Timber Cupola and Its Tie Ring

What is modelled

A thin dome carries its load mostly by forces in its own surface, in two directions: along the meridians (from crown to rim) and around the circumference (hoop). Near the crown both are compressive. Lower down, a dome steeper than about 52 degrees from the crown needs the hoop to pull. At the rim the meridional force pushes outward and downward; the downward part goes into the walls, and the outward part must be held by a tie ring, or the walls spread and crack.

Equations

• Meridional force Nφ = −(a / sin2φ) ∫0φ pz sinφ′ dφ′.
• Hoop force Nθ = −Nφ − a pz cosφ; for roof weight q alone Nφ = −qa/(1 + cosφ).
• The hoop turns to tension at cosφ = (√5 − 1)/2, 51.8 degrees.
• Rim thrust H = −Nφ cosφ0; ring force T = H a sinφ0; snow acts as s cosφ per shell area.

Checks run in the page

• Numerical forces against the textbook self-weight formulas at four angles.
• The 51.8 degree hoop switch by bisection.
• Vertical equilibrium of the whole cupola.
• The shallow-dome limit of the ring force.

Honest simplifications

Membrane theory ignores bending, which matters at the rim and under uneven snow or wind; the cupola is a perfect, uniform spherical shell rather than a boarded and ribbed timber structure; the timber strength is a single simplified value for dry pine along the grain, with the scarf joints reduced by one efficiency number; and creep, moisture and the stiffness of the drum are not modelled. The sizes are illustrative, not a measurement of the real church.

• Press Steep onion-like crown: a large part of the shell turns red and the rim thrust becomes mostly vertical.
• Press Heavy snow and watch the ring utilisation rise past 100 %.
• Press Weak joints to see how much the scarf joints cost.