therefore substituted something more prosaic. At least that’s what I like to think. 

[1] An earlier post titled ‘On Howe Trusses Work [yet again]’ describes u-frame action in more detail.

On Cruciform Columns

Why put up with torsional strut buckling?

The photograph above shows some columns that I came across at an old industrial site that was to be converted into a modern mixed use development. It was interesting because the columns, which were clearly cast iron, had a cruciform shape rather than the conventional circular hollow form. This is relatively unusual and made them older than their circular cousins.

Today cruciform columns are perhaps even less common than they were in the past. The form is certainly not included within modern codes of practise. This is relevant, because the cruciform shape has an unusual buckling mode that does not apply to other shapes. Subjected to excessive compressive load it will exhibit torsional strut buckling. The modern engineer is familiar with conventional strut buckling and lateral-torsional buckling, which afflict more common shapes, but much less so with torsional buckling. 

This isn’t intended to be a post about torsional strut buckling per se, except to say that torsional buckling reduces the capacity of a column and causes it to fail before other modes of failure. This is why its important to know. The question for this post is why designers in the past would choose a cross section that has a reduced capacity? Did they not know what they were doing?

It is certainly true that modern methods of analysis were not available at the time, though there were column sizing formulae that were used. Typically however, cast iron columns were proof tested and therefore manufacturers and engineers had seen and did understand the failure of columns, at least in a practical or empirical sense.

If we are to understand the existence of cruciform columns we must therefore look elsewhere. We must first understand the material from which they are made and then the way in which they are made. 

Cast-iron has many useful properties. It is strong in compression, it is mouldable, it is resistant to corrosion and crucially it is non-combustible. This last property was considered vital in the context of its early use in the construction of mills. It is well documented that there had been many catastrophic fires.

The challenges of cast iron were its brittleness, its low tensile strength and the tendency for flaws and blowholes to appear. These last two issues are primarily the result of manufacturing.

As the name would suggest cast-iron sections are not rolled or extruded, like other metals. It is like concrete in the sense that it is cast in a mould. Of course unlike concrete it sets hard by cooling rather than by chemical reaction. Also, like concrete it will shrink in the mould, though by a much smaller amount.

Since hardening is by cooling the rate of cooling is vitally important. If one part of the structure cools faster than another then the internal structure of the iron will be different. The faster it cools the better the tensile strength. Similarly, if one part has cooled, and therefore shrunk, while an adjacent part has still to cool and shrink then internal restraint will cause stresses to be built into the column before it has been loaded. In some cases such restraint might even cause fracture during the casting process. The shape of the casting is therefore important.

Another important factor is section thickness. The thicker the member the more likely the surface of the member is to cool before the interior. This would again cause internal restraint and internal stresses to develop.

It follows that a shape had to be developed that was straightforward to mould and allowed iron to flow inside quickly and easily. It had to be symmetrical to promote balanced cooling and the section could not be too thick to stop the interior from cooling too slowly.

The earliest attempts were a crude star shape with a solid centre, however it is not difficult to see that the next logical step would be to extend the points of the star to create ribs, thus forming a cruciform shape. It fulfilled all of the criteria for casting.

Interestingly the ribs were often cast with classical proportions, being wider at the center of the column than at the ends. From this we might conclude that the designers new full well that columns were prone to buckle in the centre and it was an advantage to have more material at that point.

While some modern engineers may look back at early columns and dismiss their designers for having a primitive understanding of buckling behaviour there is of course a deeper truth. The designers from that era understood that columns can buckle, but they also knew about the perils of casting iron and at that point in time it was a bigger factor in the safety of columns than buckling was.

A supplementary question might be why circular columns were not manufactured from the beginning, after all they have a good resistance to buckling and a shape that encourages rapid, even cooling.

I think the answer is probably rather prosaic. Molten iron is extremely hot and is therefore cast in moulds of sand. I imagine nobody had yet worked out how to produce a mould from sand with a void in the middle.

Something else that is perhaps worthy of comment is the relatively large projecting tables at the column heads on which the timber beams are supported. Given what we know about the low strength of cast iron in tension, and by extension flexure, these cantilever projections would appear to be a significant weakness.

