EngCurves

Curves in Engineering

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Catenary and parabola

Aiguille du Midi – beautiful Alpine photograph showing catenary cables

This picture shows the suspension bridge across the River Severn from Aust to Beachley.

This picture shows the Clifton suspension bridge near Bristol. The suspension bridge is easy to understand. Strong cables hang from massive towers. Smaller cables hang from the main cables and support a deck which carries a road. When the main cables have been laid, they hang in a close approximation to a curve called a catenary.

To make a catenary, imagine a compound interest or exponential curve, imagine reflecting it left-to-right in a vertical mirror, and imagine adding the two curves together. That makes a curve called a hyperbolic cosine, or cosh. A catenary is a special case of this curve. A spider web loaded uniformly with dew-drops illustrates the shape beautifully, as below.

But the final shape is not a catenary, because of the weight of the suspended structure. If the deck were vastly heavier than the cable, making the load per horizontal metre the same throughout, the correct curve would be a parabola. In fact this is not strictly true – if the cable were very light, the connection points of the hangers would lie on a parabola, but the main cable would be stretched almost straight in between them. If the curve does not get too steep, a parabola looks very similar to a catenary. You can see this by writing the polynomial series for the two exponentials of the catenary, and then using only small values on the horizontal scale. Only the x² term remains, which generates a parabola.

In an actual bridge, neither the cable or the deck is vastly heavier than the other. So the actual curve is a compromise between a catenary and a parabola. As stated above, the cable does not actually form a smooth curve. It is slightly kinked at each hanger attachment, and between attachments it follows catenaries which are less curved than than the curve on which the attachments lie. A flexible cable can never be perfectly straight unless it is vertical, since an infinite tension is impossible.

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cosh x = 0.5 (e^x + e^-x), so

cosh x = 0.5(1 + x + x²/2! + x³/3! + x⁴/4! + . . . . .

+ 1 – x + x²/2! – x³/3! + x⁴/4! – . . . . .

The odd terms cancel, so

cosh x = 1+ x²/2! + x⁴/4! + x⁶/6! + . . . . .

When x is small enough, the terms beyond x²/2! are small, leaving a curve that is close to a parabola. In fact, for short enough arcs, a catenary, and parabola, and a circle are very similar. For approximate calculations, you can then choose the one that is simplest to work with. The diagram below shows parts of all three curves, circle, catenary and parabola, al with the same initial curvature. You seldom see this much of any curve in a suspension bridge, but you can get a good idea of what a larger piece of catenary looks like from the Gateway Arch in St Louis.

Parabolas do exist in technology. The reflectors of radio and radar aerials are often paraboloids, like those of some electric heaters and lamps. The parabola is a special case, between ellipses and hyperbolas, of a whole class of curves.

It also appears as the trajectory of an object near an airless object such as the moon. In principle the path is actually an ellipse with one focus at the centre of the moon, but for a golf ball travelling a short distance this can be ignored.

Yet more curves are found in engineering. The curve of a uniform sagging beam, resting on one support at each end has a point of inflexion at each support, where the bending moment is zero. Consequently it cannot by any the the curves described above. In fact it includes a square term and a quartic term, and is the black curve below. A beam which has multiple supports, changes in cross-section, or changes in material, will have discontinuities in its bending moment, and therefore in curvature, at theses points. This is discussed under Beams. What if you push the ends of a piece of piano wire until it bends. What is the curve? Is it one of those shown below, or a different one?

Yet more curves can be generated quite simply. Here, a plastic ruler is pushed between two tools. The curve is none of the ones discussed above, because the influence of gravity is negligible. What type of curve is it? How does the bending moment vary along the ruler? The ruler has taken up the position of minimum energy. The rules for generating the curve are simpler than the equation which describes it.

In the previous curve there was only one constraint – the distance between the ends of the ruler. In this picture we control the directions at the ends as well, choosing these and the distance to create a circular arc. This is a special case of a family of curves that could be created using the three variables.

Here are two pictures made using a flexible plastic ruler, a bulldog clip, and a flat-bed scanner. They are very smooth curves, generated by a simple rule, yet they are not circles or ellipses or any other simple curve that you might learn about at school. Luckily, in structural engineering, you don’t need to know about these bendy curves: if your structure has deflected that much, your customers will probably have already voted with their feet. But if you are designing something that must store energy, such as a clockwork radio or an archery bow, you are in a different regime altogether. Designers of weapons like the ballista and trebuchet had to think about these problems. They probably discovered that some organic materials could store a lot of energy.