Dome vs. Flat Roof 2D — Snow/Wind Load Capacity Comparison

Snow / wind load 12 kN/m

Dome geometry

Rise / span ratio 0.50

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comparing load paths

Flat roof — beam depth needed

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Dome — shell thickness needed

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Material efficiency (beam ÷ dome thickness)

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Beam stress type

bending

Dome stress type

compression

How it works

Same span, same load — two different load paths. The flat roof can only resist load by bending: the top fibre of the beam is squeezed and the bottom fibre is stretched, and the required depth grows with the square root of the bending moment M = wL²/8. The dome instead channels the same load down its curve as pure compression thrust, the way a chain hung under its own weight and flipped upside-down carries load with no bending at all:

M = w·L²/8                     (flat roof bending moment per metre width)
h_beam = √(6·M / σ_tension)     (required beam depth)
N = w·L²/(8·h_rise)             (dome/arch horizontal thrust)
t_dome = N / σ_compression       (required shell thickness)

Because unreinforced masonry and stone resist compression roughly 50× better than tension, the same load needs a dome only a fraction as thick as the beam it replaces — which is why the Pantheon's shell is thin and cathedrals span wide halls with vaults instead of flat lintels.

Load — raising it grows the beam's bending moment quadratically-ish (via the required depth) while the dome's thrust grows linearly, widening the efficiency gap.
Rise/span ratio — a taller dome has a shorter lever arm for the same span, so it needs less horizontal thrust and a thinner shell; a shallow dome behaves more like a flat roof.

Tension Compression Compression flow

Stress magnitude across the span (normalized)