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