Page 332 - Aircraft Stuctures for Engineering Student

9.5 Torsion of closed section beams 3 13

4
Fig. 9.33 Warping distribution produced by selecting an arbitrary origin for s.

Substituting in Eq. (vii) for wi2 and )vi3 from Eqs (iv) and (vi) respectively and
evaluating gives

(viii)

Subtracting this value from the values of w:(= 0) and d’(= -T(b/tb - a/tU)/4abG)
we have

as before. Note that setting wo = 0 in Eq. (i) implies that wo, the actual value of
warping at the origin for s, has been added to all warping displacements. This
value must therefore be subtracted from the calculated warping displacements (i.e.
those based on an arbitrary choice of origin) to obtain true values.
It is instructive at this stage to examine the mechanics of warping to see how it
arises. Suppose that each end of the rectangular section beam of Example 9.7 rotates
through opposite angles 8 giving a total angle of twist 28 along its length L. The
corner 1 at one end of the beam is displaced by amounts a8/2 vertically and b8/2
horizontally as shown in Fig. 9.34. Consider now the displacements of the web and
cover of the beam due to rotation. From Figs 9.34 and 9.35(a) and (b) it can be
seen that the angles of rotation of the web and the cover are, respectively
4b = (ae/2)/(~/2) ae/L
=
and
4, = (b8/2)/(L/2) = bB/L

The axial displacements of the corner 1 in the web and cover are then
a be
b a8
__ __
2L’ 2L
respectively, as shown in Figs 9.35(a) and (b). In addition to displacements produced by
twisting, the webs and covers are subjected to shear strains ’yb and corresponding to