Page 487 - Aircraft Stuctures for Engineering Student

468 Structural constraint

t’

Fig. 11.26 Torsion of an open section beam fully built-in at one end.

The theory, originally developed by Wagner and Kappus, is most generally known
as the Wagner torsion bending theory. It assumes that the beam is long compared
with its cross-sectional dimensions, that the cross-section remains undistorted by
the loading and that the shear strain T~~ of the middle plane of the beam is negligible
although the stresses producing the shear strain are not. From similar assumptions is
derived, in Section 9.6, an expression for the primary warping w of the beam, viz.

In the presence of axial constraint, de/dz is no longer constant so that the longitudi-
nal strain aw/az is not zero and direct (also shear) stresses are induced. Thus

(11.54)

The or stress system must be self-equilibrating since the applied load is a pure torque.
Therefore, at any section the resultant end load is zero and
1
IC or,tds = 0 ( IC denotes integration around the beam section

or, from Eq. (11.54) and observing that d29/d? is a function of z only

2A~t = 0 (11.55)
ds
The limits of integration of Eq. (1 1.55) present some difficulty in that AR is zero when
w is zero at an unknown value of s. Let
2.4~ = 2A~,o - 2Ak
where AR.0 is the area swept out from s = 0 and A; is the value of AR!O at w = 0 (see
Fig. 11.27). Then in Eq. (1 1.55)