A press-fit joint that slips under load is a catastrophic failure — but one that is over-interfered can crack the hub during assembly; calculating the correct interference range using Lame’s equations is the only reliable way to design these joints.
Press-fit (interference fit) connections transmit torque and axial loads through friction generated by the contact pressure between mating parts. They are used for gear hubs on shafts, bearing housings, wheel assemblies, and anywhere a permanent, self-locking mechanical joint is needed without fasteners. This guide covers interference selection, Lame’s thick-cylinder equations for contact pressure, required assembly force, thermal insertion, and standard ISO fit designations.
What is an Interference Fit?
An interference fit (also called a press fit or force fit) exists when the shaft (inner member) nominal diameter is larger than the bore (outer member) nominal diameter. The interference δ is the difference between the shaft outer diameter and the hub inner diameter before assembly:
δ = dshaft − dbore
When the two parts are forced together (or thermally expanded), the shaft compresses radially and the hub expands radially until equilibrium is reached. The resulting elastic deformation creates a contact pressure p at the interface, which in turn creates friction that transmits the applied torque or axial load.
ISO Fit Designation for Interference Fits
ISO 286-1 (JIS B 0401) defines standard fits using letter-number codes. For interference fits on a shaft-in-hub assembly with basic hole system (H7 bore):
| Fit | ISO Designation | Type | Typical Use |
|---|---|---|---|
| Locational interference | H7/p6 | Light interference | Gears, pulleys — removable with press |
| Medium drive | H7/r6 | Medium interference | General drive fits, wheels on axles |
| Force fit | H7/s6 | Heavy interference | Permanent assemblies, shrink fits |
| Heavy force | H7/u6 | Very heavy | Large locomotive wheels, heavy press fits |
For a 50 mm nominal diameter H7/p6 fit: The H7 bore tolerance is +0/+25 μm (bore size 50.000 to 50.025 mm). The p6 shaft tolerance is +42/+51 μm (shaft size 50.042 to 50.051 mm). Minimum interference = 50.042 − 50.025 = 17 μm. Maximum interference = 50.051 − 50.000 = 51 μm. The actual interference within this range depends on which individual parts are paired. Lame’s equations must be checked for both minimum interference (to ensure sufficient joint capacity) and maximum interference (to ensure the hub does not yield or crack).
Lame’s Thick-Cylinder Equations for Contact Pressure
For a hub (outer cylinder) with inner radius ri = d/2 and outer radius ro, and a solid shaft (inner cylinder) with radius ri, the contact pressure p resulting from interference δ is:
For a solid shaft with hub (most common case):
p = δ / (ri × [(1/Ehub) × ((ro² + ri²)/(ro² − ri²) + νhub) + (1/Eshaft) × (1 − νshaft)])
For the common case where both hub and shaft are steel (E = 206 GPa, ν = 0.3):
p = (δ × E) / (2 × ri) × (ro² − ri²) / ro²
This simplification applies only when both members have identical elastic properties. For steel hub on aluminum shaft (common in aerospace and automotive), the full equation with different E and ν values must be used.
Hub Stress Check
The contact pressure p creates hoop (circumferential) tensile stress in the hub and radial compressive stress. The maximum hoop stress in the hub occurs at the inner surface (r = ri):
σθ,hub,max = p × (ro² + ri²) / (ro² − ri²)
This stress must remain below the material’s yield strength divided by the safety factor: σθ,hub,max ≤ Sy / n, where n = 1.5 for ductile materials with well-known loads, 2.0 for general design. If this condition is not satisfied, the hub will yield at the bore and the interference fit will lose its designed contact pressure (permanent set occurs).
The shaft is in biaxial compression at the interface. For a solid shaft, the compressive hoop stress equals −p at the surface, and the radial stress also equals −p. The von Mises equivalent stress at the shaft surface is √3 × p. For steel shafts, this is rarely the governing constraint unless the interference is very large relative to shaft diameter.
