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

Using the Pythagorean theorem, distance r is related to distances x and y by r 2 = x 2 + y 2 .
Accordingly, Equation (A.5) can be expressed as
and thus,
∫
J = 2
r dA = 2
( x + 2
y ) dA = 2 2
x dA + y dA
A
∫
A
∫
A
∫
J = I + I
(A.6)
y
x
A
795
MOMENT OF INERTIA
FOR AN AREA
Notice that the x and y axes can be any pair of mutually perpendicular axes intersecting
at O.
From the definitions given in Equations (A.3), (A.4), and (A.5), moments of inertia
are always positive terms that have dimensions of length raised to the fourth power (L 4 ).
Common units are mm 4 and in. 4 .
Area moments of inertia for several common plane shapes are summarized in Table A.1.
parallel-Axis Theorem for an Area
When the moment of inertia of an area has been determined with respect to a given axis, the
moment of inertia with respect to a parallel axis can be obtained by means of the parallel-axis
theorem, provided that one of the axes passes through the centroid of the area.
The moment of inertia of the area in Figure A.3 about the b reference axis is
y
The integral
2 2 2
I = ( y + d) dA = y dA + 2d ydA + d dA
b
∫
A
2
= I + 2d ydA + d A
x
∫
A
∫
A
∫ ydA is simply the first moment of area A about the x axis. From Equation (A.1),
A
∫
A
∫
A
(a)
O
d r
c
dA
Area
A
y
d
x
∫ ydA = yA
A
If the x axis passes through the centroid c of the area, then y _ = 0 and Equation (a)
reduces to
2
I = I + d A
(A.7)
b
c
where I c is the moment of inertia of area A about the centroidal axis that is parallel to the
reference axis (i.e., the b axis in this instance), and d is the perpendicular distance between
the two axes. In a similar manner it can be shown that the parallel-axis theorem is also
applicable for polar moments of inertia:
Jb = Jc + dr 2 A
(A.8)
The parallel-axis theorem states that the moment of inertia for an area about an axis is equal
to the area’s moment of inertia about a parallel axis passing through the centroid plus the
product of the area and the square of the distance between the two axes.
B
FIGURE A.3
Reference axis
b
composite Areas
It is often necessary to calculate the moment of inertia for an irregularly shaped area. If
such an area can be subdivided into a number of simple shapes such as rectangles, triangles,
and circles, then the moment of inertia for the irregular area can be conveniently