Mechanical Design & Machine Elements
Lame Shrink-Fit Contact Pressure and Slip-Free Torque Formula
Calculate interference press/shrink-fit interface pressure p_if = (E * delta_r / (2 * R_if)) * (1 - (R_if / R_out)^2) and slip-free torque capacity T_max in Excel and Arte Sheets.
Convert radial interference delta_r into clamping pressure and shaft torque capacity
When an outer hub of outer radius R_out is thermally shrunk or press-fitted over a solid shaft of interface radius R_if (both of identical elastic modulus E and Poisson ratio nu) with radial interference delta_r, Poisson’s ratio nu cancels identically across hub expansion and shaft compression! Lame’s thick-cylinder equations simplify to exact interface contact pressure p_if = (E * delta_r / (2 * R_if)) * (1 - (R_if / R_out)^2), transmitting maximum slip-free torque T_max_Nm = (2 * PI() * R_if^2 * L_hub * mu * p_if) / 1000.
Audit a steel gear hub on a 50 mm shaft (R_if = 25 mm in B2, hub outer radius R_out = 50 mm in B3, radial interference delta_r = 0.025 mm in B4, E = 200,000 MPa in B5, friction mu = 0.15 in B6, engagement length L_hub = 60 mm in B7): p_if = ((200,000 * 0.025) / 50) * (1 - 0.25) = 100 * 0.75 = 75.0000 MPa, and slip-free torque reaches T_max = 843.75 * PI() = 2,650.7188 N*m!
| Shrink-Fit Quantity (R_if=25, R_out=50, delta_r=0.025) | Formula | Sample Output |
|---|---|---|
| Wall-Ratio Factor 1 - (R_if / R_out)^2 | 1 - (25 / 50)^2 | 0.7500 |
| Interface Contact Pressure (p_if) | (E * delta_r / (2 * R_if)) * (1 - (R_if/R_out)^2) | 75.0000 MPa |
| Inner Hub Bore Hoop Stress (sigma_t) | p_if * (R_out^2 + R_if^2) / (R_out^2 - R_if^2) | 125.0000 MPa |
| Max Slip-Free Torque (T_max) | 2 * PI() * R_if^2 * L_hub * mu * p_if / 1000 | 2,650.7188 N*m |
Same-Material Solid-Shaft Lame Interface Pressure p_if (MPa) Formula
Caution: machining blueprints often specify DIAMETRAL interference (2 * delta_r); ensure B4 stores RADIAL interference delta_r and guard R_out > R_if > 0.
=LET(r_if, B2, r_out, B3, del_r, B4, e_mod, B5, IF(OR(r_if <= 0, r_out <= r_if, del_r < 0, e_mod <= 0), "Invalid Shrink-Fit Geometry", ROUND(((e_mod * del_r) / (2 * r_if)) * (1 - (r_if / r_out)^2), 4)))Maximum Slip-Free Shrink-Fit Torque Capacity T_max (N*m) Formula
Divide N*mm torque 2 * PI() * R_if^2 * L_hub * mu * p_if by 1,000 to report Newton-meters directly.
=LET(r_if, B2, r_out, B3, del_r, B4, e_mod, B5, mu_f, B6, l_hb, B7, IF(OR(r_if <= 0, r_out <= r_if, del_r < 0, e_mod <= 0, mu_f < 0, l_hb <= 0), "Invalid Shrink-Fit Geometry", LET(p_int, ((e_mod * del_r) / (2 * r_if)) * (1 - (r_if / r_out)^2), ROUND((2 * PI() * (r_if^2) * l_hb * mu_f * p_int) / 1000, 4))))Try the example
Use the original values or change them to test your understanding.