On Pyramids & Ziggurats

Smarter engineering than you might think

It is self evident that ancient structures were not conceived using modern codes of practice and that their designs were based on rules of thumb evolved from a process of trial and error, however there is perhaps some misunderstanding as to what this means in practise. 

It was not, as you might suppose, a process of edging successive designs slowly towards failure with fingers crossed; hoping to stop before you get there.

A rule of thumb, by its mere existence, presupposes the existence of mathematical relationships between objects. How else might proportions and limits be implemented. Furthermore, evidence suggests that trial and error was purposeful and based on underlying principles of logic. As we shall see ancient structures are more sophisticated than you might think.

Ziggurats were built in Mesopotamia on the great plain that lies between the Tigris and Euphrates rivers in what is today part of Iraq. These two great rivers carry vast amounts of silt, often depositing it along the way in times of flood. This created thick deposits of alluvial soil, which are ideal for agriculture, but much less so for constructing large, heavy buildings.

This was not the only geological issue to be overcome. With the Mesopotamian plain being covered to depth with alluvial soil, there is little building stone from which to construct monumental structures.

For this reason Ziggurats were constructed of bricks, made of locally available mud baked in the sun. Sadly, without a protective stone skin, most surviving examples are heavily eroded. That said, on account of their exposed condition, those which remain provide us with clues about how they were constructed.

 It would have quickly become apparent to the Mesopotamians that soft alluvial soils would undergo significant settlement when subjected to the weight of a ziggurat and their steep sided construction would have a tendency to spread at the base. 

Successive layers of construction would suggest that they paused and restarted the works on many occasions until the settlements and spreading eventually ceased. In this way the lower layers became layers of fill below a wide plinth or temenos, on which a great temple could be constructed.

This process is not unlike the modern technique of preloading soft ground with great berms of earth. The same technique has been used to improve sites adjacent to the River Clyde in Glasgow.

The Mesopotamians were not, however, satisfied with the pace of construction that this method afforded and they soon conceived another ingenious plan, which engineers today might consider modern.

After every eight or nine courses of brickwork they began to add a thin layer of sand containing matts of woven reeds and cables made of plant tissue. Together these innovations allowed the Mesopotamians to create a primitive form of reinforced earth not unlike that which is achieved today using geosynthetic grids and textiles.

The great weight of construction generates friction between the mud bricks and the reinforcement; clamping them together so that they cannot move relative to each other. This allows the tensile capacity of the reinforcing matts and cables, which is not possessed by the brickwork itself, to be mobilised such that the steep walls of the ziggurat are prevented from spreading laterally. If this were not clever enough the layers of sand in which the reinforcement was laid had two ingenious roles. Firstly, it would have helped to bed the bricks evenly onto the reed matts helping to ensure an even distribution of load and to prevent sharp or uneven edges from causing unwanted damage. Secondly, the sand would suck moisture from the mud-bricks and provide a route for it to escape. This leads to consolidation, increased density and greater strength.

The evidence is clear; Mesopotamian ziggurat builders were not simply stacking bricks until failure was reached. These innovations demonstrate a knowledge of complex engineering principles. 

The Pyramid’s of Egypt are built between the Libyan dessert and the western bank of the river Nile, as it flows towards its Mediterranean delta. On the face of it they appear to have much in common with Ziggurats. They are both large, heavy structures, with steep sides, constructed from masonry. They both impose massive loads at their base and are subject to lateral spreading forces.

This, however, is where the similarities end, because the great pyramid designer Imhotep came up with some rather different solutions. Perhaps the most obvious difference is that Imhotep, and those who followed him, adopted locally available limestone en lieu of mud bricks. It is a much stronger material, which requires a different treatment.

Perhaps the first thing to note is that a pyramid’s weight is not evenly distributed. The maximum pressure is exerted below its centre, reducing towards the edges. This means that a pyramid’s core and perimeter will settle differentially. 

We know that Imhotep understood this because he devised a clever method of preventing the rigid stone blocks from being fractured by said settlement. 

Pyramids are not solid structures. Examples investigated at Saqqarah, Meidum and Dahshur consist of a solid stone core laid at a steep angle, which is surrounded by independent concentric squares of masonry. The inner portion of each square, roughly 4/5, is of roughly cut stone laid in mortar while the final 1/5 is of dressed stone with smooth contact surfaces. The central core also has an outer facing of dressed zone.

Each independent square can slip relative to its neighbour, thus accommodating differential settlement. The efficacy of this process is enhanced by the smooth surface of the facing stones. 

Nevertheless, cutting and dressing smooth stone surfaces is difficult, time-consuming, expensive work, particularly using bronze age tools. It therefore made sense to minimise this type of work by using rough cut stone as the backing, although this does have consequences. While the dressed facing stones have good contact surfaces that distribute load evenly and provide a solid stable base, the rough cut stones have poor contact surfaces resulting in greater potential for consolidation and outward movement.