Joint Capacity: Torque and Axial Load
The maximum transmissible torque T and axial force F from a press-fit joint are:
T = μ × p × (π × d × L) × (d/2) = μ × p × π × d² × L / 2
Faxial = μ × p × π × d × L
Where μ = coefficient of friction at the interface (typically 0.10–0.15 for steel-on-steel press fit, lubricated; 0.15–0.20 for dry press fit; higher for knurled or serrated surfaces), d = nominal interface diameter, and L = hub length (axial engagement length). The friction coefficient must be chosen conservatively — use the lower end of the range for design, as the actual friction can be reduced by oil contamination, surface damage during assembly, or fretting over time.
Required Press-In Force
The force required to press the shaft into the hub (or vice versa) is:
Fpress = μassembly × p × π × d × L
Where μassembly is the assembly friction coefficient. During pressing, a thin oil film is typically present (μ ≈ 0.06–0.12 for machine oil), which is lower than the dry friction coefficient used for load capacity calculations. This means the assembly force is relatively manageable even when the service friction coefficient is high. Calculate Fpress to select the correct press capacity — a 50 mm × 60 mm long steel H7/s6 press fit typically requires 50–150 kN of press force depending on interference and lubrication.
Thermal Insertion (Shrink Fit)
For large interference fits or delicate assemblies where press-in force would damage components, thermal insertion is preferred. The hub is heated until its bore expands sufficiently to allow hand (slip) assembly onto the shaft, then cooled to create the interference grip. The required temperature rise of the hub is:
ΔT = (δ + clearance) / (α × d)
Where α = coefficient of thermal expansion (12 × 10⁻⁶ /°C for steel), d = bore diameter, and the clearance term is the additional expansion needed to allow easy slip assembly (typically 0.1–0.2 mm for small bores, 0.2–0.5 mm for large). For example, to install a 100 mm bore hub with 0.08 mm interference onto a shaft with 0.2 mm assembly clearance: ΔT = (0.08 + 0.20) / (12 × 10⁻⁶ × 100) = 0.28 / 0.0012 = 233°C. Heat hub to ~250°C (use oven or induction heater). An industrial oven set to 200–250°C for 30–60 minutes is the standard method. Avoid open flame heating, which creates temperature gradients and oxide layers that reduce the effective friction coefficient.
Worked Example: Gear Hub on Shaft
A steel gear hub (OD = 80 mm, ID = 40 mm) is pressed onto a steel shaft (diameter 40 mm). Required torque transmission: 500 N·m. Hub length: 50 mm. Determine the required interference.
Step 1 — Required contact pressure: T = μ × p × π × d² × L / 2. Using μ = 0.12 (conservative for dry steel fit in service): 500,000 = 0.12 × p × π × 40² × 50 / 2. p = 500,000 / (0.12 × π × 80,000) = 500,000 / 30,159 = 16.6 MPa.
Step 2 — Required interference (both steel, E = 206,000 MPa, ri = 20 mm, ro = 40 mm): δ = p × 2ri × ro² / (E × (ro² − ri²)) = 16.6 × 40 × 1600 / (206,000 × (1600 − 400)) = 1,062,400 / 247,200,000 = 0.0043 mm = 4.3 μm. This is very small.
Step 3 — ISO fit selection: For d = 40 mm, H7/p6 gives minimum interference of 17 μm — more than sufficient. Check maximum interference of 42 μm: pmax = 16.6 × (42/4.3) = 162 MPa. Hub hoop stress: σ = 162 × (1600 + 400)/(1600 − 400) = 162 × 1.667 = 270 MPa. For S235 steel Sy = 235 MPa — this exceeds yield! Use H7/p6 minimum interference and material with Sy ≥ 300 MPa, or reduce hub OD to ID ratio.
Conclusion
Press-fit design requires checking four conditions: minimum interference produces adequate contact pressure for the required torque and axial load capacity; maximum interference does not cause yielding of the hub; assembly press force is within available equipment capacity; and for thermal insertion, the required temperature rise is achievable without material degradation. ISO fit designations (H7/p6 through H7/u6) provide a convenient starting framework, but always verify the specific interference range against Lame’s equations for the actual geometry and materials involved.
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