Machinery's Handbook, 31st Edition
Radius of Gyration
235
Radius of Gyration The radius of gyration with reference to an axis is that distance from the axis at which the entire mass of a body may be considered as concentrated, the moment of inertia, mean while, remaining unchanged. If W is the weight of a body; J M its moment of inertia with respect to some axis; and k o the radius of gyration with respect to the same axis, then: k W J g J g Wk and o M M o 2 = = When using metric SI units, the formulas are: = where k o = the radius of gyration in meters, J M = moment of inertia in kilogram- meter 2 (kg·m 2 ), and M = mass in kilograms. To find the radius of gyration of an area, such as for the cross section of a beam, divide the moment of inertia of the area by the area and extract the square root. k M J J Mk and M o 2 o M = When the axis, the reference to which the radius of gyration is taken, passes through the center of gravity, the radius of gyration is the least possible and is called the principal radius of gyration. If k is the radius of gyration with respect to such an axis passing through the center of gravity of a body, then the radius of gyration k o with respect to a parallel axis at a distance d from the gravity axis is given by: k k d o 2 2 = + Tables of radii of gyration for various bodies and axes follows. Formulas for Radius of Gyration Bar of Small Diameter:
l
A A
. l 05773 3
. l 02886 12 1 2
l
k
l
k
l
1 2 2 = =
= =
A
k
2
k
k
k Axis at end
Axis at center
Bar of Small Diameter Bent to Circular Shape:
A
A
k
A
k r k r 2 2 = =
. r 07071 2
k
r
1 2 2 = =
r
k
r
k
A
A
Axis, a diameter of the ring
Axis through center of ring
Parallelogram (Thin Flat Plate):
. h 05773 3
k
h
. h 02886 12 1 2
k
h
1 2 2 = =
h k
= =
A
A
h k
k
2
A
A
k
Axis at mid-height
Axis at base
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