Rules · Arches · Structure · Proportion
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)
Alberti distinguishes three arches: the full or “right” arch, a semicircle; the “diminished” arch, a segment less than a semicircle; and the “composite” arch of two arcs meeting at a point. He regards the semicircle as the most secure; the flatter arch relies more on the strength of the walls or ties that hold its ends.
The formula is the modern statement of the same experience. A shallow segmental arch or a flat jack arch concentrates a large horizontal force at its springings; a tall pointed arch carries its load down more steeply and asks less of its abutments. Change the rise of the model and watch where the line of thrust enters the piers.
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)
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
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.
Draw a pointed arch from two centres on the springing line. With the centres at the springings, the arch is equilateral; with centres beyond them, a lancet; with centres between them, a drop arch. Late Gothic added arches of four centres and the reversed curve of the ogee.
r = s (equilateral) · r > s (lancet) · s⁄2 < r < s (drop)
Classify arches by the geometry of the intrados: the number and position of the centres (one for round and segmental, two for pointed, three or four for basket and Tudor arches), or a single continuous curve (ellipse, parabola, catenary), and distinguish true arches, with voussoirs, from corbelled “arches” of oversailing courses.
Cite this page Chris Paton, “The flatter the arch, the harder it pushes”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/arch-thrust-rise. Text, drawings and models licensed under CC BY-NC 4.0.