Rules · Arches · Structure · Geometry
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
Robert Hooke published the idea in 1676 as a Latin anagram, whose solution appeared only after his death: “Ut pendet continuum flexile, sic stabit contiguum rigidum inversum”: as hangs the flexible line, so, but inverted, will stand the rigid arch. Christopher Wren and Hooke used the principle in thinking about the dome of St Paul’s; Antoni Gaudí designed the Colònia Güell church by hanging weighted strings.
The medieval masons had no such theory, but their pointed arches lie much closer to the catenary than a semicircle does, which is one reason Gothic arches can be so slender. Choose the catenary in the model and compare its thrust line with the semicircle’s.
Build an arch of wedge-shaped stones (voussoirs) whose joints all point towards the centre of the curve: each stone can then fall only by forcing its neighbours apart, and so the ring holds itself up once the last stone, the keystone, is in place.
joints ⟂ intrados (radial to the centre)
An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
Trace the path of the resultant force through the stones from crown to foundations. If a line can be found that stays everywhere within the masonry, the arch will stand; where it touches an edge, a hinge will open, and four hinges make it collapse.
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
A round arch must rise half its span, so arches of different spans on one springing line have crowns at different heights. A pointed arch can be given any rise at least half its span, so arches of different widths can all be made to reach the same height.
round: f = s⁄2 (fixed) · pointed: r = (s²⁄4 + f²) ⁄ s for any f ≥ s⁄2
Cite this page Chris Paton, “As hangs the chain, so stands the arch”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/arch-inverted-chain. Text, drawings and models licensed under CC BY-NC 4.0.