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Bending of a thin beam element with a notch on one side

Bending of a thin beam element with a notch on one side H r h M M t
Bending of a thin beam element with a notch on one side

Values for calculation

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

Calculation

Coefficient $ C_1 $

$\text{if }\ 0.5\le t/r\le 2.0$
$\text{if }\ 0.5\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.5\le t/r\le 2.0
$$C_1=1.795+1.481\cdot\left(t/r\right)-0.211\cdot\left(t/r\right)^2$$
$C_1=1.795+1.481\cdot\left(t/r\right)-0.211\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_1=1.795+1.481\cdot\left(t/r\right)-0.211\cdot\left(t/r\right)^2
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_1=2.966+0.502\cdot\left(t/r\right)-0.009\cdot\left(t/r\right)^2$$
$C_1=2.966+0.502\cdot\left(t/r\right)-0.009\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_1=2.966+0.502\cdot\left(t/r\right)-0.009\cdot\left(t/r\right)^2

Coefficient $ C_2 $

$\text{if }\ 0.5\le t/r\le 2.0$
$\text{if }\ 0.5\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.5\le t/r\le 2.0
$$C_2=-3.544-3.677\cdot\left(t/r\right)+0.578\cdot\left(t/r\right)^2$$
$C_2=-3.544-3.677\cdot\left(t/r\right)+0.578\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_2=-3.544-3.677\cdot\left(t/r\right)+0.578\cdot\left(t/r\right)^2
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_2=-6.475-1.126\cdot\left(t/r\right)+0.019\cdot\left(t/r\right)^2$$
$C_2=-6.475-1.126\cdot\left(t/r\right)+0.019\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_2=-6.475-1.126\cdot\left(t/r\right)+0.019\cdot\left(t/r\right)^2

Coefficient $ C_3 $

$\text{if }\ 0.5\le t/r\le 2.0$
$\text{if }\ 0.5\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.5\le t/r\le 2.0
$$C_3=5.459+3.691\cdot\left(t/r\right)-0.565\cdot\left(t/r\right)^2$$
$C_3=5.459+3.691\cdot\left(t/r\right)-0.565\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_3=5.459+3.691\cdot\left(t/r\right)-0.565\cdot\left(t/r\right)^2
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=8.023+1.253\cdot\left(t/r\right)-0.020\cdot\left(t/r\right)^2$$
$C_3=8.023+1.253\cdot\left(t/r\right)-0.020\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_3=8.023+1.253\cdot\left(t/r\right)-0.020\cdot\left(t/r\right)^2

Coefficient $ C_4 $

$\text{if }\ 0.5\le t/r\le 2.0$
$\text{if }\ 0.5\le t/r\le 2.0$
Equations in LaTeX code
\text{if }\ 0.5\le t/r\le 2.0
$$C_4=-2.678-1.531\cdot\left(t/r\right)+0.205\cdot\left(t/r\right)^2$$
$C_4=-2.678-1.531\cdot\left(t/r\right)+0.205\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_4=-2.678-1.531\cdot\left(t/r\right)+0.205\cdot\left(t/r\right)^2
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-3.572-0.634\cdot\left(t/r\right)+0.010\cdot\left(t/r\right)^2$$
$C_4=-3.572-0.634\cdot\left(t/r\right)+0.010\cdot\left(t/r\right)^2$
Equations in LaTeX code
C_4=-3.572-0.634\cdot\left(t/r\right)+0.010\cdot\left(t/r\right)^2

Stress concentration factor with the nominal stress based on net area

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

Nominal or reference normal stress

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