Radius of Gyration

Area Radius of Gyration Definition

The Radius of Gyration of an Area about a given axis is a distance k from the axis. At this distance k an equivalent area is thought of as a line Area parallel to the original axis. The moment of inertia of this Line Area about the original axis is unchanged.

The Radius of Gyration kx of an Area (A) about an axis (x) is defined as:
		equ. (1)
	kx	equ. (2)
Where Ix is the Moment of Inertia about the axis (x), and A is the area. If no axis is specified the centroidal axis is assumed.
Using the Perpendicular Axis Theorem and equ. (1) from above it can be shown that:
		equ. (3)

Mass Radius of Gyration Definition

The Radius of Gyration of a Mass about a given axis is a distance k from the axis. At this distance k an equivalent mass is thought of as a Point Mass. The moment of inertia of this Point Mass about the original axis is unchanged.

The Radius of Gyration kxx of a Mass (m) about an axis (x) is defined as:
		equ. (4)
	k	equ. (5)
Where I is the Moment of Inertia about the axis (x), and m is the mass.

Radius of Gyration Concept

Figure 1
It is possible for two EQUAL areas to have equal MOMENT'S OF INERTIA about the same axis while having different dimensions.
Figure 1 contains two rectangles. Setting the Area's and Moment's of Inertia equal, the following relationships can be found.
		equ. (1)
		equ. (2)
	h1 and b1 are constants.
	b2 depends on h2
d(h2) is:
		equ. (3)
In equ. (3) there is a limited range of h2 that yields a real solution. d is maximized when h2 goes to zero. This corresponds to an infinitely long, infinitely thin rectangle. The maximum value of d is the Radius of Gyration.
		equ. (4)
This is equivalent to using the general Radius of Gyration equation on a rectangle.
The mass Radius of Gyration is the same concept just applied to a mass.