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
The line of thrust must stay inside the arch
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
The line of thrust is the hanging chain for the arch’s actual loads, turned upside down and scaled so that it passes through chosen points of the crown and springings. In the figure it is computed for each arch from the weight of the ring itself and drawn through the centres of the crown and of both springings. For the catenary it coincides with the centre line; for a parabola of the same rise it deviates only slightly; for the semicircle it escapes the ring near the haunches unless the ring is made thick. The classical engineering result is that a semicircular arch needs a thickness of roughly a ninth of its radius.
This is why Gaudí’s arches at the Palau Güell and the Casa Milà attic are parabolic or catenary, and why Roman and Romanesque semicircular arches carry heavy spandrel fill over their haunches, which pushes the thrust line back into the ring.
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.
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.
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, “The line of thrust must stay inside the arch”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/thrust-line-inside. Text, drawings and models licensed under CC BY-NC 4.0.