Shaft Press-Fit Interference Stress & Tonnage Calculator (Lamé)
To calculate the radial contact pressure (p) and maximum tangential hoop stress (σ_θ) in a cylindrical shaft-and-hub interference press fit under DIN 7190 and Lamé thick-walled cylinder theory, divide the diametral interference (δ) by the combined elastic compliance of the shaft and hub, then compute axial assembly force as F = π × d × L × p × μ.
⚙️ Interactive Shaft & Hub Press-Fit Interference Stress, Tonnage & Shrink Temp Calculator (Lamé)
Calculate interface contact pressure (p), maximum hub hoop stress (σ_θ), Von Mises yield safety factor, hydraulic press assembly force (kN / Tons), and thermal shrink-fit oven temperature (ΔT).
How Do Lamé's Thick-Walled Cylinder Equations Prevent Cracked Hubs in Press Fits?
When pressing a hardened steel shaft, bearing race, or dowel pin into a mating gear hub or housing bore using preferred ISO 286 Interference Fits (H7/p6, H7/r6, H7/s6, or H7/u6), the oversized shaft radially expands the inner bore of the hub while the hub radially compresses the shaft. Under Gabriel Lamé's classical elasticity equations standardized in ISO International Metrology Standard, DIN 7190, and Machinery's Handbook (Industrial Press), the radial contact pressure p generated at the mating diameter d is governed by the diametral interference δ = d_shaft - D_bore, the hub outer-to-inner diameter ratio Q = D_out / d, Young's moduli (E_s, E_h), and Poisson's ratios (ν_s, ν_h).
Crucially, at the inner bore wall of the hub, the tensile tangential hoop stress σ_θ = p × (D_out² + d²) / (D_out² - d²) is always significantly higher than the contact pressure itself, and combines biaxially with the radial compressive stress σ_r = -p to produce an elevated Von Mises equivalent stress σ_vm = √(σ_θ² + p² + σ_θ × p). If an engineer uses a naive rule-of-thumb interference of 0.001" per inch of diameter on a thin-walled cast-iron or 6061-T6 aluminum hub without checking Lamé's hoop stress ratio, the inner bore yields plastically or fractures during hydraulic pressing!
| Fit Class | Interference Ratio (δ / d) | δ for Ø 25 mm (1.0") Shaft | δ for Ø 50 mm (2.0") Shaft | Interface Pressure (p) | Hub Hoop Stress (σ_θ) | Shrink Temp ΔT (Steel) | Action |
|---|---|---|---|---|---|---|---|
| Light Press (H7/p6) | 0.0005 × d | 0.013 mm (0.0005") | 0.025 mm (0.0010") | 38.4 MPa (5,570 psi) | 64.0 MPa | +110 °C (+230 °F) | |
| Medium Drive (H7/r6) | 0.0010 × d | 0.025 mm (0.0010") | 0.050 mm (0.0020") | 76.9 MPa (11,150 psi) | 128.1 MPa | +175 °C (+347 °F) | |
| Heavy Shrink (H7/s6) | 0.0015 × d | 0.038 mm (0.0015") | 0.075 mm (0.0030") | 115.3 MPa (16,720 psi) | 192.2 MPa | +240 °C (+464 °F) | |
| Force Fit (H7/u6) | 0.0020 × d | 0.050 mm (0.0020") | 0.100 mm (0.0040") | 153.8 MPa (22,300 psi) | 256.3 MPa | +305 °C (+581 °F) |
How Do You Calculate Hydraulic Press Tonnage vs Thermal Shrink-Fit Oven Temperature?
During cold mechanical arbor pressing, the axial force required to overcome Coulomb friction across engagement length L is F_axial = π × d × L × p × μ, where the coefficient of friction μ ranges from 0.10–0.15 for oiled or molybdenum-disulfide (MoS₂) coated steel up to 0.25 for dry surfaces (see our Master Friction & Preload Chart). Furthermore, under NIST (National Institute of Standards and Technology) and NASA Fastener & Mechanical Design Manual (NASA RP-1228) surface texture guidelines, micro-peaks on the turned shaft and bored hub are sheared or ironed flat during cold pressing (reducing effective interference by 0.4 × (Rz_shaft + Rz_hub)), which is why mating press-fit bores should be reamed or ground to N6 (Ra ≤ 0.8 μm) per our Surface Finish Ra to Rz Converter.
