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 Stegosaurus

The original idea for this post was to show an animal’s skeletal structure and explain how it works. I have chosen to do this using a stegosaurus skeleton for two reasons. 

Firstly, this blog is going to be a bit speculative, because I don’t know anything about animals or their skeletons. In the event that a biologist or anatomist finds him or herself reading this blog there is a high probability I shall be politely informed that I don’t know anything. Based on this I figured that my best form of defence would be to choose an animal for which there are no living examples. This way I can argue that since there are no living examples what said animalist thinks they know is mere speculation. No-one can prove otherwise.

Secondly, stegosaurus is a dinosaur and dinosaurs are cool, every kid knows that. Of course I could have chosen T-rex or triceratops, but I think those are the obvious choices and I happen to like stegosaurus so that’s what I’ve chosen.  Here goes….

It seams to me that the spine of an animal, which supports its head and tail are analogous to a thin flexible truss; in fact a special type called a vierendeel truss. To explain this I am going to need to make a few assumptions. Self-evidently all animals walk, run, jump, climb and so forth. Okay, I don’t know whether the last two are true of Stegosaurs, but neither do you. See how effective that defence is? Anyway, movement gets a bit too complex for my stated aim of explaining how skeletal structures work. With that in mind I am going to assume, for the purposes of this blog post, that we have a stationary stegosaurus standing on all four legs.

 

Based on this the first observation we shall make concerns the legs. The fore-limbs, compared to the hind-limbs, are rarely small, however they are still stocky and have the appearance of being load bearing. The larger hind-limbs and pelvis are larger and would appear designed to carry greater load. This corresponds with the head and neck being rather small when compared to the size and bulk of the hind-quarters and tail.

The stegosaurus’ spine bridges between the two sets of legs and cantilevers beyond them at both ends in order to carry the neck and tail. To understand the structural implications of this arrangement we must first learn something about bending moments.

A bending moment is a turning force whose magnitude is the product of force (or weight) multiplied by the distance (known as the lever arm) to the nearest point of support. The greater the distance to a support; the greater the bending moment.

Consider if you will the following thought experiment. Supposing the stegosaurus’ fore-limbs were located at the end of its nose. I know that’s daft, but run with it. The distance between the nose and its legs is zero therefore no matter how heavy the nose the resulting bending moment is zero.

Now suppose the fore-limbs are located at the back of its head. The bending moment has increased from zero in proportion to the length of its head. I hope you can see where I am going with this. Now suppose the fore-limbs are back where they should be attached to the shoulders. The bending moment has now increased in proportion to the length of a stegosaurus neck and head.

If we were to portray the magnitude of the bending force graphically it would look like a triangle, being zero at the nose and increasing to a maximal point where the legs meet the shoulders. We could repeat the same process starting from the tail. The bending shape would be the same, though the height of the triangle would be bigger because the tail is longer and heavier.

We are now left with the bit in the middle, we need to join the two triangles together. Of course the  body of the stegosaur is somehow attached to the spine (I told you I am not an animalist). We’ll get back to that subject later, but for now we will simply note that the weight of the body must pull down on the spine. It follows that a line depicting the bending moment will join the two triangles, by sagging in middle. It might look a bit like the diagram below. 

The interesting bit; also the bit where I might start to get into trouble; is how the spine resists these bending forces.

The first thing we notice about a stegosaur spine is, like most spines, the odd shape of the vertebrae, which is distinguished by three parts, at least to my eye. There is the oval shaped portion located at the anterior. The lower part is of bone and the upper part has a hole where the spinal cord would be. 

The second feature is the piece of bone that projects vertically from the middle of the vertebra to form the posterior.  This is where the ligaments and muscles of the back are connected to the vertebrae.

The third feature is the two transverse projections; one either side of the vertebrae. These are where muscles and ligaments attach to the spine and are also the points from which the ribs are articulated. The fossil stegosaurus vertebrae pictured below is quite tall and for reasons that will become clear is postulated to be from the lower part of the back.

When the vertebrae are aligned in a row, as they would be to form a spine, they start to resemble a truss with a lower chord of bone and an upper chord of ligament and muscle. You have to imagine the muscle and ligament joining the vertical projections together, as self-evidently they haven’t been fossilised. The internal members of the truss are formed by the bone surrounding the spinal cord and the projection that extends from it. There are of course no diagonal members and that is why the resemblance is to a vierendeel truss.

