Page 306 - Aircraft Stuctures for Engineering Student

9.1 Bending of open and closed section beams 287

Note that if either Cx or Cy were an axis of symmetry, Ixy = 0 and Eqs (vi) and (vii)
reduce to

- WL’
Uf.e. = 0, vf.e. = -
3EIxx
the well-known results for the bending of a cantilever having a symmetrical cross-
section and carrying a concentrated vertical load at its free end.

We may exploit the thin-walled nature of aircraft structures to make simplifying
assumptions in the determination of stresses and deflections produced by bending.
Thus, the thickness t of thin-walled sections is assumed to be small compared with
their cross-sectional dimensions so that stresses may be regarded as being constant
across the thickness. Furthermore, we neglect squares and higher powers oft in the
computation of sectional properties and take the section to be represented by the
mid-line of its wall. As an illustration of the procedure we shall consider the channel
section of Fig. 9.9(a). The section is singly symmetric about the x axis so that IxJ = 0.
is
then
The second moment of area lYX given by
[2(i2 - t/2)l3
I X.Y - 2 [ (b +:2/2)t3 + (b + i) th’] + c
-
12
Expanding the cubed term we have
Ixx = 2 [ (b +:2/2)p + (b +;) th’] +A [(Z)’ (n’ - 3h-2+ 3hq -

3 t t2 t3)1
which reduces, after powers of 8 and upwards are ignored, to

(a) (b)
Fig. 9.9 (a) Actual thin-wailed channel section; (b) approximate representation of section.