Contact stress (cylinder)

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:

$σ_c$ $=0.798\sqrt{\cfrac{p}{K_DC_E}}$

$σ_c\leσ_H$

where:
$σ_c$ contact stress $\mathrm{MPa}$
$p$ load per unit length $\mathrm{N/mm}$
$K_D$ dimensional coefficient $\mathrm{mm}$
$C_E$ material coefficient $\mathrm{\cfrac{1}{MPa}}$
$σ_H$ allowable Hertz pressure $\mathrm{MPa}$

Load per unit length:

$p$ $=\cfrac{F}{L}$

where:
$p$ load per unit length $\mathrm{N/mm}$
$F$ total force $\mathrm{N}$
$L$ length $\mathrm{mm}$

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

Width contact area:

$b$ $=1.6\sqrt{pK_DC_E}$

where:
$b$ width contact area $\mathrm{mm}$
$p$ load per unit length $\mathrm{N/mm}$
$K_D$ dimensional coefficient $\mathrm{mm}$
$C_E$ material coefficient $\mathrm{\cfrac{1}{MPa}}$

Contact stress two cylinders:


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

Dimensional coefficient:

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

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

Contact stress of cylinder on flat surface:


contact-stress-of-cylinder-on-flat-surface D 2 L b F
Fig. 2 - Contact stress of cylinder on flat surface

Dimensional coefficient:

$K_D$ $=$

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

Contact stress of the cylinder in the cylindrical socket:


contact-stress-of-the-cylinder-in-the-cylindrical-socket D 2 b D 1 F L
Fig. 3 - Contact stress of the cylinder in the cylindrical socket

Dimensional coefficient:

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

where:
$K_D$ dimensional coefficient $\mathrm{mm}$
$D_1$ diameter of cylindrical socket $\mathrm{mm}$
$D_2$ cylinder 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.