Area Moment Of Inertia

Area moment of inertia also known as second area moment or 2nd moment of area is a property of a two-dimensional plane shape where it shows how its points are dispersed in an arbitrary axis in the cross-sectional plane. This property basically characterizes the deflection of the plane shape under some load.

Area moment of inertia is usually denoted by the letter I for an axis in a plane. It is also denoted as J when the axis lies perpendicular to the plane. The dimension unit of second area moment is Length to the power of four which is given as L4. If we take the International System of Units, its unit of dimension is meter to the power of four or m4. If we take the Imperial System of Units it can be inches to the fourth power, in4.

We will come across this concept in the field of structural engineering often. Here the area moment of inertia is said to be the measure of the flexural stiffness of a beam. It is an important property that is used to measure the resistance offered by a beam to bending or in calculating a beam’s deflection. Here we have to look at two cases.

  • First, a beam’s resistance to bending can be easily described or defined by the planar second moment of area where the force lies perpendicular to the neutral axis.
  • Secondly, the polar second moment of area can be used to determine the beam’s resistance when the applied moment is parallel to its cross-section. It is basically the beams ability to resist torsion.

Area Moment Of Inertia Formulas

If we take an arbitrary shape R, its second moment of area in connection with an arbitrary axis BB’ is given as;

JBB’ = p2dA

Here,

dA = differential area,

p = distance from the axis BB’ to dA.

If we consider the second moment of area for the x-axis then it is given as;

Ix = Ixx = y2 dx dy

Meanwhile, the “product” moment of area is defined by

Ixy = xy dx dy

However, while calculating the area moment of inertia students should mainly keep the following formula in mind:

Ix = y2dxdy

And

Iy = x2dxdy

Determining Area Moment of Inertia

Parallel Axis Theorem

We can make use of the parallel axis theorem to determine the area moment of inertia of a rigid body that can lie in any parallel axis. Here, we calculate the second area moment of a particular shape with respect to an x’ axis that is different from the centroidal axis of the shape.

Ix = Ix + Ad2

A = area of the shape

D = the perpendicular distance between the x and x’ axes.

Perpendicular Axis Theorem

The perpendicular axis theorem is often used in defining or determining the polar moment of inertia of a rigid object that is present in a certain plane. The determination is done with respect to the perpendicular axis. Polar moment of area is defined in terms of two area moments of inertia. Normally it is given as;

IZ = Ix + Iy

Meanwhile, in order to calculate the area moment of inertia of composite bodies, each part of the body should be divided into simpler shapes. Then we add the moment of inertia of each of its segments about the x-axis together. Here, all moments of inertia should be taken from the same axis.

Area Moments Of Inertia For Some Common Shapes

Here is a list of area moments of inertia of some shapes.

Shape/ Figure Description of the Axis Area moments of inertia
Rectangle Centroid along the Cartesian axis \(\begin{bmatrix} \frac{1}{2}ab^{3} & 0\\ 0 & \frac{1}{2}a^{3}b \end{bmatrix}\)
Square centroid along the Cartesian axis \(\begin{bmatrix} \frac{1}{12}a^{4} & 0\\ 0 & \frac{1}{12}a^{4}b \end{bmatrix}\)
Disk centroid \(\begin{bmatrix} \frac{1}{4}\Pi a^{4} & 0\\ 0 & \frac{1}{4}\Pi a^{4}b \end{bmatrix}\)
Ellipse centroid \(\begin{bmatrix} \frac{1}{4}\Pi ab^{3} & 0\\ 0 & \frac{1}{4}\Pi a^{3}b \end{bmatrix}\)
Annulus centroid \(\begin{bmatrix} \frac{1}{4}\left (a^{4} – b^{4} \right )\Pi & 0\\ 0 & \frac{1}{4}\left (a^{4} – b^{4} \right )\Pi \end{bmatrix}\)