Stupa Dome: Weight and Hoop Tension

What is modelled

A stupa can be read two ways. As a solid hemispherical mound its weight is a stack of discs, and the ground under it feels a pressure that is highest at the centre. As a thin dome shell, gravity creates meridional forces along the arcs and hoop forces around the rings. In a hemisphere the upper rings are compressed and the lower rings are in tension, which is why masonry domes tend to crack along meridians.

Equations

• Mound: pressure p(r) = γ √(R² − r²), weight W = 2πγR³/3, mean pressure 2γR/3.
• Thin shell under self-weight w = γ t: Nφ = −w a /(1 + cosφ), Nθ = w a (1/(1 + cosφ) − cosφ).
• The hoop force changes sign where cosφ = (√5 − 1)/2, at 51.83 degrees from the crown.

Checks run in the page

• Mound weight by summing rings and by stacking discs against the closed form.
• Vertical equilibrium of every cap of the shell.
• The sign change of the hoop force found by bisection.
• Normal equilibrium along the meridian and the cracking angle for your tension limit.

Honest simplifications

The shell follows membrane theory with no bending, no edge supports other than a rigid footing, uniform thickness and uniform masonry. A real stupa is a rubble-filled solid mound with a stone facing, and the crown load is spread uniformly over the base here. This is a teaching model of load paths, not a structural assessment of any monument.

• Press Thick shell, no tension capacity and watch the red band appear in the lower dome.
• Press Heavy stone, weak ground to see the ground pressure exceed the allowable line.
• Press Small stupa and note how the pressure scales with size while the hoop stress at the base is just the unit weight times the radius.