Page 321 - Aircraft Stuctures for Engineering Student

302 Open and closed, thin-walled beams

distribution (qb) around this ‘open’ section is given by

as in Section 9.3. The value of shear flow at the cut (s = 0) is then found by equating
applied and internal moments taken about some convenient moment centre. Thus,
from Fig. 9.24(a)

SxVO - SyCO = fpqh = fpqb dS + qs,O fp dS

where denotes integration completely around the cross-section. In Fig. 9.24(a)

SA = ~SSP
so that
f f
dA=$ pds

Hence
f
pds = 2A

where A is the area enclosed by the mid-line of the beam section wall. Hence

SxVO - S&O = f Pqb dS f 2Aqs,O (9.37)

If the moment centre is chosen to coincide with the lines of action of Sx and Sy then
Eq. (9.37) reduces to
f
= Pqb dS f 2Aqs,0 (9.38)

The unknown shear flow qs,o follows from either of Eqs (9.37) or (9.38).
It is worthwhile to consider some of the implications of the above process.
Equation (9.34) represents the shear flow distribution in an open section beam for
the condition of zero twist. Therefore, by ‘cutting’ the closed section beam of Fig.
9.24(a) to determine qb, we are, in effect, replacing the shear loads of Fig. 9.24(a)
by shear loads Sx and Sy acting through the shear centre of the resulting ‘open’ section
beam together with a torque T as shown in Fig. 9.24(b). We shall show in Section 9.5
that the application of a torque to a closed section beam results in a constant shear
flow. In this case the constant shear flow qs,o corresponds to the torque but will
have different values for different positions of the ‘cut’ since the corresponding
various ‘open’ section beams will have different locations for their shear centres.
An additional effect of ‘cutting’ the beam is to produce a statically determinate
structure since the qb shear flows are obtained from statical equilibrium considera-
tions. It follows that a single cell closed section beam supporting shear loads is
singly redundant.