Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

mecmovies
ExAmpLES
A.2 Animated example of the centroid calculation procedure
for a tee shape.
A.3 Animated example of the centroid calculation procedure
for a U-shape.
A.2 moment of Inertia for an Area
The term moment of inertia is
applied to the second moment
of area because of its similarity
to the moment of inertia of the
mass of a body.
The terms∫x dA and ∫y dA appear in the definition of a centroid [Equation (A.1)], and these
terms are called first moments of area about the y and x axes, respectively, because x and y
are first-order terms. In mechanics of materials, several equations are derived that contain
integrals of the form ∫x 2 dA and ∫y 2 dA, and these terms are called second moments of area
because x 2 and y 2 are second-order terms. However, the second moment of area is more
commonly called the moment of inertia of an area.
In Figure A.2, the moment of inertia for area A about the x axis is defined as
I
x
∫
2
= y dA
(A.3)
Similarly, the moment of inertia for area A about the y axis is defined as
A
y
Area A
dA
y
r
O x
FIGURE A.2
794
x
I
y
∫
2
= x dA
(A.4)
A
A second moment expression can also be stated for a reference axis that is normal to the
plane (such axes are called poles). In Figure A.2, the z axis that passes through origin O is
perpendicular to the plane of the area A. An integral expression called the polar moment of
inertia J can be written in terms of the distance r from the reference z axis to dA:
∫
2
J = r dA
(A.5)
A