Rules · Domes · Materials · Structure
Build a great concrete dome thickest and heaviest at its haunches and thinnest and lightest toward the crown, grading the aggregate from dense stone at the base to light volcanic pumice at the top.
Pantheon: thickness ≈ 6 m at the springing to ≈ 1.5 m at the oculus; aggregate travertine and tuff → brick → pumice and scoria
Weight near the crown pushes the dome outward at its base; weight near the base, by contrast, steadies it. The Pantheon’s builders therefore reduced the weight of every layer as they rose, both by thinning the shell and by using lighter caementa in the mortar. The coffers and the open oculus take away still more of the load where it would do harm.
Outside, the lower part of the dome is hidden by the drum, which rises well above the springing of the interior hemisphere and is loaded with a series of stepped rings. This extra mass at the haunches, where the dome would otherwise tend to crack and spread, is the Roman answer to a problem later builders solved with chains and buttresses.
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.
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.
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.
Cite this page Chris Paton, “Thick and heavy below, thin and light above”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/dome-lightened-upward. Text, drawings and models licensed under CC BY-NC 4.0.