| name | pressure-vessel |
| description | Pressure vessel analysis — thin/thick wall cylinders and spheres, ASME BPVC Section VIII, hoop/longitudinal/radial stress, head design, nozzle reinforcement, fatigue, hydrostatic test, safety factors. |
| metadata | {"priority":8,"promptSignals":{"phrases":["pressure vessel","thin wall","thick wall","hoop stress","ASME","internal pressure","cylinder pressure","vessel design"],"minScore":4}} |
Pressure Vessel Design — Complete Skill (ASME BPVC Sec. VIII Div. 1)
Thin Wall (t/r < 0.1, or t/D < 0.05)
Cylindrical vessel:
σ_hoop = pr/t = pD/2t (circumferential, governs)
σ_long = pr/2t = pD/4t (longitudinal, axial)
σ_radial = -p (inner) to 0 (outer) — neglected for thin wall
σ_hoop = 2·σ_long (cylinder is twice as strong longitudinally → failure by splitting lengthwise)
Spherical vessel:
σ = pr/2t (equal in all directions — most efficient shape)
Required thickness:
t = pR/(SE-0.6p) [ASME formula, cylinder]
t = pR/(2SE-0.2p) [ASME formula, sphere]
Where: S = allowable stress (from ASME tables), E = joint efficiency (0.65-1.0)
Thick Wall (t/r ≥ 0.1) — Lamé Equations
Radial stress: σ_r = A - B/r²
Hoop stress: σ_θ = A + B/r²
Where A = pi·ri²/(ro²-ri²), B = pi·ri²·ro²/(ro²-ri²)
At inner radius (r = ri):
σ_θ,max = pi·(ro²+ri²)/(ro²-ri²) (max hoop, tensile)
σ_r = -pi (compressive, = applied pressure)
At outer radius (r = ro):
σ_θ = 2·pi·ri²/(ro²-ri²) (tensile)
σ_r = 0
Wall ratio (Lamé design):
ro/ri = √((S+pi)/(S-pi)) [for closed-end cylinder, Tresca]
t = ri·(ro/ri - 1)
Distortion energy (von Mises) for closed cylinder:
σ_eff = √(σ_θ² - σ_θ·σ_r + σ_r²) ≤ Sy/n
ASME BPVC Allowable Stresses
S = Sut/3.5 (tensile criterion) or Sy/1.5 (yield criterion) — take minimum
Typical values at 20°C:
- SA-516 Gr. 70 (carbon steel): S = 138 MPa (20 ksi)
- SA-240 304SS: S = 138 MPa
- SA-240 316L: S = 115 MPa
- SA-335 P22: S varies with temperature (see ASME Table 1A)
Joint Efficiency E
E = 1.0 (full radiography, Type 1 joint)
E = 0.85 (spot radiography)
E = 0.70 (no radiography)
E = 0.60 (lap joints)
Heads (End Closures)
Hemispherical head: t = pR/(2SE-0.2p) — most efficient
Ellipsoidal (2:1 ratio): t = pD/(2SE-0.2p) — common standard
Torispherical (ASME flanged & dished):
t = 0.885pL/(SE-0.1p) where L = crown radius
Flat head: t = d·C·√(p/S·E) — C = 0.17-0.33 depending on attachment
Conical head: t = pD/(2cos(α)(SE-0.6p)) where α = half-apex angle ≤ 30°
Nozzle Reinforcement
Area removed by nozzle hole must be replaced within reinforcement zone:
A_required = d·t_req (area removed from shell)
A_available = (t_shell - t_req)·d + (t_nozzle - t_nozzle,req)·2·h + A_welds
Must have: A_available ≥ A_required
Reinforcement limit: width = max(d, t_shell+t_nozzle+d/2) from centerline
External Pressure (Vacuum/Jacketed)
Critical pressure: p_cr = 2E/(1-ν²) · (t/D)³ / (L/D·factor) — use ASME charts (UG-28)
Design: p_working < p_cr/3 (safety factor 3)
Fatigue (Pressure Cycling)
Number of design cycles N:
At p_design: σ_range = σ_max - σ_min (full fluctuation)
Apply S-N curve for vessel material (ASME Sec. VIII Div. 2, App. 5)
If N_design > N_allow: increase t or change material
Hydrostatic Test
p_test = 1.3 · p_design · (S_test/S_design) [ASME Sec. VIII Div. 1]
p_test = 1.5 · p_design [older code requirement]
Duration: 30 minutes minimum. Inspect all joints.
Stress Relief/PWHT
Required when t > 38mm (1.5") for carbon steel (UCS-56)
Or when thickness exceeds code-specified limits for material
Output
Provide: t_min required, t_selected (round up to standard plate), σ_hoop, σ_long, safety factors, test pressure, ASME edition referenced.
