Aluminium Design and Construction John Dwight

where A E =area of the element (normally taken positive), and Y E =distance
of the element’s centroid E from XX taken positive above XX and
negative below.
The summations are performed for all the elements. The correct position
for the neutral axis is obtained, however XX is chosen, provided the
sign of Y E is taken correctly. A negative value of Y G would indicate that
the neutral axis lies below XX. There are then two possible formulae
for I xx which both give the same answer:
Either
or
(10.9)
(10.10)
Expression (10.9) is the one taught to students. Expression (10.10) is
more convenient for complex sections because a small design change to
one element does not invalidate the quantities for all the others.
10.3.2 Inertias for a section with no axis of symmetry
When the section is skew-symmetric or asymmetric (Figure 10.5) we usually
need to know the inertias about the principal axes (Gu, Gv), known as
the principal inertias (Iuu , Ivv ). The procedure for finding the orientation
of the principal axes and the corresponding inertias is as follows:
1. Select convenient preliminary axes Gx and Gy through the centroid
G, with Gy directed 90° anti-clockwise from Gx.
2. Calculate the inertia lxx . using equation (10.9) or (10.10). Use an
equivalent expression to calculate Iyy .
3. Calculate the product of inertia Ixy (Section 10.3.3).
4. The angle a between the major principal axis Gu and the known axis
Gx is then given by:
(10.11)
Figure 10.5 Elastic bending, principal axes.
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