FORMULAS FOR INTERFERENCE (PRESS & SHRINK) FIT CALCULATIONS


The formulas used in Interference Fit Calculator are given below. For more information on interference fits calculations, please refer to pages 387 - 399 of the Precision Machine Design.

Calculation steps and list of equations for Interference (Press & Shrink)  Fit:

Parameter/Condition/Equation
Dimensional Parameters
Differential radial interference due to Poisson’s effect of axial force (upoisson)
$${ u }_{ poisson }=\frac { 2F{ v }_{ h }{ D }_{ hi } }{ \pi ({ D }_{ ho }^{ 2 }-{ D }_{ hi }^{ 2 }){ E }_{ h } } -\frac { 2F{ v }_{ s }{ D }_{ so } }{ \pi ({ D }_{ so }^{ 2 }-{ D }_{ si }^{ 2 }){ E }_{ s } } $$
Differential thermal radial interference due to different operation and assembly temperatures for different materials (uthermal)
$${ u }_{ thermal }=\Delta T({ \alpha }_{ s }-{ \alpha }_{ h }){ D }_{ hi }/2 $$
Hub radial displacement due to rotation (uh,cfg)
$${ u }_{ h,cfg }=\frac { { \rho }_{ h }{ w }^{ 2 }(1-{ v }_{ h }^{ 2 }) }{ { 8E }_{ h } } \left[ -{ (\frac { { D }_{ hi } }{ 2 } ) }^{ 3 }+(3+{ v }_{ h })\left\{ \frac { ({ D }_{ ho }^{ 2 }+{ D }_{ hi }^{ 2 }) }{ 4(1+{ v }_{ h }) } (\frac { { D }_{ hi } }{ 2 } )+\frac { { D }_{ ho }^{ 2 }{ D }_{ hi } }{ 8(1-{ v }_{ h }) } \right\} \right] $$
Shaft radial displacement due to rotation (us,cfg)
$${ u }_{ s,cfg }=\frac { { \rho }_{ s }{ w }^{ 2 }(1-{ v }_{ s }^{ 2 }) }{ { 8E }_{ s } } \left[ -{ (\frac { { D }_{ so } }{ 2 } ) }^{ 3 }+(3+{ v }_{ s })\left\{ \frac { ({ D }_{ so }^{ 2 }+{ D }_{ si }^{ 2 }) }{ 4(1+{ v }_{ s }) } (\frac { { D }_{ so } }{ 2 } )+\frac { { D }_{ si }^{ 2 }{ D }_{ so } }{ 8(1-{ v }_{ s }) } \right\} \right] $$
Maximum diametrical interference (∆)
$$\Delta =({ D }_{ so }+{ \Delta }_{ s,+tol })-({ D }_{ hi }-{ \Delta }_{ h,-tol })+2({ u }_{ poisson }+{ u }_{ thermal }+{ u }_{ s,cfg }-{ u }_{ h,cfg })$$
Pressure Parameter
Interference pressure as a result of diametrical interference (p)
$$P=\frac { \Delta }{ \frac { { D }_{ hi } }{ { E }_{ h } } (\frac { { D }_{ ho }^{ 2 }+{ D }_{ hi }^{ 2 } }{ { D }_{ ho }^{ 2 }-{ { D }_{ hi }^{ 2 } } } +{ v }_{ h })+\frac { { D }_{ so } }{ { E }_{ s } } (\frac { { D }_{ so }^{ 2 }+{ D }_{ si }^{ 2 } }{ { D }_{ so }^{ 2 }-{ { D }_{ si }^{ 2 } } } -{ v }_{ s }) }$$
Pressure & Stress Parameters (Hub)
Radial stress on hub due to interference pressure (σr,pressure)
$${ σ }_{ r,pressure }=-P$$
Circumferential stress on hub due to interference pressure (σθ,pressure)
$${ σ }_{ \theta ,pressure }=P(\frac { { D }_{ ho }^{ 2 }+{ D }_{ hi }^{ 2 } }{ { D }_{ ho }^{ 2 }-{ D }_{ hi }^{ 2 } } )$$
Axial stress on hub due to axial force (σz)
$${ σ }_{ z }=\frac { 4F }{ { \pi (D }_{ ho }^{ 2 }-{ D }_{ hi }^{ 2 })} $$
