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

y
y′
y
O
x
x
dA
θ
θ
cos θ
x′
x
y′
sin θ
y sin θ
y cos θ
x′
x
The sets of axes for which the moments of inertia are maximum
and minimum are called the principal axes of the area through point
O and are designated as the p1 and p2 axes. The moments of inertia
with respect to these axes are called the principal moments of inertia
for the area and are designated I p1 and I p2 . There is only one set of
principal axes for any area unless all axes have the same second
moment, such as the diameters of a circle.
A convenient way to determine the principal moments of inertia
for an area is to express I x′ as a function of I x , I y , I xy , and θ, and then set
the derivative of I x′ with respect to θ equal to zero to obtain the value
of θ that gives the maximum and minimum moments of inertia. From
Figure A.7,
2 2
dI = y′ dA = ( ycosθ
− xsin θ)
dA
x′
FIGURE A.7
and thus,
2 2 2 2
I = cos θ y dA − 2sinθcosθ xy dA + sin θ x dA
x′
∫ ∫ ∫
A A A
2 2
= I cos θ − 2I sinθcosθ + I sin θ
x xy y
which is commonly rearranged to the form
2 2
Ix′ = Ix cos θ + Iysin θ − 2Ixy
sinθcosθ
(A.11)
Equation (A.11) can be written in an equivalent form by substituting the following doubleangle
identities from trigonometry:
to give
2
1
cos θ = (1 + cos2 θ)
2
2
1
sin θ = (1 − cos2 θ)
2
2sinθcosθ = sin 2θ
Ix + Iy Ix − Iy
Ix′ = + cos2θ
− Ixy
sin2θ
(A.12)
2 2
The angle 2θ for which I x′ is maximum can be obtained by setting the derivative of I x′ with
respect to θ equal to zero; thus,
from which
dI ′
dθ
I
=−(2)
− I
x x y
2
tan2θ p =−
I
sin2θ
− 2I
cos2θ
= 0
2I
x
xy
− I
xy
y
(A.13)
where θ p represents the two values of θ that locate the principal axes p1 and p2. Positive
values of θ indicate a counterclockwise rotation from the reference x axis.
Notice that the two values of θ p obtained from Equation (A.13) are 90° apart. The
principal moments of inertia can be obtained by substituting these values of θ p into Equation
(A.12). From Equation (A.13),
802