Page 473 - Aircraft Stuctures for Engineering Student

454 Structural constraint

t‘

z
L
Fig. 11.8 Shear stress distributions along the beam of Fig. 1 1.5.

covers over that predicted by elementary theory and decrease the shear stress in the
webs. It may also be noted that for bt, to be greater than atb for the beam of Fig.
11.5, in which a > b, then ta must be appreciably greater than tb so that
T/2abta < T/hbtb. Also at the built-in end (z = 0), Eqs (11.21) and (11.22) reduce
to r, = T/a(bt, + atb) and 7-b = T/b(bta + atb) so that even though Tb is reduced
by the axial constraint and r, increased, rb is still greater than 7,. It should also be
noted that these values of r, and 76 at the built-in end may be obtained using the
method of Section 11.2 and that these are the values of shear stress irrespective of
whether the section has been idealized or not. In other words, the presence of inter-
mediate stringers and/or direct stress carrying walls does not affect the shear flows
at the built-in end since the direct stress gradient at this section is zero (see Section
11.2 and Eq. (9.22)) except in the comer booms. Finally, when both z and L
become large, i.e. at the free end of a long, slender beam
T T
7, -+ -
and rb +-
2abt, 2abtb
The above situation is shown in Fig. 11.8.
In the particular case when bt, = atb we see that the second terms on the right-hand
side of Eqs (1 1.21) and (1 1.22) disappear and no constraint effects are present; the
direct stress of Eqs (11.19) is also zero since wo = 0 (see Example 9.7).
The rate of twist is obtained by substituting for w from Eq. (11.18) in Eq. (11.11).
Thus
bt, - atb ’ coshp(L - z) ]
d8 - (11.23)
- dz-2a2bZG L(b+~)[l-(bt,+Utb) CoshpL
tb
in which we see that again the expression on the right-hand side comprises the rate of
twist given by elementary theory, T(b/tb + a/t,)/2a2bZG (see Section 9.5), together
with a correction due to the warping restraint. Clearly the rate of twist is always
reduced by the constraint since (bt, - atb)’ is always positive. Integration of Eq.
(1 1.23) gives the distribution of angle of twist along the length of the beam, the
boundary condition in this case being 8 = 0 at z = 0.