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.

Reference: