Rules · Proportion · Structure · Façades
Within a generation the nave vaults of the French cathedrals rose from about 37 m to 48 m while their spans stayed near 15 m: the ratio of height to width climbed from about 2.3 to over 3, until at Beauvais part of the choir vault fell.
Chartres ≈ 37 m · Reims ≈ 38 m · Amiens ≈ 42 m · Beauvais ≈ 48 m
Height was the measure of ambition. At Amiens the nave vault stands about 42 m above the floor on a span of some 14.6 m. Beauvais went further, to about 48 m; in 1284 some of its choir vaults collapsed, and the choir was rebuilt with extra piers. The collapse marked the limit of the High Gothic experiment, not because stone could not stand higher, but because the particular details of Beauvais failed.
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)
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
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.
To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.
pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
Cite this page Chris Paton, “Ever taller, ever narrower”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/gothic-height. Text, drawings and models licensed under CC BY-NC 4.0.