Page 343 - Aircraft Stuctures for Engineering Student

324 Open and closed, thin-walled beams

The centroid of area C lies on the axis of symmetry at some distance J from the
upper surface of the beam section. Taking moments of area about this upper surface
(4 x 100 x 2+4 x 200 x 2)y= 2 x 100 x 2 x 50+2 x 200 x 2 x 100
+ 200 x 2 x 200

which gives J = 75 mm.
The second moment of area of the section about Cx is given by

i.e.
Ixx = 14.5 x 106m4
The section is symmetrical about Cy so that Ixy = 0 and since Sx = 0 the shear flow
distribution in the closed section 3456 is, from Eq. (9.35)

Also the shear load is applied through the shear centre of the complete section, i.e.
along the axis of symmetry, so that in the open portions 123 and 678 the shear
flow distribution is, from Eq. (9.34)
(ii)

We note that the shear flow is zero at the points 1 and 8 and therefore the analysis may
conveniently, though not necessarily, begin at either of these points. Thus, referring to
Fig. 9.42

loo lo3 2(-25 +SI) dsl
q12 = - 14.5 x lo6 o
i.e.

q12 = -69.0 x 10-4(-50~1 +s:) (iii)
whence q2 = -34.5N/mm.
Examination of Eq. (iii) shows that q12 is initially positive and changes sign when
s1 = 50mm. Further, q12 has a turning value (dq12/ds1 = 0) at s1 = 25mm of
r
4.3 N/mm. In the wall 23
q23 = -69.0 x 10-~ 2 x 75ds2 - 34.5
i.e.
q23 = -1.04,~2 - 34.5 (iv)
Hence q23 varies linearly from a value of -34.5 N/mm at 2 to -138.5 N/mm at 3 in the
wall 23.