Page 310 - Aircraft Stuctures for Engineering Student

9.2 General stress, strain and displacement relationships 291

3
Fig. 9.13 Distribution of direct stress in Z-section beam of Example 9.3.

deform the beam section into a shallow, inverted 's' (see Section 2.6). However, shear
stresses in beams whose cross-sectional dimensions are small in relation to their
lengths are comparatively low so that the basic theory of bending may be used
with reasonable accuracy.
In thin-walled sections shear stresses produced by shear loads are not small and
must be calculated, although the direct stresses may still be obtained from the basic
theory of bending so long as axial constraint stresses are absent; this effect is discussed
in Chapter 1 1. Deflections in thin-walled structures are assumed to result primarily
from bending strains; the contribution of shear strains may be calculated separately
if required.

e 6 Istress, ^st r a i'n an d-dEplace me nt re la t i o ns h i ps
for open and single cell closed section thin-walled
beams

We shall establish in this section the equations of equilibrium and expressions for
strain which are necessary for the analysis of open section beams supporting shear
loads and closed section beams carrying shear and torsional loads. The analysis of
open section beams subjected to torsion requires a different approach and is discussed
separately in Section 9.6. The relationships are established from first principles for the
particular case of thin-walled sections in preference to the adaption of Eqs (1.6),
(1.27) and (1.28) which refer to different coordinate axes; the form, however, will
be seen to be the same. Generally, in the analysis we assume that axial constraint
effects are negligible, that the shear stresses normal to the beam surface may be
neglected since they are zero at each surface and the wall is thin, that direct and
shear stresses on planes normal to the beam surface are constant across the thickness,
and finally that the beam is of uniform section so that the thickness may vary with
distance around each section but is constant along the beam. In addition, we ignore
squares and higher powers of the thickness t in the calculation of section constants.