What is Sang’s Logarithmic Method?

If you’re building a bridge – for example, over a canal – you might think “I know! I’ll use an arch!”

This is a great solution, so long as the thing that needs to go over the canal is perpendicular to it. For most of the history of canals, that was quite easy to make the case: a road could usually be turned to cross the river, and a bit of a bend was a vast improvement over wet feet.

But then some fool invented the railway. It’s quite hard to make a train turn sharply – so engineers had to learn to build skew arches.

My other half spotted one solution on a walk in Poole – I have a thread about that here.

I am not a civil engineer ((I’m not even civil about engineers.)) so I might have the details wrong here. My understanding is that the traditional skew arch method has two problems:

  • first, not all of the arches are grounded on both sides. The more oblique the crossing the worse the problem, and once you get to 45 degrees, none of the arches reach the ground on both sides of the river.
  • second, the forces exerted by the weight of the arch act in vertical planes parallel to the faces – which means there’s a shear force between the arches, which weakens the structure yet more.

Edward Sang had another solution.

His insight: it must be possible to build the arches so that the forces act without shear – so that the brick courses ((I think they’re called voussoirs)) are perpendicular to the forces exerted.

If you unroll the cylinder, the weight forces act along a family of curves of the form $y=k-a\cos(x)$. ((The parameter $a$ is $\cot(\theta)$, where $\theta$ is the angle between the canal and the railway.))

The gradient of these curves is $\dydx = a \sin(x)$. The perpendicular gradients are the negative reciprocal of this, $\diff{y_\perp}{x}=\frac{\csc(x)}{a}$.

And we can integrate that to get the curves we want: $y_\perp = b - \ln\left|\csc(x)-\cot(x)\right|$ .

This (as I understand it) is Sang’s logarithmic method.

Why is it interesting?

Points at the bridge. Lookit!

Who was Edward Sang?

Edward Sang was born in Kirkcaldy((Always nice to see a Fifer in the DOME.)), Scotland on January 30th, 1805; he died in Edinburgh late in 1890.

He’s best-known for computing enormous tables of logarithm and for foreshadowing Foucault’s pendulum by showing how a spinning top could demonstrate the Earth’s rotation.