Page 334 - Aircraft Stuctures for Engineering Student

9.5 Torsion of closed section beams 31 5

whence
TL

The total angle of twist from end to end of the beam is 28, therefore
28 T /2a 2b\

or

as in Eq. (9.52).
Substituting for 8 in either of the expressions for the axial displacement of the
corner 1 gives the warping wl at 1. Thus

ab TL fa b\ T a

i.e.
wl=-(----) T b a
8abG tb
as before. It can be seen that the warping of the cross-section is produced by a com-
bination of the displacements caused by twisting and the displacements due to the
shear strains; these shear strains correspond to the shear stresses whose values are
fixed by statics. The angle of twist must therefore be such as to ensure compatibility
of displacement between the webs and covers.

9.5.2 Condition for zero warping at a section

The geometry of the cross-section of a closed section beam subjected to torsion may
be such that no warping of the cross-section occurs. From Eq. (9.53) we see that this
condition arises when

or

(9.54)

Differentiating Eq. (9.54) with respect to s gives