# 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