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

804
gEOMETRIC PROPERTIES
OF AN AREA
Notice that this expression is the same as Equation (A.13), which gives the orientation of
the principal axes. Consequently, the product of inertia is zero with respect to the principal
axes. Since the product of inertia is zero with respect to any axis of symmetry, it follows
that any axis of symmetry must also be a principal axis.
ExAmpLE A.6
40 mm
Determine the principal moments of inertia for the zee shape considered
in Example A.4. Indicate the orientation of the principal axes.
6 mm
y
x
6 mm
30 mm
6 mm
Plan the Solution
Using the moments of inertia and the product of inertia determined in
Examples A.3 and A.4, Equation (A.14) will give the magnitudes of
I p1 and I p2 , and Equation (A.13) will define the orientation of the principal
axes.
SolutioN
From Examples A.3 and A.4, the moments of inertia and the product
of inertia for the zee shape are
40 mm
40 mm
p2 axis
I
I
I
x
y
xy
= 170,460 mm
= 203,260 mm
= 146,880 mm
4
4
4
The principal moments of inertia can be calculated from Equation (A.14):
6 mm
y
x
41.8°
6 mm
30 mm
6 mm
I
p1, p2
Ix + Iy ⎛ Ix − Iy
= ±
2 ⎝
⎜
2
170,460 + 203,260
=
2
= 186,860 ± 147,793
2
⎞
I
⎠
⎟ +
±
2
xy
⎛ 170,460 − 203,260⎞
(146,880)
⎝
⎜
2 ⎠
⎟ +
2
2
40 mm
p1 axis
= 335,000 mm ,39,100 mm
4 4
The orientation of the principal axes is found from Equation (A.13):
Ans.
2Ixy
2(146,880)
tan2θ p =− =−
I − I 170,460 − 203,260
x
y
∴ 2θ p = 83.629°
= 8.9561
Therefore, θ p = 41.8°. Since the denominator of this expression (i.e., I x − I y ) is negative,
the value obtained for θ p gives the orientation of the p2 axis relative to the x axis. The
positive value of θ p indicates that the p2 axis is rotated 41.8° counterclockwise from the
x axis.
The orientation of the principal axes is shown in the sketch.