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)
Two shells tied by ribs
Build a great dome as two shells (a thick inner one that carries, a thin outer one that protects and gives the silhouette) joined by ribs and locked at the top by a ring under the lantern.
Florence: inner shell ≈ 2 m at the base; outer ≈ 0.9 m; 8 angle ribs + 16 intermediate ribs
Brunelleschi’s dome is not a single vault but two, with a space between them wide enough for the stairs by which the builders, and visitors today, climb to the lantern. The inner shell is the structural one; the outer is lighter and keeps the weather off it while giving the dome its great swelling profile. Eight white marble ribs mark the angles outside; between them, hidden between the shells, run sixteen further ribs, two on each face, tied by horizontal rings of stone and timber that act like the hoops of a barrel.
The double shell reduced the weight on the drum and allowed a dome of unprecedented size to look taller from the city than a single shell of the same thickness could. Michelangelo’s dome for St Peter’s and Wren’s for St Paul’s both develop the idea.
Related rules
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
In a row of arches the thrusts of neighbouring arches cancel over each intermediate pier, which may therefore be slender; at the ends of the row nothing balances them, so the outermost piers must be made wider or buttressed.
To set a round dome on a square bay, imagine a sphere passing through the four corners of the square; cut it by the four vertical planes of the sides and by the horizontal plane of their crowns. The four spherical triangles left in the corners are the pendentives; the circle at their top carries the dome.
sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2
If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.
crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)
Lift the dome on a cylindrical or polygonal drum pierced with windows between the ring of the pendentives and the springing of the dome, to light the centre of the church and to make the dome visible from afar.
drum height ≈ ⅓–1 × its radius (Byzantine); much taller in Russian churches
Cite this page Chris Paton, “Two shells tied by ribs”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/double-shell-dome. Text, drawings and models licensed under CC BY-NC 4.0.