# Moment of Inertia and Radius of Gyration

**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:

Moment of inertia about the y-axis:

**Polar Moment of Inertia**:

Polar moment of inertia is the moment of inertia about about the z-axis.

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

**Radius of Gyration**

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

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

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

**Transfer Formula for Moment of Inertia**

$x'$ = 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

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

**Product of Inertia**

**Moment of Inertia of Common Shapes**

Shape | Moment of Inertia | Radius of Gyration |
---|---|---|

Rectangle | $\bar{I}_x = \dfrac{bh^3}{12}$
$I_x = \dfrac{bh^3}{3}$ |
$\bar{k}_x = \dfrac{h}{\sqrt{12}}$
$k_x = \dfrac{h}{\sqrt{3}}$ |

Triangle | $\bar{I}_x = \dfrac{bh^3}{36}$
$I_x = \dfrac{bh^3}{12}$ |
$\bar{k}_x = \dfrac{h}{\sqrt{18}}$
$k_x = \dfrac{h}{\sqrt{6}}$ |

Circle | $\bar{I}_x = \dfrac{\pi r^4}{4}$
$\bar{J} = \dfrac{\pi r^4}{2}$ |
$\bar{k}_x = \dfrac{r}{2}$
$\bar{k}_z = \dfrac{r}{\sqrt{2}}$ |

Semicircle | $I_x = \bar{I}_y = \dfrac{\pi r^4}{8}$
$\bar{I}_x = 0.11r^4$ |
$k_x = \bar{k}_y = \dfrac{r}{2}$
$\bar{k}_x = 0.264r$ |

Quarter circle | $I_x = I_y = \dfrac{\pi r^4}{16}$
$\bar{I}_x = \bar{I}_y = 0.055r^4$ |
$k_x = k_y = \dfrac{r}{2}$
$\bar{k}_x = \bar{k}_y = 0.264r$ |

Ellipse | $\bar{I}_x = \dfrac{\pi ab^3}{4}$
$\bar{I}_y = \dfrac{\pi a^3b}{4}$ |
$\bar{k}_x = \dfrac{b}{2}$
$\bar{k}_y = \dfrac{a}{2}$ |

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