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

806
gEOMETRIC PROPERTIES
OF AN AREA
I xy I x + I y
2
I x −I y
2
A
O
I p2
B′
C
2θ
p
A′
I p1
I xy
I x , I y
−I xy
B
I y
I x
FIGURE A.8 Mohr’s circle for moments of inertia.
horizontal axis, and products of inertia are plotted along the vertical axis. Moments of
inertia are always positive and are plotted to the right of the origin. Products of inertia
can be either positive or negative. Positive values are plotted above the horizontal axis.
The horizontal distance OA′ is equal to I x , and the vertical distance A′A is equal to I xy .
Similarly, horizontal distance OB′ is equal to I y and vertical distance B′B is equal to −I xy
(i.e., the algebraic negative of the product of inertia value, which can be either a positive
or a negative number). The line AB intersects the horizontal axis at C, and line AB is the
diameter of Mohr’s circle. Each point on the circle represents I x′ and I x′y′ for one particular
orientation of the x′ and y′ axes. As in Mohr’s circle for stress analysis, angles in
Mohr’s circle are double angles 2θ. Thus, all angles on Mohr’s circle are twice as large
as the corresponding angles for the particular area.
Since the horizontal coordinate of each point on the circle represents a particular value
of I x′ , the maximum and minimum moments of inertia are found where the circle intersects
the horizontal axis. The maximum moment of inertia is I p1 and the minimum moment of
inertia is I p2 . The center C of the circle is located at
I
C =
x
and the circle radius is the length of CA, which can be found from the Pythagorean theorem:
CA = R =
The maximum moment of inertia I p1 is thus
⎛ I
⎜
⎝
x
+ I
2
y
− I
2
y
2
⎞
⎟ + I
⎠
I + I ⎛ I − I
Ip1
= C + R = + ⎜
2 ⎝ 2
and the minimum moment of inertia I p2 is
2
xy
x y x y
I + I ⎛ I − I
Ip2
= C − R = −
2 ⎝
⎜
2
These expressions agree with Equation (A.14).
x y x y
2
⎞
⎟ + I
⎠
2
2
xy
⎞
I
⎠
⎟ +
2
xy