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Stepped tension bar of circular cross section with shoulder fillet

Stepped tension bar of circular cross section with shoulder fillet D r t P P d
Stepped tension bar of circular cross section with shoulder fillet

Values for calculation

$P$ $\mathrm{N}$
$D$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$r$ $\mathrm{mm}$

Calculation

Depth of groove, notch

$$t=\cfrac{D-d}{2}$$
$t=\cfrac{D-d}{2}$
Equations in LaTeX code
t=\cfrac{D-d}{2}
$$0.1\le t/r\le 20$$
$0.1\le t/r\le 20$
Equations in LaTeX code
0.1\le t/r\le 20

Coefficient $ C_1 $

$\text{if }\ 0.1\le t/r\le 2.0$
$\text{if }\ 0.1\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.1\le t/r\le 2.0
$$C_1=0.926+1.157\cdot\sqrt{t/r}-0.099\cdot t/r$$
$C_1=0.926+1.157\cdot\sqrt{t/r}-0.099\cdot t/r$
Equations in LaTeX code
C_1=0.926+1.157\cdot\sqrt{t/r}-0.099\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=1.200+0.860\cdot\sqrt{t/r}-0.022\cdot t/r$$
$C_1=1.200+0.860\cdot\sqrt{t/r}-0.022\cdot t/r$
Equations in LaTeX code
C_1=1.200+0.860\cdot\sqrt{t/r}-0.022\cdot t/r

Coefficient $ C_2 $

$\text{if }\ 0.1\le t/r\le 2.0$
$\text{if }\ 0.1\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.1\le t/r\le 2.0
$$C_2=0.012-3.036\cdot\sqrt{t/r}+0.961\cdot t/r$$
$C_2=0.012-3.036\cdot\sqrt{t/r}+0.961\cdot t/r$
Equations in LaTeX code
C_2=0.012-3.036\cdot\sqrt{t/r}+0.961\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-1.805-0.346\cdot\sqrt{t/r}-0.038\cdot t/r$$
$C_2=-1.805-0.346\cdot\sqrt{t/r}-0.038\cdot t/r$
Equations in LaTeX code
C_2=-1.805-0.346\cdot\sqrt{t/r}-0.038\cdot t/r

Coefficient $ C_3 $

$\text{if }\ 0.1\le t/r\le 2.0$
$\text{if }\ 0.1\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.1\le t/r\le 2.0
$$C_3=-0.302+3.977\cdot\sqrt{t/r}-1.744\cdot t/r$$
$C_3=-0.302+3.977\cdot\sqrt{t/r}-1.744\cdot t/r$
Equations in LaTeX code
C_3=-0.302+3.977\cdot\sqrt{t/r}-1.744\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=2.198-0.486\cdot\sqrt{t/r}+0.165\cdot t/r$$
$C_3=2.198-0.486\cdot\sqrt{t/r}+0.165\cdot t/r$
Equations in LaTeX code
C_3=2.198-0.486\cdot\sqrt{t/r}+0.165\cdot t/r

Coefficient $ C_4 $

$\text{if }\ 0.1\le t/r\le 2.0$
$\text{if }\ 0.1\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.1\le t/r\le 2.0
$$C_4=0.365-2.098\cdot\sqrt{t/r}+0.878\cdot t/r$$
$C_4=0.365-2.098\cdot\sqrt{t/r}+0.878\cdot t/r$
Equations in LaTeX code
C_4=0.365-2.098\cdot\sqrt{t/r}+0.878\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-0.593-0.028\cdot\sqrt{t/r}-0.106\cdot t/r$$
$C_4=-0.593-0.028\cdot\sqrt{t/r}-0.106\cdot t/r$
Equations in LaTeX code
C_4=-0.593-0.028\cdot\sqrt{t/r}-0.106\cdot t/r

Stress concentration factor

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

$$σ_{nom}=\cfrac{4\cdot P}{π\cdot d^2}$$
$σ_{nom}=\cfrac{4\cdot P}{π\cdot d^2}$
Equations in LaTeX code
σ_{nom}=\cfrac{4\cdot P}{π\cdot d^2}

Maximum normal stress

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