pressure-vessel

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.