Shear stress on hub caused by torque (τ)
$${ τ }=\frac { 16T{ D }_{ hi } }{ { \pi (D }_{ ho }^{ 4 }-{ D }_{ hi }^{ 4 }) } $$
Circumferential stress on hub due to centrifugal effect (σθ,cfg)
$${ σ }_{ θ,cfg }=\frac { p{ w }^{ 2 }(3+{ v }_{ h }) }{ 8 } \left( \frac { { D }_{ hi }^{ 2 } }{ 4 } +\frac { { D }_{ ho }^{ 2 } }{ 2 } -\frac { 1+3{ v }_{ h } }{ 3+{ v }_{ h } } \frac { { D }_{ hi }^{ 2 } }{ 4 } \right) $$
Von Mises stress at the hub surface (σVM)
$${ σ }_{ VM }=\sqrt { \frac { { ({ σ }_{ r }-{ σ }_{ θ,pressure }-{ σ }_{ θ,cfg }) }^{ 2 }+{ ({ σ }_{ θ,pressure }+{ σ }_{ θ,cfg }-{ σ }_{ z }) }^{ 2 }+{ ({ σ }_{ r }-{ σ }_{ z }) }^{ 2 } }{ 2 } +3{ \tau }^{ 2 } } $$
Pressure & Stress Parameters (Shaft)
Radial stress on shaft due to interference pressure (σr,pressure)
$${ σ }_{ r,pressure }=P$$
Circumferential stress on shaft due to interference pressure (σθ,pressure)
$${ σ }_{ \theta ,pressure }=P(\frac { { D }_{ so }^{ 2 }+{ D }_{ si }^{ 2 } }{ { D }_{ so }^{ 2 }-{ D }_{ si }^{ 2 } } )$$
Axial stress on shaft due to axial force (σz)
$${ σ }_{ z }=\frac { 4F }{ \pi ({ D }_{ so }^{ 2 }-{ D }_{ si }^{ 2 }) } $$
Shear stress on shaft caused by torque (τ)
$$\tau =\frac { 16T{ D }_{ so } }{ \pi ({ D }_{ so }^{ 4 }-{ D }_{ si }^{ 4 }) } $$
Circumferential stress on shaft due to centrifugal effect (σθ,cfg)
$${ σ }_{ θ,cfg }=\frac { p{ w }^{ 2 }(3+{ v }_{ s }) }{ 8 } (\frac { { D }_{ si }^{ 2 } }{ 2 } +\frac { { D }_{ so }^{ 2 } }{ 4 } -\frac { 1+3{ v }_{ s } }{ 3+{ v }_{ s } } \frac { { D }_{ so }^{ 2 } }{ 4 } )$$
Von Mises stress at the shaft surface (σVM)
$${ σ }_{ VM }=\sqrt { \frac { { ({ σ }_{ r }-{ σ }_{ θ,pressure }-{ σ }_{ θ,cfg }) }^{ 2 }+{ ({ σ }_{ θ,pressure }+{ σ }_{ θ,cfg }-{ σ }_{ z }) }^{ 2 }+{ ({ σ }_{ r }-{ σ }_{ z }) }^{ 2 } }{ 2 } +3{ \tau }^{ 2 } }$$

List of Parameters :

Symbol Parameter
F Axial force to be transmitted
T Torque to be transmitted
νh Poisson’s ratio of hub
νs Poisson’s ratio of shaft
Dhi Hub inner diameter
Dho Hub outer diameter
Dsi Shaft inner diameter
Dso Shaft outer diameter
Eh Hub elastic modulus
Es Shaft elastic modulus
ΔT Temperature difference between operating and assembly conditions
αs Thermal expansion coefficient of shaft
αh Thermal expansion coefficient of hub
ρ Density of the shaft/hub material

Supplements:


Link Usage
Interference (press & shrink) fit calculator Calculates press fit force, required temperatures for shrink fit, fit stresses and other parameters necessary for interference fit design

Reference:

  • Slocum, A. H., Precision Machine Design, © 1995, Society of Manufacturing Engineers, Dearborn, MI. (first published by Prentice Hall in 1992), pp 387-399