Lego towers and the Menai Straits Bridge

You’ve probably seen this BBC News Magazine article on the maximum height of a tower of Lego bricks. I read it yesterday through a link from the indispensable Seth Brown (@DrBunsen), and was immediately reminded of this 1826 paper on suspension bridges by Davies Gilbert.

That Gilbert paper is my white whale. I first saw it three years ago when the Royal Society put up its Trailblazing site of historical papers to commemorate its 350th anniversary. It was a paper I knew about from Timoshenko’s History of Strength of Materials1 but had never read. When I did read it, I wanted to write one or two blog posts about it, but I never figured out the right approach. It’s been haunting me ever since.

Gilbert wrote the paper as Thomas Telford was building the Menai Straits Suspension Bridge, and he shared his analysis with Telford. It’s an excellent mix of mechanics theory and practical engineering. But what made it fascinating to me was the state of analysis and structural engineering in the early 1800s, especially in England. Had that paper been written a couple of decades later, it would’ve been very different.

First, there’s the math. Gilbert sets up and solves the differential equation for the catenary shape of the suspension chains. But instead of using the standard differential notation we all use today, he uses Newton’s fluxion notation. Seeing a space derivative, like the slope of the chain, expressed as the ratio of two time derivatives just seems really weird today. I assume Gilbert did this because he was a proper Englishman and had been taught Newton’s calculus instead of the calculus of that horrid Continental thief, Leibniz.

It’s sometimes argued that mathematical analysis in the 18th century flourished in Europe and languished in England because the English stubbornly held on to fluxions, while the Bernoullis, Euler, Lagrange, etc. happily adopted differentials. If you fight through the math in Gilbert’s paper, you’ll appreciate that argument. I knew that England was late to change but was surprised to see fluxions still being used as late as 1826. It didn’t last much longer.

As for the engineering, here’s how Gilbert describes the strength of the wrought iron used for the suspension chains:

Let the span proposed for a suspension bridge be 800 feet, and let the adjunct weight of the suspension rods, road-way, &c. be taken at one-half of the weight of the chains; then, if the full tenacity of iron is represented by the modulus of 14800 feet, the virtual modulus for the whole weight must be reduced in the proportion of 2+1:2, or to 9867 feet; and let it be determined to load the chains at the point of their greatest strain, that is at the points of suspensions, with one-sixth part of the weight they are theoretically capable of sustaining.

A lot of this makes perfect sense to the modern reader. He’s assuming that the weight of the chains themselves makes up ⅔ of the total suspended weight, and he wants to use a factor of safety of 6. But what the hell is this “full tenacity of iron is represented by the modulus of 14800 feet” stuff? He’s obviously talking about the strength of the iron, but how and why is he expressing it in feet?

The answer lies in the fact that our modern ideas of stress and strain were in their infancy. They were defined by Augustin-Louis Cauchy in a paper written in 1822, just four years before Gilbert’s paper. It’s unlikely that Cauchy’s ideas were in common currency yet, especially among practitioners. (You may have noticed that Gilbert refers to strain in the above passage. I think he’s just using the term colloquially; there’s no evidence in the paper that he understands strain as we do today.)

OK, so that explains why Gilbert didn’t express the strength of the iron in pounds per square inch (psi) as we would today. But how can a strength be expressed in feet? For that, we return to the article about the maximum height of a Lego tower (you thought I’d forgotten, didn’t you?).

In the Lego article, Ian Johnston, a lecturer at the Open University, loads a 2×2 Lego brick to failure in compression. He then takes that information, along with the height and weight of the brick, to calculate the height of a tower of bricks whose own weight would squash the one at the bottom. He gets a height of 3500 meters.2

This is, more or less, what Gilbert means when he says the tenacity of the iron has a modulus of 14,800 feet. The only difference is that instead of expressing the compressive strength as the maximum height of a tower, he’s expressing the tensile strength as the maximum length of a vertical bar suspended from its top. 14,800 feet, he’s saying, is the length of wrought iron bar that would fail under the pull of its own weight.

We can express this strength in modern terms with a little algebra. The weight of a bar of length \(L\), cross-sectional area \(A\), and unit weight \(\gamma\) is

\[W = \gamma L A\]

If this bar is hung vertically, the tensile stress on the cross-section at the top is

\[\sigma = \frac{W}{A} = \frac{\gamma L A}{A} = \gamma L\]

Wrought iron has a unit weight of about 480 pounds per cubic foot. A bar 14,800 feet long would have a stress of

\[480\;\mathrm{\frac{lb}{\;\;ft^3}} \cdot 14{,}800\; \mathrm{ft} = 7{,}104{,}000\;\mathrm{psf} = 49{,}300\;\mathrm{psi}\]

This is in the ballpark for the strength of wrought iron, but it’s a little optimistic, especially for the early 1800s. Good thing Gilbert was using a factor of safety of six.

By the way, through the magic of Google Maps, you can visit the Menai Straits Bridge without leaving the warmth of your computer. It is a beautiful bridge.

Menai Straits Bridge

The wrought iron links of the suspension chain have been replaced with steel, but the towers and the roadway layout are basically the same as they were nearly two hundred years ago. Follow the link and take a virtual drive across the bridge. Be careful—the archways through the towers are a tight squeeze.

  1. I don’t know why I bother linking—I’m sure you all have that book. But maybe you want a second copy for the office? 

  2. Two things about this calculation: first, some of the numbers look suspicious to me, but I haven’t had the time to check them out; second, Dr. Johnston knows perfectly well that a tower of 2×2 bricks would buckle long before it reaches 3.5 km—he’s calculating a theoretical maximum based on the strength only.