Page 309 - Aircraft Stuctures for Engineering Student

290 Open and closed, thin-walled beams

The section properties are calculated as follows

h3 t

Substituting these values in Eq. (i)
X
M
uz = - (6.863, - 10.30~) (ii)
h3 t
On the top flange y = h/2,0 < x < h/2 and the distribution of direct stress is given by
uz = - (3.43h - 10.30~)
M
X
h3 t
which is linear. Hence
1.72Mx
UZJ = -~ (compressive)
h3 t
uz,2 = +- 3.43Mx (tensile)
h3 t
In the web h/2 < y < -h/2 and x = 0. Again the distribution is of linear form and is
given by the equation

uz = %6.86y
ht
whence
cz,2 = +- 3.43M, (tensile)
h3 t
and
3.43Mx
a,,3 = -- (compressive)
h3 t
The distribution in the lower flange may be deduced from antisymmetry; the complete
distribution is then as shown in Fig. 9.13.

9.1.8 Applicability of bending theory

The expressions for direct stress and displacement derived in the above theory are
based on the assumptions that the beam is of uniform, homogeneous cross-section
and that plane sections remain plane after bending. The latter assumption is strictly
true only if the bending moments M, and My are constant along the beam. Variation
of bending moment implies the presence of shear loads and the effect of these is to