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Contact Stress Formulas |
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Calculation for spherical contact |
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Contact radius (a) |
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$$a=\sqrt [ 3 ]{ \frac { 3F }{ 8 } \frac { (1-{ \nu }_{ 1 }^{ 2 })/{ E }_{ 1
}+(1-{ \nu }_{ 2 }^{ 2 })/{ E }_{ 2 } }{ 1/{ d }_{ 1 }+1/{ d }_{ 2 } } } $$ |
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Maximum pressure (pmax) |
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$${ p }_{ max }=\frac { 3F }{ 2\pi { a }^{ 2 } } $$ |
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Principal stress (σx)
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$${ \sigma }_{ x }=-{ p }_{ max }\left[ (1-\left| \frac { z }{ a } \right| \tan
^{ -1 }{ \frac { 1 }{ \left| z/a \right| } } )(1+\upsilon )-\frac { 1 }{
2(1+\frac { { z }^{ 2 } }{ { a }^{ 2 } } ) } \right]$$
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Principal stress (σy) |
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$$ { \sigma }_{ y }=-{ p }_{ max }\left[ (1-\left| \frac { z }{ a } \right| \tan
^{ -1 }{ \frac { 1 }{ \left| z/a \right| } } )(1+\upsilon )-\frac { 1 }{
2(1+\frac { { z }^{ 2 } }{ { a }^{ 2 } } ) } \right] $$
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Principal stress (σz) |
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$${ \sigma }_{ z }=\frac { -{ p }_{ max } }{ 1+\frac { { z }^{ 2 } }{ { a }^{ 2
} } } $$ |
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Maximum shear stress (τmax) |
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$${ \tau }_{ max }=\frac { { \sigma }_{ x }-{ \sigma }_{ z } }{ 2 } =\frac { {
\sigma }_{ y }-{ \sigma }_{ z } }{ 2 } $$ |
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Calculation for cylindrical contact |
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Contact half-width
(b) |
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$$b=\sqrt { \frac { 2F }{ \pi l } \frac { (1-{ \nu }_{ 1 }^{ 2 })/{ E }_{ 1
}+(1-{ \nu }_{ 2 }^{ 2 })/{ E }_{ 2 } }{ 1/{ d }_{ 1 }+1/{ d }_{ 2 } } } $$ |
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Maximum pressure (pmax) |
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$${ p }_{ max }=\frac { 2F }{ \pi bl } $$ |
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Principal stress (σx) |
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$${ \sigma }_{ x }=-2\nu { p }_{ max }\left[ \sqrt { (1+\frac { { z }^{ 2
} }{ { b }^{ 2 } } ) } -\left| \frac { z }{ b } \right| \right] $$ |
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Principal stress (σy) |
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$${ \sigma }_{ y }=-{ p }_{ max }\left[ \frac { 1+2\frac { { z }^{ 2 } }{ { b
}^{ 2 } } }{ \sqrt { (1+\frac { { z }^{ 2 } }{ { b }^{ 2 } } ) } } -2\left|
\frac { z }{ b } \right| \right] $$ |
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Principal stress (σz) |
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$${ \sigma }_{ z }=\frac { -{ p }_{ max } }{ \sqrt { (1+\frac { { z }^{ 2 } }{ {
b }^{ 2 } } ) } } $$ |
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Shear stress (τxz) |
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$${ \tau }_{ xz }=\frac { { \sigma }_{ x }-{ \sigma }_{ z } }{ 2 } $$ |
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Shear stress (τyz) |
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$${ \tau }_{ yz }=\frac { { \sigma }_{ y }-{ \sigma }_{ z } }{ 2 } $$ |
