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°)
The Bent Pyramid at Dahshur (c. 2600 BC) was begun very steep. Cracks and settlement in its chambers, probably caused by the soft ground beneath, seem to have persuaded the builders to change to a much gentler slope a little below half its height, reducing the weight still to come. Sneferu then built a third pyramid nearby, the Red Pyramid, at the gentle slope from the start. The Great Pyramid of his son Khufu split the difference at 5½ palms.
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)
Specify the slope of a pyramid face by its seked: the number of palms (and fingers) by which the face runs inward for every cubit of rise. The height follows from half the base divided by the seked, times seven.
seked = 7 × (½ base ⁄ height) palms · angle = arctan(7 ⁄ seked)
The Great Pyramid of Khufu was laid out 440 royal cubits square and 280 cubits high, a seked of exactly 5½ palms and a slope of about 51°50′.
½ × 440 ÷ 280 × 7 = 5½ palms → arctan(7 ⁄ 5.5) ≈ 51.84°
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, “The bent pyramid: changing the seked in mid-course”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/bent-pyramid. Text, drawings and models licensed under CC BY-NC 4.0.