In fact there is little evidence of table failure and it has therefore been conjectured that the relatively thin sections cool rapidly after casting and develop a higher tensile strength than is found in thicker castings. 

So there we have it, counter to what you might think, based on a modern mindset, cruciform columns were actually, at the time, a really good idea.

 

On Compressive Membranes

System behaviour in fire

Something that isn’t often appreciated is that when a building is designed to achieve a 90 minute fire rating it doesn’t mean the building is designed to remain standing for 90 minutes. This might seem odd, however if we were to pose the question what size of fire is to be resisted for 90 minutes, it is immediately obvious that there is complexity involved. 

In reality a 90 minute fire rating means that the fire resisting components in the building have been tested in a furnace against a standardised fire lasting for 90 minutes. This allows the relative behaviour of different materials to be tested, however it tells us absolutely nothing about how long a real building will survive subject to a real fire. 

This is in part because fire tests treat standardised components individually, however real components are not a standard size and they act as part of a system not as individual elements. If we are being really picky we might also argue that real fires are different to standardised furnace tests.

It follows that if the structural behaviour of a system subjected to fire can be understood then this can be harnesses en lieu of the rather crude prescriptive approach, which is normally applied.

The floors of many modern buildings are constructed by casting a thin slab of concrete on a corrugated metal deck. The concrete is reinforced using a light steel mesh and the metal deck spans between down-stand steel beams. Often metal studs are welded to the top of the steel beams and are embedded in the concrete. This is known as composite construction, because the steel and concrete act together.

To protect the steel from fire the prescriptive approach is to coat it with a fire resting coating, normally intumescent paint. Intumescent paint swells when it gets hot forming an insulating layer, which prevents the steel from over-heating. Without this insulation layer steel looses significant strength and stiffness at approximately 500 degrees.

If, however, the structural system is taken into account many of the beams may not require intumescent paint. This is beneficial because intumescent paint is expensive. 

In the example shown there are primary beams joining the columns together to form a series of identical bays with two secondary beams in each bay. If we were to suppose that the secondaries are unprotected then we can begin to think about the load-path in a hypothetical fire.

As the fire becomes increasingly hot the secondaries will also become hot and will start to loose strength and stiffness. Eventually they will have little residual capacity. When this happens the floor will begin to sag and instead of supporting the concrete floor the beams will hang from it due to the embedded studs. This process is likely to be accelerated by thermal expansion which causes the beams to buckle as they push against their supports.

Conversely the primary beams, located on the column lines, are protected and will remain unaffected by the ensuing fire. They continue to form a rigid frame around each structural bay. As the floor sags it begins to tug on the primary frame simultaneously pulling each side of the bay towards the middle. Much of this work is being done by the light reinforcing mesh embedded in the floor.

This effect causes a compressive ring to be set up in the concrete at the perimeter of each bay. This ring starts to resist the floor’s tugging and allows a point of equilibrium to be reached where the weight of the hanging floor is balanced by the compressive force in the concrete ring. Although the floor has displaced significantly it has not collapsed and has therefore maintained its integrity. The fact that it has displaced significantly is not materially important, as the sole aim is survival. After a major fire a building would not expect to survive completely unaffected.

This load-path means that some of the primary beams must carry additional load, which was previously supported by the unprotected secondaries. This is acceptable, because in the fire case it is permissible for the additional load to be absorbed by the their factor of safety.

It is also worth noting that the required load assumed for most buildings is in fact much greater than the load the floors will ever see. This means the actual factor of safety is normally higher than is assumed in the cold design.

This form of system behaviour is known as a compressive membrane and I have used it successfully to assess the fire resistance of buildings on several occasions. It is a more rational approach to fire safety than the rather arbitrary prescriptive approach, which has been used historically.

On Shopping Bags & Creepy Buildings

The advantage of squashing a facade

Until we thought better of it a weekly shop meant filling disposal plastic bags, provided by the supermarket, with our groceries. If you only had a few groceries to fetch and you decided to walk or if your car was parked a reasonable distance from the supermarket entrance then you may have noticed a curious property of plastic bags.

