Rules · Vaults · Structure · Geometry
Intersect two barrel vaults of equal height at right angles: the result is a groin vault, whose weight and thrust are gathered along the diagonal groins to the four corners, leaving the sides free for openings.
surface = max(barrel₁, barrel₂) · groins on the diagonals
The Romans vaulted the great halls of their baths with groin vaults in concrete, three bays in a row carried on eight colossal piers; the frigidarium of Diocletian’s baths survives as the church of Santa Maria degli Angeli. Because the loads come down at points, the bays can be lit by large lunette windows high in the side walls.
Romanesque masons built groin vaults over aisles everywhere, and over whole naves at Speyer from about 1080. Laid in rubble on full centring they are heavy, and their groins (sharp arrises where the two barrels meet) are hard to build true.
In a bonded masonry wall the courses above an opening corbel over it of their own accord, so the lintel carries roughly the triangle of wall with sides rising at 60° from its ends; everything above that triangle arches onto the jambs.
load ≈ equilateral triangle on the span: W ≈ 0.43 · L² · t · ρ g
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.
Cite this page Chris Paton, “Two barrels crossed gather their thrust at the corners”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/groin-vault. Text, drawings and models licensed under CC BY-NC 4.0.