Rules · Structure · Materials · Doors & windows
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)
A simply supported beam of span L and depth d carrying its own weight develops a bending stress at mid-span proportional to L²⁄d: make the lintel twice as long and it must be twice as deep merely to stay as safe. If a wall stands on it, the load itself grows with the span and the stress grows roughly with L³⁄d². Meanwhile the tensile strength of stone is only a few megapascals and varies from block to block with hidden flaws and bedding planes, so builders kept a large margin.
The result is visible everywhere in trabeated architecture. Clear spans between Egyptian and Greek columns are rarely more than three or four metres; the widest stone spans of antiquity, such as the central aisle of the Karnak hypostyle or the Propylaia at Athens, were achieved with deep, carefully chosen blocks, and some Greek architraves were secretly helped by iron bars. The model computes an indicative stress for the chosen span, depth and stone (indicative strengths with a factor of safety of three); raise the span and watch the lintel crack.
In a corbelled vault each course may project only a little beyond the one below, and each overhanging half must have its centre of gravity behind the inner face of its wall; load the tails of the stones with heavy hearting until the capstone is set.
stable if Σ W·x (behind the springing edge) > Σ W·x (in front)
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)
At Stonehenge each lintel is pegged to its uprights by a mortise-and-tenon joint and locked to its neighbours by tongue-and-groove joints, and each is curved to follow the circle: the joints of a timber frame reproduced in dressed sarsen.
In a megalithic chamber the room can be no wider than the largest slab the builders can raise over it; the upright stones are set to suit the capstone, not the other way round.
The royal tomb grew in about a century from a low battered mastaba, through a stack of diminishing mastabas (the step pyramid), to a smooth-faced true pyramid; a failure of stability in between forced the change of slope of the Bent Pyramid.
Cite this page Chris Paton, “A stone lintel’s span is limited by its tension”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/lintel-span-depth. Text, drawings and models licensed under CC BY-NC 4.0.