When they are initially filled they work rather well, however if the contents of the bag are reasonably heavy, for example  some drinks or a bottle of milk, then by the time you have reached your destination the bag handles have stretched. If the contents are very heavy, and the walk long enough, the handles may even have stretched to the point of breaking.

The interesting question is why this should be so? If the shopping bag performed satisfactorily when it was first picked up, why have the handles stretched by the time you get home? After all nothing additional has been added to the contents of the bag since you left the supermarket. The bag is carrying exactly the same weight as before. Why was it ok to begin with but not afterwards?

In engineering terms this would be described as increased strain at constant stress or in layman’s terms increased stretch without a corresponding increase in load. This is the definition of a phenomenon called creep. Creep happens when the internal structure of a material starts to become rearranged due to the effect of loading. Some materials, like plastic, are more prone to creep due to the nature of their internal structure, however all materials creep a bit under sustained load. It is worth noting that while our shopping bag example is based on stretching creep can also be a squashing effect.

A good example of creep that is more directly related to structural engineering would be the behaviour of old timber floors, which are often bowed in the middle. Another example would be the extension of bridge cables, which must be taken into account in their design. Intuitively it would seem materials progressively stretching or squashing over time is a bad thing. What would be interesting is an example where creep was actually a good thing.

In the late nineties I found such an example thanks to a rather demanding architect [that is not a bad trait in an architect]. He set the challenge of designing a building with a brick facade that was free from movement joints. He viewed movement joints as being ugly, a view with which I had some sympathy. If you haven’t noticed them before now, you will after this post and you will find them ugly too.

The building was shaped like a horse shoe, but with the open end enclosed by a full height glass wall. The perimeter of the horseshoe measured roughly 300 m. 

In case you are unfamiliar with common practise vertical movement joints are normally included in brickwork every 12 meters. You can therefore appreciate the nature of the challenge.

The solution to the problem was rather ingenious. I can say that because it wasn’t my solution. I was a young engineer at the time and still had much to learn. That said having a genius idea is only part of the answer and said genius usually still needs several less experienced, but enthusiastic, engineers to help him work out how to prove the solution will work. I was one of those lucky engineers.

Modern brickwork tends to consist of two thin skins with a cavity in between to keep water out. The two skins are tied together with wire ties. This is the archetypal cavity wall. For most buildings of any size the brick is supported on a floor by floor basis by the building structure and is therefore not load-bearing. The genius part in this case was to go old school and construct a reinforced concrete frame with thick load bearing walls. The concrete floors were supported on corbels embedded in the brickwork.

To understand why this was clever you need to know something about movement joints and several things about brick and concrete.

Movement joints are required because brick expands and contracts. Without relieving joints this will cause cracking. The greater the joint spacing the greater the movement. There are several reasons for brick movement. 

Firstly, bricks expand when wet and contract as they dry, however only part of the expansion is recoverable, as some moisture chemically reacts with the brick and some fills the open pores and will eventually evaporate. Secondly, bricks expand and contract due to temperature variation; they expand when warm and contract when cold. The shade, colour and type of brick affect the magnitude of this effect. 

Conversely concrete shrinks. It does so because the free water, which allows it to be poured, starts to evaporate as the concrete cures. This results in a reduction in volume that is manifest as shrinkage. One of the reasons concrete is reinforced is to control shrinkage and to prevent cracks from developing.

For our building combining the concrete with a soft brick and mortar was intended to pit brick expansion against concrete shrinkage. We worked out that concrete shrinkage could be directed via the corbels to clamp the bricks tight and prevent them from expanding due to irrecoverable moisture movement [the two biggest effects]. These actions are not instantaneous and could therefore be neutralised by creep.

This sounds simple now that it is written down, however at the time we were not sure that it would work. Many hours were spent researching, modelling and in the end testing our solution. In the end we could not make the whole wall work without joints, but the spacing was more than 100 meters.

So there you have it sometimes a creepy building is a good thing. Its never a good thing for shopping bags.

On Trussed Girders

A curious case of small changes

Several years ago when conserving an important university building I came across some interesting timber floor structures, which I quickly recognised as ‘trussed girders’. I had seen depictions in many conservation text books, one of which is shown below, and was familiar with the form.....or so I thought. It wasn’t until I studied some actual examples that I realised that something wasn’t quite right. 

