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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) 印度电影激情片完整版在线 在线播放 印度电影激情片完整版在线 在线播放 ,松岛枫无码作品封面番号_松岛枫无码作品封面番号 精彩完 松岛枫无码作品封面番号_松岛枫无码作品封面番号 精彩完 $${ p }_{ max }=\frac { 3F }{ 2\pi { a }^{ 2 } }$$ 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.

### Reference:

• Budynas.R , Nisbett.K . (2008) . Shigley's Mechanical Engineering Design . 8th edition.  McGraw-Hill
• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill