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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 $

$$e=H-c$$

$$0\le c/e\le 1$$

Coefficient $ C_1 $

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

Coefficient $ C_2 $

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

Coefficient $ C_3 $

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

Coefficient $ C'_1 $

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

Coefficient $ C'_2 $

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

Coefficient $ C'_3 $

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

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

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

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

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

Maximum normal stress $ σ_{maxB} $

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

Maximum normal stress $ σ_{maxA} $

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