Radius of Gyration

For a planar distribution of mass rotating about some axis in the plane of the mass, the radius of gyration is the distance from the axis that all mass can be concentrated to obtain the same mass moment of inertia. Thus, the radius of gyration is the ``equivalent distance'' of the mass from the axis of rotation. In this context, gyration can be defined as rotation of a planar region about some axis lying in the plane.

For a bar cross-section with area $S$ , the radius of gyration is given by

$$R_g = \sqrt{\frac{I_S}{S}} \protect$$ (B.11)

where $I_S$ is the area moment of inertia (§B.4.8) of the cross-section about a given axis of rotation lying in the plane of the cross-section (usually passing through its centroid):

$$I_S = \int_S R^2 dS,$$

where $R$ denotes the distance of the differential area element $dS$ from the axis of gyration.