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Torsion of a bar of circular cross section with a shallow U-shaped groove

Torsion of a bar of circular cross section with a shallow U-shaped groove D d r T T
Torsion of a bar of circular cross section with a shallow U-shaped groove

Values for calculation

$T$ $\mathrm{Nm}$
$r$ $\mathrm{mm}$
$D$ $\mathrm{mm}$
$d$ $\mathrm{mm}$

Calculation

Coefficient $ C_1 $

$$C_1=-35.16+67.57\cdot\left(D/d\right)-31.28\cdot\left(D/d\right)^2$$
$C_1=-35.16+67.57\cdot\left(D/d\right)-31.28\cdot\left(D/d\right)^2$
Equations in LaTeX code
C_1=-35.16+67.57\cdot\left(D/d\right)-31.28\cdot\left(D/d\right)^2

Coefficient $ C_2 $

$$C_2=79.13-148.37\cdot\left(D/d\right)+69.09\cdot\left(D/d\right)^2$$
$C_2=79.13-148.37\cdot\left(D/d\right)+69.09\cdot\left(D/d\right)^2$
Equations in LaTeX code
C_2=79.13-148.37\cdot\left(D/d\right)+69.09\cdot\left(D/d\right)^2

Coefficient $ C_3 $

$$C_3=-50.34+94.67\cdot\left(D/d\right)-44.26\cdot\left(D/d\right)^2$$
$C_3=-50.34+94.67\cdot\left(D/d\right)-44.26\cdot\left(D/d\right)^2$
Equations in LaTeX code
C_3=-50.34+94.67\cdot\left(D/d\right)-44.26\cdot\left(D/d\right)^2

Stress concentration factor for shear stress

$$K_{ts}=C_1+C_2\cdot\left(r/d\right)+C_3\cdot\left(r/d\right)^2$$
$K_{ts}=C_1+C_2\cdot\left(r/d\right)+C_3\cdot\left(r/d\right)^2$
Equations in LaTeX code
K_{ts}=C_1+C_2\cdot\left(r/d\right)+C_3\cdot\left(r/d\right)^2

Nominal or reference shear stress

$$τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot d^3}$$
$τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot d^3}$
Equations in LaTeX code
τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot d^3}

Maximum shear stress

$$τ_{max}=K_{ts}\cdot τ_{nom}$$
$τ_{max}=K_{ts}\cdot τ_{nom}$
Equations in LaTeX code
τ_{max}=K_{ts}\cdot τ_{nom}
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