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 · Geometry · Structure
The pendentive: a sphere cut by a square
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
The pendentive solves a geometric problem exactly. A sphere through the corners of the square meets each side wall in a semicircle (the four arches) and meets the horizontal plane at the arches’ crowns in the circle inscribed in the square. The curved triangles between are parts of that same sphere, so they carry the load of the ring smoothly down to the four corner piers.
The Romans had used pendentive-like forms on a small scale; the Byzantines made them monumental at Hagia Sophia. Above the ring the builder may put a hemisphere, raise a drum, or simply continue the same sphere to make a sail vault. The pendentive passed to the Ottoman mosque and, through Brunelleschi’s Florence, to the Renaissance and Baroque church.
Related rules
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
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 combine the unity of a central plan with the direction a church needs, plan the space as an oval, set across the axis to widen the view of the altar, or along it to lead the eye towards it.
Divide the inside of a dome into coffers that shrink in step with the meridians as they rise, so that their perspective deepens the dome and draws the eye to the light of the lantern.
Cite this page Chris Paton, “The pendentive: a sphere cut by a square”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/pendentive-sphere. Text, drawings and models licensed under CC BY-NC 4.0.