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Area Moments of Inertia Example

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Find the second moments of area (moments of inertia) of a rectangle about its centroidal x- and y-axes.

Concept

The moment of inertia of area (second moment) measures distribution of area about an axis. For a rectangle of base b and height h, about centroidal axes: I_{x} = \frac{b h ^{3}}{12} (about horizontal centroidal axis) and I_{y} = \frac{h b ^{3}}{12} (about vertical centroidal axis). The parallel-axis theorem gives moment about another axis: I = I_{c} + A d^{2} where d is the distance between axes. I_{x} = \frac{b h ^{3}}{12}, I_{y} = \frac{h b ^{3}}{12} (about centroid) I = I_{c} + A d^{2} Notation: I_{x}, I_{y} = moments of inertia about centroidal x and y, I_{c} = about centroid, A = area, d = distance.

A rectangle has width b = 4 in (horizontal) and height h = 6 in (vertical). Find: Moment of inertia about the horizontal centroidal axis, I_{x} . Moment of inertia about the vertical centroidal axis, I_{y} .

b = 4 in, h = 6 in

I_{x} = \frac{b h ^{3}}{12}, I_{y} = \frac{h b ^{3}}{12} (about centroid) I_{x} = \frac{( 4 ) ( 6 ^{3} )}{12} = \frac{864}{12} = 72 in^{4}

I_{y} = \frac{( 6 ) ( 4 ^{3} )}{12} = \frac{384}{12} = 32 in^{4}

I_{x} = 72 in^{4}, I_{y} = 32 in^{4} .

I_{x} = \frac{b h ^{3}}{12}, I_{y} = \frac{h b ^{3}}{12} (rectangle about centroid); I = I_{c} + A d^{2}