The inverted catenary arch
In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had found “a true mathematical and mechanical form of all manner of Arches for Building”. He did not publish the solution of this anagram in his lifetime, but in 1705 his executor provided it as Ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning “As hangs a flexible cable so, inverted, stand the touching pieces of an arch.”
The walls of the cathedral are particularly thick to avoid the need for largeflying buttresses. The windows are set into deep recesses in the walls. The upper parts of the cathedral walls are reinforced with small flying buttresses, which were a relatively late design change to give extra strength. These are concealed behind a large curtain wall, which was added to keep the building’s classical style intact.
The large crossing dome is composed of three layers. The inner and outer layers are catenary curves, but the structural integrity to support the heavy stone structure atop the dome is provided by a intermediary layer which is much steeper and more conical in shape. The dome is restrained round its base by a wrought iron chain to prevent it spreading and cracking.
Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down (i.e. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (e.g., stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shearforces are present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline.
Catenary arches are often used in the construction of kilns. In this construction technique, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.
However the conditions for a catenary to be the ideal arch are almost never fulfilled: arches usually support more than their own weight, and on the rare occasions when they are freestanding they are sometimes not of uniform thickness. The ideal shape for an arch supporting a large weight is more like a parabola than a catenary. As a result, there are very few arches that have been deliberately built as catenaries, though there quite a few incorrect claims that various arches are catenaries.
The Gateway Arch in Saint Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation y=Acosh(Bx). (A catenary would have AB=1.) A catenary is the ideal shape for a freestanding arch of constant thickness, but the gateway arch is not of constant thickness as it gets narrower near the top.