Page 327 - Aircraft Stuctures for Engineering Student

308 Open and closed, thin-walled beams

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Fig. 9.28 Determination of the shear flow distribution in a closed section beam subjected to torsion.

Note that the origin 0 of the axes in Fig. 9.28 may be positioned in or outside the
cross-section of the beam since the moment of the internal shear flows (whose resul-
tant is a pure torque) is the same about any point in their plane. For an origin outside
the cross-section the term $p ds will involve the summation of positive and negative
areas. The sign of an area is determined by the sign ofp which itself is associated with
the sign convention for torque as follows. If the movement of the foot of p along the
tangent at any point in the positive direction of s leads to an anticlockwise rotation of
p about the origin of axes, p is positive. The positive direction of s is in the positive
direction of q which is anticlockwise (corresponding to a positive torque). Thus, in
Fig. 9.29 a generator OA, rotating about 0, will initially sweep out a negative area
since PA is negative. At B, however, pB is positive so that the area swept out by the
generator has changed sign (at the point where the tangent passes through 0 and
p = 0). Positive and negative areas cancel each other out as they overlap so that as
the generator moves completely around the section, starting and returning to A
say, the resultant area is that enclosed by the profile of the beam.

Fig. 9.29 Sign convention for swept areas.