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

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

Values for calculation

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

Calculation

Coefficient $ C_1 $

$\text{if }\ 0.25\le t/r\le 2.0$
$\text{if }\ 0.25\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.25\le t/r\le 2.0
$$C_1=0.966+1.056\cdot\sqrt{t/r}-0.022\cdot t/r$$
$C_1=0.966+1.056\cdot\sqrt{t/r}-0.022\cdot t/r$
Equations in LaTeX code
C_1=0.966+1.056\cdot\sqrt{t/r}-0.022\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=1.089+0.924\cdot\sqrt{t/r}+0.018\cdot t/r$$
$C_1=1.089+0.924\cdot\sqrt{t/r}+0.018\cdot t/r$
Equations in LaTeX code
C_1=1.089+0.924\cdot\sqrt{t/r}+0.018\cdot t/r

Coefficient $ C_2 $

$\text{if }\ 0.25\le t/r\le 2.0$
$\text{if }\ 0.25\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.25\le t/r\le 2.0
$$C_2=-0.192-4.037\cdot\sqrt{t/r}+0.674\cdot t/r$$
$C_2=-0.192-4.037\cdot\sqrt{t/r}+0.674\cdot t/r$
Equations in LaTeX code
C_2=-0.192-4.037\cdot\sqrt{t/r}+0.674\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-1.504-2.141\cdot\sqrt{t/r}-0.047\cdot t/r$$
$C_2=-1.504-2.141\cdot\sqrt{t/r}-0.047\cdot t/r$
Equations in LaTeX code
C_2=-1.504-2.141\cdot\sqrt{t/r}-0.047\cdot t/r

Coefficient $ C_3 $

$\text{if }\ 0.25\le t/r\le 2.0$
$\text{if }\ 0.25\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.25\le t/r\le 2.0
$$C_3=0.808+5.321\cdot\sqrt{t/r}-1.231\cdot t/r$$
$C_3=0.808+5.321\cdot\sqrt{t/r}-1.231\cdot t/r$
Equations in LaTeX code
C_3=0.808+5.321\cdot\sqrt{t/r}-1.231\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=2.486+2.289\cdot\sqrt{t/r}+0.091\cdot t/r$$
$C_3=2.486+2.289\cdot\sqrt{t/r}+0.091\cdot t/r$
Equations in LaTeX code
C_3=2.486+2.289\cdot\sqrt{t/r}+0.091\cdot t/r

Coefficient $ C_4 $

$\text{if }\ 0.25\le t/r\le 2.0$
$\text{if }\ 0.25\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.25\le t/r\le 2.0
$$C_4=-0.567-2.364\cdot\sqrt{t/r}+0.566\cdot t/r$$
$C_4=-0.567-2.364\cdot\sqrt{t/r}+0.566\cdot t/r$
Equations in LaTeX code
C_4=-0.567-2.364\cdot\sqrt{t/r}+0.566\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-1.056-1.104\cdot\sqrt{t/r}-0.059\cdot t/r$$
$C_4=-1.056-1.104\cdot\sqrt{t/r}-0.059\cdot t/r$
Equations in LaTeX code
C_4=-1.056-1.104\cdot\sqrt{t/r}-0.059\cdot t/r

Stress concentration factor for shear stress

$$K_{ts}=C_1+C_2\cdot\left(\cfrac{2\cdot t}{D}\right)+C_3\cdot\left(\cfrac{2\cdot t}{D}\right)^2+C_4\cdot\left(\cfrac{2\cdot t}{D}\right)^3$$
$K_{ts}=C_1+C_2\cdot\left(\cfrac{2\cdot t}{D}\right)+C_3\cdot\left(\cfrac{2\cdot t}{D}\right)^2+C_4\cdot\left(\cfrac{2\cdot t}{D}\right)^3$
Equations in LaTeX code
K_{ts}=C_1+C_2\cdot\left(\cfrac{2\cdot t}{D}\right)+C_3\cdot\left(\cfrac{2\cdot t}{D}\right)^2+C_4\cdot\left(\cfrac{2\cdot t}{D}\right)^3

Nominal or reference shear stress

$$τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^3}$$
$τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^3}$
Equations in LaTeX code
τ_{nom}=\cfrac{16\cdot T\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^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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