Imhotep would have known that the inclination of the dressed facing masonry had to be optimised so that it leans into the rough stone and contains its tendency to spread. It has been found that the angle adopted corresponds to the prime numbers 2, 7 & 11. 

These observations demonstrate that Imhotep, and those who followed, had a clear understanding of structural load paths and of building materials. Furthermore, what evidence we have for design by trial and error falls within this rational framework.

The Stepped Pyramid at Medium and the ‘Bent’ Pyramid at Dahshur are good examples.

The former has a strange shape, which archaeologist originally presumed to be the result of stolen facing stones. It is not clear why one would steal from the top and not the base;  engineering appears to provide a better explanation. While the construction follows Imhotep’s settlement mitigation strategy some of the stone has been found to be of poor quality. It’s friable nature caused a local collapse by creating the conditions for a slip plane to develop thus causing the loss of several structural layers due to spreading. 

Similarly, the so called bent pyramid clearly shows that the designer realised part way through the build that the angle of inclination was too steep and had to be reduced to maintain equilibrium and thereby prevent spreading. This demonstrates that he understood something was going wrong and then knew what to do about it.  

It follows that for both the ziggurat’s of Mesopotamia and the pyramid’s of Egypt there is clear evidence of structural principles being understood and refined by purposeful trial and error and captured in rules of thumb with a basis in Maths. 

One might argue that they are good examples of qualitative design. They are certainly a reminder for modern engineers that complex sums are secondary to clear thinking about underlying load-paths and a practical knowledge of material.

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 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 Accidental Bridges

The robustness of brick walls

I came across this rather interesting photo, which shows the remains of a partially demolished nineteenth century mill. 

The picture is dominated by a large brick wall, which is suspended above ground level. The original load-path for its support appears to have been an arcade consisting of cast iron columns carried on masonry arches. 

To the left of the image one of the arches has been altered. The profile of its voussoirs has been straightened on one side and the arch infilled with modern brickwork. The infill brickwork is in turn supported on a modern beam and post structure, which is likely formed of hot rolled steelwork.

Below the centre of the wall two of the cast iron columns have been pushed over. The one on the left is clearly visible, while the one on the right is a little harder to pick out. Its head can be detected, because it is still connected to the wrought iron rods that would have been used to resist thrusts at its abutments. To the right of the photo the rods can still be seen linking the remaining iron columns.

It is also evident that the masonry arches which would have been carried by the missing columns have collapsed. It is assumed that rubble, which can be seen at the base of the picture, is what remains of them.

What is of course remarkable is that the brick wall continues to stand. It has found a way of bridging between the steel column on the left and the cast iron column on the right. This is interesting for several reasons.

Since the span is roughly equal to the height of the wall it is reasonable to assume that it has achieved this feat by acting as a deep beam. Were the deep beam made of concrete it would be normal to assess it using a ‘strut and tie’ model. This means imagining a triangular load path in the wall. The two vertices of the triangle form compressive struts, which are sat over the supporting columns, while reinforcement in the base of the wall would complete the triangle by resisting lateral thrusts from the inclined struts. An alternative way of considering this model would be to think of it as a tied arch, as the principle would be similar. Indeed, post failure the residual brickwork appears to have formed a crude arch between the column supports.

In this case there is no concrete and no reinforcement, because the wall is made of brick. Brick is good in compression, which would seem to make either an arch, or compressive struts, a viable load path, however there is no equivalent for the reinforcement, which is required to complete the triangle. 

Brickwork is poor in tension and therefore the base of the wall ought to have failed, however there seems little evidence of distress other than the smaller arches giving way to form a larger, if somewhat makeshift arch.

This means the deep beam must, on this occasion, have a different mechanism for resisting lateral thrust. Two options seem plausible, though one seems more likely than the other. 

It is possible that there are rigid load-resisting structures located just out of picture on either side of the image, however if this were the case then there would be no need for the wrought iron ties, which originally held the column heads together.

It seems more likely that the weight of the brick located above the arch supports is sufficient to divert the arch thrusts back to the vertical i.e. it behaves like heavy bridge abutments. Both headers and stretchers can be seen in the brickwork suggesting that the wall is at least a full brick thick and would therefore be relatively heavy.

Looking carefully at the image there is also evidence of pockets in the brickwork, which would have seated beams in the wall. In fact the remains of an I-shaped steel beam can still be seen embedded on the left hand side. It was presumably easier to cut the beam off at the support than to completely remove it during prior alteration works. This may be relevant because the load carried by the wall must have been reduced. This in turn means the thrust is less.

Since two columns are missing we would ordinarily expect those which remain to take 100% more load. For this reason a reduction in load carried by the wall is of significant benefit to the columns too.

So it would seem that a structure whose intended load-path has been compromised has managed to find an alternative load-path demonstrating rather well that structures will exhaust all possible ways of standing before they collapses.

This does not mean that we would necessarily wish to rely on this load path in the long-term, but nevertheless it is a reminder for engineers that there is more than one way of looking at a problem….in this case an accidental bridge.