Page 346 - Aircraft Stuctures for Engineering Student

9.8 Structural idealization 327

The torsional rigidity of the complete section is then

GJ=5000x 107+6x 1O7=5O06x 107Nmm2
In all unrestrained torsion problems the torque is related to the rate of twist by the
expression
d8
T=GJ-
dz
The angle of twist per unit length is therefore given by

de - T - 10 x lo6 = 0.0002 rad/mm
dz - - 5006 x lo7
Substituting for T in Eq. (9.49) from Eq. (9.52), we obtain the shear flow in the closed
section. Thus

from which
qcl = 250N/mm
The maximum shear stress in the closed section is then 250/1.5 = 166.7N/mm2.
In the open portion of the section the maximum shear stress is obtained directly
from Eq. (9.58) and is

T,,,,,~ = 25 000 x 2 x 0.0002 = 10 N/mm2
It can be seen from the above that in terms of strength and stiffness the closed portion
of the wing section dominates. This dominance may be used to determine the warping
distribution. Having first found the position of the centre of twist (the shear centre)
the warping of the closed portion is calculated using the method described in Section
9.5. The warping in the walls 13 and 34 is then determined using Eq. (9.67), in which
the origin for the swept area AR is taken at the point 1 and the value of warping is that
previously calculated for the closed portion at 1.

So far in this chapter we have been concerned with relatively uncomplicated struc-
tural sections which in practice would be formed from thin plate or by the extrusion
process. While these sections exist as structural members in their own right they are
frequently used, as we saw in Chapter 7, to stiffen more complex structural shapes
such as fuselages, wings and tail surfaces. Thus a two spar wing section could take
the form shown in Fig. 9.45 in which Z-section stringers are used to stiffen the thin
skin while angle sections form the spar flanges. Clearly the analysis of a section of
this type would be complicated and tedious unless some simplifying assumptions
were made. Generally, the number and nature of these simplifying assumptions deter-
mine the accuracy and the degree of complexity of the analysis; the more complex the
analysis the greater the accuracy obtained. The degree of simplification introduced is