Aluminium Design and Construction John Dwight

10.3.3 Product of inertia
The product of inertia I xy , appearing in Section 10.3.2, can be either
positive or negative. It may be calculated using the following expression
in which the summations are for all the elements of the section:
(10.13)
where I Exy =an element’s ‘own’ product of inertia, referred to parallel
axes Ex and Ey through its centroid E, A E =area of the element (taken
positive), and X E , y E= coordinates of E referred to the main axes Gx and
Gy, with strict attention to signs.
Table 10.1 includes expressions for I Exy for the common element shapes
shown in Figure 10.6. In applying these, it is important to realize that
I Exy can be either positive or negative, depending on which way round
the element is drawn. As drawn in the Figure, it is positive in every
case. But if the element is reversed (left to right) or inverted, it becomes
negative. If it is reversed and inverted, it becomes positive again.
10.3.4 Inertia of the effective section
The elastic section properties should when necessary be based on an
effective section, to allow for HAZ softening, local buckling or holes.
There are two possible methods for so doing.
In method 1, we take effective thicknesses in the nominal HAZ regions
(Section 6.6.1) and effective stress blocks in slender elements (Chapter
7), I being calculated accordingly. The HAZ softening factor kz is put
equal to kz2. If the section is semi-compact, there are no slender elements
and thus no deductions to be made for local buckling.
In method 2, we treat the effective section as being composed of all
the actual elements that constitute the gross section, on which are then
super-imposed appropriate negative elements (i.e. ‘lost areas’). In any
HAZ region, the lost area is Az (1-kz2 ). In a slender element it consists of
the ineffective material between effective stress blocks (internal elements),
or at the toe (outstands). I is obtained basically as in Section 10.3.1 or
10.3.2, combining the effect of all the positive elements (the gross section)
with that of the negative ones (the lost areas), and taking AE , IExx , IEyy and IExy with reversed sign in the case of the latter. In considering the
lost area due to softening at a longitudinal weld, there is no need to
locate the HAZ’s centroid with great precision.
Method 2 tends to be quicker for sections with small longitudinal
welds, since it only involves a knowledge of Az without having to find
the actual distribution of HAZ material.
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