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.
Djoser’s tomb at Saqqara, around 2650 BC and credited to Imhotep, began as a square mastaba and was enlarged into six stages of stone, the first large building in cut stone. Under Sneferu the steps were filled out to make true pyramids, at Meidum and then at Dahshur.
Herodotus, writing two thousand years later, was told that the Great Pyramid was raised first in steps and that the stones were lifted from step to step by contrivances of short timbers, the casing being finished from the top downwards. Whatever the machinery, the internal structure of the pyramids, a stepped core wrapped in a casing of fine limestone, confirms the sequence.
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)
When a steep pyramid shows signs of distress, flatten the slope of the upper part: Sneferu’s pyramid at Dahshur rises at about 54° for its lower part and about 43° above.
lower seked 5 palms (≈ 54.5°) → upper ≈ 7½ palms (≈ 43°)
Make the outer faces of pylons, enclosure walls and mastabas lean inward as they rise, so that the wall is thickest at its foot; mark the corners with a torus roll.
batter of pylons ≈ 1 : 8 (about 83°)
Cite this page Chris Paton, “From mastaba to true pyramid”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/pyramid-evolution. Text, drawings and models licensed under CC BY-NC 4.0.