Page 331 - Aircraft Stuctures for Engineering Student

3 12 Open and closed, thin-walled beams

2

t a 1

Fig. 9.32 Arbitrary origin for s.

Suppose now that the origin for s is chosen arbitrarily at, say, point 1. Then, from
Fig. 9.32, So, in the wall 12 = q/t, and Aos = $sIb/2 = Slb/4 and both are positive.
Substituting in Eq. (i) and setting wo = 0

so that +vi2 varies linearly from zero at 1 to
b
I T -+- a U
wp=-2
2abG (tb tu) [ 2(b/tb + u/t,)t, 4
at 2. Thus

or
b
.;=--(----) a
T
4abG tb
Similarly

The warping distribution therefore varies linearly from a value
-T(b/rb - a/tu)/4abG at 2 to zero at 3. The remaining distribution follows from
symmetry so that the complete distribution takes the form shown in Fig. 9.33.
Comparing Figs 9.31 and 9.33 it can be seen that the form of the warping distribu-
tion is the same but that in the latter case the complete distribution has been displaced
axially. The actual value of the warping at the origin for s is found using Eq. (9.46).
Thus
(vii)