Contact stress (sphere)

When two bodies with curved surfaces are pressed together, the line of contact changes to the contact surface and the stresses in the bodies become spatial. Contact stress problems occur at the point of contact of the wheel with the rail, in the valve manifolds of internal combustion engines between cams and valve tappets, in gear engagement and in rolling bearings. Characteristic disturbances that can be observed are cracks, wells or peeling of the surface layer of the material.
The most general example of contact stress occurs when each of the contacting bodies has two different radii of curvature-the radius in the rolling plane is different from the radius in the plane perpendicular thereto, both planes passing through the axes of thrust forces.

Contact stress:

$σ_c$ $=0.918\sqrt[3]{\cfrac{F}{K_D^2C_E^2}}$

$σ_c\leσ_H$

where:
$σ_c$ contact stress $\mathrm{MPa}$
$F$ total force $\mathrm{N}$
$K_D$ dimensional coefficient $\mathrm{mm}$
$C_E$ material coefficient $\mathrm{\cfrac{1}{MPa}}$
$σ_H$ allowable Hertz pressure $\mathrm{MPa}$

Material coefficient:

$C_E$ $=\cfrac{1-ν_1^2}{E_1}+\cfrac{1-ν_2^2}{E_2}$

where:
$C_E$ material coefficient $\mathrm{\cfrac{1}{MPa}}$
$ν_1$ Poisson's ratio 1 $\mathrm{ }$
$ν_2$ Poisson's ratio 2 $\mathrm{ }$
$E_1$ Young's modulus 1 $\mathrm{MPa}$
$E_2$ Young's modulus 2 $\mathrm{MPa}$

Allowable Hertz pressure:
- for non-hardened material

$σ_H$ $=\cfrac{7\cdot }{S_F}\cdot C_c$

- for hardened material
$σ_H$ $=\cfrac{4.2\cdot }{S_F}\cdot C_c$

where:
$σ_H$ allowable Hertz pressure $\mathrm{MPa}$
$HB$ hardness $\mathrm{HB}$
$R_{p0.2/T}$ the minimum yield strength $\mathrm{MPa}$
$S_F$ safety factor $\mathrm{ }$
$C_c$ coefficient according to load $\mathrm{ }$

Coefficient according to load:

load $\mathrm{}$
Static load 1
Unidirectional load, non-impact load 0.8
Unidirectional load, with a small impact load 0.7
Unidirectional load, with a big impact load 0.6
Alternating load, with a small impact load 0.45
Alternating load, with a big impact load 0.25

Diameter of sphere contact area:

$b$ $=1.442\sqrt[3]{FK_DC_E}$

where:
$b$ diameter of sphere contact area $\mathrm{mm}$
$F$ total force $\mathrm{N}$
$K_D$ dimensional coefficient $\mathrm{mm}$
$C_E$ material coefficient $\mathrm{\cfrac{1}{MPa}}$

Contact stress two sphere:


contact-stress-two-sphere F F D 2 D 1 b
Fig. 1 - Contact stress two sphere

Dimensional coefficient:

$K_D$ $=\cfrac{D_1D_2}{D_1+D_2}$

where:
$K_D$ dimensional coefficient $\mathrm{mm}$
$D_1$ sphere diameter 1 $\mathrm{mm}$
$D_2$ sphere diameter 2 $\mathrm{mm}$

Contact stress of sphere on flat surface:


contact-stress-of-sphere-on-flat-surface F F D 2 D 1 b D 2 b F
Fig. 1 - Contact stress of sphere on flat surface

Dimensional coefficient:

$K_D$ $=$

where:
$K_D$ dimensional coefficient $\mathrm{mm}$
$D_2$ sphere diameter $\mathrm{mm}$

Contact stress of the sphere in the sphere socket:


contact-stress-of-the-cylinder-in-the-sphere-socket F D 2 b D 1
Fig. 1 - Contact stress of the sphere in the sphere socket

Dimensional coefficient:

$K_D$ $=\cfrac{}{-}$

where:
$K_D$ dimensional coefficient $\mathrm{mm}$
$D_1$ diameter of spherical socket $\mathrm{mm}$
$D_2$ sphere diameter $\mathrm{mm}$

Literature:
- Warren C. Young, Richard G. Budynas: Roark’s Formulas for Stress and Strain.
- ČSN EN 13001-3-3: Jeřáby – Návrh všeobecně – Část 3-3: Mezní stavy a prokázání způsobilosti kontaktů kolo/kolejnice.
- Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas: Konstruování strojních součástí 2010.