10.3:
Radius of Gyration of an Area
The second moment of an area of an object about any axis can be estimated by concentrating its area into a thin strip parallel to the same axis.
The distance between this strip and the axis is defined as the radius of gyration. It is expressed as the square root of the ratio of the second moment of area about that axis and the area.
Similarly, the polar radius of gyration is expressed as the square root of the ratio between the polar moment and area.
Consider a rectangular beam with a cross-sectional area equal to the product of its height and width.
The second moment of area about a centroidal axis is obtained by multiplying the rectangle's width by the cube of its height and dividing it by twelve.
The radius of gyration is calculated using the area and second moment of area. The result equals the beam's height divided by the square root of twelve.
A higher radius of gyration indicates greater resistance against bending deformation due to an increased moment of inertia.
10.3:
Radius of Gyration of an Area
The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object's resistance against bending deformation, or stiffness. To accurately estimate the second moment of area along any axis, one needs to concentrate all areas associated with that object into a thin strip, which should be placed parallel to that particular axis.
As a result, the distance between this strip and the concerned axis can be determined by calculating its radius of gyration. The radius of gyration is represented as the square root of the ratio between the second moment of area and total area.
To illustrate this concept, consider a rectangular beam for which the area is equal to the product of the beam's height and width. In such a case, we can calculate its second moment of area about the centroidal axis through a simple formula: multiplying its width with the cube of its height and dividing the result by twelve.
The radius of gyration resulting from the product of area and the second moment of area gives a numerical value equal to the beam's height divided by the square root of twelve.
A higher radius of gyration directly equates to increased resistance against deformations due to greater moments of inertia occurring therein. As such, understanding and manipulating the radius of gyration is one way for mechanical engineers to create durable structures that can safely bear significant load capacities.
Suggested Reading
- F.P. Beer, E.R. Johnston, D.F. Mazurek, P.J. Cornwell, B.P. Self, Vector Mechanics For Engineers Statics and Dynamics Engineering Mechanics Statics, Mc Graw-Hill Education. Pp. 490-491
- R. C. , Hibbeler Engineering Mechanics Statics, Pearson. Pp. 531