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Bending of a thin beam with a circular hole displaced from the center line

Bending of a thin beam with a circular hole displaced from the center line H h d c e M M σ maxB σ maxA
Bending of a thin beam with a circular hole displaced from the center line

Values for calculation

$M$ $\mathrm{Nm}$
$H$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$c$ $\mathrm{mm}$

Calculation

Dimension $ e $

$$e=H-c$$
$e=H-c$
Equations in LaTeX code
e=H-c
$$0\le c/e\le 1$$
$0\le c/e\le 1$
Equations in LaTeX code
0\le c/e\le 1

Coefficient $ C_1 $

$$C_1=3-0.631\cdot\left(\cfrac{d}{2\cdot c}\right)+4.007\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C_1=3-0.631\cdot\left(\cfrac{d}{2\cdot c}\right)+4.007\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C_1=3-0.631\cdot\left(\cfrac{d}{2\cdot c}\right)+4.007\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Coefficient $ C_2 $

$$C_2=-5.083+4.067\cdot\left(\cfrac{d}{2\cdot c}\right)-2.795\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C_2=-5.083+4.067\cdot\left(\cfrac{d}{2\cdot c}\right)-2.795\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C_2=-5.083+4.067\cdot\left(\cfrac{d}{2\cdot c}\right)-2.795\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Coefficient $ C_3 $

$$C_3=2.114-1.682\cdot\left(\cfrac{d}{2\cdot c}\right)-0.273\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C_3=2.114-1.682\cdot\left(\cfrac{d}{2\cdot c}\right)-0.273\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C_3=2.114-1.682\cdot\left(\cfrac{d}{2\cdot c}\right)-0.273\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Coefficient $ C'_1 $

$$C'_1=1.0286-0.1638\cdot\left(\cfrac{d}{2\cdot c}\right)+2.702\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C'_1=1.0286-0.1638\cdot\left(\cfrac{d}{2\cdot c}\right)+2.702\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C'_1=1.0286-0.1638\cdot\left(\cfrac{d}{2\cdot c}\right)+2.702\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Coefficient $ C'_2 $

$$C'_2=-0.05863-0.1335\cdot\left(\cfrac{d}{2\cdot c}\right)-1.8747\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C'_2=-0.05863-0.1335\cdot\left(\cfrac{d}{2\cdot c}\right)-1.8747\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C'_2=-0.05863-0.1335\cdot\left(\cfrac{d}{2\cdot c}\right)-1.8747\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Coefficient $ C'_3 $

$$C'_3=0.18883-0.89219\cdot\left(\cfrac{d}{2\cdot c}\right)+1.5189\cdot\left(\cfrac{d}{2\cdot c}\right)^2$$
$C'_3=0.18883-0.89219\cdot\left(\cfrac{d}{2\cdot c}\right)+1.5189\cdot\left(\cfrac{d}{2\cdot c}\right)^2$
Equations in LaTeX code
C'_3=0.18883-0.89219\cdot\left(\cfrac{d}{2\cdot c}\right)+1.5189\cdot\left(\cfrac{d}{2\cdot c}\right)^2

Stress concentration factor with the nominal stress based on gross area at point $ B $

$$K_{tgB}=C_1+C_2\cdot\left(\cfrac{c}{e}\right)+C_3\cdot\left(\cfrac{c}{e}\right)^2$$
$K_{tgB}=C_1+C_2\cdot\left(\cfrac{c}{e}\right)+C_3\cdot\left(\cfrac{c}{e}\right)^2$
Equations in LaTeX code
K_{tgB}=C_1+C_2\cdot\left(\cfrac{c}{e}\right)+C_3\cdot\left(\cfrac{c}{e}\right)^2

Stress concentration factor with the nominal stress based on gross area at point $ A $

$$K_{tgA}=C'_1+C'_2\cdot\left(\cfrac{c}{e}\right)+C'_3\cdot\left(\cfrac{c}{e}\right)^2$$
$K_{tgA}=C'_1+C'_2\cdot\left(\cfrac{c}{e}\right)+C'_3\cdot\left(\cfrac{c}{e}\right)^2$
Equations in LaTeX code
K_{tgA}=C'_1+C'_2\cdot\left(\cfrac{c}{e}\right)+C'_3\cdot\left(\cfrac{c}{e}\right)^2

Maximum normal stress $ σ_{maxB} $

$$σ_{maxB}=\cfrac{K_{tgB}\cdot6\cdot M\cdot10^3}{H^2\cdot h}$$
$σ_{maxB}=\cfrac{K_{tgB}\cdot6\cdot M\cdot10^3}{H^2\cdot h}$
Equations in LaTeX code
σ_{maxB}=\cfrac{K_{tgB}\cdot6\cdot M\cdot10^3}{H^2\cdot h}

Maximum normal stress $ σ_{maxA} $

$$σ_{maxA}=\cfrac{K_{tgA}\cdot6\cdot M\cdot10^3}{H^2\cdot h}$$
$σ_{maxA}=\cfrac{K_{tgA}\cdot6\cdot M\cdot10^3}{H^2\cdot h}$
Equations in LaTeX code
σ_{maxA}=\cfrac{K_{tgA}\cdot6\cdot M\cdot10^3}{H^2\cdot h}
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