Interference and Press Fit: Method

Lamé Solution, Yield, Holding Capacity and Assembly

The thick-cylinder fields

A hub pressed or shrunk on a shaft is two thick cylinders squeezed together by the contact pressure p over their common surface of radius r_c. Lamé's solution for an axisymmetric cylinder in plane stress (free ends, no axial stress) is σ_r = A − B/r² and σ_θ = A + B/r², the two constants set by the pressures on its inner and outer surfaces: for a cylinder of radii a and b under inner pressure p_i and outer pressure p_o, A = (a² p_i − b² p_o)/(b² − a²) and B = (p_i − p_o) a² b²/(b² − a²). The hub carries p on its bore and any outside pressure on its outer surface; the shaft carries p on its outside and nothing on its bore (a solid shaft is a cylinder with a = 0, uniformly compressed to −p in both directions). The radial displacement follows from Hooke's law as u = r (σ_θ − ν σ_r)/E.

Contact pressure from the interference

Before assembly the shaft is larger than the bore by the diametral interference δ_d; after assembly the bore has moved out and the shaft surface in by amounts that add up to the radial interference δ_r = δ_d/2. Each displacement is proportional to p: the hub bore moves out by C_h p with C_h = (r_c/E_h)[(r_o² + r_c²)/(r_o² − r_c²) + ν_h], the shaft surface moves in by C_s p with C_s = (r_c/E_s)[(r_c² + r_i²)/(r_c² − r_i²) − ν_s] (1 − ν_s in the bracket for a solid shaft), so p = δ_r/(C_h + C_s). With equal materials and a solid shaft the Poisson terms cancel and the expression reduces to p = E δ_d (D_o² − d²)/(2 d D_o²), the form printed in NASA/TM-2008-215153 for bearing rings on shafts. A thin hub (wall t much smaller than r_c) on a rigid shaft gives the thin-ring limit p = E δ_r t/r_c², the hoop-strain relation of a ring. An outside pressure on the hub closes its bore by 2 r_o² r_c p_o/[E_h(r_o² − r_c²)]; that closure acts as extra interference and raises p.

Tolerances and surface finish

The parts are made to tolerances, so the interference lies between the loose combination (smallest shaft, largest bore) and the tight one. The tool takes the nominal interference at the mean of the tolerances and a half-band on each part, and reports all three fits: the tight fit governs the stresses and the press force, the loose fit the holding torque. On assembly the asperities of the two surfaces are flattened and a little of the interference is lost; NASA/TM-2008-215153 took 0.004 mm (160 µin) off the diametral interference of accurately ground bearing seats, and a common shop rule is about three times the roughness average of each surface. The entered finish loss is subtracted from every case.

Stresses, yield and margins

The hub's largest stresses are at its bore: a hoop tension p (r_o² + r_c²)/(r_o² − r_c²) and a radial compression −p. The shaft is in compression everywhere; a hollow shaft's largest stress is the uniaxial hoop compression −2 p r_c²/(r_c² − r_i²) at its bore. With σ_z = 0 the von Mises stress is √(σ_r² + σ_θ² − σ_rσ_θ + 3τ²), τ being the torsional shear T r/J of a transmitted torque, and the Tresca stress the largest difference of the three principal stresses. First yield of the hub by Tresca is at p_Y = F_ty(1 − r_c²/r_o²)/2, a little below the von Mises value F_ty/√(k² + k + 1) with k the hoop-to-pressure ratio; a solid shaft yields at p = F_ty. The largest elastic interference is the smaller first-yield pressure times 2 (C_h + C_s). The margins compare the yield strength of each part with the design factor times the governing von Mises stress of the tight fit, taking the worse of the assembled state at rest and the state at the operating speed.

Holding torque, axial force and press force

With the pressure taken uniform over the engaged length L, the normal force on the interface is p 2π r_c L and Coulomb friction with coefficient μ gives the holding torque T = 2π r_c² L p μ and the axial holding force F = 2π r_c L p μ. The slip check puts the resultant demand √(F_app² + (T_app/r_c)²) against the capacity of the loose fit at the operating speed. The press force is the axial friction force with the assembly coefficient and the nominal pressure; it builds up during insertion as the contact length grows, so the final value is the largest. The coefficient of friction is the least certain number in the analysis: it depends on the finish, the lubricant, whether the joint was pressed or shrunk, and the time in service, and should come from tests on the actual parts.

Shrink assembly

Heating the hub by ΔT grows its bore by α_h d ΔT. To clear the shaft at the tight fit with an assembly gap c_a, ΔT_h = (δ_d,max + c_a)/(α_h d); cooling the shaft instead needs ΔT_s = −(δ_d,max + c_a)/(α_s d), and a combination of both is often used for large interferences. The expansion coefficient is the mean value between room temperature and the assembly temperature. The hub must stay below the temperature that would draw its temper.

Rotation

A rotating ring grows outward. In plane stress the free rotating annulus of inner radius a and outer radius b has σ_r = (3 + ν) ρ ω² (a² + b² − a² b²/r² − r²)/8 and σ_θ = (3 + ν) ρ ω² [a² + b² + a² b²/r² − (1 + 3ν) r²/(3 + ν)]/8, the forms printed in NASA TM 87165, so its bore grows by u = ρ ω² a [(3 + ν) b² + (1 − ν) a²]/(4E); a solid disk's surface grows by ρ ω² b³ (1 − ν)/(4E). The hub bore grows faster than the shaft surface, the difference comes off the interference, and the contact pressure falls as the square of the speed until it reaches zero at the release speed ω₀. Because the elastic fields superpose, the stresses at speed are the Lamé fields with the reduced pressure plus the free rotating fields; the hub bore hoop stress may rise or fall with speed depending on the proportions.

What is not covered

The pressure is not uniform in reality: it peaks sharply at the ends of the hub where the shaft continues beyond it, and that peak together with the fretting it causes is the usual fatigue origin of a press-fitted shaft (Pilkey's Peterson's Stress Concentration Factors gives the factors). Plane stress is right for a hub of about the shaft's diameter in length; a very long hub is nearer plane strain. Plastic shrink fits, the loss of interference by creep or by differential thermal expansion of two materials in service, bending stresses in the shaft and the effect of keyways on the fit are outside the tool.