# FORMULAS FOR INTERFERENCE (PRESS & SHRINK) FIT CALCULATIONS

Interference fit formulas for press fit and shrink fit calculations.

 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 (∆) $$\Delta =({ D }_{ so }+{ \Delta }_{ s,+tol })-({ D }_{ hi }-{ \Delta }_{ h,-tol })+2({ u }_{ poisson }+{ u }_{ thermal }+{ u }_{ s,cfg }-{ u }_{ h,cfg })$$ Interference pressure as a result of diametrical interference (p) $$P=\frac { \Delta }{ \frac { { D }_{ hi } }{ { E }_{ h } } (\frac { { D }_{ ho }^{ 2 }+{ D }_{ hi }^{ 2 } }{ { D }_{ ho }^{ 2 }-{ { D }_{ hi }^{ 2 } } } +{ v }_{ h })+\frac { { D }_{ so } }{ { E }_{ s } } (\frac { { D }_{ so }^{ 2 }+{ D }_{ si }^{ 2 } }{ { D }_{ so }^{ 2 }-{ { D }_{ si }^{ 2 } } } -{ v }_{ s }) }$$ 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) $${ σ }_{ VM }=\sqrt { \frac { { ({ σ }_{ r }-{ σ }_{ θ,pressure }-{ σ }_{ θ,cfg }) }^{ 2 }+{ ({ σ }_{ θ,pressure }+{ σ }_{ θ,cfg }-{ σ }_{ z }) }^{ 2 }+{ ({ σ }_{ r }-{ σ }_{ z }) }^{ 2 } }{ 2 } +3{ \tau }^{ 2 } }$$ 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 :

• F: Axial force to be transmitted
• T: Torque to be transmitted
• νh:Poisson’s ratio of hub
• νs:Poisson’s ratio of shaft
• Di: 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

### Reference:

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