Moment of Inertia and Radius of Gyration

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Moment of Inertia
Moment of inertia, also called the second moment of area, is the product of area and the square of its moment arm about a reference axis.
 

Moment of inertia about the x-axis:

$ \displaystyle I_x = \int y^2 \, dA $

 

000_moment_of_inertia.gif

 

Moment of inertia about the y-axis:

$ \displaystyle I_y = \int x^2 \, dA $

 

Polar Moment of Inertia:
Polar moment of inertia is the moment of inertia about about the z-axis.

$ J = I_x + I_y $

$ \displaystyle J = \int r^2 \, dA $

 

Radius of Gyration

$ k = \sqrt{\dfrac{I}{A}} $

$ k_x = \sqrt{\dfrac{I_x}{A}} $

$ k_y = \sqrt{\dfrac{I_y}{A}} $

$ k_z = \sqrt{\dfrac{J}{A}} $

 

Transfer Formula for Moment of Inertia
 

000_transfer_of_moment_of_inertia.gif

 

$ I = \bar{I} + Ad^2 $

 

Where
$ x' $ = centroidal axis
$ x $ = any axis parallel to the centroidal axis
$ I $ = moment of inertia about the x-axis
$ \bar{I} $ = centroidal moment of inertia
$ A $ = area of the section
$ d $ = distance between x and x’

 

In the same manner, the transfer formula for polar moment of inertia and the radii of gyration are respectively

$ J = \bar{J} + Ad^2 $

$ k^2 = {\bar{k}}^2 + d^2 $

 

Product of Inertia

$ \displaystyle I_{xy} = \int xy \, dA $

 

Moment of Inertia of Common Shapes

Shape Moment of Inertia Radius of Gyration
Rectangle

000_moment_of_inertia_rectangle.gif

$ \bar{I}_x = \dfrac{bh^3}{12} $

$ I_x = \dfrac{bh^3}{3} $

$ \bar{k}_x = \dfrac{h}{\sqrt{12}} $

$ k_x = \dfrac{h}{\sqrt{3}} $

Triangle

000_moment_of_inertia_triangle.gif

$ \bar{I}_x = \dfrac{bh^3}{36} $

$ I_x = \dfrac{bh^3}{12} $

$ \bar{k}_x = \dfrac{h}{\sqrt{18}} $

$ k_x = \dfrac{h}{\sqrt{6}} $

Circle

000_moment_of_inertia_circle.gif

$ \bar{I}_x = \dfrac{\pi r^4}{4} $

$ \bar{J} = \dfrac{\pi r^4}{2} $

$ \bar{k}_x = \dfrac{r}{2} $

$ \bar{k}_z = \dfrac{r}{\sqrt{2}} $

Semicircle

000_moment_of_inertia_semi-circle.gif

$ I_x = \bar{I}_y = \dfrac{\pi r^4}{8} $

$ \bar{I}_x = 0.11r^4 $

$ k_x = \bar{k}_y = \dfrac{r}{2} $

$ \bar{k}_x = 0.264r $

Quarter circle

000_moment_of_inertia_quarter-circle.gif

$ I_x = I_y = \dfrac{\pi r^4}{16} $

$ \bar{I}_x = \bar{I}_y = 0.055r^4 $

$ k_x = k_y = \dfrac{r}{2} $

$ \bar{k}_x = \bar{k}_y = 0.264r $

Ellipse

000_moment_of_inertia_ellipse.gif

$ \bar{I}_x = \dfrac{\pi ab^3}{4} $

$ \bar{I}_y = \dfrac{\pi a^3b}{4} $

$ \bar{k}_x = \dfrac{b}{2} $

$ \bar{k}_y = \dfrac{a}{2} $

 

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