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Central circular hole in a member of rectangular cross section for elastic stress, in-plane bending

Central circular hole in a member of rectangular cross section r t D
Central circular hole in a member of rectangular cross section
Elastic stress, in-plane bending P σ A σ X M M
Elastic stress, in-plane bending

Values for calculation

$M$ $\mathrm{Nm}$
$D$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$r$ $\mathrm{mm}$

Calculation

The elastic stress concentration factor

$$K_t=2$$
$K_t=2$
Equations in LaTeX code
K_t=2

Nominal or reference normal stress

$$σ_{nom}=\cfrac{12\cdot M\cdot 10^3\cdot r}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}$$
$σ_{nom}=\cfrac{12\cdot M\cdot 10^3\cdot r}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}$
Equations in LaTeX code
σ_{nom}=\cfrac{12\cdot M\cdot 10^3\cdot r}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}

Normal stress at point $ A $

$$σ_A=K_t\cdot σ_{nom}$$
$σ_A=K_t\cdot σ_{nom}$
Equations in LaTeX code
σ_A=K_t\cdot σ_{nom}

Normal stress at point $ X $

$$σ_X=\cfrac{6\cdot M\cdot 10^3\cdot D}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}$$
$σ_X=\cfrac{6\cdot M\cdot 10^3\cdot D}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}$
Equations in LaTeX code
σ_X=\cfrac{6\cdot M\cdot 10^3\cdot D}{t\cdot\left[D^3-\left(2\cdot r\right)^3\right]}

Maximum normal stress

$$σ_{max}=\max\left(σ_A, σ_X\right)$$
$σ_{max}=\max\left(σ_A, σ_X\right)$
Equations in LaTeX code
σ_{max}=\max\left(σ_A, σ_X\right)
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