The shape of a hanging chain

Hold a chain by its two ends and let it droop. The curve it makes looks like a parabola, and it is often said that Galileo thought it was one. It is not quite. The reason is simple: where the weight sits along the chain.

A curve with a job

A chain can only pull. Push on its ends and it folds up. So when a chain hangs still, every link is in tension, and the chain has settled into the one shape in which pulling alone holds up its weight. In 1675 Robert Hooke saw what that means for builders. He published it as an anagram of a Latin phrase, ut pendet continuum flexile, sic stabit contiguum rigidum inversum: “as hangs the flexible line, so but inverted will stand the rigid arch.”

Why it matters

Stone behaves the opposite way to chain. Block, DeJong and Ochsendorf put the two side by side: “The chain can support only tension, and the masonry arch acts in compression.” Flip the hanging shape upside down, and every pull turns into a push along the same line. The shape a string takes under a set of loads, turned over, is a path along which an arch can carry those same loads by squeezing alone. They call it the funicular shape for those loads. In 1748 Giovanni Poleni used the idea on the cracked dome of St Peter’s in Rome. He hung a chain carrying “32 unequal weights”, each in proportion to one slice of the dome, and showed that the chain’s curve fitted inside the dome’s section. By the rule the authors cite from Heyman (1966), a line of force that lies everywhere inside the masonry shows the structure is safe for that set of loads.

The balance at every point

Cut the chain anywhere and ask what the rest of it must supply there. Osserman states the answer in one sentence: “the vertical component at each point is determined by the weight of the chain below it that needs to be supported, while the horizontal component is simply transmitted unchanged along the chain.” Take the lowest point as the start. The sideways pull, call it H, is the same all the way along. The upward pull at any point equals the weight of the chain between that point and the bottom. So the slope of the chain at a point is that weight divided by H.

Everything now depends on how weight builds up as you move out from the bottom. Two cases give two different curves:

• Weight spread evenly across the horizontal. A suspension bridge cable holds up a deck far heavier than itself, and the deck’s weight is laid out evenly from side to side. The weight supported grows in step with horizontal distance, so the slope does too. A slope that rises in a straight line gives a parabola. Osserman notes that this is the case for the cables between the towers of a suspension bridge.
• Weight spread evenly along the chain. A bare chain carries only itself, so its weight grows with the length of chain, measured along the curve. Out near the supports the chain is steep, so each step sideways adds more chain, and more weight, than it did near the bottom. The slope rises faster than a straight line. The curve is the catenary, from catena, chain. Osserman writes it as y = cosh x = ½(e^x + e^−x) in a suitable choice of coordinates. Jacob and Johann Bernoulli, among others, worked it out at the end of the seventeenth century.

Near the bottom a chain is nearly flat, so length along the curve and distance across are almost the same, and the two curves agree. That is why the old story about Galileo needs care. He is often said to have called the chain a parabola. Osserman finds that he treated it as an approximation, and said the chain goes “almost exactly” along the parabola when the parabola rises at less than 45°. Our arithmetic shows how close it is. A chain that sags one-tenth of its span strays from the parabola with the same span and sag by 0.03% of the span. At a sag of half the span the gap grows to 2.5%.

The Gateway Arch is a weighted chain

A catenary is the right arch only if the arch’s weight is spread evenly along it, like links of equal weight. The Gateway Arch in St Louis is not. Its cross-sections are equilateral triangles 54 ft on a side at the ground and 17 ft at the top. The finished arch is 630 ft wide and 630 ft high. A formula often quoted for the arch, y = −127.7 ft·cosh(x/127.7 ft) + 757.7 ft, is a true catenary of those dimensions. Osserman’s verdict is that “it is not the equation that was used in the construction of the arch”: the National Park Service sheet at the Arch itself gives the blueprint equation instead.

The blueprints give the centre line of the legs as y = A(cosh Bx − 1), with A = 68.7672 and B = .0100333 in feet. That is a catenary only if A = 1/B. Here A×B is .69, so it is a catenary “shrunk in the vertical direction by just under a third”. Osserman calls this a flattened catenary. It is the shape a chain takes when the links near the ends are heavier, which matches the legs of the arch thickening toward the ground. Our arithmetic from Osserman’s formulas puts that centre line 625.1 ft high and 598.5 ft across at the ground. Adding the thickness of the legs brings the outside to 630 ft by 630 ft. The widget’s second view compares it with a true catenary drawn through the same three points. The two differ by up to 22.0 ft (our arithmetic).

In short

A hanging chain carries the same sideways pull all along its length. The upward pull at each point equals the weight below it. Weight spread evenly across the span gives a parabola. Weight spread evenly along the chain gives a catenary. Weight that grows toward the ends gives a curve rounder at the top than the catenary. Turn any of these upside down and you have an arch that stands by pushing alone, for exactly the loads that shaped it.