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Filleted bars in bending

Filleted bars in bending H r h d t L M M
Filleted bars in bending

Values for calculation

$M$ $\mathrm{Nm}$
$H$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$L$ $\mathrm{mm}$

Calculation

Depth of groove, notch

$$t=\cfrac{H-d}{2}$$
$t=\cfrac{H-d}{2}$
Equations in LaTeX code
t=\cfrac{H-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
$$C_1=1.006+0.967\cdot\sqrt{t/r}+0.013\cdot t/r$$
$C_1=1.006+0.967\cdot\sqrt{t/r}+0.013\cdot t/r$
Equations in LaTeX code
C_1=1.006+0.967\cdot\sqrt{t/r}+0.013\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=1.058+1.002\cdot\sqrt{t/r}-0.038\cdot t/r$$
$C_1=1.058+1.002\cdot\sqrt{t/r}-0.038\cdot t/r$
Equations in LaTeX code
C_1=1.058+1.002\cdot\sqrt{t/r}-0.038\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
$$C_2=-0.270-2.372\cdot\sqrt{t/r}+0.708\cdot t/r$$
$C_2=-0.270-2.372\cdot\sqrt{t/r}+0.708\cdot t/r$
Equations in LaTeX code
C_2=-0.270-2.372\cdot\sqrt{t/r}+0.708\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-3.652+1.639\cdot\sqrt{t/r}-0.436\cdot t/r$$
$C_2=-3.652+1.639\cdot\sqrt{t/r}-0.436\cdot t/r$
Equations in LaTeX code
C_2=-3.652+1.639\cdot\sqrt{t/r}-0.436\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
$$C_3=0.662+1.157\cdot\sqrt{t/r}-0.908\cdot t/r$$
$C_3=0.662+1.157\cdot\sqrt{t/r}-0.908\cdot t/r$
Equations in LaTeX code
C_3=0.662+1.157\cdot\sqrt{t/r}-0.908\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=6.170-5.687\cdot\sqrt{t/r}+1.175\cdot t/r$$
$C_3=6.170-5.687\cdot\sqrt{t/r}+1.175\cdot t/r$
Equations in LaTeX code
C_3=6.170-5.687\cdot\sqrt{t/r}+1.175\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
$$C_4=-0.405+0.249\cdot\sqrt{t/r}-0.200\cdot t/r$$
$C_4=-0.405+0.249\cdot\sqrt{t/r}-0.200\cdot t/r$
Equations in LaTeX code
C_4=-0.405+0.249\cdot\sqrt{t/r}-0.200\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-2.558+3.046\cdot\sqrt{t/r}-0.701\cdot t/r$$
$C_4=-2.558+3.046\cdot\sqrt{t/r}-0.701\cdot t/r$
Equations in LaTeX code
C_4=-2.558+3.046\cdot\sqrt{t/r}-0.701\cdot t/r

Stress concentration factor

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

Nominal or reference normal stress

$$σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\cdot d^2}$$
$σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\cdot d^2}$
Equations in LaTeX code
σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\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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