You can see in the image above that trussed girders are formed from two timber beams, each of which had a groove cut into one side. Into that groove was inserted wrought iron, plate or sometimes hard wood, which was clamped to the timbers with large bolts aligned vertically. The two timbers were also joined by horizontal bolts, which passed through their cross section.

The problem was I couldn’t decipher how such a beam would work, even after reading accounts of how they were supposed to work. Most of the accounts agreed that designers at the time thought they were improving the capacity of the original timber beams, but modern understanding had demonstrated that their only effect was to instil  an upward pre-camber, which helped to control deflection.

My initial reaction had been that the beam was, from an analytical perspective, upside down. If it were inverted I could see that the timber at the top would be in compression and the iron plate at the bottom would be in tension. There wasn’t a particularly good mechanism for transferring tension into the iron plate, but I figured the large bolts at either end would be capable of something. This would be a sort of composite timber and iron truss, which would at least be a nod to the name ‘trussed girder’.

The trouble was the beam wasn’t upside down and inverting the logic really doesn’t work. The iron plates would need to behave as compression struts, which would tend to push outwards and the timber would need to behave in tension. There was literally no observable evidence for how tension would be generated in the timber. Not only that this would invert everything we know about how Victorian engineers thought about timber trusses. Timber was always in compression and the tension joints were always reinforced with wrought iron.

I also thought about what the conservation books had said. I could certainly agree that the arrangement didn’t appear to convey any additional strength, but I also could not work out by what mechanism the arrangement would apply an upward camber.

The only form of adjustment that I could see was the potential for tightening the nuts on the vertical bolts, but I could not imagine how this action would lead to an upward camber.

The answer to my conundrum was only discovered when I consulted a Victorian carpentry manual. The image below is what I found. 

This was interesting for two reasons. Firstly, there were in fact two examples of my inverted logic, but in both cases an iron shoe is visible at the end of each beam, which is clearly capable of transferring thrust into the tension rods, which project through and below the timber beams.

Secondly, the various other examples of the trussed girder all had an iron plate on the soffit of the timber which formed a tie and complete the internal truss. This is also clearly shown in the details at the bottom of the picture.

The most interesting example was the one third from top, again for two reasons. Firstly, the iron struts are torpedo shaped. This has been done because the designer new full well that thin plates placed in compression will buckle at the centre. He has elected a torpedo shape to specifically place material in the middle of the cross-section so that the tendency to buckle is more ably resisted. 

Secondly, there appears to be a pair of joints in the bottom tie member; one either side of the vertical bolts. If these joints were used to tighten the tie, causing it to shorten, this would pull the ends of the struts together and would unquestionably cause the center of the beam to rise i.e. there was a perfectly rational explanation for how camber could be imparted.

Satisfied that I had solved the puzzle of the ‘trussed girder’ my mind turned to why it had been a puzzle in the first place. Did the authors of those conservation books not know what they were doing? Possibly, but I am not entirely sure that is the whole story.

Maybe, just like me, the authors were familiar with the form, because they had also seen it in prior text books, but hadn’t had reason to stop and think about it more deeply. If so this would be a lesson for all aspiring engineers that even textbooks are not always right.

There is however a more intriguing possibility, perhaps the authors had in fact worked on various historic examples that lacked a bottom tension chord. After all this is what I had found; I would not otherwise have started on this journey.

In this case maybe the authors simply concluded the concept was flawed and moved on, after all some of the illustrations reproduced in text books do look quite old. Maybe, because the authors hadn’t seen a tied variant, they figured the original designers simply hadn’t worked it out right.

My suspicion is that trussed girders were rather well understood by the originators of the concept, however structural design was not codified at this time and people often learned by copying. It is entirely possible the someone had tried to copy an original trussed girder based on arrangements they had witnessed. Perhaps they had misremembered what they had seen or had not appreciated the purpose of the tie. Others may then have seen the amended design and copied it too. Before long it is not hard to conceive of illustrations being draw showing the faulty design.

The lesson would therefore be, when borrowing a design concept be sure that you have properly understood it.