10.7: Products of Inertia

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Key Questions

Why do we need to quantify the product of inertia for beams?
Why is the product of inertia of a symmetrical cross section zero?

The product of inertia is another integral property of area, and is defined as

\begin{equation} I_{xy} = \int_A {x}{y}\ dA\text{.}\tag{10.7.1} \end{equation}

The parallel axis theorem for products of inertia is

\begin{equation} I_{xy} = \bar{I}_{x'y'} + A \bar{x}\bar{y}\text{.}\tag{10.7.2} \end{equation}

Unlike the rectangular moments of inertia, which are always positive, the product of inertia may be either positive, negative, or zero, depending on the object's shape and the orientation of the coordinate axes. The product of inertia will be zero for symmetrical objects when a coordinate axis is also an axis of symmetry.

If the product of inertia is not zero it is always possible to rotate the coordinate system until it is, in which case the new coordinate axes are called the principle axes. When the coordinate axes are oriented in the principle directions, the centroidal moments of inertia are maximum about one axis and minimum about the other, but neither is necessarily zero. The principle directions determine the best way to orient a beam to for maximum stiffness, and how much asymmetrical beams, like channels and angles, will twist when a load is applied.