Towers that project, approaches that turn

A beam is squeezed along its top and stretched along its bottom; stone resists squeezing well but stretching badly, so a stone lintel must be deep in proportion to its span and cannot be long. Doubling the span quadruples the stress from its own weight.

σ ≈ ¾ ρ g L² ⁄ d (self-weight) · + ≈ 0.43 ρ g L³ ⁄ d² (a wall above)

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

Weight for weight, timber is many times stronger in bending than stone; long spans in trabeated buildings are therefore roofed with timber beams resting on stone walls and columns.

bending strength ÷ density: oak ≈ 70 · limestone ≈ 3 (kPa per kg ⁄ m³)

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.