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)
Rules · Domes · Structure · Materials
Building a dome without centering
Lay the brickwork of a large dome in horizontal rings, each ring locked by bricks set on end in a herringbone pattern, so that every stage of the work stands by itself without a timber frame beneath it.
Nobody in 1418 knew how to vault the crossing of Florence Cathedral: the timber needed to support such a span during construction was thought impossible to obtain. Brunelleschi’s solution, as far as it can be reconstructed from the fabric and from his early biographers, was to make the dome self-supporting while it rose. The bricks are laid in courses that sag slightly towards the centre of each face, and at intervals vertical bricks are inserted in a spiralling herringbone (spina pesce) pattern that interlocks the courses, so that each new ring is held in place while the mortar sets.
He also designed the hoisting machinery (an ox-powered reversible hoist and cranes for placing the stones), which impressed his contemporaries almost as much as the dome itself.
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, “Building a dome without centering”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/dome-without-centering. Text, drawings and models licensed under CC BY-NC 4.0.