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The material and content of Engineering Archives is designed to be subject to change and alteration at all times. Therefore, it may not be assumed that any presented material herein is static. The material is constantly subject to change for the purpose of maximizing the ability of the learner to comprehend and absorb the material. The changing and alteration is conducted based on user feedback. Therefore, effective changes cannot take place without the user feedback. Please go to the forum for questions and feedback. See more…

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Radius of Gyration of an Area

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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

Ix	moment of inertia with respect to the x-axis
Iy	moment of inertia with respect to the y-axis
JO	polar moment of inertia
A	area

Details

Consider an area A which has a moment of inertia I_x 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 k_x from the x-axis, where k_x is defined by the relation:

I_x = k_x²A

Solving for k_x:

(Eq1)

kx =  √ Ix A

The distance k_x 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 k_y and k_O can be defined. The following image represents an area with the same moment of inertia with respect to the y-axis:



For k_y :

I_y = k_y²A

Solving for k_y :

(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 J_O :

J_O = k_O²A

Solving for k_O :

(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

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The material and content of Engineering Archives is designed to be subject to change and alteration at all times. Therefore, it may not be assumed that any presented material herein is static. The material is constantly subject to change for the purpose of maximizing the ability of the learner to comprehend and absorb the material. The changing and alteration is conducted based on user feedback. Therefore, effective changes cannot take place without the user feedback. Please go to the forum for questions and feedback. See more…
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