Page 491 - Aircraft Stuctures for Engineering Student

472 Structural constraint

2AR10

cs
1-34 5 6
di2 d23 Etc.

(a) (b)
Fig. 11.29 Computation of torsion bending constant rR: (a) dimensions of flat-sided open section beam; (b)
variation of 2AR,o around beam section.

In the derivation of Eq. (1 1.56) we showed that

Suppose now that the line 1’2’3‘. . .6‘ is a wire of varying density such that the weight
of each element 6s’ is tSs. Thus the weight of length 1‘2’ is tdI2 etc. They coordinate of
the centre of gravity of the ‘wire’ is then

Comparing this expression with the previous one for 2AL, y and J are clearly analo-
gous to 2AR,o and 2Ak respectively. Further

Expanding and substituting

2Ak IC t ds for jc 2AR,ot ds

gives
I I
rR = (2A~,o)~tds (2AL)’ tds (1 1.62)
-
Thus, in Eq. (1 1.62), rR is analogous to the moment of inertia of the ‘wire’ about an
axis through its centre of gravity parallel to the s axis.

Example 11.2
An open section beam of length L has the section shown in Fig. 11.30. The beam is
M y built-in at one end and carries a pure torque T. Derive expressions for the
direct stress and shear flow distributions produced by the axial constraint (the or
and qr systems) and the rate of twist of the beam.