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Contact stress of sphere on flat surface

Contact_stress_of_sphere_on_flat_surface
Contact stress of sphere on flat surface

Values for calculation

$F$ $\mathrm{N}$
$E_1$ $\mathrm{MPa}$
$ν_1$
$HB_1$ $\mathrm{HB}$
$S_{y-1}$ $\mathrm{MPa}$
$D_2$ $\mathrm{mm}$
$E_2$ $\mathrm{MPa}$
$ν_2$
$HB_2$ $\mathrm{HB}$
$S_{y-2}$ $\mathrm{MPa}$
$C_c$
$S_F$

Calculation

Allowable Hertz pressure

- for non-hardened material$$σ_H=\min\left(\cfrac{7\cdot HB_1}{S_F}\cdot C_c, \cfrac{7\cdot HB_2}{S_F}\cdot C_c\right)$$- for hardened material$$σ_H=\min\left(\cfrac{4.2\cdot S_{y-1}}{S_F}\cdot C_c, \cfrac{4.2\cdot S_{y-2}}{S_F}\cdot C_c\right)$$

Material coefficient

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

Dimensional coefficient

$$K_D=D_2$$

Diameter of sphere contact area

$$b=1.442\cdot\sqrt[3]{F\cdot K_D \cdot C_E}$$

Contact stress

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