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Mechanics of Materials

Topic: Area Moment of Inertia

Here, you can learn what an area moment of inertia (also known as second moment of area, or second-order area moment) is, where it finds application in engineering mechanics, and how to calculate it.

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1.2.2 Definition of the Polar Area Moment of Inertia

The polar area moment of inertia is a geometric quantity used in engineering mechanics to describe a surface's ability to resist rotation about its axis of symmetry.

To calculate the polar area moment of inertia, the distance \(r\) of the infinitesimal area element \(\mathrm{d}A\) to the \(x\)-axis or the origin of the \(y\), \(z\)-coordinate system must be included in the formula. This is elegantly achieved by adding the axial area elements:

$$ \begin{align} \tag{1} I_p &= I_y + I_z \\[7pt] \tag{2} &= \int\limits_{(A)} z^2 \ \mathrm{d}A + \int\limits_{(A)} y^2 \ \mathrm{d}A \\[7pt] \tag{3} &= \int\limits_{(A)} (y^2 + z^2) \ \mathrm{d}A \end{align} $$

The relationship between \(y\), \(z\), and \(r\) is evident in Figure 6.1.2:

$$ \begin{align} \tag{4} r^2 &= y^2 + z^2 \end{align} $$

Therefore, the polar area moment of inertia is as follows:

$$ \begin{alignat}{3} I_p &= \int\limits_{(A)} r^2 \ \mathrm{d}A \end{alignat} $$

(6.3)

Since \(I_p\) is the sum of the always greater than or equal to zero \(I_y\) and \(I_z\), \(I_p\) must also always be greater than or equal to zero!

Note:

The \(I_p\) calculated here applies exclusively to circular cross-sections, i.e., only for circular and annular cross-sections!