The formulas and parameters used in Hertzian Contact Stress Calculator are given below. According to
the input parameters and selection of contact type from the spherical and cylindrical contacts,
suitable formulas are selected from the list of equations given below for the calculation of Hertzian contact stresses.
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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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$${ \tau }_{ yz }=\frac { { \sigma }_{ y }-{ \sigma }_{ z } }{ 2 } $$ |
Note: For a plane
surface, use d = ∞. For an internal surface, the diameter is expressed as a negative
quantity...