Page 337 - Aircraft Stuctures for Engineering Student

3 18 Open and closed, thin-walled beams

Referring the tangential displacement wt to the centre of twist R of the cross-section
we have, from Eq. (9.28)

(9.64)

Substituting for dwt/dz in Eq. (9.63) gives

from which

(9.65)

On the mid-line of the section wall rzs = 0 (see Eq. (9.57)) so that, from Eq. (9.65)

Integrating this expression with respect to s and taking the lower limit of integration
to coincide with the point of zero warping, we obtain

(9.66)

From Eqs (9.62) and (9.66) it can be seen that two types of warping exist in an open
section beam. Equation (9.66) gives the warping of the mid-line of the beam; this is
known as primary warping and is assumed to be constant across the wall thickness.
Equation (9.62) gives the warping of the beam across its wall thickness. This is
called secondary warping, is very much less than primary warping and is usually
ignored in the thin-walled sections common to aircraft structures.
Equation (9.66) may be rewritten in the form
(9.67)

or, in terms of the applied torque

W, = -2A (see Eq. (9.60)) (9.68)
s:
in which AR = 4 pR ds is the area swept out by a generator, rotating about the
centre of twist, from the point of zero warping, as shown in Fig. 9.37. The sign of
w,, for a given direction of torque, depends upon the sign of AR which in turn depends
upon the sign OfpR, the perpendicular distance from the centre of twist to the tangent
at any point. Again, as for closed section beams, the sign of pR depends upon the
assumed direction of a positive torque, in this case anticlockwise. Therefore, pR
(and therefore AR) is positive if movement of the foot of pR along the tangent in
the assumed direction of s leads to an anticlockwise rotation of pR about the centre
of twist. Note that for open section beams the positive direction of s may be
chosen arbitrarily since, for a given torque, the sign of the warping displacement
depends only on the sign of the swept area AR.