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

A.3 product of Inertia for an Area
The product of inertia dI xy of the area element dA in Figure A.4 with respect to the x and y
axes is defined as the product of the two coordinates of the element multiplied by the area
of the element. The product of inertia of the total area A is thus
799
PROduCT OF INERTIA
FOR AN AREA
I
xy
∫ A
= xydA
(A.9)
The dimensions of the product of inertia are the same as those of the moment of inertia
(i.e., length units raised to the fourth power). Whereas the moment of inertia is always
positive, the product of inertia can be positive, negative, or zero.
The product of inertia for an area with respect to any two orthogonal axes is zero
when either of the axes is an axis of symmetry. This statement can be demonstrated by
Figure A.5 in which the area is symmetric with respect to the x axis. The products of inertia
of the elements dA and dA′ on opposite sides of the axis of symmetry will be equal in magnitude
and opposite in sign; thus, they will cancel each other in the summation. The resulting
product of inertia for the entire area will be zero.
The parallel-axis theorem for products of inertia can be derived from Figure A.6 in
which the x′ and y′ axes pass through the centroid c and are parallel to the x and y axes. The
product of inertia with respect to the x and y axes is
I
xy
=
∫
∫
A
xydA
= ( x + x′ )( y + y′
) dA
A
c
∫
c
= xy dA + x y′ dA + y x′ dA + xy ′′ dA
c c c c
A
A
∫
∫
A
∫
A
The second and third integrals in the preceding equation are zero since x′ and y′ are centroidal
axes. The last integral is the product of inertia of the area A with respect to its centroid.
Consequently, the product of inertia is
Ixy = Ixy ′′ + xyA c c
(A.10)
The parallel-axis theorem for products of inertia can be stated as follows: The product of
inertia for an area with respect to any two orthogonal axes x and y is equal to the product of
inertia of the area with respect to a pair of centroidal axes parallel to the x and y axes added
to the product of the area and the two centroidal distances from the x and y axes.
The product of inertia is used in determining principal axes of inertia, as discussed
in the following section. The determination of the product of inertia is illustrated in
the next two examples.
y
Area A
dA
y
dA
y y′
c
dA
Area
A
y′
x′
y c
y
y
O
x
x
dA′
x
O
x c
x
x′
x
FIGURE A.4
FIGURE A.5
FIGURE A.6