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U-notch in a circular shaft for elastic stress, bending

U-notch in a circular shaft D h r
U-notch in a circular shaft

Values for calculation

$D$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$r$ $\mathrm{mm}$

Calculation

Coefficient $ C_1 $

$\text{if }\ 0.25\le h/r\le 2.0$
$$C_1=0.455+3.354\cdot\sqrt{h/r}-0.769\cdot h/r$$
$\text{else}$
$$C_1=0.935+1.922\cdot\sqrt{h/r}+0.004\cdot h/r$$

Coefficient $ C_2 $

$\text{if }\ 0.25\le h/r\le 2.0$
$$C_2=0.891-12.721\cdot\sqrt{h/r}+4.593\cdot h/r$$
$\text{else}$
$$C_2=-0.552-5.327\cdot\sqrt{h/r}+0.086\cdot h/r$$

Coefficient $ C_3 $

$\text{if }\ 0.25\le h/r\le 2.0$
$$C_3=0.286+15.481\cdot\sqrt{h/r}-6.392\cdot h/r$$
$\text{else}$
$$C_3=0.754+6.281\cdot\sqrt{h/r}-0.121\cdot h/r$$

Coefficient $ C_4 $

$\text{if }\ 0.25\le h/r\le 2.0$
$$C_4=-0.632-6.115\cdot\sqrt{h/r}+2.568\cdot h/r$$
$\text{else}$
$$C_4=-0.138-2.876\cdot\sqrt{h/r}+0.031\cdot h/r$$

The elastic stress concentration factor

$$K_t=C_1+C_2\cdot\left(\cfrac{2\cdot h}{D}\right)+C_3\cdot\left(\cfrac{2\cdot h}{D}\right)^2+C_4\cdot\left(\cfrac{2\cdot h}{D}\right)^3$$

Requirements

$$0.25 \le h/r \le 50$$