Engineering · statics

A stone arch has no glue holding it up. Take away the wooden frame it was built on and it simply stays, for centuries, held up by nothing but the shape of its curve. So which curve is it, and how do you find it?

Stone can be squeezed, not pulled

Stone is very strong when it is squeezed and very weak when it is stretched or bent. An arch works because every block is pushed into its neighbours and never pulled away from them. The loads travel down the curve as a chain of pushes until they reach the ground. The shape has to make that true for every block at once. A shape that is slightly wrong makes one joint open up, and the arch folds.

The answer came from a rope

In 1675 Robert Hooke published a riddle. Tucked into the back of a book on telescopes was a jumble of Latin letters, an anagram, and the claim that he had found “a true mathematical and mechanical form of all manner of Arches for Building.” He kept the solution secret. It was decoded in 1705, after his death: ut pendet continuum flexile, sic stabit contiguum rigidum inversum. As the flexible cord hangs, so, inverted, will the rigid arch stand.

Try it below. The string is pinned at both ends and carries nine weights. Drag a weight down to make it heavier and watch the string rearrange itself. Then press Flip into an arch.

Why a rope knows the answer

A rope can only pull along its own length. It cannot push and it does not resist bending. So wherever it hangs, each piece of it is held in perfect balance by the pulls of its neighbours, with no sideways shearing between them. That balance is the shape of the string.

Now turn the drawing upside down. Every force reverses: each pull becomes a push. The same shape, built from blocks, is a structure in which every joint is squeezed straight along the curve and nothing tries to slide or open. That is what stone needs. Engineers call the curve of those forces the line of thrust, and an arch is safe as long as some line of thrust fits inside the thickness of its stones.

Flat arches push sideways

Look at the first readout. At each end of the string the weight is carried by a pull that has a downward part and a sideways part. If the string is nearly horizontal, the sideways part has to be huge to hold the same weight. Flip it, and that becomes an outward push at the base of the arch.

For an arch carrying an even load, the sideways push at each foot is roughly the total weight times the span, divided by eight times the height. An arch half as tall as it is wide pushes sideways with about a quarter of its total weight. An arch one quarter as tall pushes with about half. One an eighth as tall pushes with all of it. This is why low arches need buttresses, or an iron tie across the feet, and why tall ones can stand on slim piers.

Now press Heavy middle. The string bends into a point, because the weight in the middle needs steep lines to hold it. Flipped, that is a pointed arch, the same family as a Gothic one. The right curve depends on the load. Only an arch carrying just its own even weight follows the smooth catenary, the curve of a plain chain, whose name comes from the Latin for chain. In 1691 Leibniz, Huygens and Johann Bernoulli each worked out its equation.

Where the trick is still used

Antoni Gaudí designed the church of Colònia Güell near Barcelona with a hanging model at one tenth scale: strings loaded with weights, which he turned upside down to read off the shapes of the arches and vaults. The same idea reaches Missouri. The Gateway Arch in St. Louis is 630 feet tall and 630 feet wide, and its centreline is a weighted catenary: a catenary adjusted because its stainless steel legs taper as they rise. It was built between 1963 and 1965.

Hooke’s riddle comes down to this. To find the shape that holds a load with nothing but squeezing, let the load pull a string into its own shape, then turn it over.

Model: a string of fixed length with nine point loads at equal horizontal spacing. Its sag at each point equals the bending moment of the loads divided by the sideways pull, and the pull is chosen so the string has the length you set. Sizes are in arbitrary units.