# 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:

${\sigma }_{c}=0,798\sqrt{\frac{p}{{K}_{D}{C}_{E}}}\le {\sigma }_{H}$

σc - contact stress - [MPa]

p - load per unit length - [N/mm]

KD - dimensional coefficient - [mm]

CE - material coefficient - [1/MPa]

σH - allowable hertz pressure - [MPa]

Load per unit length:

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

p - load per unit length - [N/mm]

F - total force - [N]

L - length - [mm]

Material coefficient:

${C}_{E}=\frac{1-{\nu }_{1}^{2}}{{E}_{1}}+\frac{1-{\nu }_{2}^{2}}{{E}_{2}}$

CE - material coefficient - [1/MPa]

ν1 - Poisson's ratio 1 - []

ν2 - Poisson's ratio 2 - []

E1 - Young’s modulus 1 - [MPa]

E2 - Young’s modulus 2 - [MPa]

Allowable hertz pressure:

- for non-hardened material

${\sigma }_{H}=\frac{7HB}{{S}_{F}}*{C}_{c}$

- for hardened material

${\sigma }_{H}=\frac{4,2{R}_{p0,2T}}{{S}_{F}}*{C}_{c}$

σH - allowable hertz pressure - [MPa]

HB - hardness - [HB]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient according to load - []

Coefficient according to load:

Width contact area:

$b=1,6\sqrt{p{K}_{D}{C}_{E}}$

b - width contact area - [mm]

p - load per unit length - [N/mm]

KD - dimensional coefficient - [mm]

CE - material coefficient - [1/MPa]

Contact stress two cylinders:

Fig. 1 contact stress two cylinders
${K}_{D}=\frac{{D}_{1}{D}_{2}}{{D}_{1}+{D}_{2}}$

KD - dimensional coefficient - [mm]

D1 - cylinder diameter 1 - [mm]

D2 - cylinder diameter 2 - [mm]

Contact stress of cylinder on flat surface:

Fig. 2 contact stress of cylinder on flat surface
${K}_{D}={D}_{2}$

KD - dimensional coefficient - [mm]

D2 - cylinder diameter - [mm]

Contact stress of the cylinder in the cylindrical socket:

Fig. 3 contact stress of the cylinder in the cylindrical socket
${K}_{D}=\frac{{D}_{1}{D}_{2}}{{D}_{1}-{D}_{2}}$

KD - dimensional coefficient - [mm]

D1 - diameter of cylindrical socket - [mm]

D2 - cylinder diameter - [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.