Beams - Mechanics of Structures: Variational and Computational Methods - page 734

FIGURE 12.6
Representation of the cross-sectional characteristics for particular axes.

These properties are displayed in Fig. 12.6b. To understand the manipulations needed to
move from Eqs. (12.18a) and (12.18b) to Eq. (12.18c), observe that

=

A (

− )( − )

I y z y y C z z C dA

y C z C = yz dA − zdA − ydA + y C z C dA

A A A A

I y
A I z

I z
A I y

I z
A

I y
A A

= − − +

I yz

I y I z
A
If the y, z axes are rotated through an angle

= I yz −

α until reaching y, z such that

I y z = 0 (12.19a)

then these are referred to as centroidal principal axes. The condition I y z = 0 leads to the

familiar formula

y = y cos α + z sin α

2 I y z
I y y

tan 2 α = (12.19b)

− I z z z = z cos α − y sin α

and the principal moments of inertia

I y y
I zz

1
2 (

1
2

= + ) ± ( − )

2

+

4 I y z

I y y I z z I y y I z z (12.19c)

Suppose the origin of the y, z axes is shifted from the centroid to the origin of the y ∗ , z ∗ axes
for which

I y ∗ ω ∗ = I z ∗ ω ∗ =

0 (12.20a)