Pressure Bulkhead

Pressure Bulkhead Dome (structures/fem/pressure-bulkhead)

Use when the task is the membrane-theory sizing of the fuselage pressure-bulkhead dome that closes a pressurized cylindrical barrel. This leaf implements the closed-form dome stress field for the three standard bulkhead geometries, a spherical cap, a hemisphere, and a 2:1 ellipsoid, the apex and equator meridional and circumferential membrane stresses of an ellipsoidal dome (including the known 2:1 knuckle compression), the dome margin against ultimate under the FAR 25.303 factor of safety of 1.5, the spherical-cap rise where the cap cuts the barrel, and the unbalanced meridional-resultant junction ring load at the dome-barrel interface with the ring area required. It is pure Python, stdlib only, deterministic and offline. It pairs with vehicle-design/structures-integration/fuselage-skin-stringer, which sizes the barrel skin and stringers that this dome closes, and with structures/fem/cylindrical-shell-buckling for the stability check of the same barrel. Standalone spacecraft tank walls sized from burst pressure are covered by the space-systems tank leaf, not here.

Domain quick reference

Conventions: cabin differential pressure p (Pa), barrel radius a (m), dome thickness t (m). SI throughout: Pa, m, N/m, N, m^2.

• Barrel membrane cross-check (barrel sizing itself belongs to vehicle-design): circumferential sigma_theta = pa/t and longitudinal sigma_long = pa/(2*t).
• Spherical dome radius R: sigma = pR/(2t) everywhere, equal meridional and circumferential membrane resultants.
• Ellipsoid, semi-axes a (barrel radius) and b (dome depth):
  • Apex, both directions equal: sigma = pa^2/(2b*t).
  • Equator meridional: sigma_phi = pa/(2t).
  • Equator circumferential: sigma_theta = (pa/t)(1 - a^2/(2b^2)). For b = a the three collapse to pa/(2t), the sphere limit. For the 2:1 dome (b = a/2) the equator circumferential stress is (pa/t)(1 - 2) = -pa/t, compressive: the known 2:1 knuckle compression that drives the knuckle check.
• Dome-cap rise of a spherical cap cutting the barrel at radius a: h = R - sqrt(R^2 - a^2); h = a for the hemisphere.
• Junction ring: the spherical cap meridional resultant pR/2 acts at the dome-barrel interface with the radial component (R - h)/R, so the unbalanced radial line load is q = (pR/2)(R - h)/R [N/m] and the ring tension is F_ring = qa [N]. The ring area against the material ultimate at the factor of safety is A_ring = F_ring*FS/sigma_ultimate. The hemisphere (h = R = a) carries zero unbalanced ring load and needs no ring.
• Margin: allowable = sigma_ultimate/FS; margin_of_safety = allowable/sigma_max - 1 and reserve_factor = allowable/sigma_max, where sigma_max is the largest absolute dome membrane stress.

Workflow

1. Fix the load case and geometry: cabin differential pressure p, the barrel radius a the dome must close, and the dome thickness t.
2. Cross-check the barrel membrane level with cylinder_membrane_stresses, which returns (sigma_hoop, sigma_longitudinal) = (pa/t, pa/(2*t)); the dome cannot be lighter than the pressure loads feeding the barrel.
3. Choose the dome type. Spherical cap: set the sphere radius R and read the uniform stress pR/(2t) from spherical_dome_stress. Hemisphere: R = a by geometry, same function. Ellipsoid: pass the semi-axes to ellipsoid_dome_stresses for the apex and equator meridional and circumferential stresses.
4. Get the cap geometry with dome_cap_rise(R, a) when the cap cuts the barrel below its equator; the rise h places the junction and sets the ring load.
5. Size the junction ring with junction_ring_load(p, a, R, h) for the radial line load and ring tension, then junction_ring_area for the cross-section area at the chosen ultimate and factor of safety.
6. Run bulkhead_summary for the consolidated dict (geometry, all stresses, sigma_max, allowable, margin_of_safety, reserve_factor, and the ring q, tension and area) and confirm it against the individual function results.
7. Confirm the deterministic checks with the contract test scripts/test_pressure_bulkhead.py.

Worked example

Narrowbody barrel: a = 1.88 m, DeltaP = 0.0593 MPa, 7075 dome t = 2 mm, Ftu = 469 MPa, FS = 1.5. Real module outputs:

• cylinder_membrane_stresses: barrel hoop 55.742 MPa, longitudinal 27.871 MPa (magnitudes 55.7 and 27.9 MPa).
• Spherical cap R = 3.76 m (2*a): 55.742 MPa everywhere; bulkhead_summary margin_of_safety 4.609 (about 4.61) and reserve_factor 5.609 against 469/1.5 = 312.7 MPa allowable.
• 2:1 ellipsoid b = 0.94 m: apex +55.742 MPa, equator meridional +27.871 MPa, equator circumferential -55.742 MPa (compression).
• Hemisphere R = 1.88 m: 27.871 MPa everywhere, junction q = 0, ring tension 0, ring area 0 exactly.
• Cap rise dome_cap_rise(3.76, 1.88) = 0.503744 m (about 0.504 m).
• Junction ring for the cap: q = 96.548 kN/m (about 96.5 kN/m), ring tension 181.510 kN (about 181.5 kN), ring area 5.805e-4 m^2 = 580.5 mm^2 (about 581 mm^2) at 1.5 x 469 MPa.

