Area Moment of
inertia

The area moment of inertia is the second moment of area around a
given axis. For example, given the axis *O-O*
and the shaded area shown, one calculates the second moment of the area by
adding together _{} for all the elements
of area *dA* in the shaded area.

The area moment of inertia, denoted by *I*, can, therefore, be calculated from

_{}

If we have a rectangular coordinate system as shown, one can
define the area moment of inertial around the *x*-axis, denoted by *I _{x}*

_{}

The polar area moment of inertia, denoted by *J*_{O}, is the area moment of inertia about the *z*-axis given by

_{}

Note that since _{} one has the relation

_{}

The radius of gyration is the distance *k *away
from the axis that all the area can be concentrated to result in the same
moment of inertia. That is,

_{}

For a given area, one can define the radius of gyration around the
*x*-axis, denoted by *k*_{x}, the radius of gyration
around the *y*-axis, denoted by *k*_{y}, and the radius of
gyration around the *z*-axis, denoted
by *k*_{O}. These are
calculated from the relations

_{}

It can easily to show from _{} that

_{}

The parallel axis theorem is a relation between the moment of
inertia about an axis passing through the centroid and the moment of inertia
about any parallel axis.

Note that from the picture we have

_{}

Since _{} gives the distance of
the centroid above the *x*'-axis, and
since the this distance is zero, one must conclude that the integral in the
last term is zero so that the parallel axis theorem states that

_{}

where *x*' must pass
through the centroid of the area. In this same way, one can show that

_{}

In general, one can use the parallel axis theorem for any two
parallel axes as long as one passes through the centroid. As shown in the
picture, this is written as

_{}

where _{} is the moment of
inertia about the axis *O'-O' *passing
through the centroid, *I* is the moment
of inertia about the axis *O-O, *and *d* is the perpendicular distance between
the two parallel axis.

The moment of inertia of composite bodies can be
calculated by adding together the moment of inertial of each of its sections.
The only thing to remember is that all moments of inertia must be evaluated
bout the same axis. Therefore, for example,

_{}

To calculate the area moment of inertia of the composite body
constructed of the three segments shown, one evaluates the moment of inertial
of each part about the *x*-axis and
adds the three together.

ã Mehrdad Negahban and the University of Nebraska, 1996-2000.

All rights reserved

Copy and distribute freely for personal use only

Department of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526

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