 

For this information to be of use we now need to describe how a truss works. We shall begin by considering a beam, which is altogether a simpler and humbler form than the truss. 

If a beam spans between two supports, one at either side, it will deflect in the direction of the bending force to form a curve. As the beam deflects the outer edge of the curve is stretched and is evidently in tension. Conversely the inside edge is squashed and must be in tension. The centre of the beam, known as the neutral axis is neither in compression nor tension; it is at rest.

If we apply the same logic to a truss the outer chord of the truss carries a tensile force and the inner chord carries compression. The internal members of the truss transfer load between the outer and inner chords. 

In the case of our stegosaur based truss the ligaments and muscle at the posterior are ideal for carrying tension and the boney part at the anterior is ideal for resisting compression. This is remarkably good luck, because as nature would have it, the bending moments imparted by the tail and neck of the stegosaur, as shown in our earlier diagram, impart tension on the posterior side and compression on the anterior.

If that is not remarkable enough the depth of the vertebrae increase towards the hind limbs to match the increasing magnitude of the bending forces. Nature has actually fine tuned the size and shape of the vertebrae so that the resulting truss matches the shape of the bending forces to which they are subject.

Having made this observation it is tempting to end this post satisfied that our work is complete, but I am not quite ready to do that. There is one more thing that is just too interesting not to pass comment. I also noted earlier in the post that I would return to this subject.

It is not lost on me that the stegosaur spine has a very distinct curve, which given the apparent fine tuning of the vertebrae, cannot be without reason. Based on that it is, I think, worth a further speculation. 

It strikes me when looking at the bulk of a stegosaur’s body, how is that great big, heavy fleshy part of the animal supported from the spine. Presumably it somehow hangs? I imagine the animal’s flank muscles, perhaps reinforced by the ribcage, are responsible for transferring the load.

It then strikes me that an arch is a rather efficient way of supporting the load and transferring it to the legs. There is of course a potential issue with this load path. Arches generally have large abutments whose purpose is to resist the lateral thrusts generated at the base of an arch and thus prevent it from spreading. It is these same thrusts that make it so difficult to build a house of cards. Of course stegosaurus have no abutments.

There is a potential solution, which is to be found in the form of a bow string arch. Sometimes when there is no opportunity to provide bridge abutments the engineer will instead join the arch supports together with a tie member. This works by causing the two sides of the arch to pull against each other. Since the pull is equal and opposite equilibrium is maintained and the arch is prevented from spreading.

My final speculation is therefore that the stegosaurus’ sternum and chest muscles provide a tie, which form’s a bow string truss made of meat and bone.

Now I can finish the post and wait to be shot down by people that know what they’re talking about. I shall be lying in wait with my pre-prepared speculation defence.

On Samson & Goliath

Samson and Goliath are a pair of gantry cranes at the Harland and Wolff shipyard in Belfast. They were built by the German engineering firm Krupp. Harland and Wolff was of course the firm made famous by their design and construction of the passenger liner RMS Titanic. That being said, Goliath was completed in 1969 and Samson in 1974, both long after the Titanic was lost at sea. 

The cranes are named after two characters found in the Bible. Goliath was a fearsome Giant, reported to be 9.5 feet tall and Samson was a man of enormous strength. Both names are surely appropriate, because of the cranes’ enormous size and great lifting power, however perhaps Samson is the more appropriate of the two. 

The Old Testament book of Judges tells us that there was a secret to Samson’s great strength. A secret that he had kept from Delilah; at least to begin with. This suggests to me that contrary to the popular image of Samson portrayed in artwork, he must have been a slight man. For if he were a large powerful man surely no one would have thought there was a secret to his great strength.

When observing both cranes one cannot help but notice the oddness of their form and wonder whether there is also a secret to their great strength. They are supported on one side be two slender tubular columns, which are attached to the large box girder that forms the gantry with a small connection. On the other side there is a large box column with a substantial connection to the gantry box girder. The question arises, why is this so?

The secret of Samson and Goliath is subtle. It relies on understanding a simple, yet at the same time complex, structural load path. If you are unfamiliar with engineering terminology then a couple terms need to be explained to understand what is going on.

