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⛪ Masonry Dome Restoration
A historic masonry dome has vertical cracks running up from its base. Your job is to place an iron tension ring — but where? Sweep the angle and watch the hoop force change sign. Masonry takes compression happily and cracks in tension, so the crack zone is wherever the hoop force turns positive. The crossover angle is the same for every spherical dome ever built.
Dome
Radius a = 10.0 m
Self-weight q = 5.0 kN/m²
Springing angle: 90° (hemispherical)
Your Tension Ring
Live Result
—
Hoop Force Nθ (kN/m)
—
Meridional Nφ (kN/m)
🎯 Goal: find the exact angle where hoop force crosses zero — that's where the cracking starts, and where the ring belongs.
Powered by classical shell membrane theory
(
engine/dome-membrane.js). The critical angle was verified two
independent ways before publishing: analytically (setting Nθ=0 gives
cos²φ+cosφ−1=0, whose root is exactly the golden ratio conjugate 0.6180) and
by numerical bisection on the hoop-force expression itself — both give
51.827°. Note this is membrane theory: it assumes the shell carries load
purely in-plane, with no bending. Real masonry domes also develop bending
near supports and at discontinuities.Method
Membrane theoryNφ = −aq/(1+cosφ); Nθ = aq[1/(1+cosφ) − cosφ] (Timoshenko & Woinowsky-Krieger).
φ_crit = 51.83°Where cos²φ + cosφ − 1 = 0 — independent of dome radius and self-weight, a purely geometric result.