Area Moment of Inertia

The area moment of inertia is a property of a two-dimensional plane shape which characterizes its deflection under loading. It is also known as the second moment of area or second moment of inertia. The area moment of inertia has dimensions of length to the fourth power. Unfortunately, in engineering contexts, the area moment of inertia is often called simply "the" moment of inertia even though it is not equivalent to the usual moment of inertia (which has dimensions of mass times length squared and characterizes the angular acceleration undergone by a solids when subjected to a torque).

The second moment of area about the about the -axis is defined by

(1)

while more generally, the "product" moment of area is defined by

(2)

Here, the positive sign convention is used (e.g., Pilkey 2002, p. 15).

More generally, the area moment of inertia tensor is given by

			(3)
			(4)

by analogy with the moment of inertia tensor, which has negative signs on the off-diagonal elements and, unlike the moment of inertia tensor, is not expressed in term of mass of the lamina.

For a closed lamina of uniform density with boundary specified by for and the lamina on the left as the curve is traversed, Green's theorem can be used to compute the components of the area moment of inertia tensor as

			(5)
			(6)
			(7)

The following table summarizes some area moments of inertia for some common shapes.

shape	axis
annulus	centroid
disk	centroid
ellipse	centroid
half-disk	along lower boundary
hexagon	diameter
pentagon	axis through center and vertex
quarter-disk	Cartesian axis
rectangle	centroid along Cartesian axis
square	centroid along Cartesian axis

The area moments of inertia about axes along an inradius and a circumradius of a regular polygon with sides (for ) are given by

			(8)
			(9)
			(10)
			(11)

(Roark 1954, p. 70).