Page 335 - Aircraft Stuctures for Engineering Student

316 Open and closed, thin-walled beams

or
2A
pRGt = - constant (9.55)
=
6
A closed section beam for which pRGt = constant does not warp and is known as a
Neuber beam. For closed section beams having a constant shear modulus the condi-
tion becomes
PRt = constant (9.56)
Examples of such beams are: a circular section beam of constant thickness; a
rectangular section beam for which ath = ht, (see Example 9.7); and a triangular
section beam of constant thickness. In the last case the shear centre and hence the
centre of twist may be shown to coincide with the centre of the inscribed circle so
that pR for each side is the radius of the inscribed circle.

An approximate solution for the torsion of a thin-walled open section beam may be
found by applying the results obtained in Section 3.4 for the torsion of a thin
rectangular strip. If such a strip is bent to form an open section beam, as shown in
Fig. 9.36(a), and if the distance s measured around the cross-section is large compared
with its thickness t then the contours of the membrane, i.e. lines of shear stress, are
still approximately parallel to the inner and outer boundaries. It follows that the
shear lines in an element 6s of the open section must be nearly the same as those in
an element SJJ of a rectangular strip as demonstrated in Fig. 9.36(b). Equations

-n

(a) (b)
Fig. 9.36 (a) Shear lines in a thin-walled open section beam subjected to torsion; (b) approximation of
elemental shear lines to those in a thin rectangular strip.