On Century Tower

Ductility and seismic design

Century tower is an interesting building with a striking facade. The question is whether it has been designed this way for aesthetic reasons or whether there are any engineering reasons for its appearance? 

The building is located in Tokyo and this fact provides us with several clues. The external structure is redolent of Japanese calligraphy and that is surely not accidental. We also know that Tokyo is in region of seismic activity and it turns out this is also important.

For minor events buildings designed to resist earthquakes are expected to survive without damage, however for an earthquake of moderate size some damage to the cladding and fittings is considered acceptable. The real design challenge is what happens when a major event takes place. In these circumstances permanent deformation is expected to principal structural members. Not only is this expected it is actually required; it is part of the design strategy and this is why buildings with seismic resistance look and feel different to those which do not.

That said, one of the reasons Century Tower is interesting is because it does not look like a traditional seismic design. To understand why this is we need to take a couple of steps backwards.

Ordinary buildings tend to be kept stable by triangulated bracing, however this is not a good solution for earthquake resistance, because of the potential failure mechanisms if the design load were to be exceeded. Fracture of a tension brace or buckling of a compression brace would be catastrophic for a building’s stability.

A better way to survive an earthquake is to ensure that deformation occurs instead. This has the advantage of being a non-catastrophic failure mode and is also a useful way of absorbing seismic energy.

As we saw in my last post ‘On Ductility’ steel is a ductile material, which permits plastic strains [deformation] to develop without failure occurring. The art is to make the deformations take place where you want them to and avoid those locations where you don’t.

The first step in this process is to give the structure a vierendeel frame rather than a braced one. We have met the vierendeel frame before in earlier blog posts. Its primary characteristic is rigid joints which permit no rotation at the junctions between beams and columns. This forces deformations to occur within the members themselves due to bending. If sufficient bending stresses are developed then plastic hinges will form and these happen to be rather effective at absorbing energy.

Conventional wisdom is to adopt a tight structural grid so as to maximise the number of opportunities for plastic hinges to form and also to avoid overly large members. The down side to this approach is that modern open plan floor plates and open facades, which let in light, are difficult to achieve.

The next step is to ensure that plastic hinges occur in the beams rather than the columns. This is achieved by making the latter much stiffer than the former; the so called strong column-weak beam approach. The reason for this is self-evident, plastic hinges in the columns will render them incapable of supporting the weight of the building and will cause it to topple. Conversely, plastic hinges in the beams will lead to deformation, but not collapse.

Century Tower is interesting because it retains the strong column-weak beam approach, however it does so while adopting an open two storey structure. This is achieved with an eccentrically braced frame [EBF] en lieu of a vierendeel frame. EBF’s are essentially a modified system of K bracing. Each bay consists of two rigid braces, which are sized to avoid yielding, connected to a ductile beam, which is intended to develop plastic hinges where the braces apply their thrusts.

Thus, although the EBF is not strictly a vierendeel it effectively behaves in the same way. Rigid braces lock the beam-column junction and thereby force plastic hinges to develop in the gap between them. Providing hinges form before the braces are able to yield they cannot fail catastrophically, as they would in a orthodox frame.

It is a rather clever system, although it requires a more careful analysis than a conventional vierendeel, because there are fewer locations for plastic hinges to form i.e. the structure needs to behave as intended and must successfully mobilise each bay. This is not an easy thing to work out, as the precise nature of a given earthquake is difficult to predict.

Now, returning to the question with which we started, has this rather striking facade been designed for aesthetic reasons or is it for engineering reasons?

I think it would be fair to say that there are strong architectural reasons for the design. Firstly, the ability to have a modern open plan building with large windows and perhaps secondly to hint at Japanese culture. It is however quite clear that the structural form also plays a rather important role in resisting seismic forces.

The answer to the question is therefore that the facade has been designed with both aesthetic and engineering requirements in mind. It is a good example of the symbiotic relationship that exists between architect and engineer. 

I rather suspect that the design went through many iterations before the final solution was settled upon and that both parties had a significant role in its development. 

On Ductility

The benefit of elastoplastic materials

One of the most useful structural properties a material can have is ductility. In order to understand why this is so it is first necessary to explain some related concepts. We will start with stress and strain.

Stress is a measure of load intensity; more precisely it is the ratio of force divided by area. This concept can be visualised by considering how it is that someone can lie down on a bed of nails without being harmed, yet if the same person were to tread on a single nail it would pierce their foot. The reason a bed of nails causes no harm is because the person’s weight is spread over the cumulative area of many nails whereas a single nail concentrates a person’s weight over a very small area i.e. the bed of nails applies low stress, while the single nail applies high stress.

