Aluminium Design and Construction John Dwight

10.5.3 Evaluation of warping
When the ends of a thin-walled twisted member are free to warp (i.e.
under no longitudinal restraint, Figure 10.7(a)), each point in the crosssection
undergoes a small longitudinal movement z known as the
‘warping’, which is constant in value down the length of the member,
z is measured relative to an appropriate transverse datum plane, and
is related to the rate of twist � by the equation:
(10.24)
in which w, known as the unit warping, is purely a function of the
section geometry. In order to locate S and determine H, we have to
study how w varies around the section.
Figure 10.15 shows an element LN forming part of the cross-section
of such a member. The twist is assumed to be in a right-handed spiral
(as in Figure 10.7(a)), and the warping movement z (and hence w) is
measured positive out of the paper. It can be shown that the increase
dw in w, as we proceed across the element in the direction of flow from
L to N, is given by:
�w=bd (10.25)
where b=element width, and d=perpendicular distance from the shearcentre
S to the median line of the element, taken positive when the
direction of flow is clockwise about S, and negative when the flow is
anti-clockwise.
For any element r, we define a quantity w m equal to the value of w
at its midpoint M. This is given by:
wm =wo +ws (10.26)
where wo =value of w at the start-point O, and ws =increase in w as we
proceed from O to M given by:
(10.27)
Figure 10.15 Plate element in a member twisted about an axis through the shear-centre S.
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