| name | thick-cylinders |
| description | Thick-walled cylinders and spheres — Lamé equations, hoop/radial/axial stress, Tresca/VM failure, autofrettage, interference fit, shrink fit cylinder design. |
| metadata | {"priority":7,"promptSignals":{"phrases":["thick cylinder","Lamé","hoop stress","thick wall","autofrettage"],"minScore":3}} |
Thick-Walled Cylinders — Complete Skill
Lamé Equations (Internal Pressure p_i, External p_o)
Radial and Hoop Stresses
σ_r = A - B/r²
σ_θ = A + B/r²
Constants from boundary conditions:
A = (p_i r_i² - p_o r_o²) / (r_o² - r_i²)
B = r_i² r_o² (p_i - p_o) / (r_o² - r_i²)
Internal pressure only (p_o = 0):
σ_r = p_i r_i² (1 - r_o²/r²) / (r_o² - r_i²) [compressive at inner, 0 at outer]
σ_θ = p_i r_i² (1 + r_o²/r²) / (r_o² - r_i²) [tensile, max at r = r_i]
Maximum hoop stress (at inner radius):
σ_θ,max = p_i (r_o² + r_i²) / (r_o² - r_i²)
Axial stress (closed ends):
σ_a = p_i r_i² / (r_o² - r_i²) [uniform across cross-section]
Wall Thickness Design
Tresca Criterion (max shear stress)
At inner radius: τ_max = (σ_θ - σ_r)/2 = p_i r_o² / (r_o² - r_i²) ≤ S_y/2
t/r_i = √((S_y + p_i)/(S_y - p_i)) - 1 [solving for wall thickness]
Von Mises Criterion
(σ_θ - σ_r)² + (σ_r - σ_a)² + (σ_a - σ_θ)² = 2 S_y²
ASME Code (not exact Lamé):
t = p_i D_o / (2SE + 0.8p_i) [ASME VIII thin-cylinder, conservative]
Use Lamé for t/r_i > 0.1 (thick wall criterion)
Radial Displacement
u_r = r[(1-ν)(σ_θ + σ_r) - 2ν σ_a] / (2E) [general]
At inner radius (internal pressure only):
u_r,i = (p_i r_i)/(E) × [r_i²(1-ν) + r_o²(1+ν)] / (r_o² - r_i²)
Interference Fit (Compound Cylinder)
Contact Pressure at Interface
δ = u_r,outer - u_r,inner [interference = diametral]
p_c = δ × E_hub/(2r_c) × [(r_o² - r_c²)/r_o²] × [(r_c² - r_i²)/r_c²] / [(r_o² - r_c²)(r_c² - r_i²)/r_c²(r_o² - r_i²)]
Simplified (same material, hub outer r_o, inner r_i, shaft solid or inner r_i=0):
p_c = (δ/r_c) × E/2 × (r_o² - r_c²)(r_c² - r_i²) / [r_c²(r_o² - r_i²)]
δ = diametral interference [mm], r_c = interface radius [mm]
Autofrettage
Apply pressure to yield inner layers → compressive residual stress at bore
After unloading: inner bore has compressive hoop stress → higher fatigue life
Yield pressure (elastic limit):
p_y = S_y × (1 - r_i²/r_o²) / 2 [Tresca, internal pressure only]
Autofrettage overpressure: p_a > p_y → plastic zone extends to radius r_p
Residual stress benefit: σ_θ,residual,i ≈ -S_y (compressive) after unloading
Effective pressure capacity doubles for same yield-onset: p_y,autofrettaged ≈ 2 × p_y,plain
Thick Spherical Shell
σ_r = A - 2B/r³
σ_θ = A + B/r³
At inner radius (p_o=0):
σ_θ,max = p_i(2r_i³ + r_o³) / (2(r_o³ - r_i³))
Output
Provide: σ_θ and σ_r at inner and outer radii [MPa], maximum stress location, required wall thickness t [mm] for given S_y and SF, contact pressure p_c [MPa] for interference fit, autofrettage pressure p_y [MPa].
