Page 336 - Aircraft Stuctures for Engineering Student

9.6 Torsion of open section beams 3 17

(3.27), (3.28) and (3.29) may therefore be applied to the open beam but with reduced
accuracy. Referring to Fig. 9.36(b) we observe that Eq. (3.27) becomes

dB
rzs = 2Gn -, r,,, = 0 (9.57)
dz
Eq. (3.28) becomes

(9.58)

and Eq. (3.29) is
3 Lt
st3 1
J=C- or J=- t3ds (9.59)
3
In Eq. (9.59) the second expression for the torsion constant is used if the cross-section
has a variable wall thickness. Finally, the rate of twist is expressed in terms of the
applied torque by Eq. (3.12), viz.
d0
T=GJ- (9.60)
dz
The shear stress distribution and the maximum shear stress are sometimes more COR-
veniently expressed in terms of the applied torque. Therefore, substituting for de/&
in Eqs (9.57) and (9.58) gives
2n tT
rZs = - T, T ~ = f - , ~ ~ ~ (9.61)
~
J J
We assume in open beam torsion analysis that the cross-section is maintained by
the system of closely spaced diaphragms described in Section 9.2 and that the beam
is of uniform section. Clearly, in this problem the shear stresses vary across the thick-
ness of the beam wall whereas other stresses such as axial constraint stresses which we
shall discuss in Chapter 11 are assumed constant across the thickness.

---
9.6.1 Warping of the cross-section
.,...I--- Ll_-___lP---
We saw in Section 3.4 that a thin rectangular strip suffers warping across its thickness
when subjected to torsion. In the same way a thin-walled open section beam will warp
across its thickness. This warping, wt, may be deduced by comparing Fig. 9.36(b) with
Fig. 3.10 and using Eq. (3.32), thus
d0
wt = ns- (9.62)
dz
In addition to warping across the thickness, the cross-section of the beam will warp in
a similar manner to that of a closed section beam. From Fig. 9.16

(9.63)