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]{\frac{F}{K_{D}^{2} C_{E}^{2}}} \leq σ_{H}

σ_c - contact stress - [MPa]

F - total force - [N]

K_D - dimensional coefficient - [mm]

C_E - material coefficient - [1/MPa]

σ_H - allowable hertz pressure - [MPa]

Material coefficient:

C_{E} = \frac{1 - ν_{1}^{2}}{E_{1}} + \frac{1 - ν_{2}^{2}}{E_{2}}

C_E - 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

σ_{H} = \frac{7 H B}{S_{F}} * C_{c}

- for hardened material

σ_{H} = \frac{4,2 R_{p 0,2 T}}{S_{F}} * C_{c}

σ_H - allowable hertz pressure - [MPa]

HB - hardness - [HB]

R_p0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

S_F - safety factor - []

C_c - coefficient according to load - []

Coefficient according to load:

load	[]
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 circular contact area:

b = 1,442 \sqrt[3]{F K_{D} C_{E}}

b - diameter of circular contact area - [mm]

F - total force - [N]

K_D - dimensional coefficient - [mm]

C_E - material coefficient - [1/MPa]

Contact stress two sphere:

Fig. 1 contact stress two sphere

K_{D} = \frac{D_{1} D_{2}}{D_{1} + D_{2}}

K_D - dimensional coefficient - [mm]

D₁ - diameter sphere 1 - [mm]

D₂ - diameter sphere 2 - [mm]

Contact stress of sphere on flat surface:

Fig. 2 contact stress of sphere on flat surface

K_{D} = D_{2}

K_D - dimensional coefficient - [mm]

D₂ - diameter sphere - [mm]

Contact stress of the sphere in the spherical socket:

Fig. 3 contact stress of the sphere in the spherical socket

K_{D} = \frac{D_{1} D_{2}}{D_{1} - D_{2}}

K_D - dimensional coefficient - [mm]

D₁ - diameter spherical socket - [mm]

D₂ - diameter sphere - [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.

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Contact stress (sphere).pdf

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