Radius of gyration
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The radius of gyration of an area with respect to a particular axis is the square root of the quotient of the moment of inertia divided by the area. It is the distance at which the entire area must be assumed to be concentrated in order that the product of the area and the square of this distance will equal the moment of inertia of the actual area about the given axis.
In other words, the radius of gyration describes the way in which the total cross-sectional area is distributed around its centroidal axis. If more area is distributed further from the axis, it will have greater resistance to buckling. The most efficient column section to resist buckling is a circular pipe, because it has its area distributed as far away as possible from the centroid.
If a cross-section has more than one radius of gyration (it is not circular) it will tend to buckle around the axis with the smallest value.
The numerical value of the radius of gyration, r, is given by the following formula in which I is the moment of inertia and A is the area:
- <math>r = \sqrt{I \over A}<math>
The radius of gyration of a mass is similar except that the moment of inertia of the mass is involved. The numerical value of the radius of gyration of a mass, k, is given by the following formula in which I is the mass moment of inertia and m is the mass
- <math>k = \sqrt{I \over m}<math>