# FORMULAS FOR CONTACT STRESS CALCULATIONS

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.

### Hertzian Contact Stress Formulas:

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

Note: For a plane surface, use d = ∞. For an internal surface, the diameter is expressed as a negative quantity...

### List of Parameters :

 Symbol Parameter F Applied force ν1 Object-1 Poisson’s ratio E1 Object-1 elastic modulus v2 Object-2 Poisson’s ratio E2 Object-2 elastic modulus d1 Object-1 diameter d2 Object-2 diameter z Depth below the surface l Contact length of cylinders

### Supplements:

 Link Usage Hertzian contact calculator Calculates contact parameters such as contact pressure, shear and Von Misses stresses for spherical and cylindrical contact cases.