Rules · Roofs · Geometry · Structure
Over the vaults of a northern church raise a timber roof of about 55–60°, its section close to an equilateral triangle, so that snow slides off and the slates or tiles overlap generously.
equilateral roof: pitch = 60°, rafter = span
The roofs of the great French cathedrals are as tall as a house: the timber “forest” of Notre-Dame in Paris rose about ten metres above the vaults. The steepness had several reasons (climate, the slate and lead coverings, the need to clear the crowns of high vaults), and it gave the northern Gothic silhouette its sharp gables. When the Gothic Revival returned to it in the nineteenth century, the equilateral pitch became a sign of the style.
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.
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.
Cite this page Chris Paton, “The steep northern roof”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/gothic-steep-roof. Text, drawings and models licensed under CC BY-NC 4.0.