### 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 , the radius of gyration is given by

where 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):

#### Rectangular Cross-Section

For a rectangular cross-section of height and width , area , the area moment of inertia about the horizontal midline is given by*e.g.*, rectangular), have at least two radii of gyration. A circular cross-section has only one, and its radius of gyration is equal to half its radius, as shown in the next section.

#### Circular Cross-Section

For a circular cross-section of radius , Eq.(B.11) tells us that the squared radius of gyration about any line passing through the center of the cross-section is given by*tube*in which the mass of the cross-section lies within a circular

*annulus*having inner radius and outer radius , the radius of gyration is given by

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