When a structural member is exposed to stress it will change in length. If the stress is tensile it will become longer and if it is compressive the reverse is true. If the material is uniform then the elongation or shortening is spread evenly along the length of the member. For example, a member that is half as long will change in length by half as much. The ratio of elongation or shortening per unit length is defined as strain.

Stress and strain are of course related; higher stress will cause higher strain. A useful way of expressing this relationship is to plot a stress-strain graph. The convention is to measure stress on the vertical axis and strain on the horizontal axis.

If a material is ductile, like steel, the relation of stress to strain will be directly proportional resulting in a straight line on our graph [1]. The inclination of the straight line is know as the elastic modulus, for reasons we will see shortly.

There will, however, come a point where strain will increase more quickly than stress and the graph is no longer a straight line. The proportional limit of the material has thus been breached. Soon after this point the graph will become horizontal, which means that strain will increase without a corresponding increase in stress. This threshold is known as the yield stress.

Beyond yield the internal structure of the material will begin to change at an atomic level. This process is known as strain hardening and it results in additional strength, and therefore higher stress, in return for further strains.

Eventually the stress-strain curve will again flatten leading to increased strains, but this time with decreasing stress. The point where this occurs is know as the ultimate stress.

Increased strain accompanied by decreasing stress is a rather curious effect. Why would increasing strain result from a reduced stress? There is of course a rational answer to this question. 

When any material is stretched there is a corresponding narrowing to facilitate the stretch. Generally this is too small to notice, however beyond the point at which the stress-strain curve flattens the effect becomes more pronounced and ‘necking’ occurs.

If we were to calculate true stress, based on a necked [reduced] cross-section, rather than continuing with a nominal stress, based on the original cross-section, then the stress-strain plot would in fact continue to grow. After a period of necking fracture will eventually occur.

Returning to the beginning of our stress-strain plot we can modify our approach. This time instead of continually increasing the stress applied to our test member we can load it to a known stress and then unload it again. We can then increase the stress to a slightly higher value and then unload again. This sequence of loading and unloading may be repeated.

When we do this we find that as long as we remain below the proportional limit the unloaded member will fully recover its original shape and will follow the straight line portion of the stress-strain graph. Beyond the proportional limit a full recovery will not be made; there will be some residual strain left behind.

When the member fully recovers its shape it is said to be elastic. The point at which it loses its elasticity is known as the elastic limit. Beyond the elastic limit materials are considered to be plastic. For many materials, like steel. The proportional limit, yield stress and elastic limit are relatively closer together and are in practise treated as if they were the same. For some materials, such as rubber, the elastic limit lies well beyond the proportional limit.

When large permanent strains start to occur within the plastic zone the term plastic flow is adopted. 

There are several reasons why ductility, incorporating both elastic and plastic behaviour, is important. We shall discuss one of them below and save another for the next post.

Supposing we wished to design a series of floor beams for a new building. The owners would not be terribly happy if they were to permanently deform and sag while the building was in use. For this reason we would want them to remain within the elastic range. We would therefore perform our design based on limiting stresses in the beams to the yield stress of the material.

This would satisfy the requirements for every day use of the building, however supposing an unforeseen or extreme event occurred, which caused the floor to become overloaded. In such circumstances you would not wish the floor beams to fail suddenly and without warning, for this would inevitably lead to a loss of life. 

Self-evidently it would be preferable for there to be a visible warning that something was wrong so that the occupants could vacate the floor safely and remedial action could be taken. This opportunity is provided by plastic behaviour in the floor beams. Although permanent deformation will occur within the plastic range the floor will at least remain stable and safe.

It is fairly obvious how this principle will work if the floor beams are made of steel, but perhaps less so for concrete. After all concrete is not a ductile material. It is weak in tension and fails explosively in compression.

The first problem is solved by casting steel reinforcement into the concrete in zones where the concrete is in tension. The reinforcing bars, rather than the concrete, resist the applied tensile stresses.

The addition of steel reinforcing bars also provides the means to deal with concrete’s lack of ductility. The reinforcement is deliberately designed to have a smaller capacity in tension than the concrete has in compression. This way as the reinforced concrete bends the reinforcing bars in the tension zones will reach their elastic limit before the concrete reaches its brittle limit in the compression zones. This principle is known as under-reinforcement and it is key to all reinforced concrete design. 

Thus reinforced concrete becomes a ductile material.

[1] In this example we are of course assuming the application of tensile stress

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.