To eliminate surface galling and hydraulic press tonnage completely on heavy H7/s6 or H7/u6 fits, machine shops use Thermal Shrink Fitting: heating the outer hub in an induction heater or oven by ΔT = (δ + c_clearance) / (α_th × d) (or chilling the inner shaft in liquid nitrogen at -196°C) expands the bore so the shaft drops in with zero axial force. When supplementary positive torque locking is required for shock loads, combine your interference fit with our ANSI B17.1 & DIN 6885 Keyway Calculator and Retaining Ring Groove Chart per ANSI Engineering Standards Webstore standards.
| Engineering Alloy | Young's Modulus (E) | Poisson Ratio (ν) | Yield Strength (S_y) | Thermal Expansion (α_th) | Dry Press Friction (μ) |
|---|---|---|---|---|---|
| AISI 1045 Carbon Steel | 205 GPa (29.7 Mpsi) | 0.29 | 530 MPa (76.8 ksi) | 11.5 × 10⁻⁶ /°C | 0.15 – 0.22 |
| AISI 4140 Pre-Hard Steel | 210 GPa (30.5 Mpsi) | 0.29 | 655 MPa (95.0 ksi) | 12.2 × 10⁻⁶ /°C | 0.14 – 0.20 |
| 304 / 316 Stainless Steel | 193 GPa (28.0 Mpsi) | 0.30 | 290 MPa (42.0 ksi) | 17.2 × 10⁻⁶ /°C | 0.18 – 0.28 (Anti-seize req) |
| 6061-T6 Aluminum Alloy | 68.9 GPa (10.0 Mpsi) | 0.33 | 276 MPa (40.0 ksi) | 23.1 × 10⁻⁶ /°C | 0.15 – 0.25 |
| SAE 660 Bearing Bronze | 100 GPa (14.5 Mpsi) | 0.34 | 140 MPa (20.3 ksi) | 18.0 × 10⁻⁶ /°C | 0.10 – 0.16 |
Surface roughness plays a critical role in the effective retention torque of an interference press fit. When two turned or ground cylinders are pressed together axially, the microscopic surface peaks (asperities) are sheared and flattened, reducing the effective diametral interference by approximately Δ_loss = 1.2 × (Rz_shaft + Rz_hub). Specify a ground surface finish of Ra 0.4 to 0.8 µm (Rz 2.5 to 5.0 µm) on both mating diameters to prevent fretting corrosion and preserve full Lamé contact pressure.
When designing hollow shafts or thin-walled bearing sleeves with an outside-to-inside diameter ratio below D_o / d < 1.45, always check the inner bore radial contraction using the thick-wall Lamé displacement equation. Pressing a rolling-element bearing into a thin aluminum housing causes the bearing outer raceway to contract inward by 70% to 85% of the diametral interference, which can eliminate internal radial bearing clearance (C0) and cause premature thermal seizure at high spindle RPM.
Frequently Asked Questions About Press-Fit Interference and Stress Calculations
How much assembly clearance should I add when calculating shrink-fit heating temperature?
Always add an extra diametral assembly clearance of at least 0.001" to 0.002" (0.025 to 0.050 mm) beyond the maximum diametral interference δ when calculating ΔT so the heated hub does not seize halfway down the cold shaft during manual alignment.
Why do leading chamfers on press-fit shafts prevent bore scoring and galling?
Machining a shallow 5° to 10° lead-in chamfer (1.5 to 2.5 mm long) on the nose of the shaft—combined with high-pressure MoS₂ paste—guides the shaft coaxially into the bore so the sharp edge cannot shave metal from the hub wall.