Learning Goal:

To section a composite shape into simple shapes so the

moment$-$of$-$inertia tables and the parallel$-$axis theorem can

be used to find the moment of inertia of the composite

shape.

Tables listing moments of inertia cannot contain the formula

for every possible shape that can be built. However, many

shapes can be sectioned into pieces that have entries in

the tables. Consider the shape in the following figure,

$($Figure $1)$

This shape can be sectioned into three rectangles as so$,$

$($Figure $2)$

Because the moment of inertia of a rectangle is a well$-$

known formula, the moment of inertia for the full composite

can be found by calculating the individual moments of

inertia of the three sections and adding them. Because the

centroids of the vertical sections do not align with the $x$ or

$y$ axes, we will need to use the parallel$-$axis theorem to

calculate the moments of inertia of those sections. The

moment of inertia for a cross$-$section where the centroid of

Figure

Part A $-$ Moment of Inertia of a Composite Beam about the $x$ axis

For the built$-$up beam shown below, calculate the moment of inertia about the $x$ axis.

$($Figure $7)$

The dimensions are $d_1=6\mathrm{.}5in,d_2=13\mathrm{.}5in,d_3=7\mathrm{.}5in,$ and $t=0\mathrm{.}80in.$

Express your answer to three significant figures and include the appropriate units.

View Available Hint$($s$)$

$I_x=$

Part B $-$ Moment of Inertia of a Composite Beam about the $y$ axis

For the beam from Part A $($shown again here for reference$),$ calculate the moment of inertia about the $y$ axis.

$($Figure $7)$

The dimensions are $d_1=6\mathrm{.}5in,d_2=13\mathrm{.}5in,d_3=7\mathrm{.}5in,$ and $t=0\mathrm{.}80in.$

Express your answer to three significant figures and include the appropriate units.