Inertia is simply a body’s property to always be in a continuous state of rest or continuous uniform motion until an external force acts upon it. Naturally, the given body must also show resistance when rotated. This measure of the resistance offered by the body to the angular acceleration is called Moment of Inertia or Mass Moment of Inertia.

What is the Area Moment of Inertia?

Imagine a two-dimensional plane body under loading. What sort of deflection will it go through? And more importantly, how do we measure it?

 The area moment of inertia is a body’s characteristic that helps us exactly with this. Area moment of inertia characterises the deflection of a plane body shape when an external load is applied. 

There are many names for the moment of inertia. Some of them are second moment of area, quadratic moment of area or second area moment. The area moment of inertia is essentially an area’s geometrical property that reflects how the points are distributed with respect to an axis. 

 The area moment of inertia is symbolised by an I or by a J. I is used when the axis lies in the plane of the body and J is used when the axis lies perpendicular to the body. However, the calculation remains the same which is by using multiple integrals over the body in the picture. The dimension of the area moment of inertia is length to the power of four.

Applications of Area Moment of Inertia

The area moment of inertia has important structural engineering applications, especially when calculating a beam’s deflection and the stress that a moment applied to the beam causes. 

 The area moment of inertia (planar) gives us an insight into how the beam is resisting when bending occurs due to an applied moment or a force. While the polar area moment gives us an insight into how the beam resists a torsional deflection.

Parallel Axis Theorem

The Huygens–Steiner theorem, also known as Parallel Axis Theorem, is used to determine the area moment of inertia of a rigid body about any given axis, provided that the object’s moment of inertia is calculated about a parallel axis passing through the object’s centre of gravity and the perpendicular distance between the axes is known.

Let’s see how:

Consider a body with an area A. For example, the body’s centroid or centre of gravity lies at C and the axis BB’ is the centroidal axis. AA’ is the axis around which we want to calculate the area moment of inertia, and the distance between the centroidal axis (BB’) and AA’ is d.

Now,

Ix = ∫ (dy + y’)2dA

= ∫ (dy)2 dA + 2 * ∫ (dy*y’) dA + ∫ (y’)2 dA

But, ∫ y’dA = 0

 This is because the centre of gravity lies on the x’ axis itself.

 Thus,

 Ix = ∫ (dy)2 dA + (y’)2 * ∫ dA

 Ix = Ix’ + A*(dy)2

Perpendicular Axis Theorem

For a two-dimensional planer body, the inertia about an axis perpendicular to the plane of the body is the addition of the moments of inertia of the body about the other two axes that are perpendicular to each other in the plane of the body where all three axes intersect each other.

From the figure, CC’ is the perpendicular axis around which we wish to calculate inertia. AA’ and BB’ are mutually perpendicular axes that intersect with CC’ in the plane of the body.

ICC’ = IAA’ + IBB’

Area Moments Of Inertia For Some Common Shapes

 Conclusion

The letter ‘I’ represents the ‘mass moment of inertia’ and ‘area moment of inertia.’ At times this might confuse you, but we can always figure it out by applying the formulas. 

The importance of the area moment of inertia is felt in the area of engineering mechanics, where calculating bendings and stresses in a beam is required.

To calculate the quadratic moment of area for complex areas, we must divide the given area into multiple simple areas or shapes. The quadratic moment of area would then be the sum of the quadratic moments of all other simpler shapes calculated about a common axis. The shapes that are absent like hollow shapes or holes are to be subtracted and not added to calculate the total quadratic moment of area of the given planar body. This means that the quadratic moment of area for the missing parts is treated as negative.