Page 71 - Aircraft Stuctures for Engineering Student

56 Torsion of solid sections

't

Fig. 3.4 Rigid body displacement in the cross-section of the bar.

Rearranging and substituting for u and from Eqs (3.9)
dw rzx dB dw r7 dB
-
-A_-
--
- - (3.10)
- +-y,
ax G dz dy G dzX
For a particular torsion problem Eqs (3.10) enable the warping displacement w of
the originally plane cross-section to be determined. Note that since each cross-section
rotates as a rigid body 0 is a function of z only.
Differentiating the first of Eqs (3.10) with respect toy, the second with respect to x
and subtracting we have

Expressing rzx and rzy in terms of 4 gives

or, from Eq. (3.4)
de
-2G- = V2$ = I; (constant) (3.11)
dz
It is convenient to introduce a torsion constant J defined by the general torsion
equation
dB
T = GJ- (3.12)
dz
The product GJ is known as the torsional rigidity of the bar and may be written, from
Eqs (3.8) and (3.1 1)

(3.13)