Page 468 - Aircraft Stuctures for Engineering Student

----- - 11.3 Thin-walled rectangular section beam 449

.%-^-"
11.3 Thir ;-walled rectangular section beam subjected
to t orsion

In Example 9.7 we determined the warping distribution in a thin-walled rectangular
section beam which was not subjected to structural constraint. This free warping
distribution (wo) was found to be linear around a cross-section and uniform along
the length of the beam having values at the corners of

The effect of structural constraint, such as building one end of the beam in, is to
reduce this free warping to zero at the built-in section so that direct stresses are
induced which subsequently modify the shear stresses predicted by elementary torsion
theory. These direct stresses must be self-equilibrating since the applied load is a pure
torque.
The analysis of a rectangular section beam built-in at one end and subjected to a
pure torque at the other is simplified if the section is idealized into one comprising
four corner booms which are assumed to carry all the direct stresses together with
shear-stress-only carrying walls. The assumption on which the idealization is based
is that the direct stress distribution at any cross-section is directly proportional to
the warping which has been suppressed. Thus, the distribution of direct stress is
linear around any cross-section and has values equal in magnitude but opposite in
sign at opposite corners of a wall. This applies at all cross-sections since the free
warping will be suppressed to some extent along the complete length of the beam.
In Fig. 1 1.4(b) all the booms will have the same cross-sectional area from anti-
symmetry and, from Eq. (9.70) or Eq. (9.71)
at, hth 1
B = -(2 - 1) +-(2 - 1) =-(at, + hth)
6 6 6
To the boom area B will be added existing concentrations of area such as connecting
angle sections at the corners. The contributions of stringers may be included by allow-
ing for their direct stress carrying capacity by increasing the actual wall thickness by
an amount equal to the total stringer area on one wall before idealizing the section.
We have seen in Section 9.8 that the effect of structural idealization is to reduce
the shear flow in the walls of a beam to a constant value between adjacent booms.

R

a
(a) (b)
Fig. 11.4 Idealization of a rectangular section beam subjected to torsion: (a) actual; (b) idealized.