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Values for calculation

$T$ $\mathrm{Nm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$c$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$ϕ$ $\mathrm{rad}$
$L$ $\mathrm{mm}$
$G$ $\mathrm{MPa}$

Calculation

Profile area

$$A=2\cdot\left(a\cdot b+c\cdot d+r^2-\cfrac{π\cdot r^2}{4}\right)$$

Polar moment of inertia coefficient $ K_1 $

$$K_1=a\cdot b^3\cdot \left[\cfrac{1}{3}-0.21\cdot\cfrac{b}{a}\cdot\left(1-\cfrac{b^4}{12\cdot a^4}\right)\right]$$

Polar moment of inertia coefficient $ K_2 $

$$K_2=c\cdot d^3\cdot\left[\cfrac{1}{3}-0.105\cdot\cfrac{d}{c}\cdot\left(1-\cfrac{d^4}{192\cdot c^4}\right)\right]$$

Coefficient $ α $

$$α=\cfrac{d}{b}\cdot\left(0.07+0.076\cdot\cfrac{r}{b}\right)$$

Diameter of largest inscribed circle

$$D=2\cdot\left[d+b+3\cdot r-\sqrt{2\cdot\left(2\cdot r+b\right)\cdot\left(2\cdot r+d\right)}\right]$$

Coefficient $ C $

$$C=\cfrac{D}{1+\cfrac{π^2\cdot D^4}{16\cdot A^2}}\cdot\left[1+\left[0.118\cdot\ln{\left(1+\cfrac{D}{2\cdot r}\right)+0.238\cdot\cfrac{D}{2\cdot r}}\right]\cdot \tanh\left(\cfrac{2\cdot ϕ}{π}\right)\right]$$

Polar moment of inertia

$$K=K_1+K_2+α\cdot D^4$$

Angle of twist

$$θ=\cfrac{T\cdot 10^3\cdot L}{K\cdot G}$$

Torsion stress

$$τ_{max}=\cfrac{10^3\cdot T}{K}\cdot C$$

Requirements

$$d<2\left(b+r\right)$$