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Key
Concept: You may wish to find the moment of
inertia of an area about axes
parallel but different from the centroidal axes.
The parallel axis theorem provides
a method to accomplish this transformation.
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In
a Nutshell: Use the parallel
axis theorem to transform the moment of inertia through
the centroid to any other
parallel axis. Let x , y, z
be any axes parallel to the centroidal
axes x1 , y1, and z1
. Then the moments of inertia Ixx
, Iyy, or Izz
equal the moments of inertia calculated
about the centroidal axes plus the area times the
distance between the parallel axes as shown in the table below.
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Ixx =
Ixx1 + A ybar2 |
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Iyy =
Iyy1 + A xbar2 |
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Izz =
Izz1 + A d2 |
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In the figure below C refers
to the centroid of the area, A is the total area, x-y-z
and
x1˗y1˗z1 are parallel axes, x1˗y1˗z1 are axes through the centroid, C, of the area A, xbar ,
ybar and d are
the distances between the parallel axes.
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The parallel axis theorem is frequently applied
to composite areas once the area, centroid,
and distances between the parallel axes are
known. Once again, if the area
happens to be
a void, then the negative value applies to the moment
of inertia calculation.
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Click here for an example.
Click here to continue with a discussion of
composite areas.
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