# Pressure Bulkhead > Use when you must size a fuselage pressure-bulkhead dome: compute the membrane stresses of a spherical cap, a hemisphere, or a 2:1 ellipsoid closing a pressurized cylindrical barrel with membrane theory, including the apex and equator meridional and circumferential stresses of an ellipsoidal dome, the dome margin against ultimate strength under the FAR 25.303 1.5 factor of safety, the spherical-cap rise, and the unbalanced meridional-resultant junction ring load at the dome-barrel interface with the ring area required. Produces the dome stresses, the margin, and the junction ring area that gate the bulkhead design. Trigger: pressure bulkhead, ellipsoidal bulkhead, spherical cap dome, dome margin, junction ring, dome to barrel ring, pressurized fuselage dome, 2:1 ellipsoid. - Skill: `ashfordeou/pressure-bulkhead` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/pressure-bulkhead` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/pressure-bulkhead/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/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 = p*a/t and longitudinal sigma_long = p*a/(2*t). - Spherical dome radius R: sigma = p*R/(2*t) everywhere, equal meridional and circumferential membrane resultants. - Ellipsoid, semi-axes a (barrel radius) and b (dome depth): - Apex, both directions equal: sigma = p*a^2/(2*b*t). - Equator meridional: sigma_phi = p*a/(2*t). - Equator circumferential: sigma_theta = (p*a/t)*(1 - a^2/(2*b^2)). For b = a the three collapse to p*a/(2*t), the sphere limit. For the 2:1 dome (b = a/2) the equator circumferential stress is (p*a/t)*(1 - 2) = -p*a/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 p*R/2 acts at the dome-barrel interface with the radial component (R - h)/R, so the unbalanced radial line load is q = (p*R/2)*(R - h)/R [N/m] and the ring tension is F_ring = q*a [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) = (p*a/t, p*a/(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 p*R/(2*t) 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 p*a^2/(2*b*t) to the equator meridional p*a/(2*t) and circumferential (p*a/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 = 2*a reproduces the barrel hoop p*a/t (55.742 MPa in the worked example), but the hemisphere runs at p*a/(2*t) = 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 = 2*a reproduces the barrel hoop stress exactly; the ellipsoid with b = a returns p*a/(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.