This is a property of a section. It is also a function of the second moment of area. The radius of gyration gives the **stiffness** of a section. It is based on the shape of the cross-section. Normally, we use this for compression members such as a column.

As shown in the diagram, the member bends in the thinnest plane.

Using the radius of gyration, we can compare the behavior of various structural shapes under compression along an axis. It can be used to predict buckling in a compression member such as a column.

**The Formula for the Radius of Gyration – r**

Where *I* = the second moment of area

*A* = cross-sectional area of the member

The unit of measurement is mm.

The **smallest** value of the radius of gyration is considered for the calculations of the structural stiffness of the member. That is the plane in which the member is most likely to fail or buckle.

Square or circular shapes have the same radius of gyration about any plane. There is no smallest value. Therefore, these sections are ideal selections for columns.

## Calculating the radius of gyration

The plan view of a column is shown below.

## First, we have to calculate the I value about *x-x* and *y-y* axes.

*I** _{xx}* = 33.3 x 10

^{6}mm

^{4}

*I*_{yy}_{ }= 2.08 x 10^{6} mm^{4}

*A* = cross sectional area = 50 mm x 200 mm = 10,000 mm^{2}

Substituting the value of *I _{xx}* and cross-sectional area

*A*in the above formula we can get r

_{xx}.

This is the value of the r about the *x-x* axis.

This is the value of the r about the *y-y* axis.

Since r_{yy} is smaller, probable failure occurs about the *y-y* plane.