Expect a hemispherical dome to be in compression along its meridians everywhere but in tension round its horizontal rings below about 52° from the crown; restrain that lower zone with weight, thickness, rings or chains, or it will crack along the meridians.
thin hemisphere under its own weight: hoop force changes sign at ≈ 51.8° from the crown
A dome can be thought of as a family of arches (the meridians) held together by rings (the parallels). Near the crown the rings squeeze inward and help the arches; lower down the meridians try to spread, and the rings are stretched. Masonry and unreinforced concrete resist tension poorly, so the lower part of an ancient dome usually cracks into segments along its meridians (the Pantheon’s did) and then stands as a ring of arches leaning on a compressed crown.
Builders met the problem in different ways: the Pantheon’s thick drum and step rings, the buttresses round the window ring of Hagia Sophia, and later the iron and timber chains embedded in Brunelleschi’s dome in Florence and the iron rings added to St Peter’s in Rome in the eighteenth century.
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.
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
An arch cannot support itself until it is complete. Build it on a timber centring shaped to the intrados, strike the centring only when the keystone is set and the mortar has taken, and reuse it from bay to bay, which is one reason to keep the spans of an arcade equal.
Cite this page Chris Paton, “The lower dome pulls apart”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/hoop-tension. Text, drawings and models licensed under CC BY-NC 4.0.