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Bending of a stepped bar of circular cross section with a shoulder fillet

Bending of a stepped bar of circular cross section with a shoulder fillet D r t d M M
Bending of a stepped bar of circular cross section with a shoulder fillet

Values for calculation

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

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
$$C_1=0.947+1.206\cdot\sqrt{t/r}-0.131\cdot t/r$$
$C_1=0.947+1.206\cdot\sqrt{t/r}-0.131\cdot t/r$
Equations in LaTeX code
C_1=0.947+1.206\cdot\sqrt{t/r}-0.131\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=1.232+0.832\cdot\sqrt{t/r}-0.008\cdot t/r$$
$C_1=1.232+0.832\cdot\sqrt{t/r}-0.008\cdot t/r$
Equations in LaTeX code
C_1=1.232+0.832\cdot\sqrt{t/r}-0.008\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
$$C_2=0.022-3.405\cdot\sqrt{t/r}+0.915\cdot t/r$$
$C_2=0.022-3.405\cdot\sqrt{t/r}+0.915\cdot t/r$
Equations in LaTeX code
C_2=0.022-3.405\cdot\sqrt{t/r}+0.915\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-3.813+0.968\cdot\sqrt{t/r}-0.260\cdot t/r$$
$C_2=-3.813+0.968\cdot\sqrt{t/r}-0.260\cdot t/r$
Equations in LaTeX code
C_2=-3.813+0.968\cdot\sqrt{t/r}-0.260\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
$$C_3=0.869+1.777\cdot\sqrt{t/r}-0.555\cdot t/r$$
$C_3=0.869+1.777\cdot\sqrt{t/r}-0.555\cdot t/r$
Equations in LaTeX code
C_3=0.869+1.777\cdot\sqrt{t/r}-0.555\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=7.423-4.868\cdot\sqrt{t/r}+0.869\cdot t/r$$
$C_3=7.423-4.868\cdot\sqrt{t/r}+0.869\cdot t/r$
Equations in LaTeX code
C_3=7.423-4.868\cdot\sqrt{t/r}+0.869\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
$$C_4=-0.810+0.422\cdot\sqrt{t/r}-0.260\cdot t/r$$
$C_4=-0.810+0.422\cdot\sqrt{t/r}-0.260\cdot t/r$
Equations in LaTeX code
C_4=-0.810+0.422\cdot\sqrt{t/r}-0.260\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-3.839+3.070\cdot\sqrt{t/r}-0.600\cdot t/r$$
$C_4=-3.839+3.070\cdot\sqrt{t/r}-0.600\cdot t/r$
Equations in LaTeX code
C_4=-3.839+3.070\cdot\sqrt{t/r}-0.600\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

$$σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot d^3}$$
$σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot d^3}$
Equations in LaTeX code
σ_{nom}=\cfrac{32\cdot M\cdot 10^3}{π\cdot d^3}

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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