On Britannia Bridge

An empirical masterpiece

When I studied engineering at university one of the structures that was frequently put forward by Professors, as an excellent example of structural design was Britannia Bridge. Throughout my time at university and probably for a time afterwards I didn’t really understand why. It didn’t seem a particularly great design to me. I don’t suppose for one moment that was my Professors’ fault; I had good teachers. I suspect it was more likely that I wasn’t paying sufficient attention.

Though I was unquestionably wrong, if I give my younger self the benefit of the doubt, my view was not entirely without merit. Firstly, I had not seen lots of other bridges old or modern that resembled Britannia; suggesting that despite its supposed brilliance few saw fit to copy it[1]. Secondly, passing steam trains through a tubular structure, as Britannia did, didn’t seem like a good idea to me. The smoke and soot must have been pretty unbearable. Furthermore, the integrity of the bridge was compromised by fire in 1970, which brought about its eventual demise. It therefore failed in its primary purpose of carrying trains from Wales to Anglesey. 170 years of service is by most accounts rather good, but I reasoned this was luck. What would have happened had the fire occurred earlier in its life? 

Notwithstanding all of this, Britannia Bridge was, in my humble opinion, ugly. My architect friends might well quip that this is exactly what to expect from a bridge designed by an engineer. I beg to differ, but that debate is for another day.

Having got that historical baggage off my chest I am now, as you might have guessed, wholly convinced of Britannia’s brilliance and it’s aesthetic has sort of grown on me. I’m not saying it has become easy on the eye; rather an appreciation of how it works and how it came to be has resulted in a greater appreciation of its aesthetic.

So why did I change my view of Britannia’s structural design and what was it that persuaded me? Much like students today, my younger self lacked an appreciation of context and when judging the achievements of the past context is everything.

Thomas Telford, the first president of the Institution of Civil Engineers, completed the Menai Strait suspension bridge in 1826. At that time it held the record for the longest suspended span in the world and it is an elegant and beautiful bridge. That said Telford’s masterpiece [perhaps the subject of a future post] was not without problems. It suffered aerodynamically and was damaged on several occasions before being stiffened. 

When Robert Stephenson was faced with the problem of bridging the Menai Strait for the Chester & Holyhead railway a suspension bridge, like Telford’s, was not considered stiff enough to carry the weight of steam locomotives [the suspension form was still in its infancy].

The obvious answer would have been to construct intermediate piers from which arch centerings could be erected, however the currents in the strait were considered to be treacherous and therefore the Admiralty insisted that there be no obstructions placed in the channel, which would make it more difficult to navigate. The only place an intermediate support could be found was at Britannia rock, which was located almost halfway between either shore. 

An alternative idea was to construct an arch on centring built from floating pontoons, however this was also ruled out as there would have been insufficient clearance for ships near the abutments at high tide.

Since a suspension bridge was deemed too flexible and the Admiralty’s intervention had ruled out an arch, Stephenson had to invent something altogether new. 

To achieve the required strength and stiffness Stephenson had some notion that the answer would involve creating large wrought iron plate girders by riveting. He later settled on the idea of a wrought iron tube through which the trains would pass. He had originally thought of the tube as being suspended from chains, but decided instead that it should behave as a hollow beam spanning from end to end.

In 1845 there was no structural theory that could be used to design Stephenson’s hollow tube, albeit British engineers had little time for the theories of their European neighbours. This was not unreasonable as theory was unable to justify the structures that British Engineers were already producing at that time. Telford himself was on occasion disdainful of theory, his Menai Bridge was entirely justified by rigorous testing of its components.

Stephenson turned to William Fairbairn, who was known for his investigation of iron and his experience of ship hulls, which seemed analogous to his hollow tube idea. He engaged Fairbairn to investigate a hollow tube design by conducting experiments on scale models.

Fairbairn realised however, that extrapolating behaviour observed in models to full scale was not straightforward. He therefore recommended Eaton Hodgkinson to Stephenson for the purpose of deriving empirical relationships from his models that could be used to size the bridge. We have of course met this pair before in a prior blog post ‘On Fish-bellies’. 

And so it was that three great pioneers Stephenson, Fairbairn and Hodgkinson set about designing Britannia bridge.

Amongst their findings the trio discovered the mechanism of plate buckling, which overturned the idea that wrought iron was weaker in compression than tension. They also discovered that rivets work by clamping and friction, when previously it was assumed their behaviour was governed by dowel action. They looked at the effects of continuity over an intermediate structural support, which enabled a more efficient design to be realised. They also explored thermal movements and wind effects. Their collaboration was a remarkable piece of work.

That said, it wasn’t all plain sailing. They argued about whether suspension chains were in fact required so late into the process that the bridge piers had to be built full height just incase they were needed. Fairbairn took the position, based on his experiments, that the chains were not required, while Hodgkinson, based on the empirical theory he was developing, took the position that they were. In the end Fairbairn’s argument prevailed; Stephenson omitted the chains and Fairbairn was proved right. 

The resulting bridge was a mixed success. It performed as intended, however it was exceedingly expensive, owing to the cost of iron in the mid nineteenth century. This is probably the main reason its form was not adopted more widely. Nevertheless, Stephenson’s design methodology was exceedingly influential, as were the empirical insights gleaned from Fairbairn’s models.

In this sense Britannia Bridge is archetypal of the empirical approach to design that was practised by British Engineers at that time.

Ironically the fire, which caused its eventual demise in 1970 was caused by ignition of the wooden roof covering that was later added to protect the bridge from corrosion. It wasn’t even part of the original bridge, so my younger self got that completely wrong too!

