Product of Inertia | Radius of Gyration

**Parallel-Axis Theorem: **In many instances the moment of inertia of an area about an axis that is parallel to its centroidal axis is sought. The equations for moments of inertia about such parallel axes are given as

(5) | |

(6) |

where I_{xc} and I_{yc} are the moments of inertia of the area about the centroidal axes x_{c} and y_{c}, respectively. d_{y} is the perpendicular distance between the centroidal axis x_{c} and the parallel axis denoted by x', while d_{x} is the perpendicular distance between y_{c} and y'. The x and y axes in this case serve as reference axes for finding the centroidal location of the area.

The parallel-axis theorem also applies to the polar moment of inertia

(7) |

where J_{o} is the polar moment of inertia about point o, J_{c} is the polar moment of inertia about the centroid, and d^{2} = d_{x}^{2} + d_{y}^{2}.

Similarly, the product of inertia with respect to x'y' axes can be found using the parallel-axis theorem as

(8) |

The parallel-axis theorem is used in calculating the moments of inertia of a composite shape, one made up of a collection of elementary shapes. In that case, the moment of inertia of each elementary shape about its own centroidal axes are obtained first, then the corresponding parallel axis terms are added to determine the moments of inertia of the composite area. This procedure is demonstrated in the following example.