Page 311 - Aircraft Stuctures for Engineering Student

292 Open and closed, thin-walled beams

(a) (bl
Fig. 9.14 (a) General stress system on element of a closed or open section beam; (b) direct stress and shear
flow system on the element.

The parameter s in the analysis is distance measured around the cross-section from
some convenient origin.
An element 6s x 6z x t of the beam wall is maintained in equilibrium by a system of
direct and shear stresses as shown in Fig. 9.14(a). The direct stress a, is produced by
bending moments or by the bending action of shear loads while the shear stresses are
due to shear and/or torsion of a closed section beam or shear of an open section beam.
The hoop stress us is usually zero but may be caused, in closed section beams, by inter-
nal pressure. Although we have specified that t may vary with s, this variation is small
for most thin-walled structures so that we may reasonably make the approximation
that t is constant over the length 6s. Also, from Eqs (1.4), we deduce that
rrs = rsz = r say. However, we shall find it convenient to work in terms of shear
flow q, i.e. shear force per unit length rather than in terms of shear stress. Hence, in
Fig. 9.14(b)
q = rt (9.21)
and is regarded as being positive in the direction of increasing s.
For equilibrium of the element in the z direction and neglecting body forces (see
Section 1.2)

(a, +z6r)*6s - azt6s + (2) - qsz = 0
q+-&
sz
which reduces to
a4 aaz
-+t-=O (9.22)
as az
Similarly for equilibrium in the s direction

(9.23)

The direct stresses a, and us produce direct strains E, and E,, while the shear stress r
induces a shear strain y(= T~~ = T,,). We shall now proceed to express these strains in
terms of the three components of the displacement of a point in the section wall (see
Fig. 9.15). Of these components v, is a tangential displacement in the xy plane and is
taken to be positive in the direction of increasing s; w,, is a normal displacement in the