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

Bending of a bar of circular cross section with a U-shaped groove D r t M M
Bending of a bar of circular cross section with a U-shaped groove

Values for calculation

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

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
$$C_1=0.594+2.958\cdot\sqrt{t/r}-0.520\cdot t/r$$
$C_1=0.594+2.958\cdot\sqrt{t/r}-0.520\cdot t/r$
Equations in LaTeX code
C_1=0.594+2.958\cdot\sqrt{t/r}-0.520\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=0.965+1.926\cdot\sqrt{t/r}$$
$C_1=0.965+1.926\cdot\sqrt{t/r}$
Equations in LaTeX code
C_1=0.965+1.926\cdot\sqrt{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
$$C_2=0.422-10.545\cdot\sqrt{t/r}+2.692\cdot t/r$$
$C_2=0.422-10.545\cdot\sqrt{t/r}+2.692\cdot t/r$
Equations in LaTeX code
C_2=0.422-10.545\cdot\sqrt{t/r}+2.692\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-2.773-4.414\cdot\sqrt{t/r}-0.017\cdot t/r$$
$C_2=-2.773-4.414\cdot\sqrt{t/r}-0.017\cdot t/r$
Equations in LaTeX code
C_2=-2.773-4.414\cdot\sqrt{t/r}-0.017\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
$$C_3=0.501+14.375\cdot\sqrt{t/r}-4.486\cdot t/r$$
$C_3=0.501+14.375\cdot\sqrt{t/r}-4.486\cdot t/r$
Equations in LaTeX code
C_3=0.501+14.375\cdot\sqrt{t/r}-4.486\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=4.785+4.681\cdot\sqrt{t/r}+0.096\cdot t/r$$
$C_3=4.785+4.681\cdot\sqrt{t/r}+0.096\cdot t/r$
Equations in LaTeX code
C_3=4.785+4.681\cdot\sqrt{t/r}+0.096\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
$$C_4=-0.613-6.573\cdot\sqrt{t/r}+2.177\cdot t/r$$
$C_4=-0.613-6.573\cdot\sqrt{t/r}+2.177\cdot t/r$
Equations in LaTeX code
C_4=-0.613-6.573\cdot\sqrt{t/r}+2.177\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-1.995-2.241\cdot\sqrt{t/r}-0.074\cdot t/r$$
$C_4=-1.995-2.241\cdot\sqrt{t/r}-0.074\cdot t/r$
Equations in LaTeX code
C_4=-1.995-2.241\cdot\sqrt{t/r}-0.074\cdot t/r

Stress concentration factor with the nominal stress based on net area

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

$$σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^3}$$
$σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^3}$
Equations in LaTeX code
σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot\left(D-2\cdot t\right)^3}

Maximum normal stress

$$σ_{max}=K_{tn}\cdot σ_{nom}$$
$σ_{max}=K_{tn}\cdot σ_{nom}$
Equations in LaTeX code
σ_{max}=K_{tn}\cdot σ_{nom}

Normal stress

$$σ=\cfrac{32\cdot M\cdot 10^3}{π\cdot D^3}$$
$σ=\cfrac{32\cdot M\cdot 10^3}{π\cdot D^3}$
Equations in LaTeX code
σ=\cfrac{32\cdot M\cdot 10^3}{π\cdot D^3}

Stress concentration factor with the nominal stress based on gross area

$$K_{tg}=\cfrac{σ_{max}}{σ}$$
$K_{tg}=\cfrac{σ_{max}}{σ}$
Equations in LaTeX code
K_{tg}=\cfrac{σ_{max}}{σ}
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