A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
Rules · Arches · Structure · Geometry
As hangs the chain, so stands the arch
Hang a chain (or a model of strings loaded like the building) from two points; the curve it takes, turned upside down, is the ideal shape of an arch carrying the same loads, in pure compression.
A hanging chain can only pull, so the curve it adopts under its own weight, the catenary, is the line along which that weight is carried in pure tension. Invert the curve and every tension becomes a compression of the same size: an arch of that shape carries its own weight with no tendency to bend. Hooke published the idea as an anagram in 1676, which decoded reads, in translation, “as hangs the flexible line, so but inverted will stand the rigid arch”.
Gaudí turned the principle into a design method. For the church of the Colònia Güell he built a hanging model of strings weighted with small bags of lead shot, photographed it, and turned the photographs upside down: the result was a structure of leaning columns and vaults in which every member follows its line of thrust.
Related rules
A thin curved shell carries load by the membrane forces in its surface, not by bending; so shape it to its loads and supports, and it can span great distances with a thickness of a few centimetres.
span ÷ thickness of a good shell ≈ 400–1000 (an egg: ≈ 100)
A fabric or cable-net roof is stable only if it is prestressed into an anticlastic (saddle) shape, curved upward in one direction and downward in the other, so that wind from either side meets a surface already in tension.
A masonry arch stands if a line of thrust (the path of the compressive force through it) can be drawn entirely within its thickness; the catenary arch contains its own line of thrust exactly, the semicircle only if it is thick enough.
catenary: thrust line = centre line · semicircle: needs t ⁄ R ≳ 0.11
On an equator-facing window, make the overhang deep enough to shade the whole window from the high noon sun of summer, and no deeper, so that the low winter sun passes beneath it into the room.
noon sun altitude = 90° − latitude ± 23.44° · overhang D = window height ⁄ tan(summer altitude)
A new material calls for new forms: iron should be used in slender members that show its strength in tension and bending, not cast in imitation of stone columns and vaults.
Cite this page Chris Paton, “As hangs the chain, so stands the arch”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/catenary-inverted. Text, drawings and models licensed under CC BY-NC 4.0.