Page 470 - Aircraft Stuctures for Engineering Student

11.3 Thin-walled rectangular section beam 451

I

Fig. 11.6 Shear distortion of (a) an element of the top cover; (b) an element of the right hand web.

The elements of length Sz of the covers and webs of the beam will warp into the shapes
shown in Fig. 11.5 if T is positive (anticlockwise) and b/tb > a/t,. Clearly there must
be compatibility of displacement at adjacent edges of the elements. From Fig. 11.6(a)

(11.8)

and from Fig. 1 1.6(b)
aw w
-
- (11.9)
as -b/2
Substituting for aw/ds and av,/az in Eq. (9.39) separately for the covers and webs, we
obtain

(1 1.10)

Now substituting for q, and qb in Eq. (1 1.4) we have

Rearranging
_- 2T (1 1.1 1)
de 4w(bt, - atb)
-
dz ab(bt, + atb) + abG(bt, + atb)
If we now substitute for dO/dz from Eq. (1 1.11) into Eqs (1 1.10) we have
-4WGtbt, Tta 4WGtbta Ttb (11.12)
= bt, + atb + a(bt, + atb) ' qb = bt, + atb + b(bt, + atb)

Equations (1 1.1 1) and (1 1.12) give the rate of twist and the shear flows (and hence
shear stresses) in the beam in terms of the warping w and the applied torque T.
Their derivation is based on the compatibility of displacement which exists at the
cover/boom/web junctions. We shall now use the further condition of equilibrium
between the shears in the covers and webs and the direct load in the booms to