[1] In a very pure sense this isn’t strictly true, because modern bridges do make use of box girders, but not in the same way as Britannia. 

 

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 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 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.

On Fish Bellies

The search for a ‘more rational’ beam

The picture below was taken from a building I conserved and altered. It’s interesting because it’s a rare example of fish-belly beams. They are made of cast-iron and are supported on circular columns, also made of cast iron.

 

Cast Iron columns are common, but fish-belly beams were only manufactured for a short period of time in the nineteenth century. They were simultaneously the culmination of engineering knowledge at the time and a dead end technology that marked the end of an era. There are several reasons why this is the case and I hope to explain some of them in this post.

To understand the paradox it is first necessary to know something about building design and construction materials in the nineteenth century. One of the issues with describing construction history is that it doesn’t always fit into neat periods of time when one technology starts and another ends. In reality developments overlap and there are differences between countries and even within regions of the same country. I don’t really want to devote this post to unpicking historical subtleties so we are going to make some broad generalisations.

For many years traditional buildings had been constructed with load bearing masonry walls and timber floors. This resulted in cellular room layouts with short floor spans that were vulnerable to fire. As industrialisation became more common factories and mills wanted buildings with a more open plan format that were also fire proof.

Engineers responded by replacing the internal walls with columns made of cast iron. Floors were initially still made of timber, but were gradually replaced with ‘jack arches’ made of brick and supported on iron beams. Iron was not fire proof, but it was at least non-combustible and would therefore not contribute to or spread fire.

While these developments were first introduced in mills and factories in the UK they also represent the intellectual origin of high-rise building in the United States. Perhaps that would be a good subject for a later post. 

Today we think of iron and steel, as being strong reliable materials, which are relatively cheap to mass produce, but this was not the case in the nineteenth century. 

Cast iron was strong in compression, however it was much weaker in tension and was also quite brittle. This meant that it was a good material with which to make columns, providing they were loaded concentrically, but beams were more of a challenge. 

When a beam bends it starts to take up a curved profile. The inside surface of the curve, the top of the beam, gets shorter and the outside surface, the bottom of the beam, gets longer. This means that the top of the beam is subject to a compressive force, while the bottom is subject to tension. The tensile and compressive forces are of equal magnitude but act in opposite directions.

Being stronger in compression than tension the governing factor for designing a cast iron beam was therefore its tensile capacity at the bottom of the section. Another important factor was cast iron’s brittle behaviour which meant that an overloaded beam would fracture quickly and without warning. By contrast, modern steel is equally strong in tension and compression and more importantly it is ductile. This means steel beams will deform rather than fracture, thus providing a period of warning to building occupants before failure occurs.

Of course steel was not available when factories and mills were being designed. Wrought iron has similar behavioural properties to steel and was available, but was very expensive and could not be manufactured in large section sizes. That said, while cast iron cost less than wrought iron, it was not exactly cheap either. It follows that finding the most efficient design for cast iron beams was, for a time, the holy grail.

In the 1820’s the person who provided the necessary impetus was William Fairbairn, who owned a large ironworks in Manchester. He enlisted the help of mathematician Eaton Hodgkinson to plan a series of experiments in order to establish a ‘more rational beam’ cross section.

An inverted T beam with a large bottom flange to prevent tensile failure was the first logical step. A smaller top flange was also introduced to prevent compression buckling at the top of the web. Hodgkinson advocated a ratio of 6 to 1. This is counter intuitive to the modern engineer, who is primarily concerned about the top flange of a steel beam buckling. It is however a perfectly sensible approach based on the properties of cast iron.

The next logical step was to consider the distribution of bending force in beams. A bending moment is the product of a force multiplied by the distance to the nearest support. This means that at the supports bending moment is zero rising to a maximum at the centre of the span. If the beam is uniformly loaded then the force in between follows a curved profile.

This meant that if a beam’s flanges were curved on plan, so that they were wider in the middle of the span, they would match the distribution of bending moment along the length of the beam. 

It was also recognised that, while the capacity of a beam is proportional to its width, it is also proportional to the square of its depth. This means that a beam’s depth is actually more significant than its breadth. Matching a beam’s longitudinal profile to the distribution of bending moment would therefore also result in ‘more rational’ cross section. Such beams came to be known as fish-belly beams.

With this a ‘more rational’ beam had truly been realised. It maximised the capacity of a beam while allowing Fairbairn to make them with 20-30 percent less iron. The first building to benefit from the new approach was Orrell’s Mill in 1834. 

Ironically not long after the ‘more rational’ beam had been created it fell out of use. The reason; manufacture of wrought iron and then steel had suddenly become technically and economically viable. This completely changed the parameters of what made a beam economic. Cast-iron, as the name would suggest, is formed by a casting process. A beam can be made to any shape for which a mould can be made. Conversely, wrought iron and steel are rolled into shape from larger billets of metal. Creating a fish-belly profile in wrought iron or steel would therefore require additional fabrication steps. For this reason it was, and remains, more economic to have a mass produced profile that is easy to make than a more efficient profile that is more expensive to make.

And so it was that the rather elegant cast-iron fish-belly beam was redundant almost as soon as it had arrived. It was the nineteenth century equivalent of the Sony mini-disc; a super piece of design that could not compete with the unexpected arrival of MP3 players.

This is of course why it was a joy to discover these rare examples of the fish belly profile and to be given the opportunity to conserve them, although in this case the horizontal profile was uniform.