When two structural members are joined together, but can rotate relative to each other a ‘pinned joint’ is formed. For example, the two parts of a pair of scissors are held together by a pin joint. If the same connection is formed, but the joints are instead held rigidly so that there can be no rotation, a ‘fixed joint’ is formed. This would make a hopeless pair of scissors. It also important to know that pin joints relieve bending forces by rotating instead of offering resistance while fixed joints attract them.

It is clear that one side of the crane gantry is connected to the supporting column with a rigid joint and the other side is connected with a pinned joint. Understanding the effect these joints have on the structure is the key to understanding the structure’s secret. 

Perhaps the best way to do this is to consider what would happen if the connections were jointed differently. Let us consider in the first instance the gantry box girder being fixed to the columns with pin joints, one at either side. 

If the columns on both sides of the crane could rotate relative to the gantry box girder then a mechanism would be formed and a horizontal load applied to the side of the crane would cause the columns to rotate thus allowing the gantry box girder to slide sideways. The crane is unstable and would topple.

Conversely the gantry could be supported by fixed joints, one at either side. In this instance the columns cannot rotate, because they are fixed rigidly to the gantry box girder. The structure is apparently stable, except that it isn’t.

Since the fixed joints do not allow rotation they attract bending forces out of the gantry girder and transfer them into the columns. This is a perfectly satisfactory situation where the supporting column is large, however the two slender tubes will simply buckle again causing the crane to topple.

The obvious solution to this problem is to use a large stiff column on both sides, so that the structure behaves like the portal frames found in a large supermarket or warehouse. There is of course a good reason why this has not been done. 

One of the immutable characteristics of bending forces is that they are always accompanied by shear forces. One cannot exist without the other. In the case of our imagined postalised frame bending forces at the connection between the columns and girder will necessarily co-exist with horizontal shear forces at the column heads and vertical shear forces at either end of the girder.

Horizontal shear forces at the column heads are a problem. The reason they are a problem is that equilibrium demands an equal and opposite shear force at the base of the columns. The two column bases are on wheels, which allow the crane to run back and forth along a set of rails. Horizontal shear forces at the base of the columns are therefore most unwelcome, because they will tend to press the wheels into the side of the rails causing either the rails to buckle or the wheels to jam or both.

It is now time to return to the actual cranes Samson and Goliath, which have one pin joint and one fixed connection at the column heads. It is of course very tempting to assume that when the crane is lifting the pinned connection is free from bending forces and the fixed connection attracts bending from the girder span. It follows that no bending forces are transferred into the slender tubular columns, which makes sense, and some bending forces are transferred into the box column. These after all are the rules and therefore, some engineers will make this assumption. Of course in this case they would be wrong.

Those who make these assumptions have forgotten about equilibrium. A bending force at the head of a column must, as we know, be accompanied by a horizontal shear force. The horizontal shear must in turn have an equal and opposite partner to maintain overall equilibrium, except in this case there can’t be an equal and opposite partner, because the opposite column has a pin joint i.e. if a shear is present the pin joint must carry a bending force which it cannot do.

The solution to this riddle is of course that the fixed connection does not carry any bending forces and behaves as if it was a pinned connection. How is this possible; as that is not the rule? This is the subtle part of the solution that is often missed. 

Since the large column is not infinitely stiff it will, when loaded, start to deflect ever so slightly due to the bending force its fixed joint would like to impart. This deflection is just sufficient to rotate the head of the column, but without changing the angle between the column and the girder i.e. rotation occurs in the column rather than at the joint, which causes the structure to behave as if there were a pinned connection. Only a very small movement is required for this effect to occur.

Now, since there are no bending forces at the head of either column there are no horizontal shears in the column heads and therefore equilibrium demands no shears at the base of the columns either. This means the cranes are free to travel up and down their rails without becoming stuck.

Conversely, if a horizontal load is applied, say the wind, the box column is sufficiently stiff, in combination with the fixed joint, to deflect only a small amount before resisting the horizontal load and keeping the crane stable. This is the reason for having a large box column on one side only. It is the secret of the Samson and Goliath Cranes.

Incidentally, the reason there are two rather than one slender column is to ensure that the cranes are stable out of plane. The keen observer will notice that the tubular columns rake in opposite directions to provide the required stable platform.