Aluminium Design and Construction John Dwight

Figure 10.2 Asymmetric plastic bending.
(C) from the tensile material (T), will not be parallel to the axis mm
about which M acts. The object is to determine S m , the plastic modulus
corresponding to a given inclination of mm.
Consider first a bisymmetric section having axes of symmetry Ox
and Oy, with mm inclined at � to Ox (Figure 10.2(a)). The inclination of
the neutral axis nn can be specified in terms of a single dimension w as
shown. Before we can obtain the plastic modulus S m for bending about
mm, we must first find the corresponding orientation of nn (i.e. determine
w). To do this, we split up the area on the compression side into convenient
elements, and obtain expressions in terms of w for the plastic moduli
S x and S y about Ox and Oy, as follows:
(10.3)
where A E =area of an element, and x E , y E =coordinates of the element’s
centroid E referred to the axes Ox and Oy.
In each equation, the actual summation is just performed for the
compression material (i.e. for elements lying on one side only of nn).
The correct inclination of nn, corresponding to the known direction of
mm, is then found from the following requirement (which provides an
equation for w):
S y =S x tan �. (10.4)
Having thus evaluated w, the required modulus S m is obtained from
(10.5)
The procedure is similar for a skew-symmetric section (Figure 10.2(b)),
where a single parameter w is again sufficient to define the neutral axis
nn. In this case, the axes Ox, Oy are selected having any convenient
orientation. The above equations (10.3)–(10.5) hold good for this case,
and may again be used to locate nn and evaluate S m .
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