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

and
cos2θ =
p
∓
( I − I )/2
x
⎛ ( Ix
− Iy)
⎞
I
⎝
⎜
2 ⎠
⎟ +
y
2
2
xy
803
PRINCIPAL MOMENTS
OF INERTIA
sin2θ =±
p
I
xy
⎛ ( Ix
− Iy)
⎞
I
⎝
⎜
2 ⎠
⎟ +
2
2
xy
When these expressions are substituted in Equation (A.12), the principal moments of
inertia reduce to
I
p1, p2
=
I + I ⎛ I − I
±
2 ⎝
⎜
2
x y x y
2
⎞ 2
Ixy
⎠
⎟ + (A.14)
Equation (A.14) does not directly indicate which principal moment of inertia, either I p1 or
I p2 , is associated with the two values of θ that locate the principal axes [Equation (A.13)].
The solution of Equation (A.13) always gives a value of θ p between +45° and −45° (inclusive).
The principal moment of inertia associated with this value of θ p can be determined
from the following two-part rule:
• If the term I x − I y is positive, θ p indicates the orientation of I p1 .
• If the term I x − I y is negative, θ p indicates the orientation of I p2 .
The principal moments of inertia determined from Equation (A.14) will always be positive
values. In naming the principal moments of inertia, I p1 is the larger value algebraically.
The product of inertia of the element of area in Figure A.7 with respect to the x′ and y′
axes is
dIxy ′′ = xydA ′′ = ( xcosθ + ysin θ)( ycosθ − xsin θ)
dA
and the product of inertia for the area is
2 2 2 2
I = (cos θ − sin θ ) xydA + sin θ cos θ y dA − sin θ cos θ x dA
xy ′′
∫
A
2 2
= I (cos θ − sin θ) + I sinθcosθ − I sinθcosθ
xy x y
which is commonly rearranged to the form
∫
A
2 2
Ixy ′′ = ( Ix − Iy)sinθcos θ + Ixy(cos θ − sin θ)
(A.15)
An equivalent form of Equation (A.15) is obtained with the substitution of double-angle
trigonometric identities:
Ix
− Iy
Ixy
′′ = sin2θ
+ Ixy
cos2θ
(A.16)
2
The product of inertia I x′y′ will be zero for values of θ given by
2I
xy
tan2θ =−
I − I
x
y
∫
A