press-fit

name	press-fit
description	Press and shrink fit analysis — interface pressure, hub/shaft stresses, torque capacity, insertion force, thermal assembly, thick-wall cylinder theory, fit selection.
metadata	{"priority":6,"promptSignals":{"phrases":["press fit","interference fit","shrink fit","hub stress","fit torque","assembly force"],"minScore":4}}

Press/Shrink Fit Analysis — Complete Skill

Interference and Fit Selection

δ = R_shaft - R_hole (radial interference, must be > 0 for press fit) Or: δ = (D_shaft - D_hole)/2

Fit selection (ISO 286): Light press: H7/p6 (δ ≈ 0.015-0.040mm for 50mm diameter) Medium press: H7/s6 (δ ≈ 0.027-0.052mm) Shrink: H7/u6 (larger δ — heat hub to assemble) Force required determines selection — see below

Interface Pressure (Lamé Theory)

For hub (outer) and shaft (inner), both same material (E, ν):

Interface pressure: p = (δ/R) × E / (((R_o²+R²)/(R_o²-R²)) + ((R²+R_i²)/(R²-R_i²)) - 2ν/(1) ... )

For solid shaft (R_i = 0): p = (E × δ/R) / ((R_o²+R²)/(R_o²-R²) + 1) [simplified from thick wall]

For general case: p = δ × E / (2R × ((R_o²+R²)/(R_o²-R²))) [hub only, solid shaft — most common] More precisely with both thick-wall equations: p = (δ/R) / ((1/E_h × (R_o²+R²)/(R_o²-R²) + ν_h/E_h) + (1/E_s × (R²+R_i²)/(R²-R_i²) - ν_s/E_s))

For same material (E_h=E_s=E, ν_h=ν_s=ν) and solid shaft (R_i=0): p = (δ × E) / (2R × (R_o²/(R_o²-R²))) p = (δ/R) × E × (R_o²-R²)/(2R_o²)

Hub Stresses (Thick Wall Cylinder Under Internal Pressure p)

Hoop stress at inner surface (r = R, hub bore): σ_θ,i = p × (R_o²+R²)/(R_o²-R²) [tensile, maximum]

Hoop stress at outer surface (r = R_o): σ_θ,o = p × 2R²/(R_o²-R²) [tensile, smaller]

Radial stress at inner surface: σ_r,i = -p [compressive = contact pressure]

Shaft Stresses (External Pressure p): For solid shaft: σ_r = σ_θ = -p (uniform biaxial compression, no variation) For hollow shaft (R_i > 0): worst at bore σ_θ = -p × 2R²/(R²-R_i²) (compressive at R_i — could yield)

Design Check

Hub: σ_θ,i < Sy_hub/n (tensile yield at bore, n ≥ 1.5) Shaft: |σ_r| = p < Sy_shaft/n (compressive — biaxial compression is good) Von Mises at hub bore: σ_vm = √(σ_θ² - σ_θ×σ_r + σ_r²) < Sy/n

Torque Capacity

T = μ × p × 2πR² × L Where:

• μ = coefficient of friction (0.10-0.15 dry steel, 0.06-0.10 oiled)
• p = interface pressure
• R = interface radius
• L = contact length (axial)

Axial holding force (pressing direction): F_axial = μ × p × 2πRL

Safety factor for torque: n_T = T_capacity / T_applied ≥ 1.5 (minimum), ≥ 2.0 recommended

Press-In Force (Assembly Without Heat)

F_press = μ_press × p × 2πRL × C_friction μ_press ≈ 0.10-0.15 (steel on steel, dry) This is the force to slide the shaft into the hub bore. C_friction = lead-in chamfer factor (≈1.0 with good chamfer)

Hydraulic press capacity must exceed F_press with margin.

Thermal Assembly (Shrink Fit)

Heat hub to expand bore above shaft: ΔT_hub = δ/(α_hub × R) [°C above ambient] Typical: ΔT = 100-200°C for steel (heat in oven, not torch for uniform expansion)

Cool shaft (optional, liquid nitrogen -196°C or dry ice -78°C): ΔT_shaft = -δ/(α_shaft × R) [below ambient] Combined: smaller ΔT needed for each component

Assembly gap (clearance at assembly temperature): Clearance = α_hub×R×ΔT_hub + |α_shaft×R×ΔT_shaft| ≥ δ + 0.002×R [for easy assembly]

Complete Example

Steel hub (OD=100mm, ID=50mm) on solid steel shaft (D=50.04mm): R = 25mm, R_o = 50mm, δ = 0.04/2 = 0.02mm (radial) E = 200 GPa

p = δ/R × E/(1+(R_o²+R²)/(R_o²-R²)) = 0.02/25 × 200,000 × (50²-25²)/(2×50²) = 30 MPa

σ_θ,i = 30 × (50²+25²)/(50²-25²) = 30 × 3125/1875 = 50 MPa (well below Sy=250) T_capacity = 0.12 × 30 × 2π × 0.025² × 0.040 = 1,414 N·m

Output

Provide: δ [mm], p [MPa], σ_θ,i [MPa], safety factor, T_capacity [N·m], F_press [kN], ΔT for thermal assembly.