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Bending of a flat beam with opposite U notches

Bending of a flat beam with opposite U notches H r h M M t
Bending of a flat beam with opposite U notches

Values for calculation

$M$ $\mathrm{Nm}$
$r$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$H$ $\mathrm{mm}$
$h$ $\mathrm{mm}$

Calculation

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
$$C_1=1.024+2.092\cdot\sqrt{t/r}-0.051\cdot t/r$$
$C_1=1.024+2.092\cdot\sqrt{t/r}-0.051\cdot t/r$
Equations in LaTeX code
C_1=1.024+2.092\cdot\sqrt{t/r}-0.051\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=1.113+1.957\cdot\sqrt{t/r}$$
$C_1=1.113+1.957\cdot\sqrt{t/r}$
Equations in LaTeX code
C_1=1.113+1.957\cdot\sqrt{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
$$C_2=-0.630-7.194\cdot\sqrt{t/r}+1.288\cdot t/r$$
$C_2=-0.630-7.194\cdot\sqrt{t/r}+1.288\cdot t/r$
Equations in LaTeX code
C_2=-0.630-7.194\cdot\sqrt{t/r}+1.288\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-2.579-4.017\cdot\sqrt{t/r}-0.013\cdot t/r$$
$C_2=-2.579-4.017\cdot\sqrt{t/r}-0.013\cdot t/r$
Equations in LaTeX code
C_2=-2.579-4.017\cdot\sqrt{t/r}-0.013\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
$$C_3=2.117+8.574\cdot\sqrt{t/r}-2.160\cdot t/r$$
$C_3=2.117+8.574\cdot\sqrt{t/r}-2.160\cdot t/r$
Equations in LaTeX code
C_3=2.117+8.574\cdot\sqrt{t/r}-2.160\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=4.100+3.922\cdot\sqrt{t/r}+0.083\cdot t/r$$
$C_3=4.100+3.922\cdot\sqrt{t/r}+0.083\cdot t/r$
Equations in LaTeX code
C_3=4.100+3.922\cdot\sqrt{t/r}+0.083\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
$$C_4=-1.420-3.494\cdot\sqrt{t/r}+0.932\cdot t/r$$
$C_4=-1.420-3.494\cdot\sqrt{t/r}+0.932\cdot t/r$
Equations in LaTeX code
C_4=-1.420-3.494\cdot\sqrt{t/r}+0.932\cdot t/r
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-1.528-1.893\cdot\sqrt{t/r}-0.066\cdot t/r$$
$C_4=-1.528-1.893\cdot\sqrt{t/r}-0.066\cdot t/r$
Equations in LaTeX code
C_4=-1.528-1.893\cdot\sqrt{t/r}-0.066\cdot t/r

Stress concentration factor with the nominal stress based on net area

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

$$σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\cdot\left(H-2\cdot t\right)^2}$$
$σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\cdot\left(H-2\cdot t\right)^2}$
Equations in LaTeX code
σ_{nom}=\cfrac{6\cdot M\cdot 10^3}{h\cdot\left(H-2\cdot t\right)^2}

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

$$σ=\cfrac{6\cdot M\cdot 10^3}{H^2\cdot h}$$
$σ=\cfrac{6\cdot M\cdot 10^3}{H^2\cdot h}$
Equations in LaTeX code
σ=\cfrac{6\cdot M\cdot 10^3}{H^2\cdot h}

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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