Pressure vessels and hoop stress

Sizing the pressure hull and the rotating structure.

Key results

Basic equations

  • Thin-walled cylinder: hoop stress σ = P·r/t, longitudinal σ/2. Sphere or spherical cap: σ = P·r/(2t) lamontagne, 2022-10-13; lamontagne, 2026-09-15.
  • Treat pressure and rotation separately. The rotating ring stress is σ = ω²ρ(r₁² + r₁r₂ + r₂²)/3, with an equivalent density equal to station mass per metre ÷ structural cross-section. Alternatively, treat the station as a hoop with "pressure" = mass × acceleration ÷ area lamontagne, 2022-10-13; lamontagne, 2022-10-20.
  • For rotating structures the "pressure" P includes air pressure + hull areal mass × g + internal loads (soil, walls, equipment) at the rim lamontagne, 2026-09-15.
  • Air at 1 atm ≈ 10 t/m². A radiation shield lighter than ~10 t/m² adds less load than the air Paul451, 2023-09-29.
  • Interior mass acting over the major radius R adds P_t = ρ·a·R, which at R = 175 m and 100 kg/m³ is already ~1.7 atm Twark_Main, 2023-09-29. Paul451 and Twark_Main argued over whether interior floors pass their loads to the shell Paul451, 2023-09-30; Twark_Main, 2023-10-01.
  • End loads. 1 bar on a 4.4 m module end is ~691 kN (~70 t), far more than the 1 g loads of small stations rklaehn, 2014-02-17. 3.5 atm on a ~64 m² Starship dome is >2000 t of force DistantTemple, 2019-04-08.

Torus-specific results

  • Hoop stress in a torus is set by the minor radius (Roark's formulas). Meridional stress averages P·r/t and varies slightly from the inner to the outer side lamontagne, 2023-09-26.
  • Half-circle cross-sections with a flat floor are bad. The flat floor becomes a cylinder of the major radius and needs huge tension, or it must act as a beam. Brace the corners with an equal opposite arch, which gives a circular section Barley, 2023-09-26; Barley, 2023-09-26.
  • Self-supporting torus limit. Using only the "free" longitudinal half of the wall strength to carry rotating mass, the maximum major radius is ~50–100 m before shielding, at 100–200 kg/m³ Twark_Main, 2022-11-13.
  • Squared-off tori ("squorus") need more tension per unit interior volume. "I doubled the concrete in my client's skyscraper, but in exchange I made all the light switches interchangeable" Twark_Main, 2023-09-29.
  • Stability in all four states. It is desirable for the structure to be stable spinning and pressurized, non-spinning and pressurized, spinning and unpressurized, and non-spinning and unpressurized Twark_Main, 2023-09-30.

Worked examples

  • JohnFornaro's steel "bicycle chain" armature: ~128 MPa (19 ksi) against 400 MPa (58 ksi) steel, safety factor 3. Cross-section 14 m², about one 200 mm steel floor lamontagne, 2022-10-25.
  • lamontagne's Kalpana: carbon fibre composite at 1500 MPa, SF 3, 5 m shield → ~0.3 m pressure structure; end-cap elements ~1.5 m on 20% of the area lamontagne, 2023-01-11.
  • lamontagne's small 100 m Avalon-type station: 3 cm of CFRP at 1500 MPa, SF 4, × 1.8 for secondary structure lamontagne, 2023-11-18.
  • Barbell vs torus in steel at 100 kPa and SF 3: the sphere costs 12 kg per m³ of enclosed volume against 18 kg/m³ for a cylinder, so the barbell is ~33% lighter lamontagne, 2023-09-21.
  • Bigelow skins withstand ~10 t/m² internal pressure; walls ≥6 inches; "rigid as concrete" at 1 atm Paul451, 2015-10-23; Paul451, 2015-03-03.
  • Hatches opening inward are held shut by ~10 t/m² (a 1.5 m hatch ≈ 18 t) and cannot be opened under pressure. That makes manual hatches safe; overpressure vents should be small, spring-closed and alarmed Paul451, 2022-10-24.

