Radius of Gyration of an Area


Equations
(Eq1)    
kx
Ix
A
radius of gyration of an area with respect to the x-axis
(Eq2)    
ky
Iy
A
radius of gyration of an area with respect to the y-axis
(Eq3)    
kO
JO
A
radius of gyration of an area about the origin
(Eq4)    
kO2 = kx2 + ky2
equation relating radii of gyration


Nomenclature
Ixmoment of inertia with respect to the x-axis
Iymoment of inertia with respect to the y-axis
JOpolar moment of inertia
Aarea


Details

Consider an area A which has a moment of inertia Ix with respect to the x-axis as shown.



Imagine that the area is concentrated into a thin strip parallel to the x-axis as shown:



If the area A, thus concentrated, is to have the same moment of inertia with respect to the x-axis, the strip should be placed at a distance kx from the x-axis, where kx is defined by the relation:

Ix = kx2A

Solving for kx:

(Eq1)    
kx
Ix
A

The distance kx is referred to as the radius of gyration of the area with respect to the x-axis. In a similar way, the radii of gyration ky and kO can be defined. The following image represents an area with the same moment of inertia with respect to the y-axis:



For ky :

Iy = ky2A

Solving for ky :

(Eq2)    
ky
Iy
A

Following is an image of a circular area centered about the x- and y-axes that has the same moment of inertia with respect to the x- and y-axes:



Similarly, for JO :

JO = kO2A

Solving for kO :

(Eq3)    
kO
JO
A

If Eq2 from the Polar Moment of Inertia lesson is rewritten in terms of the radii of gyration, it is found that:

(Eq4)    
kO2 = kx2 + ky2