Deep-Sea Pressure Hull

What is modelled

Under the sea a closed hull has air at normal pressure inside and the full water pressure outside, so the whole wall is in compression. A thick wall fails by yielding, a thin one by sudden elastic buckling, at a pressure far below the yield load. Real shells are never perfectly round, so the buckling pressure is cut by a knock-down factor.

Equations

• Seawater dp/dz = ρ0 g (1 + p/K) so p = K (eρ0 g z/K − 1), K = 2.3 GPa.
• Thin shell stress σ = p R / 2t (sphere), p R / t (cylinder); yield collapse py = 2 σy t/R or σy t/R.
• Elastic buckling pcr = 2E / √(3(1−ν²)) (t/R)² (sphere), E/(4(1−ν²)) (t/R)³ (long cylinder), times the knock-down.
• Thick wall (Lame) σr, σθ from equilibrium dσr/dr + k (σr − σθ)/r = 0, k = 2 sphere, 1 cylinder.

Checks run in the page

• Runge-Kutta integration of the compressible pressure against the closed form.
• The Lame stresses against the equilibrium equation and both boundary pressures.
• The lowest eigenvalue of a discretised ring against the classical 3EI/R³.
• The wall ratio at which yielding and buckling coincide, by bisection.

Honest simplifications

Only perfect, uniform walls are analysed: no welds, openings, stiffening rings, creep, fatigue or temperature effect on the seawater, the knock-down is one number, and the buckling formulas are the classical small-deflection ones. Use it to see how thickness, shape and material trade off, never to certify a hull.

• Press Cylinder tube, same wall: a tube needs a much thicker wall than a sphere for the same depth.
• Press Steel sphere, thin and watch it buckle into lobes below its collapse depth.
• Raise the knock-down factor toward 1 to see how much a perfectly round shell would gain.