Redundant hulls

  • BIS Avalon triple hull: 100/66/33 kPa; three ~18 cm steel shells with 3 cm gaps; each carries ~33 kPa at SF 5; the outer shell alone can take 100 kPa lamontagne, 2023-09-29.
  • Twark_Main's criticism: all three layers accumulate cycles at the same rate, so failures are correlated. Better to put the cycles on one layer, as with aircraft windows, or pump down the interspace Twark_Main, 2023-09-29; Twark_Main, 2023-09-29.
  • Double hull with a pumped interspace (~1/1000 atm) cuts leakage about 500× Twark_Main, 2024-02-11.

Open questions

  • What is the optimal filament-winding angle for a composite torus that combines pressure and spin loads? Twark_Main, 2023-09-21
  • How much internal structure (floors, grid) can share the shell's spare longitudinal strength?

Sources

Forum posts

  • rklaehn, 2014-02-17 (reply #22, 4 likes): 1 bar on 4.4 m end ~691 kN. Post.
  • Paul451, 2015-03-03 (reply #481): Walls handle ~1 t/m² loads; inflated floors. Post.
  • Paul451, 2015-10-23 (reply #728, 1 like): Bigelow walls ≥6 in, 10 t/m². Post.
  • DistantTemple, 2019-04-08 (reply #1504, 1 like): 3.5 atm dome > 2000 t. Post.
  • lamontagne, 2022-10-13 (reply #2046): Hoop stress equations; separate pressure and rotation. Post.
  • lamontagne, 2022-10-13 (reply #2049): Rotating ring stress; spokes vs hub. Post.
  • lamontagne, 2022-10-20 (reply #2161): Continuous ring = thin tank; no bending. Post. Attachments: Ring-Structural-p2-10-20-22ML.png.
  • Paul451, 2022-10-24 (reply #2201, 1 like): Inward hatches held shut by pressure. Post.
  • lamontagne, 2022-10-25 (reply #2204): Armature stress check SF 3. Post. Attachments: Station (1).xlsx.
  • Twark_Main, 2022-11-13 (reply #2377): Torus self-support limit. Post.
  • lamontagne, 2023-01-11 (reply #2866): Kalpana CFRP 0.3 m hull. Post.
  • Twark_Main, 2023-09-21 (reply #4001): Filament winding angle; no free lunch. Post.
  • lamontagne, 2023-09-21 (reply #4007): Barbell vs torus steel calc. Post.
  • Barley, 2023-09-26 (reply #4025): Flat floor = cylinder of radius R; brace with arch. Post.
  • Barley, 2023-09-26 (reply #4029): Clarification: flat section stress. Post.
  • lamontagne, 2023-09-26 (reply #4031): Roark: torus hoop stress set by minor radius (1 like). Post. Attachments: roak1.PNG, roak2.PNG.
  • Twark_Main, 2023-09-29 (reply #4065): Critique: correlated cycle fatigue. Post.
  • lamontagne, 2023-09-29 (reply #4066): Avalon triple hull details. Post.
  • Twark_Main, 2023-09-29 (reply #4075): Layers reach end of life together. Post.
  • Twark_Main, 2023-09-29 (reply #4078): Squorus: more tension per volume. Post.
  • Paul451, 2023-09-29 (reply #4081): Air pressure ≈ 10 t/m²; shield <10 t/m² less load. Post.
  • Twark_Main, 2023-09-29 (reply #4083): P_t = ρaR. Post.
  • Twark_Main, 2023-09-30 (reply #4085): Stable in all four states. Post.
  • Paul451, 2023-09-30 (reply #4089): Floors carry own loads; density debate. Post.
  • Twark_Main, 2023-10-01 (reply #4093): Use density, not raw mass. Post.
  • lamontagne, 2023-11-18 (reply #4359, 2 likes): 3 cm CFRP hull. Post.
  • Twark_Main, 2024-02-11 (reply #4464): Double hull + pump reclaim 500×. Post.
  • lamontagne, 2026-09-15 (reply #5107, 3 likes): Thin-wall checks: P·r/t and P·r/2t (3 likes). Post. Attachments: Image51_002.jpg.

External references

  • Roark's Formulas for Stress and Strain (torus section).
  • engineeringtoolbox.com, thin-walled pressure vessel equations.