The ellipsoidal dome matches the spherical-cap peak stress at the apex but trades the junction ring for a compressive knuckle band at the equator, which is why the margin and the knuckle check, not the ring, tend to gate the 2:1 design.

Pitfalls

• Reading the 2:1 equator circumferential stress as tension: the equator hoop of the 2:1 ellipsoid is -p*a/t, compressive (the knuckle compression drives the knuckle check), and it only turns negative for b < a/sqrt(2), so the margin must use the largest ABSOLUTE stress.
• Assuming every dome needs a ring: the hemisphere (h = R = a) carries zero unbalanced radial line load, ring tension and ring area exactly; only caps that cut the barrel below their equator need the junction ring.
• Using the apex stress for the whole ellipsoid: the ellipsoid field varies from the apex pa^2/(2bt) to the equator meridional pa/(2t) and circumferential (pa/t)(1 - a^2/(2*b^2)); checking only the apex misses the knuckle.
• Confusing dome stress with barrel stress: the spherical cap at R = 2a reproduces the barrel hoop pa/t (55.742 MPa in the worked example), but the hemisphere runs at pa/(2t) = 27.871 MPa; cross-checking the dome against the barrel membrane level is part of the workflow, not a substitute.
• Sizing a spacecraft tank dome here: standalone tank walls sized from burst pressure belong to the space-systems propellant tank leaf; this leaf closes a pressurized fuselage barrel under cabin differential pressure.
• Feeding a cap that cannot close the barrel: a barrel radius larger than the sphere radius, non-positive pressure, radius, thickness, ultimate or factor of safety, and unknown dome types all raise ValueError.

Verification

• Confirm the worked magnitudes: barrel hoop and spherical cap 55.742 MPa (55.7 MPa), ellipsoid apex 55.742 MPa, equator meridional 27.871 MPa, equator circumferential -55.742 MPa, hemisphere 27.871 MPa, cap rise 0.503744 m, q 96.548 kN/m, ring tension 181.510 kN, ring area 580.5 mm^2, margin about 4.61.
• Confirm the identities: barrel hoop is exactly twice the longitudinal stress; a spherical cap with R = 2a reproduces the barrel hoop stress exactly; the ellipsoid with b = a returns pa/(2*t) at the apex and equator in both directions (sphere limit); the hemisphere junction ring load and tension are exactly zero; the cap rise obeys h = R - sqrt(R^2 - a^2).
• Confirm the 2:1 equator circumferential stress is negative (compressive) for b < a/sqrt(2).
• Confirm every non-positive pressure, radius, thickness, ultimate or factor of safety raises ValueError, as does a barrel radius larger than the sphere radius and an unknown dome type.
• Confirm dict key contracts: ellipsoid_dome_stresses returns exactly sigma_apex, sigma_equator_meridional and sigma_equator_hoop; junction_ring_load returns exactly q_n_per_m and ring_tension_N; bulkhead_summary returns the documented ten keys.
• Deterministic: no RNG, identical float results run to run.
• Run the contract test offline: python3 scripts/test_pressure_bulkhead.py (32 tests).

Related leaves

• vehicle-design/structures-integration/fuselage-skin-stringer: the barrel skin and stringer sizing this dome closes.
• structures/fem/cylindrical-shell-buckling: buckling check of the same barrel wall.
• structures/fem/plate-buckling: flat panel stability, the alternative stability check for the dome knuckle region when it is treated as a curved panel.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_pressure_bulkhead.py

The 32 tests cover the worked example within the spec magnitude bounds (55.7 MPa barrel and spherical cap, 27.9 MPa hemisphere, 2:1 apex +55.7 and equator -55.7 MPa compressive, rise 0.504 m, q 96.5 kN/m, ring tension 181.5 kN, ring area 581 mm^2, spherical-cap margin 4.61), the cylinder hoop-to-longitudinal identity, the sphere limit of the ellipsoid, the hemisphere zero-ring case, the cap-rise closed-form identity, dict key contracts, ValueError rejection of non-positive inputs and of barrel radius above sphere radius, the ellipsoid axes tuple contract, and run-to-run determinism.

Compliance

• Standards referenced, not reproduced: FAR 25.303 (factor of safety of 1.5 on the limit loads) and CS-25 provide the airworthiness context; the membrane and junction-ring relations above are standard engineering methodology, summary-only per standards-map.yaml. No regulatory text is reproduced.
• compliance: STANDARDS-REF, gated: false.